# A general quantum circuit framework for Extended Wigner's Friend Scenarios: logically and causally consistent reasoning without absolute measurement events

V. Vilasini¹,²,* and Mischa P. Woods¹,²,³,†

¹Institute for Theoretical Physics, ETH Zurich, 8093 Zürich, Switzerland

²Université Grenoble Alpes, Inria, 38000 Grenoble, France

³ENS Lyon, Inria, LIP, 69342 Lyon, France

Extended Wigner's Friend Scenarios (EWFSs) go beyond the standard usage of quantum theory where agents are treated classically, and model agents as unitary evolving quantum systems. This has been the subject of several no-go results: Frauchiger and Renner (FR) suggested that quantum agents reasoning using quantum theory will arrive at logical paradoxes, while other works, e.g. the Local-Friendliness theorem, highlight challenges for having an objective notion of measurement events and for causal reasoning in EWFSs. This raises the question: Is it possible to reliably make and test scientific predictions, and consistently reason about the world when applying quantum theory universally, and without assuming that observed measurement outcomes are absolute? We give a positive answer by developing a general quantum circuit framework for EWFSs. We formalise the concept of Heisenberg cuts by mapping them to distinct channels in a quantum circuit, and prove that FR-type paradoxes can be fully resolved by making explicit the conditioning on the quantum channels that are used in the reasoning process. We also provide concrete rules by which quantum agents can reason and make predictions in a logically and causally consistent manner. Our framework describes all perspectives and predictions of an EWFS within a single, well-defined causal structure, although it allows events to be fundamentally subjective. Moreover, we show that an objective notion of measurement events nevertheless emerges in real-world experiments. Our work demonstrates the possibility of a relational yet operational framework overcoming challenges to scientific reasoning in EWFSs, without modifying the Born rule, quantum unitarity or the axioms of classical logic and probability theory applied to measurement outcomes. This enables analysis and comparison of different EWFS arguments and yields a formal platform to extend existing quantum information methods and studies consistently to the domain of Wigner's Friend Scenarios.

## CONTENTS

<table><tr><td>I. Introduction</td><td>2</td></tr><tr><td>II. Background information</td><td>5</td></tr><tr><td>A. Conditional probabilities in operational theories</td><td>5</td></tr><tr><td>B. Wigner's friend scenarios: a tale of two evolutions</td><td>5</td></tr><tr><td>III. A general circuit framework for EWFS</td><td>6</td></tr><tr><td>A. Extended Wigner's friend scenarios and quantum predictions</td><td>6</td></tr><tr><td>B. Formulating an EWFS within a single quantum circuit</td><td>8</td></tr><tr><td>IV. Completeness, consistency and causality without absolute events</td><td>11</td></tr><tr><td>A. Properties of general quantum predictions in EWFS</td><td>11</td></tr><tr><td>B. Application to agents' reasoning</td><td>13</td></tr><tr><td>C. Reason for apparent inconsistencies</td><td>14</td></tr><tr><td>V. A simple resolution to the FR apparent paradox</td><td>15</td></tr><tr><td>A. Augmented circuit</td><td>15</td></tr><tr><td>B. Explicit version of the statements that resolve the paradox</td><td>16</td></tr></table>

<table><tr><td>C. Setting-dependence: a refined interpretation of the FR paradox</td><td>18</td></tr><tr><td>VI. Emergence of absolute measurement events</td><td>20</td></tr><tr><td>A. Causal criteria for superagency: distinguishing standard and genuinely Wigner's Friend scenarios</td><td>20</td></tr><tr><td>B. Recovering Heisenberg cut independence in standard quantum experiments</td><td>22</td></tr><tr><td>VII. Discussions</td><td>22</td></tr><tr><td>A. Sound scientific reasoning in EWFS and analogies to classical multi-agent reasoning</td><td>22</td></tr><tr><td>B. Interpretations of quantum theory</td><td>25</td></tr><tr><td>C. Physical interpretation of settings</td><td>25</td></tr><tr><td>VIII. Conclusions and outlook</td><td>27</td></tr><tr><td>Acknowledgments</td><td>29</td></tr><tr><td>A. Generality of the definition of EWFSs</td><td>30</td></tr><tr><td>B. Distinguishing predictive and observational statements: refining consistency</td><td>30</td></tr><tr><td>C. Overview of the Frauchiger-Renner apparent paradox</td><td>32</td></tr><tr><td>1. The FR no-go theorem</td><td>32</td></tr><tr><td>2. Entanglement version of the FR experiment</td><td>33</td></tr><tr><td>3. Prepare and measure version of the FR</td><td>33</td></tr></table>

<footnote>* vilasini@inria.fr</footnote>

<footnote>† mischa.woods@gmail.com</footnote>
<table><tr><td>experiment</td><td>34</td></tr><tr><td>D. Derivation of predictions in the augmented EWFS</td><td>36</td></tr><tr><td>E. Reduction of the augmented circuit in standard quantum scenarios</td><td>38</td></tr><tr><td>F. Detailed analysis of the FR experiment</td><td>39</td></tr><tr><td>1. Entanglement version of the FR experiment</td><td>39</td></tr><tr><td>2. Prepare and measure version of the FR experiment</td><td>42</td></tr><tr><td>3. Setting-dependence in FR's experiment</td><td>47</td></tr><tr><td>G. Classical example reproducing certain features of the FR correlations</td><td>49</td></tr><tr><td>H. Relation to Hardy's logical proof of contextuality</td><td>50</td></tr><tr><td>I. Relation to previous works: a more unified picture</td><td>52</td></tr><tr><td>1. Previous works rejecting one of FR's assumptions</td><td>52</td></tr><tr><td>2. Previous suggestions for consistent reasoning</td><td>54</td></tr><tr><td>3. Previous works discussing the validity of FR's claim</td><td>55</td></tr><tr><td>J. Proofs of all results</td><td>56</td></tr><tr><td>1. Proofs of results from the main text</td><td>56</td></tr><tr><td>2. Proofs of results from the Appendix</td><td>61</td></tr><tr><td>References</td><td>63</td></tr></table>

# I. INTRODUCTION

Quantum theory is among the most successful physical theories, accurately describing microscopic phenomena. In recent years, efforts have been underway to observe quantum phenomena in larger systems, both for scalable quantum computing and for fundamental tests of physical laws. Hence, it is crucial to consider the implications if quantum theory were universally valid. It is natural to expect a complete and universally valid theory of the physical world to be able to consistently model observers or agents as physical systems of the theory, at least in principle.

Wigner was among the first to concretely consider this question back in the 1960s, through an intriguing thought experiment [1]. Wigner's thought experiment highlights challenges in applying quantum theory to agents: an agent (the Friend) measures a quantum system and observes a classical outcome, which is at odds with the view of an outside agent (Wigner, the “super-agent”) who models the Friend's lab as a closed quantum system evolving unitarily, and can perform quantum operations on the lab's quantum superposition state, including “undoing” the unitary evolution of

the Friend's measurement. This tension lies at the core of the quantum measurement problem, and Wigner's scenario beautifully highlights how this can have empirical consequences for observers in quantum theory.

More recent works (e.g., [2–4]) explore Extended Wigner’s Friend Scenarios (EWFSs) involving additional agents, suggesting even more radical implications for physics. Frauchiger and Renner (FR) suggested that in EWFSs, agents who model each other as quantum systems and, at the same time, reason about each other’s knowledge, would arrive at logical contradictions [2]. Specifically, they claimed that the following assumptions cannot all hold simultaneously: the physical predictions follow the quantum Born rule<fnref n="1" /> (Q), agents (reasoning using the same theory) can inherit each other’s conclusions (C), and measurements yield single non-contradictory outcomes in each run (S). In [5], additional assumptions involved in the FR argument were made explicit: agents’ labs can evolve unitarily (U) and the distributive axiom of logic holds for classical outcomes (D). Thus, FR’s result suggests that Q, U, C, and D together imply a violation of S, i.e., a paradox where a measurement yields contradictory outcomes.

Other no-go results, starting from the work by Brukner [3] and including the more recent Local-Friendliness (LF) theorem [4], address more ontological aspects of quantum theory through EWFSs. These suggest that observed measurement events cannot be regarded as absolute and objective (under certain compelling metaphysical assumptions relating to causality and free choice). This challenges the notion that objective events, for instance “the light is on in this room”, may only hold true relative to something (such as an agent), complicating causal reasoning in EWFSs [6, 7].

FR’s claim raises concerns about the consistent usability of quantum theory in EWF scenarios where agents are modelled as quantum systems. The non-absoluteness of events in EWFSs raises concerns about whether the results of scientific experiment can be considered objective facts about the world. More generally, it therefore becomes imperative to formally address the following question:

Q: How can we continue to reliably do science, that is, consistently reason about the world, make and test physical predictions, if unitary quantum theory were valid at the level of agents who have full quantum control over each others’ labs and where one does not assume an absolute notion of measurement events?

This issue is pertinent as a sufficiently large quantum computer could act as an agent in these EWFS arguments, necessitating a clear resolution.

These EWFS no-go theorems have been met with several responses, with different interpretations of

<footnote n="1">¹ Strictly speaking, FR require a weaker version of the Born rule, restricted to 0 or 1 probabilities.</footnote>
quantum theory suggesting distinct resolutions, espe-
cially for FR's paradox. Responses range from con-
ceptual discussions on the implications of FR's results
for physics (e.g., [5, 8, 9]), suggestions for additional
reasoning rules to avoid the paradox in FR's scenarios
(e.g., [10–12]), to arguments against the validity of FR's
theorem due to implicit assumptions (e.g., [13–16]).

However, there is no concrete framework for identi-
fying which implicit assumptions (if any) are necessary
to recover the apparent FR paradox. In the context
of previous reasoning rules, it is often suggested (e.g.,
[10, 12, 17]) that the validity of an agent Alice's pre-
diction or the ability to reason about Alice's outcome
should depend on whether a super-agent undoes Al-
ice's measurement in the future. While this can ensure
logical consistency, it raises concerns for causality prin-
ciples and the efficiency of reasoning. How many possi-
ble future operations must be tracked when reasoning
about present measurement outcomes? If Bob is space-
like separated from Alice, how does the possibility of a
super-agent acting on Alice's lab affect the validity of
Bob's conclusions about Alice?

Causality issues are also central to LF-type no-go
arguments. Challenges for causality from LF indicate
that current quantum causal modeling frameworks can-
not account for non-absolute measurement events [6, 7].
These formalisms impose absoluteness of events by as-
suming the existence of a single global probability dis-
tribution over the outcomes of all agents in a given sce-
nario. In [7], the authors show that causal models in
any theory assuming absoluteness of events, relativistic
causality, and free choice cannot explain LF inequali-
ties' violations. This holds even when allowing cyclic
causal structures, raising the challenge of whether a
consistent causal modeling formalism exists in quantum
theory gives up absoluteness of events, while preserv-
ing the other fundamental relativistic and operational
principles.

FR and LF-type EWFS arguments are often stud-
ied separately, with FR seen as agent-centric and LF
as more metaphysical. Despite extensive literature,
these often focus on the specific 4-agent EWFS of FR
and LF, and there is a lack of a comprehensive and
unified framework that addresses both relational as-
pects from non-absoluteness and operational aspects of
agents' reasoning in general EWFSs.

We observe that at the core of all the no-go argu-
ments lies the ambiguity in how a measurement is to be
modelled: as an irreversible evolution leading to classi-
cal records or a reversible unitary evolution of a closed
system (the agents’ lab). In quantum theory, this can
also be understood as a choice of Heisenberg cut, that
distinguishes which parts of an experiment are mod-
elled as classical vs quantum. So far, this has remained
more of a philosophical concept that does not formally
appear in formalisms for quantum theory. However,
Wigner’s thought experiment highlights the need to
take this seriously and to make explicit how measure-
ments are modelled.

Desiderata for a consistent formalism for

EWFSs Keeping these discussions in mind, we mo-
tivate some important desiderata for a formalism to
address such questions.

Firstly, the formalism must clearly formulate the pre-
dictions that quantum theory implies in a Wigner’s
Friend Scenario, as both FR and LF-type arguments
stem from such predictions. It must do so while for-
mally and explicitly specifying the assumptions about
how measurements are modelled and what knowledge
about measurement outcomes is known, when deriving
said predictions.

Secondly, it must be applicable to general EWFSs
and must allow for the possibility of modeling agents’
labs as unitarily evolving systems on which another
agent can have full quantum control. Since this al-
lows a measurement to be “undone” by reversing its
unitary evolution, this means that an absolute notion
of measurement events and records is not assumed by
the formalism.

Thirdly, the formalism must guarantee consistency
of the predictions as well as logical statements agents
can make using said predictions, within a clear causal
semantics: the rules of logical reasoning must be com-
pliant with causality principles such as the impossibility
of signaling faster than light.

Fourthly, it is crucial for such a framework which
does not assume absolute events to explain how ob-
jectivity of measurement outcomes emerges in existing
real-world experiments, and reproduce the observable
predictions of quantum theory in such experiments.

Finally, it is desirable to have an interpretation-
independent formalism, such that the identification of
relevant assumptions, and resolution of paradoxes will
apply across interpretations of quantum theory, facili-
tating agreement on the matter.

Within the standard usage of quantum theory, where
agents are not regarded as quantum systems, the quan-
tum circuit framework satisfies several of these desider-
ata. It provides a clear causal semantics, an unambigu-
ous manner to compute predictions using the Born rule
while ensuring that these predictions are well-defined
and form the basis of consistent reasoning. The circuit
implies an information-theoretic causal structure that
tells us about the flow of information between different
systems, which is compatible with the direction of time.
Moreover, choices of future quantum operations do not
influence outcomes of earlier measurements. Further-
more, a subscriber of any interpretation of quantum
theory can at the least, use such a circuit description
as a tool to make and test empirical predictions and rea-
son about the world, as this is independent of whether
they believe the circuit to be a true representation of
an “ontological state of reality”.

Here, we develop a framework incorporating all the
desiderata motivated above, by consistently generaliz-
ing the quantum circuit formalism to general EWFSs
where agents’ labs/memories are explicitly included as
wires in the circuit. In particular, this yields a concrete
and constructive solution to the question Q, among sev-
eral other results.
**Overview of contributions** We provide an overview of the main contributions of this work below.

* **Quantum circuit framework for EWFSs:** In Section II, we review Wigner's thought experiment, then in Section III, we build a comprehensive circuit framework for all EWFSs in quantum theory, accommodating any number and configuration of agents and super-agents. This framework formalises Heisenberg cuts by mapping them to different channels, labeled by a parameter (the setting) that distinguishes whether one refers to the classical outcome of a measurement vs whether one regards it as a purely quantum, unitary evolution. Thus we show that every EWFS in quantum theory can be represented in terms of a single augmented circuit which allows to compute well-defined normalised probabilities, relative to a choice of settings.

* **Completeness, consistency, and causality:** In Section IV A, we prove three key properties of the augmented circuit: completeness (all quantum predictions in EWFSs can be recovered within the single augmented circuit), consistency (no contradictory predictions can arise in EWFSs using the augmented circuit), and causality (outcomes depend only on past choices of Heisenberg cuts relative to the causal order of the protocol). These results hold without assuming absolute measurement events or the existence of a unique joint probability distribution for all agents' outcomes.

* **Resolution and root of EWF reasoning paradoxes:** In Section IV B, we apply our formalism to agents' reasoning, demonstrating that the augmented circuit allows agents in any EWFS to consistently reason while simultaneously using the quantum Born rule, unitary evolution of closed quantum systems, classical logic and probability theory applied to observed outcomes. In Section IV C, we prove that any apparent inconsistencies can only arise in a scenario when an additional assumption I, which provably fails in that scenario, is imposed. I captures that physical predictions are independent of Heisenberg cuts (or how a measurement is modelled), and the result concretely identifies the failure of this assumption as a core reason for apparent EWFS paradoxes.

* **Detailed analysis of the FR scenario and claims:** Our above results establishing the general consistency of quantum theory and logical reasoning, are contrary to FR's claim that "Quantum theory cannot consistently justify the use of itself" [2]. We apply our framework to analyse the FR arguments in full detail, considering the entanglement version of their scenario in Section V (and the original prepare and measure version in Appendix F 2). We show that our formalism yields a simple resolution of the FR paradox even though it reproduces, in an explicit form, every statement made in FR's arguments, and without placing any restrictions on agents' reasoning. In Section V C, we provide a more refined understanding of the message of FR's result, by discussing the role of the I assumption in FR's scenario, and more generally in relation to the meta-physical concept of absoluteness of events considered in other EWFS no-go theorems (e.g., [3, 4]).

* **Emergence of objective measurement events and role of causality:** In Section VI, we address how subjective events in EWFS reconcile with objective measurement results in real-world experiments. We distinguish between standard and Wigner's Friend type experiments by identifying concrete criteria for "super-agency" based on the causal structure. We prove that in standard experiments (where agents do not measure each other's memories/labs in a non-trivial manner), predictions become Heisenberg-cut independent, and objectivity of measurement events emerges.

* **Discussions on multi-agent reasoning and physical interpretations:** In Section VII A, we discuss classical multi-agent scenarios that can lead to inconsistencies when agents overlook common knowledge and implicit assumptions. This helps us contrast the genuinely quantum aspects of EWFS arguments from classical ones. Based on these insights, we outline a general paradigm for scientific reasoning to ensure consistency in multi-agent contexts, showing how this is incorporated into our formalism. We then comment on the generalisation of our approach to ensure logically and causally consistent, and efficient reasoning in scenarios where agents only have partial knowledge of the protocol. In Section VII C, we discuss the physical interpretations of the concept of settings introduced in our work. In particular, we outline how setting choices in reasoning can be updated over time, similar to Bayesian updates, in light of new observations. Finally, in Section VII B, we discuss the interpretation-independence of our results.

* **A more unified picture:** Our framework provides a more unified platform for several aspects of EWFSs while shedding light on their relations. Specifically, in Appendix H, we discuss the links between FR's argument and Hardy's logical proof of Bell non-locality, highlighting the relations between Heisenberg cuts (given by our settings) and measurement contexts. In Appendix I we provide a more unified view of the relations between our framework and different classes of previous responses to FR's results, and we also discuss how different interpretations of quantum theory could apply our formalism consistently. Although we have focused more on the application of our general framework to resolving FR-type reasoning paradoxes, the framework can also describe the LF scenario. In forthcoming work [18], we apply the same formalism to analyse the LF scenario and derive further insights on the (non-)absoluteness of observed events.
## II. BACKGROUND INFORMATION

In Section II A, we start with a brief overview on the role of conditional probabilities in ensuring consistency in theories described in terms of circuits. We then review Wigner's original argument in Section II B. In both cases, we highlight the salient features and insights which are core to understanding our general framework and solution to EWFS paradoxes.

### A. Conditional probabilities in operational theories

Operational procedures in any theory involve state preparations, transformations, and measurements on physical systems. These procedures are often represented through circuit diagrams and can be formalised within several existing frameworks such as generalised probabilistic theories and process theories (e.g., [19–21]), the details of which are not pertinent here. The common feature across all these frameworks is that they provide rules for computing the probabilities of measurement outcomes.

These probabilities are conditioned on the relevant preparations, transformations, and measurements in the circuit. For example, if a system $S$ is prepared in state $ρ$, evolved using transformation $U$, and measured with $M$ to obtain outcome $a$, the probability of $a$ taking value $a$ is $P(a = a|ρ, U, M)$. This probability generally depends on the specific choices of $ρ, U$, and $M$; for instance, $P(a = a|ρ, U, M) \neq P(a = a|ρ, U', M)$ if $U$ and $U'$ are different transformations.

Consider a classical scenario where $S$ is a bit, $ρ = 0$, and $M$ measures the bit's value, yielding $a \in \{0, 1\}$. Let $U$ be the identity transformation and $U'$ be a bit flip. Then (1) $P(a = 1|ρ, U, M) = 0$ while (2) $P(a = 1|ρ, U', M) = 1$. Ignoring the conditioning on $U$ and $U'$ would lead to a paradox where $a = 1$ both never occurs (according to (1)) and certainly occurs (according to (2)). This demonstrates the necessity of considering the conditioning information to avoid contradictions, even in simple classical scenarios.

**Remark II.1.** It is not always necessary to condition on all the information in a circuit when computing probabilities. If some preparations or transformations are fixed and unchanging during the analysis, conditioning on them can be safely omitted without causing inconsistencies. In a multi-agent protocol, this corresponds to fixed elements that are common knowledge.

Here and in the rest of Section II, we use the names of the preparations, transformations, and measurements in the conditional probabilities. Later, when developing our framework, we will simplify this by labeling certain transformations with binary labels, $x \in \{0, 1\}$, as only two choices will be relevant in the scenarios of interest.

### B. Wigner's friend scenarios: a tale of two evolutions

The postulates of quantum theory propose two types of evolutions: unitary evolution of closed quantum systems and the projection postulate for the evolution associated with observing a measurement outcome. However, the theory does not specify when to apply each type of evolution. While this ambiguity does not affect our ability to apply quantum theory successfully in real world experiments (see Section VI for further discussion on this point), Wigner's 1967 thought experiment highlights beautifully why we should be concerned about this ambiguity.

The thought experiment assumes quantum theory is universally applicable to measurement devices, agents performing the measurements, and their laboratories. Notably, an agent in this context need not be a conscious human (although Wigner speculated about this aspect in his work); a sufficiently advanced quantum computer capable of measuring another system, storing the outcome in quantum memory, and performing basic computations can serve as an agent in Wigner's Friend no-go theorems and arguments.

Suppose Alice (Wigner's friend) measures a quantum system $S$ prepared in the state $|ψ⟩ = α|0⟩_S + β|1⟩_S$ in the computational basis, obtaining a classical outcome $a$ with values $a \in 0, 1$. She stores this outcome in her memory $A$, initialised to the state $|0⟩_A$. Treating $A$ as representing the rest of Alice's lab, $SA$ represents Alice's entire lab. If Alice's lab is a closed quantum system, it evolves unitarily according to the initial premise of the universal validity of quantum theory. Thus, according to the unitarity postulate, we would describe the evolution of Alice's lab as

$$ \mathcal{M}_{\text{unitary}}^{\text{A}} : |\psi\rangle_S \otimes |0\rangle_A \mapsto \alpha |00\rangle_{\text{SA}} + \beta |11\rangle_{\text{SA}}, \quad (1) $$

where $|aa⟩_{SA}$ represents the state of Alice's lab observing the outcome $a = a$. When the memory is initialised to $|0⟩_A$, the unitary evolution $\mathcal{M}_{\text{unitary}}^A$ for the computational basis measurement on $S$ (storing the outcome value in memory) is a CNOT gate with $S$ as control and $A$ as target.

Alternatively, applying the projection postulate, when Alice obtains outcome $a = 0$, her lab's evolution is $|ψ⟩_S|0⟩_A \mapsto |0⟩_S|0⟩_A$, and for $a = 1$, it is $|ψ⟩_S|0⟩_A \mapsto |1⟩_S|1⟩_A$. These are trace-decreasing evolutions, but summing over all possible outcomes gives the trace-preserving evolution:

$$ \mathcal{M}_{\text{projection}}^{\text{A}} : |\psi\rangle\langle\psi|_S \otimes |0\rangle\langle 0|_A \mapsto \\ |\alpha|^2 |00\rangle\langle 00|_{SA} + |\beta|^2 |11\rangle\langle 11|_{SA} \quad (2) $$

This describes the evolution of Alice's lab when using the projection postulate without considering specific outcomes, such as if Alice forgets the outcome after measurement.

This highlights that depending on whether a measurement is regarded as producing classical records (as
in the projection postulate) or as a purely unitary evolution of quantum systems, one would describe it through distinct evolutions of the same initial state. Wigner's thought experiment shows that this ambiguity can have observable consequences if quantum theory is universally valid.

Consider an outside agent, Wigner, who has full quantum control over SA (Alice's lab). Such a Wigner, a “superagent” can perform arbitrary quantum operations on SA. Since these evolutions result in distinct states on SA, Wigner can operationally distinguish them by measuring SA in a suitable basis. That is, the probability of Wigner's measurement outcome $w$, will generally depend on how the friend, Alice's measurement is modelled: using the unitarity (Equation (1)) or the projection postulate (Equation (2)).

Deutsch's version of the thought experiment [22] adds a twist. Suppose Alice leaves a note saying “I observed a definite outcome” without specifying the value. This note is then unentangled from Alice's memory which stores the outcome value, and remains unaffected after Wigner measures Alice's lab. If Wigner's measurement confirms the superposition state of Alice's system and memory, it is at conflict with Alice's note that confirms a definite outcome was observed.

Frauchiger-Renner's work [2] elevated this ambiguity into an apparent logical paradox, by extending Wigner's original set-up to include two friends and two Wigners (or superagents). They propose a no-go theorem which claims that when agents model each others' labs as quantum mechanical systems and reason about each others' knowledge of measurement outcomes using classical logic, they arrive at logical contradictions. They demonstrate this through a particular thought experiment with four agents, where agents reasoning in this manner arrive at an apparent paradox. A review FR's claimed theorem, the associated assumptions and scenario can be found in Appendix C.

On the other hand, the Local-Friendliness theorem [4] extends Wigner's original thought-experiment in a similar manner with four agents, but to address a different aspect than agents' reasoning, proving that a set of seemingly reasonable metaphysical assumptions about a physical theory cannot mutually hold. One of these assumptions is about the absoluteness of observed events, originally discussed in [3].

### III. A GENERAL CIRCUIT FRAMEWORK FOR EWFS

We now develop our general circuit framework for EWFS that meets all the desiderata motivated in the introduction. We do so by carefully distinguishing between, and also connecting the predictions of quantum theory, agents' knowledge and statements, while defining what it means for a theoretical model to make consistent predictions in an EWFS.

#### A. Extended Wigner's friend scenarios and quantum predictions

The term Extended Wigner's Friend Scenario (EWFS) is commonly used in the literature to describe specific scenarios such as the Frauchiger-Renner and Local-Friendliness scenarios. However, a general definition has been lacking. We propose a general definition of EWFSs within quantum theory, encompassing all finite multi-agent quantum protocols where agents' memories (in which they store the measurement outcome) or equivalently agents' labs are modelled as quantum systems, and where one agent can have full quantum control over the labs of other agents in the scenario. The formal definition is provided below, and its generality is justified in Appendix A.

**Definition III.1** (Extended Wigner's Friend Scenario (EWFS)). An EWFS is a quantum protocol that consists of

1. A finite set $\mathcal{S} := \{S_1, ..., S_m\}$ of systems,

2. A finite set $\mathcal{A} := \{A_1, ..., A_N\}$ of agents,

3. A set $\mathcal{M} := \{M_1, ..., M_N\}$ of memory systems, one for each agent $A_i$ where they store the outcome of their measurement.

4. For each agent $A_i$, a subset $S_i \subseteq \mathcal{S} \cup \mathcal{M} \setminus \{M_i\}$ of systems (which can include the memories of other agents) that they measure at time $t_i$ according to a measurement $\mathcal{M}^{A_i}$, obtaining a measurement outcome $a_i$ that they store in their memory system $M_i$. Moreover, $t_i < t_j$ for $i < j$.

5. For each agent $A_i$, a finite set $\mathcal{O}_i := \{0, 1, ..., d_{S_i} - 1\}$ in which the value $a_i$ of their outcome $a_i$ belongs.

6. For each agent $A_i$, a fixed operation $\mathcal{E}^i$ acting on their measured system and memory $S_i \cup \{M_i\}$ that captures the possibility of performing a further operation on the system $S_i$ after the measurement, depending on the measurement outcome.

7. A joint initial state $\rho_{S_1, ..., S_m}$ of all the systems $\mathcal{S}$ at time $t_0 < t_i$ for all $i \in \{1, ..., N\}$

The protocol takes the form of Figure 1, and we can consider projective measurements $\mathcal{M}^{A_i} = \{\pi_{a_i}^{S_i} = |a_i\rangle\langle a_i|_{S_i}\}_{a_i \in \mathcal{O}_i}$ without loss of generality.

The general form of an EWFS is illustrated in Figure 1. It is useful to note that we can assume without loss of generality that each agent performs only one measurement. Any scenario where an agent performs multiple measurements can be transformed into this form by modeling it as multiple agents, each performing a single measurement. Similarly, scenarios where an agent can choose between multiple measurements can be modelled as a single measurement by encoding the measurement choice in an initial state, as we show
<figure><img src="image_1.png" /><figcaption>FIG. 1: General form of an EWFS involving N agents and m systems described in the main text. All agents agree on the initial state $\rho_{S_1,...,S_m}$ of the systems and initialise their memories to $|0\rangle$. Each agent $A_i$ performs a measurement on some subset of the systems and on a subset of the memories of all agents who acted before, and they store the outcome of the measurement in their own memory $M_i$. After the measurement, each agent may perform a fixed transformation $\mathcal{E}^i$ (that is previously agreed upon by all agents) before the next agent measures.</figcaption></figure>

However, the agents need not necessarily agree on how each measurement is modelled, depending on their perspective some agents may describe a measurement as a purely unitary evolution as in Equation (3) while others may describe the same measurement as a decoherent process by assigning projectors as in Equation (8). Due to this ambiguity how a measurement is modelled, this diagram is not a fully specified quantum circuit.

explicitly in an upcoming work [18] for the LF scenario, which includes different measurement choices.

In an EWFS, the unitary description $\mathcal{M}_{\text{unitary}}^{A_i}$ of a measurement $\mathcal{M}^{A_i}$ corresponds to a CNOT from $S_i$ to the memory $M_i$ (chosen to be of appropriate dimensions) in the basis $\{|a_i\rangle_{S_i}\}_{a_i \in \mathcal{S}_i}$ of the measurement. For any state $|\psi\rangle_{S_i} = \sum_{a_i} c_{a_i} |a_i\rangle_{S_i}$ expressed in this basis, we have<fnref n="2" />

$$ \mathcal{M}_{\text{unitary}}^{A_i} : |\psi\rangle_{S_i} \otimes |0\rangle_{M_i} \mapsto \sum_{a_i} c_{a_i} |a_i\bar{a}_i\rangle_{S_i M_i} . \quad (3) $$

As highlighted by Wigner's thought experiment, in Wigner's Friend Scenarios (WFS), the ambiguity in modeling a measurement—whether through unitarity or the projection postulate—leads to observably different predictions. Quantum theory itself does not offer a clear set of rules to resolve this ambiguity or determine the correct predictions in an EWFS. Consequently, various sets of rules can be proposed for making predictions or statements about observed outcomes

in a given EWFS. Different interpretations of quantum theory yield different predictions for observable correlations in an EWFS, even though they agree on predictions for currently realisable quantum experiments.

Given this ambiguity and the potential for multiple approaches to making predictions and reasoning in an EWFS, we provide general definitions of predictions and statements. These definitions will offer a unified framework for discussing the wide range of previous results and responses to EWFS arguments, and their relation to our main results. In the following, whenever we have a subset $\mathcal{M}^{A_{j_1}},...,\mathcal{M}^{A_{j_p}}$ of measurements, it will be useful to denote the corresponding set of outcomes and set of values using vectors

$$ \begin{aligned} \vec{a}_j &:= a_{j_1}, \dots, a_{j_p} \\ \vec{a}_j &:= \bar{a}_{j_1}, \dots, \bar{a}_{j_p}. \end{aligned} \quad (4) $$

Then a value assignment to a set of outcomes, such as $a_{j_1} = a_{j_1}, \dots, a_{j_p} = a_{j_p}$ becomes $\vec{a}_j = \vec{a}_j$.

**Definition III.2** (Prediction and scenario parameters). Consider an EWFS along with a set $\mathcal{R}$ of rules for computing predictions in the scenario. A prediction in the EWFS is conditional probability $P(\vec{a}_j = \vec{a}_j | \vec{a}_l = \vec{a}_l, k = k)$, where $\vec{a}_j$ and $\vec{a}_l$ represent the set of outcomes associated with any two disjoint subsets $\mathcal{M}^{A_{j_1}},...,\mathcal{M}^{A_{j_p}}$ and $\mathcal{M}^{A_{l_1}},...,\mathcal{M}^{A_{l_p}}$ (latter possibilities

<footnote n="2">² In the following equation, we have used a pure initial state $|\psi\rangle_{S_i}$ to make the equations more concise, but it is easy to verify that analogous equations and the same arguments hold for mixed initial states $\rho_{S_i}$.</footnote>
bly empty) of measurements in the EWFS, and k is a (possibly empty) set of random variables whose values k encode additional information about the scenario (the exact description of which is to be specified by the rules R). Whenever P(ā_j = ā_j|ā_l = ā_l, k = k) takes values in {0, 1}, we refer to it as a logical prediction.

We have already seen an example of the k parameters in Section II A, where they represented the states, transformations and measurements we were conditioning on. We will see more examples later.

**Remark III.1.** In probability theory, a conditional probability is by definition “well-defined”. This means it is uniquely determined by the event space and is normalised, ensuring no contradictions arise when applying them. In the above definition, however, we do not require conditional probabilities to be uniquely defined based on the rules for calculating them (e.g., if the rules are ambiguous) or to sum to one (e.g., if the rules are inconsistent). We merely assume they are numbers in [0, 1], which can lead to “paradoxes” under rules that do not yield correctly normalised probabilities. We will see examples where the rules produce both well-defined and not well-defined conditional probabilities.

**Definition III.3** (Statements associated with predictions). Every prediction P(ā_j = ā_j|ā_l = ā_l, k = k) can be associated with a corresponding statement.

"If the outcomes ā_l take values ā_l and the additional parameters of the scenario take the value k = k, then the outcomes ā_l take values ā_l with a probability P(ā_j = ā_j|ā_l = ā_l, k = k)."

The set of all statements associated with predictions made using a set of reasoning rules in a given EWFS will be denoted as Σ (when the scenario and rules are evident from context).

**Definition III.4** (Logical statements). Statements associated with logical predictions are called logical statements. When we have a logical prediction P(ā_j = ā_j|ā_l = ā_l, k = k) = 1 (or = 0), the corresponding statement has the same conditional part as above, and the latter part of the statement becomes "...then it is certain that the outcomes ā_l (do not) take values ā_l." For logical predictions, we can also express the statements using logical operators AND (and), ⇒ (implies) and ¬ (negation). Specifically, the statements associated with P(ā_j = ā_j|ā_l = ā_l, k = k) = 0 and P(ā_j = ā_j|ā_l = ā_l, k = k) = 1 would respectively be

$$ \begin{aligned} \bar{a}_l &= \bar{a}_l \wedge k = k \Rightarrow \neg(\bar{a}_j = \bar{a}_j), \\ \bar{a}_l &= \bar{a}_l \wedge k = k \Rightarrow \bar{a}_j = \bar{a}_j. \end{aligned} \qquad (5) $$

The set Σ_L of all such logical statements of an EWFS is a subset of Σ.

In examples, we may have ā_l and k being empty sets, for instance we can have a prediction P(a_i = a_i) = 1. Then the associated logical statement is simply a_i = a_i

or in words “it is certain that the outcome a_i takes the value a_i.”

We then have the following definition of consistency for any set of predictive statements.

**Definition III.5** (Consistency for Σ). A set Σ of statements obtained in an EWFS from a set of reasoning rules is said to be consistent iff Σ contains no pairs of statements S and S' associated with predictions P(ā_j = ā_j|ā_l = ā_l, k = k) and P'(ā_j = ā_j|ā_l = ā_l, k = k) respectively where P ≠ P'.

**Lemma III.1.** If Σ is a set of consistent predictive statements, then

$$ S \in \Sigma \Rightarrow \neg S \cap \Sigma = \emptyset, \qquad (6) $$

where ¬S denotes the negation of the statement S.

At this point, it is important to note that there are two distinct ways in which one can be certain of an outcome value, as illustrated by the following simple example.

* **Scenario 1:** Alice has a fair coin yielding outcome c taking values in {heads, tails}. She flips the coin and observes c = heads in a round and is then certain that c = heads.

* **Scenario 2:** Alice has a biased coin that reads heads on both sides, P(c = heads) = 1 and she is certain that c = heads in every round (without observing the outcome).

Clearly, these scenarios are different. To ensure reliability and consistency when communicating with other agents, it is necessary to distinguish these cases. Our previous definition of statements associated with predictions only covers Scenario 2. To cover Scenario 1, we need to include a set of statements associated with observations that agents make in a given experimental run, denoted as Σobs. We define this case in Appendix B and also discuss a more refined definition of consistency that accounts for both predictive and observational statements there. For all the main results and discussions, it will be sufficient to consider the predictive statements Σ and the generalisation of the results to include Σobs is presented in the appendix. This is because a theory only permits observation of certain outcomes when its predictions assign a non-zero probability to that outcome, thus the main features of the theory can be understood by studying its predictions and associated statements as defined here.

## B. Formulating an EWFS within a single quantum circuit

As stressed before, in EWFS, there is ambiguity in applying the postulates of quantum theory to compute probabilities. The postulates can be applied in different ways (using chosen “rules”) to obtain different probabilities for the same outcomes, while distinct rules could
still lead to the same probabilities in certain cases. In previous literature on EWFS, there is a common pattern in the probabilities considered used in EWFS arguments, such as in the FR and LF results. However, the rules for arriving at these probabilities are often not explicitly specified.

We consider the probabilities conventionally used in previous literature when they refer to “predictions of quantum theory” in the context of EWFS, where it is assumed that unitary evolutions can be applied to agents’ labs.<fnref n="3" /> These probabilities typically do not condition on additional variables *k* or channels in the circuit, their expressions only refer to the outcomes, e.g., *P*(*a* = 0, *b* = 1) denotes the probability of the measurement outcomes *a* = 0 and *b* = 1 (which can be associated with agents Alice and Bob) in a given scenario. Below we formalise one way to arrive at these type of probability expressions in an EWFS, which recovers the probabilities used in the FR and LF scenarios.

**Definition III.6** (Conventional predictions in an EWFS). A conventional prediction in an EWFS is a conditional probability $P_{\text{conv}}(\vec{a}_j = \vec{a}_j|\vec{a}_l = \vec{a}_l)$ evaluated by applying the following rules.

1. In the given EWFS (see also Figure 1), all the measurements $M^{A_i} \notin \{M^{A_{j_1}}, \dots, M^{A_{j_p}}\} \cup \{M^{A_{l_1}}, \dots, M^{A_{l_q}}\}$ (those not appearing in the prediction) are modelled as unitary evolutions of their labs (according to the channel Equation (3)).

2. For each $M^{A_i} \in \{M^{A_{j_1}}, \dots, M^{A_{j_p}}\} \cup \{M^{A_{l_1}}, \dots, M^{A_{l_q}}\}$, the projective measurement $\{\pi_{\vec{a}_i}^{S_i} = |a_i\rangle\langle a_i|_{S_i}\}_{a_i \in \mathcal{O}_i}$ is applied on the corresponding system $S_i$, followed by a CNOT in the same basis from the system $S_i$ to the memory $M_i$ which models the procedure of making the measurement on the system and storing the outcome in the memory.

3. This fixes all the channels in the circuit, and the joint probability $P_{\text{conv}}(\vec{a}_j = \vec{a}_j, \vec{a}_l = \vec{a}_l)$ is then calculated by applying the Born rule to this circuit, using the measurement projectors given above.<fnref n="4" />

4. The prediction $P_{\text{conv}}(\vec{a}_j = \vec{a}_j|\vec{a}_l = \vec{a}_l)$ is then obtained through the usual conditional probability rule.

A conventional prediction corresponds to the case where *k* is the empty set, meaning the prediction is not conditioned on any additional information about the scenario (such as states and channels used). Such predictions often lead to FR-type paradoxes, highlighting that the above rules for computing probabilities in EWFS are not generally consistent and do not yield valid joint probabilities for measurement outcomes in EWFSs.

In standard quantum theory outside the context of WFS, where agents' labs are not typically modelled as quantum systems in a circuit, all predictions for a given experiment are defined relative to a single circuit, ensuring consistent and well-defined joint probabilities. In EWFS, it is initially unclear which circuit is associated with a given scenario due to the ambiguity in the channels modelling the measurements (as discussed in Section II B), which has observable consequences in these scenarios. We now explain how every EWFS can be mapped to a single quantum circuit which we call the *augmented circuit* of the EWFS, from which all the predictions of the EWFS can be computed and used consistently, while allowing measurements to be modelled as unitary evolutions of agents' labs and also preserving the validity of the Born rule.

For this, recall that for a measurement $M^{A_i}$ in a basis defined by the projectors $\{\pi_{\vec{a}_i}^{S_i} = |a_i\rangle\langle a_i|_{S_i}\}_{a_i \in \mathcal{O}_i}$ acting on a state $|\psi\rangle_{S_i} = \sum_{a_i} c_{a_i} |\vec{a}_i\rangle_{S_i}$, the associated unitary evolution $M_{\text{unitary}}^{A_i}$ is as given in Equation (3). On the other hand, applying the projection postulate for the measurement on $S_i$ and the fact that the outcome is copied to the memory $A_i$, one would obtain the final state $|a_i\vec{a}_i\rangle\langle a_i\vec{a}_i|_{S_i M_i}$ when the outcome is $a_i$. If we consider the view that such a classical outcome is obtained but one lacks knowledge of its value, we would obtain the following trace-preserving evolution.

$$ M_{\text{projection}}^{A_i}(|\psi\rangle\langle\psi|_{S_i} \otimes |0\rangle\langle0|_{M_i}) = \sum_{a_i} |c_{a_i}|^2 |\vec{a}_i\vec{a}_i\rangle\langle a_i\vec{a}_i|_{S_i M_i} \quad (7) $$

Notice that $M_{\text{projection}}^{A_i}(|\psi\rangle\langle\psi|_{S_i} \otimes |0\rangle\langle0|_{M_i})$ above can be equivalently expressed as

$$ \sum_{a_i} \pi_{\vec{a}_i M_i}^{S_i M_i} (M_{\text{unitary}}^{A_i} (|\psi\rangle\langle\psi|_{S_i} \otimes |0\rangle\langle0|_{M_i}) M_{\text{unitary}}^{A_i \dagger}) \pi_{\vec{a}_i M_i}^{S_i M_i} \quad (8) $$

where $\{\pi_{\vec{a}_i}^{S_i M_i} := |a_i\vec{a}_i\rangle\langle a_i\vec{a}_i|_{S_i M_i}\}_{a_i \in \mathcal{O}_i}$, and we still have a one-to-one correspondence between the measurement outcomes $\vec{a}_i$ and an element from the above set of projectors. Therefore the two possible (trace-preserving) evolutions associated with a measurement $M^{A_i}$ are now given by Equation (8) and Equation (3). They differ is whether or not one views the measurement as having produced classical records.

We explicitly account for this choice by introducing a classical binary variable for each measurement that takes values $x_i \in \{0, 1\}$ (which we call the *setting*), such that $x_i = 0$ and $x_i = 1$ correspond to the evolutions Equation (3) and Equation (8) respectively.

<footnote n="3">³ Collapse theories, for instance, would violate this assumption and prescribe different probabilities and rules for computing them.</footnote>

<footnote n="4">⁴ One can formally write down the probability expressions obtained in steps 3 and 4 through the Born rule. However, in the interests of conciseness, we will decline from doing so at this point, as we will show that these conventional predictions since are recovered as a special case of our to-be-described and rigorously defined general formalism (Theorem IV.1).</footnote>
This is captured through the following enlarged set of projectors, obtained by appending a trivial outcome value $⊥$ to the outcome set $0_i$.

$$ \Pi_{x_i}^{A_i} := \begin{cases} \{\pi_{a_i, x_i}^{A_i}\}_{a_i \in \{\perp\}} = \{\pi_{\perp, 0}^{A_i}\} := 1_{S_i M_i} & \text{if } x_i = 0 \\ \{\pi_{a_i, x_i}^{A_i}\}_{a_i \in 0_i} = \{\pi_{a_i, 1}^{A_i} := \pi_{s_i M_i}^{A_i}\}_{a_i \in 0_i} & \text{if } x_i = 1. \end{cases} \tag{9} $$

That is, for $x_i = 0$ the value of the outcome $a_i$ is fixed uniquely to $a_i = \perp$ and the corresponding projector is the identity operator $1_{S_i M_i}$. Meanwhile, for $x_i = 1$, $a_i$ takes values $a_i$ in the set $0_i$ (reflecting the different possible classical outcomes of the measurement, see Definition III.1). We will only use the terminology *outcome*, to refer to the value of $a_i$ when $x_i = 1$. In the case $x_i = 0$, $a_i = \perp$ and we will not regard this as an outcome (we will sometimes refer to this case as the *trivial outcome*).

Then we can model each measurement $M^{A_i}$ as the corresponding unitary $M_{unitary}^{A_i}$, followed by a (setting-dependent) projective measurement $\Pi_{x_i}^{A_i}$. The pure unitary picture is always recovered by setting $x_i = 0$, in which case only $M_{unitary}^{A_i}$ unitary is applied. The set of choices of settings for all $N$ agents can be represented by a vector $\vec{x} = (x_1, \dots, x_N)$. With this explicit model, we can transform any EWFS into a standard temporally ordered quantum circuit parametrised by the settings $\vec{x}$ such that all the predictions made by different agents having different perspectives can be derived from the single circuit by fixing the settings $\vec{x}$ (as shown in Theorem IV.1). We call this the *augmented circuit* of the EWFS. This is illustrated in Figure 2, and summarised in the following definition.

**Definition III.7 (Augmented circuit of an EWFS).** Given an EWFS of the form of Figure 1, we can associate with it an augmented circuit of the form of Figure 2 which is obtained from the EWFS by replacing each measurement $M^{A_i}$ by the corresponding unitary description $M_{unitary}^{A_i}$ (Equation (3)), followed by the setting-dependent enlarged set of projectors $\Pi_{x_i}^{A_i}$ (Equation (9)). The setting takes binary values $x_i \in \{0, 1\}$, where we have only the trivial outcome $⊥$ whenever $x_i = 0$ and the non-trivial measurement outcome $a_i \in 0_i$ whenever $x_i = 1$. We will refer to this as an augmented EWFS, in short.

We now make a crucial observation. While we have so far given full freedom in choosing how to model the measurements, with both $x_i = 0$ (pure unitary evolution) and $x_i = 1$ (also assigning non-trivial projectors) being allowed, we notice that the form of the prediction being computed using the Born rule fixes some of these choices. For example, consider a prediction $P(a_i = a_i|\vec{x})$. To compute this using the Born rule, we must apply the projector associated with the outcome $a_i = a_i$, which is needed to identify that outcome. Therefore for this prediction, $x_i = 1$. The other components of the setting vector $\vec{x}$ can generally vary according to the rules of reasoning being applied.

When we say “applying a projector”, this is not to be conflated with “collapsing” the state. Even when we

do not explicitly refer to the post-measurement state of an agents’ measurement, when reasoning about that agents’ outcome using the Born rule, we apply knowledge of the measurement basis (given here by the projectors which identify the outcome). This is further discussed in Remark V.1, where we show how our framework can also be applied to EWF scenarios where the Born is rule is applied without need for a particular state update rule. The main observation above is that given a prediction, the measurements whose outcomes appear in that prediction are always regarded as producing classical records (associated with $x_i = 1$ in our framework) when computing the prediction.

Applying this simple observation about quantum probabilities to agents’ reasoning, we can consider an example. If $A_i$ reasons about $A_j$’s outcome based on their own outcome, they set $x_i = x_j = 1$ in order to identify the classical outcomes they are reasoning about, but they may choose $x_k = 0$ for all $k \neq i, j$ and model all other agents unitarily. How these remaining settings are to be chosen will need to be specified by a set of reasoning rules $\mathcal{R}$, which we do not yet fix. This generality will become relevant when discussing the different responses to FR’s no-go theorem (Appendix I).

Having defined the augmented circuit, we explain how one can compute predictions here. This can be done by applying the Born rule along with the conditional probability rule, as long as we are given a choice of initial values for the setting vector $\vec{x}$ (as this is required for the full specification of the circuit).

Different interpretations of quantum theory would generally propose different rules $\mathcal{R}$ to fully specify $\vec{x}$ and can arrive at different predictions for the same EWFS. As we will see in Theorem IV.1, formalising conventional predictions (Definition III.6) in our augmented circuit yields an explicit rule for choosing the settings which will be relative to a subset of systems of measurements being considered, and can therefore allow subjective choices of settings when we consider reasoning agents (see also Section VII A).

Until then, all our results regarding the augmented circuit apply to all possible rules for choosing the settings and hence apply to several interpretations of quantum theory (both relational and non-relational ones).

**Definition III.8 (Setting-conditioned prediction).** Given an EWFS, a setting-conditioned prediction is a prediction $P(\vec{a}_j = \vec{a}_j|\vec{a}_l = \vec{a}_l, \vec{x} = \vec{\xi})$ (Definition III.2) where the setting vector $\vec{x}$ represents the set $k$ of scenario parameters. A setting-conditioned prediction is computed by mapping the EWFS to an augmented circuit, then applying the quantum Born rule together with the rule for conditional probabilities for the given choice $\vec{x} = \vec{\xi}$ of the settings. This is explicitly shown in Appendix D. Whenever $P(\vec{a}_j = \vec{a}_j|\vec{a}_l = \vec{a}_l, \vec{x} = \vec{\xi}) \in \{0, 1\}$, we will refer to it as a logical setting-conditioned prediction.

**Definition III.9 (Statements associated with setting-conditioned predictions).** Every setting-conditioned
prediction $P(\vec{a}_j = \vec{a}_j|\vec{a}_l = \vec{a}_l, \vec{x} = \vec{\xi})$ can be associ-
ated with a corresponding statement, in the same way
that Definition III.3 assigns statements to predictions
$P(\vec{a}_j = \vec{a}_j|\vec{a}_l = \vec{a}_l, k = k)$, but for setting-conditioned
predictions, the scenario parameter values $k = k$ are
replaced with the setting values $\vec{x} = \vec{\xi}$. Statements as-
sociated with logical setting-conditioned prediction can
be expressed using logical operators as given in Defini-
tion III.4 but with $k = k$ replaced by $\vec{x} = \vec{\xi}$.

**Definition III.10** (Set of all statements in an augmented EWFS). *Consider the set of all setting-conditioned predictions that can be made in an augmented circuit of an EWFS, obtained under all possible setting choices. The set of all statements $\Sigma^{\text{aug}}$ in that EWFS is obtained by mapping each setting-conditioned prediction to a corresponding statement as per Definition III.9. Similarly, restricting to the set of all logical predictions, we have the corresponding set of all logical statements of the scenario which will be denoted as $\Sigma_L^{\text{aug}} \subseteq \Sigma^{\text{aug}}$.*

**Remark III.2.** In setting-conditioned predictions, the setting variables are fixed to some specific value deterministically. More generally, we can consider arbitrary prior distribution over the settings $\vec{x}$ in our framework. This allows to compute a probability distribution such as $P(\vec{a}_j = \vec{a}_j)$ which does not explicitly feature any settings, which is obtained by choosing some prior $P(\vec{x})$ and averaging over the settings, $P(\vec{a}_j = \vec{a}_j) = \sum_{\vec{\xi}} P(\vec{a}_j = \vec{a}_j|\vec{x} = \vec{\xi}) P(\vec{x} = \vec{\xi})$.

We have established that for statements made us-
ing setting-conditioned predictions, the specific setting
value assumed must be explicitly specified. Analogously,
our framework and results can be generalised to accom-
modate arbitrary priors by ensuring that when using
such priors to compute outcome probabilities or pre-
dictions, the corresponding statements about the mea-
surement outcomes specify the chosen prior over the
settings, as this is an additional choice that an agent
must make in the reasoning process. However, we will
not delve into the case of general prior distributions
further, as it is not pertinent to the main results and
insights of our work.

IV. COMPLETENESS, CONSISTENCY AND CAUSALITY WITHOUT ABSOLUTE EVENTS

In this section, we formalise and prove several key
properties of our framework that are pertinent to log-
ical and causal reasoning in EWFS, without imposing
an absolute notion of measurement events.

A. Properties of general quantum predictions in EWFS

We begin by noting that our framework does not as-
sume an absolute and objective notion of measurement

events, unlike the majority of existing frameworks for
describing quantum information protocols. It does not
require the existence of a single objective joint prob-
ability distribution $P(a_1, .., a_N)$ over the (non-trivial)
outcomes $a_i \in \mathcal{O}_i$ of all measurements in the scenario.
Rather, by introducing settings $\vec{x}$, which as we will dis-
cuss later Section VII C, model choices of Heisenberg
cuts, the framework allows outcome probabilities to be
fundamentally relational.

For example, suppose we have an EWFS with two
measurements $\mathcal{M}^{\text{A}_1}$ and $\mathcal{M}^{\text{A}_2}$. If we treat the joint
system $S_1M_1$ (modelling the lab of the first agent) as
storing classical outcome records after the measure-
ment while regarding the $S_2M_2$ as a purely unitarily
evolving quantum system, then $x_1 = 1$ and $x_2 = 0$
and we can only compute probabilities involving the
non-trivial measurement outcomes $a_1 \in \mathcal{O}_1$. If we treat
both measurements as being associated with classical
records, then we have $x_1 = x_2 = 1$ and can compute
joint probabilities of $a_1$ and $a_2$, the probability for $a_1$
in the two cases need not generally agree since we use
a different setting $x_2$ in computing this probability in
the two cases.

We now show that even though our augmented cir-
cuit formalism does not a-priori impose absoluteness of
events, it provides a complete representation of all pre-
dictions that can be made in an EWFS in a way that
is consistent and respects causality principles. Before
stating the formal theorem, we clarify what we mean
by causality principles. To formalise this, we begin by
defining a directed acyclic graph (DAG)<fnref n="5" /> that corre-
sponds to every EWFS. This DAG captures the poten-
tial information flow within the EWFS protocol, ad-
hering to the time-ordered sequence of operations. We
will refer to this as the causal structure of the EWFS.

Recall that by definition, an EWFS with $N$ agents
is modelled as having $2N$ operations w.l.o.g., as each
agent performs a measurement $\mathcal{M}^{\text{A}_i}$ followed by some
quantum channel $\mathcal{E}_i$, both associated with a time step
$t_i$. Let $\mathcal{O}$ denote any one of these $2N$ operations we will
denote by $S_{\mathcal{O}}$ the subset of all systems and memories
$S_{\bigcup \mathcal{M}}$ such that the operation $\mathcal{O}$ acts non-trivially on all
systems in $S_{\mathcal{O}}$ and as the identity on the rest of $S \cup \mathcal{M}$.

**Definition IV.1 (Causal structure of an EWFS).** The causal structure of an EWFS with $N$ agents is a directed acyclic graph (DAG) $G$ with the following properties

1. $G$ has $2N$ vertices given by the set $\text{Vert}(G) := \{V_i^M, V_i^\mathcal{E}\}_{i=1}^N$, where $V_i^M$ and $V_i^\mathcal{E}$ are respectively associated with the operations $\mathcal{M}^{\text{A}_i}$ and $\mathcal{E}_i$, and both pairs of such vertices are associated with the time step $t_i$, for each $i$.

<footnote n="5">⁵ In simple terms, a DAG is a graph with additional structure: 1) The edges have a direction associated with them. 2) By "following the edges in the indicated direction" one can never get back to the starting point, i.e. no "directed loops".</footnote>
<figure><img src="image_2.png" /><figcaption>FIG. 2: Augmented circuit for the general form of an EWFS illustrated in Figure 1 that makes explicit the implicit setting choices needed to model the measurement of each agent. This circuit makes it clear that each agent has a choice in how they describe each measurement in the scenario, when the setting $x_i = 0$, the corresponding measurement $\mathcal{M}^{A_i}$ is modelled as a unitary evolution $\mathcal{M}_{\text{unitary}}^{A_i}$ (in this case the projector in the blue box implements an identity operation and $a_i = \perp$ deterministically) and when $x_i = 1$, the same measurement is modelled as the unitary $\mathcal{M}_{\text{unitary}}^{A_i}$ followed by non-trivial projectors that identify the classical measurement outcome $a_i \in \{0, 1, \dots, d_{S_i} - 1\} := 0_i$. In order for any agent to reason about the measurement outcome $a_i$ of an agent $A_i$, they must necessarily choose $x_i = 1$ in order to identify and calculate probabilities for the classical outcome that they are reasoning about. They may however choose to model all other agents $A_j$ unitarily by choosing $x_j = 0$.</figcaption></figure>

2. G contains a directed edge $V \to V'$ for $V, V' \in \text{Vert}(G)$ whenever $t_V < t_{V'}$ and $S_{O_V} \cap S_{O_{V'}} \neq \emptyset$, where $O_V$ and $O_{V'}$ are the operations, and $t_V$ and $t_{V'}$ are the time steps associated with the vertices $V$ and $V'$ respectively.

**Definition IV.2 (Directed paths and partial order).**
The DAG G associated with an EWFS in Definition IV.1 defines a partial order relation $\prec$ on agents in the EWFS, with $A_i \prec A_j$ if and only if there is a directed path from the measurement vertex $V_i^M$ of $A_i$ to the measurement vertex $V_j^M$ of $A_j$. Furthermore, we will write $A_i \prec^S A_j$ whenever there is such a directed path and the system S is included in at least one of the set intersections $S_{O_V} \cap S_{O_{V'}}$, involved in that path. We use $A_i \not\prec A_j$ and $A_i \not\prec^S A_j$ to denote the absence of directed paths with the above-defined properties.

Crucially, notice that the above graph G and induced partial order $\prec$ represent objective properties inherent to the EWFS, independent of the specific settings chosen to model the measurement $\mathcal{M}^{A_i}$ in the augmented EWFS. This is because the definition only relies on the time order of operations and the set of systems on which the operations act. These aspects are included in the description of the original EWFS, Definition III.1, and

the settings do not affect them.

Given this definition, a natural causality principle is that an outcome $a_j$ should not depend on a setting $\xi_i$ of a measurement whenever $A_i \not\prec A_j$. As our circuits are acyclic and operations therein have a clear time ordering, we have $A_i \prec A_j$ implies $t_i < t_j$, and it is easy to see that this ensures that G is indeed a directed acyclic graph and that $\prec$ is a partial order relation. Moreover, this causality principle ensures that there is no retrocausal dependence of outcomes on future settings, despite settings in our formalism representing Heisenberg cuts rather than actual experimental measurement choices (further elaborated in Section VII C).

Finally we note that $t_i < t_j$ does not necessarily imply $A_i \not\prec A_j$ as it is possible to have two measurements acting at different times but on non-overlapping sets of systems, then G will not contain any directed paths between the measurements of these agents. Thinking of different subsystems as being embedded at different “spatial locations”, we can regard such disjoint sets of systems as being space-like separated. Therefore, if we consider the circuit as being embedded in a spacetime, with an output of one operation connected to the input of another only if the former is in the past light-cone of the latter, then $\prec$ is compatible with the causal structure.
ture of the spacetime [23, 24].

**Theorem IV.1.**

1. Completeness: In any given EWFS, all conventional predictions in that EWFS can be derived within the single augmented circuit of that EWFS. More explicitly, each conventional prediction $P_{conv}(\vec{a}_j = \vec{a}_j|\vec{a}_l = \vec{a}_l)$ in the EWFS equals a particular setting conditioned prediction $P(\vec{a}_j = \vec{a}_j|\vec{a}_l = \vec{a}_l, \vec{x} = \vec{\xi}^*)$ of the augmented circuit where the setting choice $\vec{x} = \vec{\xi}^*$ is such that $x_i = 1$ for all $i \in \{j_1, ..., j_p, l_1, ..., l_q\}$ and $x_i = 0$ for all $i \notin \{j_1, ..., j_p, l_1, ..., l_q\}$.

2. Consistency: For any EWFS, the set of all statements $\Sigma^{aug}$ obtained in the corresponding augmented circuit (Definition III.10) are consistent according to Definition III.5.

3. Causality: For every setting-conditioned prediction $P(\vec{a}_j = \vec{a}_j|\vec{a}_l = \vec{a}_l, \vec{x} = \vec{\xi})$, and every $i$ such that $A_i \not\in A_k$ for all $k \in \{j_1, ..., j_p, l_1, ..., l_q\}$, the prediction is independent of the setting $x_i$. That is, for all such $i$, we have the following, where we denote $P(a=a)$ as $P(a)$ for short and note that $\vec{x} = (x_1, ..., x_N)$.

$$ \forall \xi_i, \xi_i', \quad P(\vec{a}_j|\vec{a}_l, (\xi_1, \dots, \xi_i, \dots, \xi_N)) = P(\vec{a}_j|\vec{a}_l, (\xi_1, \dots, \xi_i', \dots, \xi_N)). \quad (10) $$

Whenever a setting-conditioned prediction is independent of a setting $x_i$, i.e., satisfies Equation (10), we will simply drop $x_i$ from that prediction and denote it as follows.

$$ P(\vec{a}_j|\vec{a}_l, (\xi_1, \dots, \xi_N)) = P(\vec{a}_j|\vec{a}_l, (\xi_1, \dots, \xi_{i-1}, \xi_{i+1}, \dots, \xi_N)) \quad (11) $$

**B. Application to agents’ reasoning**

So far our results have been about the properties of predictions (probabilities) that can be computed in an EWFS and showing that our formalism yields a complete, logical and causally consistent way to make predictive statements in such scenarios. We now apply our general formalism to the subject of agents’ reasoning in EWFS. One immediate corollary of our general consistency result of Theorem IV.1, for agents’ reasoning is the following.

**Corollary IV.1.** If any two agents use the same choice of settings $\vec{x}$ for all measurements $\{M^{A_i}\}_i$ in an augmented EWFS then they make all the same predictions in that scenario.

This corollary concerns a rather restricted case where all agents have a fixed and common choice of Heisenberg cut (here formalised through the settings). More generally, agents can choose different settings depending on their perspective or prediction they wish to

compute. Below, we show that our formalism enables agents in an EWFS to reason consistently even when they apply typical axioms of classical logic to observed classical outcomes, and can model each others’ labs as unitarily evolving quantum systems and have fundamentally subjective perspectives. In particular, we will consider the logical axioms used in FR’s argument, which relate to the inheritance of knowledge (of other trusted agents) and the distributivity of logical statements. We first formalise the relevant logic axioms and the quantum theory dependent assumptions and in the context of our framework.

One of the assumptions (or logical axioms) used in the FR argument is of the form “If Alice is certain that Bob is certain that the outcome $a = 1$”, then “Alice is certain that $a = 1$”. This is called the C assumption there, and is about the ability of Alice to inherit Bob’s knowledge. In [5], this assumption C was formalised within the framework of modal logic. Here $K_{A_i}(S)$ is used to denote that “agent $A_i$ knows that the statement $S$ is true”, where $K^{A_i}$ is known as a knowledge operator (see [5] for a formal definition of the knowledge operators in terms of Kripke structures in modal logic). The assumption C can then be succinctly expressed as an inference of the form

$$ K_{A_i} K_{A_j}(S) \Rightarrow K_{A_i}(S), \quad (12) $$

More specifically, it is pointed out in [5] that such an inference only needs to be made when the agent $A_i$ trusts the agent $A_j$ (for otherwise $A_i$ may not believe in everything that $A_j$ claims to know, and may not want to inherit $A_j$’s knowledge). In Wigner’s Friend scenarios such a that of FR, [5] instantiate the trust structure by considering pairs of agents performing compatible measurements, and only apply C for such pairs. The trust structure will not be relevant for the general solution that we propose here because, as we will show (c.f. Theorem IV.1), the inclusion of the settings in our framework ensures the general validity of Equation (12) in an augmented EWFS independently of the trust structure.

**Definition IV.3 (Assumption C).** For any two agents $A_i$ and $A_j$, Equation (12) holds for all statements $S \in \Sigma^{aug}$.

As noted in [5], the FR argument also uses the distributive axiom of logic. We instantiate this in our formalism below.

**Definition IV.4 (Assumption D).** For any set of agents reasoning using the augmented circuit, if $S_1 \in \Sigma_L^{aug}$, and $S_1 \Rightarrow S_2$, then $S_2 \in \Sigma_L^{aug}$ holds and we have

$$ K_{A_i}(S_1 \wedge (S_1 \Rightarrow S_2)) \Rightarrow K_{A_i}(S_2), \quad (13) $$

that is if an agent $A_i$ knows $S_1$ and also that $S_1$ implies $S_2$, then the agent knows $S_2$.

Finally, we consider the S assumption of FR which states that an agent cannot be certain of two opposite
values—say $a = 0$ and $a = 1$—of a measurement outcome. In our framework, this is a weaker version of our general consistency condition of Definition III.5 as the latter applies to all predictions while the former only to logical predictions.

**Definition IV.5 (Assumption S).** For any subset $\vec{a}_j$ of measurement outcomes, if $P(\vec{a}_j = \vec{a}'_j) = 1$, then it is impossible to have $P(\vec{a}_j = \vec{a}'_j) = 1$ for any outcome values $\vec{a}_j \neq \vec{a}'_j$.

Now, for the quantum theory dependent assumptions which independently capture the validity of the Born rule and of unitary evolution of closed quantum systems (including agents’ labs).

**Definition IV.6 (Assumption Q).** Consider a statement $S :=$ “If the outcomes $\vec{a}_i$ take values $\vec{a}_i$ and the settings take the value $\vec{x} = \xi$, then the outcomes $\vec{a}_i$ take values $\vec{a}_i$ with a probability $P$.” Agents in an EWFS can regard such a statement $S$ as true if and only if the corresponding setting-conditioned prediction $P(\vec{a}_j = \vec{a}'_j|\vec{a}_l = \vec{a}'_l, \vec{x} = \vec{\xi}) = P$ can be derived by applying the Born rule to the EWFS (as detailed in Definition III.8).

We note that by construction, the set of all statements $\Sigma^{aug}$ constructed from setting-conditioned predictions in the augmented circuit has this property. In our formalisation of the **U** assumption, we make explicit another implicit assumption in the previous literature, which relates to quantum control over other agents’ labs.

**Definition IV.7 (Assumption U).** Agents can choose the setting $x_i = 0$ (i.e., pure unitary description) for the measurement of any agent $A_i$ whose outcome they are not logically reasoning about i.e., for every setting-conditioned prediction $P(\vec{a}_j = \vec{a}'_j|\vec{a}_l = \vec{a}'_l, \vec{x} = \vec{\xi})$, both choices $x_i \in \{0,1\}$ are allowed for all $i \notin \{j_1, ..., j_p, l_1, ..., l_q\}$. Moreover, agents can have full quantum control over the labs of other agents, i.e., the measurement of an agent $A_i$ can act non-trivially on the total system $S_jM_j$ comprising the lab of another agent $A_j$.

We then obtain the following corollary, which follows by construction of our framework along with the general consistency result of Theorem IV.1. We nevertheless include a proof in Appendix J for completeness.

**Corollary IV.2.** If agents in an EWFS reason about each other's knowledge using the augmented circuit for the scenario, then they can never arrive at a logical contradiction even if they reason using all five assumptions **Q**, **U**, **C**, **D** and **S**.

We have given a formalisation **Q**, **U**, **C**, **D**, **S** of the FR assumptions **Q**, **U**, **C**, **D** and **S** within our framework, showing our version of the 5 assumptions to be perfectly consistent in an EWFS, despite FR's claim that the original version of these assumptions lead to

contradictions. FR's assumptions, although motivated as capturing “quantum theory” and “logical axioms” do not appear to be fully and rigorously formalised, allowing room for interpretation (see also Appendix I for references to previous responses to FR's arguments). See Section VC for further discussion on the interpretation of FR's claims in light of our results.

Physically, our assumptions **Q**, **U**, **C**, **D**, **S** still encompass the same essential physical requirements highlighted by FR: the universal applicability of quantum theory (Born rule + unitarity), the inheritance of agents' knowledge, and the validity of classical logic applied to knowledge of measurement outcomes. However, mathematically, within the Kripke structure of modal logic, the set of statements $\Sigma$ to which our assumptions apply differs from that considered in the original modal logic formulation of FR's result (as given in [5]), due to the additional structure provided by the setting labels in our framework.

In the forthcoming section, we identify an additional assumption within our framework necessary to reproduce apparent inconsistencies akin FR's. This underscores that, even at a physical level, an implicit assumption concerning the independence of predictions from the choice of Heisenberg cuts is required to establish an FR-type no-go theorem in quantum theory.

### C. Reason for apparent inconsistencies

Despite satisfying a formal version of each of the FR assumptions (Theorem IV.1 and Corollary IV.2), our framework remains logically consistent. This raises the question: what additional assumption is needed to reproduce logical contradictions such as the apparent FR paradox within our framework?

The main difference between our framework and previous analyses of EWFSs is in the explicit introduction of the settings. Here we say “explicit” since the settings are indeed present in the conventional computations of quantum predictions, as a choice about how measurements are modelled has to be made when computing the probabilities. The difference between these conventional predictions and statements, and ours is that the former do not specify this in the probability expressions or at the level of statements being made while our formalism does so. Dropping this choice corresponds to an assumption regarding the ability to ignore the setting choice or the Heisenberg cut. We formalise this assumption below within our framework, and show that this is necessary to recover a paradox. This in turn yields a more precise and refined interpretation of FR type apparent paradoxes (as discussed in Section VC after analysing the example of the FR scenario).

**Definition IV.8 (Assumption I: Independence).** A setting-conditioned prediction $P(\vec{a}_j = \vec{a}'_j|\vec{a}_l = \vec{a}'_l, \vec{x} = \vec{\xi})$ is said to be setting-independent if $P(\vec{a}_j = \vec{a}'_j|\vec{a}_l = \vec{a}'_l, \vec{x} = \vec{\xi}) = P(\vec{a}_j = \vec{a}'_j|\vec{a}_l = \vec{a}'_l, \vec{x} = \vec{\xi}')$ for all allowed
values $\xi$ and $\xi'$ of the settings.<fnref n="6" /> Then the prediction can be consistently represented by dropping the settings, as $P(\vec{a}_j = \vec{a}'_j | \vec{a}_l = \vec{a}'_l, \vec{x} = \vec{\xi}) = P(\vec{a}_j = \vec{a}'_j | \vec{a}_l = \vec{a}'_l)$.

We can then immediately obtain the following corollary of our results.

**Corollary IV.3.** In order to obtain an apparent logical contradiction (i.e., a violation of S) in an augmented EWFS where agents reason using assumptions Q, U, C and D, it is necessary to assume I on at least one logical setting-conditioned prediction that is not setting-independent.

This corollary follows because firstly, using Corollary IV.2 we have that the augmented EWFS is perfectly consistent with all five assumptions Q, U, C, D and S. Further, the consistency of our framework captured by Theorem IV.1 and Corollary IV.1 implies that any apparent violation of S (i.e., $P(\vec{a}_j = \vec{a}'_j) = 1$ and $P(\vec{a}_j = \vec{a}'_j) = 1$ for $\vec{a}'_j \neq \vec{a}_j$) that might be obtained in an EWFS (for instance, the violation obtained by FR [2]), necessarily arises by computing the probability of the outcomes under two distinct choices of settings and then ignoring this choice by identifying the two predictions as the same (which is equivalent to applying I to the two setting-conditioned predictions).

In other words, an apparent violation of S such as $P(\vec{a}_j = \vec{a}'_j) = 1$ and $P(\vec{a}_j = \vec{a}'_j) = 1$ when formulated explicitly within our framework, always translates to $P(\vec{a}_j = \vec{a}'_j | \vec{x} = \vec{\xi}) = 1$ and $P(\vec{a}_j = \vec{a}'_j | \vec{x} = \vec{\xi}') = 1$ (where $\vec{\xi}$ and $\vec{\xi}'$ are two distinct setting values), which is not paradoxical as it refers to two distinct conditional probability distributions. Note however that this necessarily violates I since the predictions are indeed dependent on the setting, which is why the apparent paradox is recovered when I is imposed.

V. A SIMPLE RESOLUTION TO THE FR APPARENT PARADOX

Having developed a fully general framework, here we apply it to an example to show it in action. In particular, we provide a simple resolution to the FR paradox. We focus here on the entanglement version of the FR protocol illustrated in Figure 3 (and reviewed in detail in Appendix C 2), describing the main idea behind the resolution. This version of FR’s protocol was proposed by Luis Masanes and Matthew Pusey in their talks. In Appendix F 1, a detailed analysis of the entanglement version of the FR scenario can be found, which explicitly shows all the calculations backing the main points.

Furthermore, while many consider the entanglement version to be equivalent to the original prepare and

measure version, it has been suggested by the authors of the FR paper that the entanglement version misses important subtleties regarding the timing information involved in the reasoning process, to which a lot of care has been given in the original FR formulation of the experiment. Our results are fully general and resolve (in particular) the apparent paradoxes arising in both these situations. In Appendix F 2, we resolve the original prepare and measure version of the FR paradox, while giving a statement-by-statement analysis and comparison to show how the paradox completely disappears in our framework even though all the agents can freely reason using unitary quantum theory, the standard Born rule and classical logic, and all the individual statements of FR’s original argument can be reproduced.

A. Augmented circuit

We first formulate the entanglement version of the FR protocol within our framework by giving its augmented circuit. A typical circuit associated with this scenario, also found in the previous literature, is shown in Figure 3. The caption of the figure gives a quick recap of the protocol sufficient to follow this discussion. It is one which models the measurements of the agents Alice and Bob purely unitarily. As these are computational basis measurements, the corresponding unitary is the CNOT. However, this circuit alone does not allow us to calculate the probabilities of Alice and Bob’s classical outcomes. In the previous literature (see for instance [9]), multiple different circuits are considered for calculating the probabilities from the perspective of each agent. In our framework, we have shown that all such reasoning can in fact be captured with a single circuit—the augmented circuit. This circuit includes a setting $x_i$ for each agent $A_i$, which when set to $x_i = 0$ models their measurement as the unitary $M_{unitary}^{A_i}$ and when set to $x_i = 1$ models the measurement as the unitary $M_{unitary}^{A_i}$ followed by a set of projectors associated with the measurement outcome $a_i$.

In the present case, we have four agents. In the full augmented circuit, we would therefore have four setting variables, one for each agent. Moreover, in the FR protocol, Ursula and Wigner announce their classical outcomes $u$ and $w$ in each run of the protocol and halt when they obtain $u = w = 0$. Therefore in the full augmented circuit this communication channel between Ursula and Wigner which captures this announcement will also be present. However, for the purpose of our discussion here and to see the resolution of the paradox, it suffices to work with a much simplified augmented circuit, where we fix Ursula’s and Wigner’s settings to 1 (as they are effectively classical from the perspective of all agents involved)⁷ and ignore the explicit communication channel for the announcement as this is a post-processing. Then we only need to assign settings $x_1$ and $x_2$ to the agents Alice and Bob.

<footnote n="6">⁶ Recall that in our framework, the settings $x_i = 1$ for all outcomes $a_i$ belonging to the outcome sets $\vec{a}_j$ or $\vec{vec}_a$ appearing in the given prediction $P(\vec{a}_j = \vec{a}'_j | \vec{a}_l = \vec{a}'_l, \vec{x} = \vec{\xi})$, the remaining components of $\vec{x}$ can be varied.</footnote>
<figure><img src="image_3.png" /><figcaption>FIG. 3: Circuit that describes the entanglement version of the FR protocol, from the view of the superagents Ursula and Wigner who describe the measurement of Alice and Bob as unitary evolutions. The protocol proceeds as follows: Alice and Bob share a bipartite state $|\psi\rangle_{RS} = \frac{1}{\sqrt{3}}(|00\rangle + |10\rangle + |11\rangle)_{RS}$, Alice measures R and Bob measures S, both in the computational basis, and the agents store the outcome of the measurement in their memories A and B respectively (the unitary description of these measurements is a CNOT). Ursula then measures RA (Alice's lab) and Wigner measures SB (Bob's lab), both agents measure in the $\{|ok/fail\rangle := \frac{1}{\sqrt{2}}(|00\rangle \mp |11\rangle)\}$ basis to obtain the outcomes $u, w \in \{\text{ok, fail}\}$. Note that this circuit alone does not allow the superagents to reason about the classical measurement outcome of Alice and Bob as these are modelled purely unitarily, and hence no classical measurement outcomes $a$ and $b$ are identified.</figcaption></figure>

We can now write the enlarged set of projectors $\Pi_{x_1}^A$, and $\Pi_{x_2}^B$ (c.f. Equation (9)) as

$$ 
\begin{aligned}
\Pi_0^A &:= \{\pi_{0,\perp}^A = 1_{RA}\}, \\
\Pi_1^A &:= \{\pi_{1,0}^A = |00\rangle\langle00|_{RA}, \pi_{1,1}^A = |11\rangle\langle11|_{RA}\} \\
\Pi_0^B &:= \{\pi_{0,\perp}^B = 1_{SB}\}, \\
\Pi_1^B &:= \{\pi_{1,0}^B = |00\rangle\langle00|_{SB}, \pi_{1,1}^B = |11\rangle\langle11|_{SB}\}
\end{aligned} 
\quad (14) 
$$

These capture the fact that when Alice's setting $x_1 = 0$, her measurement is modelled as a unitary evolution and the $a$ is set deterministically to the trivial value $\perp$. When $x_1 = 1$, Alice's measurement is first modelled unitarily and then her measurement outcomes $a = 0$ and $a = 1$ are identified by the projectors $\pi_{1,0}^A$ and $\pi_{1,1}^A$. The case for Bob is similar.

With this we obtain the (simplified) augmented circuit for the entanglement version of the FR protocol which is illustrated in Figure 4. Note that here Alice and Bob act at the same time $t = 2$ and Ursula and

Wigner act at $t = 3$, but we could always have the agents' operations occur at different times without affecting the circuit structure, such that this circuit fits within the general form of Figure 2. We do not do this here for simplicity, but it is easy to see that this transformation would not affect any of the arguments.

## B. Explicit version of the statements that resolve the paradox

As reviewed in Appendix C 2, the four logical statements involved in the entanglement version of the FR paradox (see Figure 3 for a quick recap of the protocol) are as follows.

$$ 
\begin{aligned}
u &= ok \wedge w = ok \\
u &= ok \Rightarrow b = 1 \\
b &= 1 \Rightarrow a = 1 \\
a &= 1 \Rightarrow w = fail
\end{aligned} 
\quad (15) 
$$

The paradox ensues since the last three statements can be combined using classical logic to yield $u = ok \Rightarrow w = fail$, which contradicts the first statement. These statements follow due to the following four conventional predictions (Definition III.6) of the 4-agent

<footnote n="7">⁷ We need not consider the case where Ursula and Wigner's measurements are modelled unitarily (setting 0) as this is only the case if there were further super-super agents who measured Ursula and Wigner's labs.</footnote>
<figure><img src="image_4.png" /><figcaption>FIG. 4: Augmented circuit for the entanglement version of the FR protocol. Unlike the circuit of Figure 3, this circuit allows agents to model the measurements of Alice and Bob as unitary evolutions and at the same time also reason about their classical measurement outcomes. To model Alice unitarily, one must set $x_1 = 0$, in which case the trivial outcome $a = \perp$ is obtained deterministically and the blue box acts as an identity operation. To reason about Alice's classical outcome, one must set $x_2 = 1$, in which case the blue box applies the projectors corresponding to the classical outcome $a \in \{0, 1\}$. All possible reasoning in this FR set-up can be derived from this circuit under different choices of settings, but these statements can no longer be combined to yield a paradox as explained in the main text.</figcaption></figure>

EWFS specified by this protocol, while post-selecting on an experimental run where $u = w = ok$ is obtained by the superagents Ursula and Wigner. The first prediction in the list below guarantees this post-selection will eventually succeed.

$$ 
\begin{aligned}
P_{conv}(u = ok, w = ok) &= \frac{1}{12} \\
P_{conv}(b = 1|u = ok) &= 1 \\
P_{conv}(a = 1|b = 1) &= 1 \\
P_{conv}(w = fail|a = 1) &= 1
\end{aligned} 
\qquad (16) $$

From our main theorem, Theorem IV.1, it follows that these conventional predictions are equivalent to the following four setting conditioned predictions (that can be computed from the augmented circuit of Figure 4).

$$ 
\begin{aligned}
P(u = ok, w = ok|(x_1, x_2) = (0, 0)) &= \frac{1}{12} \\
P(b = 1|u = ok, (x_1, x_2) = (0, 1)) &= 1 \\
P(a = 1|b = 1, (x_1, x_2) = (1, 1)) &= 1 \\
P(w = fail|a = 1, (x_1, x_2) = (0, 1)) &= 1
\end{aligned} 
\qquad (17) $$

The 3 logical statements associated with the last three setting-conditioned predictions, together with the corresponding original 3 logical statements of FR are given in Table I for comparison.

<table><caption>TABLE I: Explicit versions of the FR statements as given in our framework, which provides a logically consistent resolution to the FR apparent paradox without giving up any of the original assumptions of FR. While the original statements can be chained together to yield $u = ok \Rightarrow w = fail$ which yields a contradiction along with the fact that $P(u = w = ok) > 0$, our explicit version of the statements cannot be chained together to yield this conclusion even using the axioms of classical logic.</caption><thead><tr><th>original formulation</th><th>our explicit formulation</th></tr></thead><tbody><tr><td>$u = ok \Rightarrow b = 1$</td><td>$(x_1, x_2) = (0, 1) \land u = ok \Rightarrow b = 1$</td></tr><tr><td>$b = 1 \Rightarrow a = 1$</td><td>$(x_1, x_2) = (1, 1) \land b = 1 \Rightarrow a = 1$</td></tr><tr><td>$a = 1 \Rightarrow w = fail$</td><td>$(x_1, x_2) = (1, 0) \land a = 1 \Rightarrow w = fail$</td></tr></tbody></table>

Then we can immediately see that while the original statements can be chained together using classical logical rules to yield $u = w = ok \Rightarrow w = fail$ (Equation (C7)), the explicit version of the statements obtained in our framework cannot be chained together in the same manner, even under the standard rules of classical logic.

The fact that the probabilities of Equation (16) and Equation (17) indeed match can be seen by explicitly writing out the expression of the conventional prediction using the Born rule, which we show in full detail in Appendix F 1. We illustrate this here for some of these
cases. Consider the conventional prediction $P_{conv}(u = ok, w = ok)$. This is computed in the circuit of Figure 3 i.e., applying the two CNOT gates to the initial state of $|\psi\rangle_{RS}$ and memories initialised to $|0\rangle_A$ and $|0\rangle_B$, one obtains $|\psi\rangle_{RASB} = \frac{1}{\sqrt{3}}(|0000\rangle + |1100\rangle + |1111\rangle)_{RASB}$. The measurements of Ursula and Wigner act on this state. Indeed our augmented circuit of Figure 4 for the case $(x_1, x_2) = (0, 0)$ is equivalent to the circuit of Figure 3. Thus writing our the two predictions, we have

$$ 
\begin{aligned}
P_{conv}(u = ok, w = ok) \\
&= P(u = ok, w = ok | (x_1, x_2) = (0, 0)) \quad (18) \\
&= |\langle ok|_{RA} \otimes \langle ok|_{SB} \cdot |\psi\rangle_{RASB}|^2
\end{aligned}
$$

For comparison, consider the conventional prediction $P_{conv}(a = 1|b = 1)$. This is fully specified by $P_{conv}(a = 1, b = 1)$, and it will be more illustrative to consider that. This is obtained by applying the computational basis measurement on R and on S to the initial state $|\psi\rangle_{RS}$, which is equivalent to applying the $\{|00\rangle, |11\rangle\}$ basis measurement to $|\psi\rangle_{RASB}$. This is exactly what one would get when using the settings $(x_1, x_2) = (1, 1)$ in the augmented circuit of Figure 4. We have

$$ 
\begin{aligned}
P_{conv}(a = 1, b = 1) \\
&= P(a = 1, b = 1 | (x_1, x_2) = (1, 1)) \quad (19) \\
&= |\langle 11|_{RA} \otimes \langle 11|_{SB} \cdot |\psi\rangle_{RASB}|^2.
\end{aligned}
$$

The equivalence of the remain predictions can be similarly shown, as detailed in Appendix F 1.

**Remark V.1 (On the role of the projection postulate).** As seen in Section IIB, Wigner's Friend Scenarios generally involve an interplay of three aspects of quantum theory: unitary evolution, projection postulate and the Born rule. While all three aspects are present in textbook quantum theory, one may however question the role of the projection postulate in arguments based on measurement probabilities.

Notice that the apparent FR paradox in the entanglement version discussed in this section, arises through the combination of the logical statements in Equation (15) which are fully implied by the conventional predictions of Equation (16). Computing these predictions (which are measurement probabilities) requires applying the Born rule and unitary modelling of agents' measurements, but does not rely on the projection postulate which also specifies a post-measurement state. However, our resolution of the apparent paradox is also entirely at the level of measurement probabilities given by the setting-conditioned predictions Equation (17). Therefore, the resolution given in Table I also need not invoke the projection postulate.

An important but subtle point here is that even though the setting-conditioned predictions involve the setting 1 (which was described in the augmented circuit as associated with the state update of the projection postulate), one does not require this state update rule for computing the predictions. Nevertheless, the projectors associated

with the setting 1 description are necessary to identify the measurement outcome and compute its probability via the Born rule (they specify the measurement basis).

For instance, for the prediction concerning Ursula and Bob's outcomes u and b, Alice's measurement is modelled unitarily (setting $x_1 = 0$) and the basis in which we describe this unitary does not matter for the predictions, however the basis information for Bob (encoded in the projectors $\{\pi_{1,0}^B = |00\rangle\langle00|_{SB}, \pi_{1,1}^B = |11\rangle\langle11|_{SB}\}$ of Equation (14) associated with $x_2 = 1$) is needed for identifying the outcome b and computing its probability. Therefore the setting 1 case need not be thought of as modelling an objective “collapse”, but can be regarded as encoding knowledge about a measurement outcome and the basis needed to identify that classical record (independently of the post-measurement state). This can be perfectly consistent with a unitary description of the measurement by another agent, for whom the original measurement is regarded as an evolution of a closed system and the classical record is unknown (associated with trivial outcome $\perp$).

This highlights that even versions of the FR argument that do not invoke the projection postulate or associated state update rule can be resolved in a similar manner within our approach, without invoking these assumptions, but by being careful about conditioning on the relevant knowledge used in the reasoning.

### C. Setting-dependence: a refined interpretation of the FR paradox

In Section IVC we identified a new assumption I (setting-independence) and showed that inconsistent quantum predictions in EWFSs only arise when assuming I in a situation where predictions do depend on the setting. Analysing this assumption for the FR scenario sheds light on the root cause of such FR paradoxes yielding more refined physical interpretation.

In Appendix F 3, we show the setting-dependence of predictions in the FR scenario (i.e., a violation of I), by explicitly computing them. In particular, recall that we recovered the conventional predictions of the FR scenario equivalently as specific setting conditioned predictions in Equation (17). For instance it follows from the detailed analysis of Appendix F 1 that $P_{conv}(w = \text{fail}|a = 1) = P(w = \text{fail}|a = 1, (x_1, x_2) = (1, 0)) = 1$. $P(w = \text{fail}|a = 1, (x_1, x_2) = (1, 0))$ is indeed setting independent, as one can verify (see Appendix F 3) that $P(w = \text{fail}|a = 1, (x_1, x_2) = (1, 1)) = \frac{1}{2} \neq 1$. Similarly, the setting-dependence of other predictions of the FR scenario can also be verified in our framework.

Using the general results of Section IV and the above analysis of the FR experiment, we can readily prove the following theorem regarding our assumptions applied to the FR scenario. This applies to both the entanglement version (that was the focus here) and the prepare and measure versions (described in Appendix F 2).

**Theorem V.1.** There exists a consistent description of the FR protocol (both versions) that satisfies all five
assumptions **Q**, **U**, **C**, **D** and **S** but violates **I** for certain logical setting-conditioned predictions. Furthermore, when simultaneously assuming **Q**, **U**, **C**, **D** and **S** in the FR protocol, additionally imposing **I** on at least one logical setting-conditioned prediction is a necessary condition for reproducing the apparent FR paradox, while imposing **I** on all logical setting-conditioned predictions is a sufficient condition for the same.

A refined interpretation FR have claimed that their assumptions **Q**, **U**, **C**, **D** and **S** lead to a contradiction in a physical theory that reproduces the quantum predictions of the FR scenario, while we have formalised a version **Q**, **U**, **C**, **D** and **S** of these, showing that they can always be applied consistency even while reproducing the FR predictions. This calls for closer examination of FR's claim to understand this apparent mismatch. FR's work suggests that their assumptions should be interpreted as capturing the validity of unitary "quantum theory" and "classical logic" applied to the knowledge of agents. However, the assumptions are not sufficiently rigorously formalised, due to ambiguities in defining "agents' knowledge", and especially how agents model measurements in each statement they make.

Specifically our results show that if the FR assumptions are interpreted as just capturing the validity of quantum theory and of classical logic, then FR's claimed theorem would be wrong, as we have shown rigorously the general consistency of these assumptions within our framework by developing an explicit consistent model for reasoning in quantum theory where all these assumptions are satisfied.

In order for FR's theorem to be correct, FR's assumptions should be interpreted as imposing a version of quantum theory that ignores choices of Heisenberg cuts (as allowed by our **I** assumption). This distinction necessitates recognizing two versions of quantum theory in the context of EWFSs: (1) *Heisenberg cut independent* and (2) *Heisenberg cut dependent* versions.

In essence, FR's result can be interpreted as revealing a contradiction between version (1) of quantum theory and classical logic within a specific EWFS, while our findings establish the general consistency between version (2) of quantum theory and classical logic across all EWFSs. Then the two sets of results are mutually consistent.

Moreover, the violation of **I**—the dependence of predictions on Heisenberg cuts in EWFSs—is not unexpected once the concept of such cuts is formalised in terms of different channels describing a measurement. Not only is the **I** assumption violated in the FR scenario as shown here, this is also the case in Wigner's original thought experiment. Indeed, Wigner clearly points to this effect in the original paper where the thought experiment was introduced, although evidently not in the language of the settings we use here. As we can see from the review of Wigner's experiment in Section II B, the core message is that the ambiguity in how a measurement is modelled, in light of the unitarity vs projection postulates (which in our framework is labeled

by the settings), does have empirical consequences in Wigner's scenario.

Explicitly, recall that the two evolutions of an initial state $\sqrt{\frac{1}{2}}(|0\rangle + |1\rangle)_S$ measured by an agent Alice in the computational basis, lead to the following final states of her system **S** and memory **A**. Here we label the two cases with the corresponding settings of our framework, where **x** denotes the setting of Alice's measurement.

$$ \sqrt{\frac{1}{2}}(|0\rangle + |1\rangle)_S \underset{x=0}{\longrightarrow} \sqrt{\frac{1}{2}}(|00\rangle + |11\rangle)_{SM}, \quad (20) $$

$$ \sqrt{\frac{1}{2}}(|0\rangle + |1\rangle)_S \underset{x=1}{\longrightarrow} \frac{1}{2}(|00\rangle \langle 00| + |11\rangle \langle 11|)_{SM}. $$

If the superagent Wigner now measures **SA** in the $|\text{ok}\rangle\langle\text{fail}| := \{\frac{1}{\sqrt{2}}(|00\rangle \mp |11\rangle)\}$ basis to obtain the outcome **w**, clearly, we have $P(\text{w} = \text{ok}|\text{x} = 0) = 0$ and $P(\text{b} = \text{ok}|\text{x} = 1) > 0$. Therefore **I** is violated, and if we nevertheless ignore the settings, we obtain an apparent paradox with $P(\text{w} = \text{ok}) = 0$ and $P(\text{b} = \text{ok}) > 0$.

This highlights that in EWFSs, when assuming the universal validity of unitary quantum theory, it is natural to consider version (2) of quantum theory, which incorporates the Heisenberg cut dependence of predictions. And we have shown that this version of quantum theory, appropriately formalised, is perfectly consistent in all EWFSs. Both Wigner's original result as well as FR's result can be regarded as a cautionary note on the dangers of insisting to use version (1) of quantum theory in such EWFSs. However, FR's version arrives at the apparent contradiction while only requiring agents performing compatible measurements to reason about each other in each statement, while the apparent paradox obtained as above in Wigner's experiment requires a super-agent (Bob) to issue statements about an agent (Alice) who performs an incompatible measurement.

**Comment on absoluteness of events** Although FR's arguments center of agents' reasoning, our analysis here takes a step back and focuses on the more fundamental aspect of the predictions of quantum theory for the scenario. This allows us the draw insights on the failure of an absolute notion of events here, and its relation to the settings we have introduced in this work.

Absoluteness of observed events (AoE) entails that the predictions of the scenario can be derived from a single joint probability distribution on the observed (non-trivial) outcomes of all agents. However, the quantum predictions of the FR scenario given in Equation (16) are not compatible with a single joint distribution $P(u, w, a, b)$ on the observed outcomes of all four agents.

We have shown that these conventional predictions of Equation (16) are equivalent to the setting-conditioned predictions of Equation (17). The setting choices are present but not made explicit in the conventional representation.

Once we account for the setting-dependence of these predictions using their explicit form given in Equa-
tion (17), their incompatibility with a single, well-defined joint distribution $P(u, w, a, b)$ independent of any settings, becomes immediate. This is because those predictions arise from different conditional probability distributions but by ignoring the conditioning information (on which the prediction depends), therefore whenever there is such setting-dependence in a scenario (violation of $I$), we cannot expect AoE to hold for the corresponding conventional predictions of that scenario.

Due to space considerations, we leave a detailed discussion of other no-go results for EWFSs relating to the absoluteness of events [3, 4], particularly the Local-Friendliness (LF) theorem [4], and the relation between AoE and $I$ to a follow-up paper [18]. However, we note that our $I$ assumption plays a very different role in the FR vs LF scenarios. While we have shown that the violation of $I$ is sufficient to fully evade the conclusion of FR's paper that "Quantum theory cannot consistently justify the use of itself", the violation of $I$ in the LF scenario cannot be used to evade their conclusions, rather it sheds deeper light on the structure and meaning of their absoluteness of events assumption.

## VI. EMERGENCE OF ABSOLUTE MEASUREMENT EVENTS

In this section, we address how our formalism covers the perceived objectivity of measurement outcomes and the standard predictions of quantum theory in present real-world experiments, even though the formalism can be used in a fundamentally relational manner for general EWFSs where absoluteness of events may not hold.

In Wigner's Friend Scenarios, relational approaches typically propose to avoid paradoxes by demanding that measurement outcomes are to be defined relative to an agent, a context, a world or some other new concept introduced within the framework (see also the discussions in [2, 5, 9]). However, this leaves open the crucial question of how one can recover predictions of realistic quantum experiments where we perceive measurement outcomes and probabilities to be non-relational and objective. To address this question, we develop criteria to distinguish genuinely Wigner's Friend type experiments from standard quantum scenarios, when agents do not measure each other's memories/labs in a non-trivial manner (or more colloquially, when they do not "Hadamard each others' brains").

Recall that the memory $M_i$ of each agent $A_i$ plays the role of "their entire lab except the system that they measure" i.e., the quantum system $S_i$ which $A_i$ measures together with their memory $M_i$ constitute an idealised model of $A_i$'s lab. We now proceed to formally define what is meant by "acting non-trivially on an agent's memory".

Here it is important to note that even in everyday scenarios, agents do act on the memories of each other in the sense that an agent can consult their memory, which stores a classical outcome, and communicate it

to another agent. This may influence the operations and reasoning process of the second agent. Therefore, we need a definition that is not too restrictive to forbid this kind of "trivial" or "standard" way of acting on each others' memories, while still strong enough to identify "non-trivial" or "non-standard" ways in which the operation performed by one agent in a Wigner's Friend Scenario can act on the memory/lab of another agent.

We provide such a formal definition, and then show that indeed the settings of the augmented circuit (which are the only perspectival/relational part of our general framework) can be safely dropped in standard quantum experiments while preserving the predictions of the augmented circuit.

### A. Causal criteria for superagency: distinguishing standard and genuinely Wigner's Friend scenarios

We apply the notions of causal structure and directed paths introduced in Definition IV.1 and Definition IV.2 to establish a criterion distinguishing when one agent does not act as a superagent to another in an EWFS. This criterion helps differentiate parts of a general EWFS as standard quantum sectors as opposed to genuine Wigner's Friend experiments.

We term this concept a *non-superagent structure (nSA)*. The main idea is that for an agent $A_j$ not to act non-trivially on the memory of another agent $A_i$, it suffices to know either that $A_j$ does not act after $A_i$ in the causal structure (in which case they do not act on $A_i$'s memory at all[^8]), or that all operations in the augmented circuit occurring after $A_i$'s measurement, including $A_j$'s operations, can be simulated by equivalent operations that act trivially on $A_i$'s memory.

A simple example illustrates this concept, showing that it does not exclude scenarios where agents may communicate different information based on the outcomes stored in their memories. The post-measurement state of an agent's memory $M_i$ and measured system $S_i$ after a measurement $M^{A_i}$ is symmetric in the exchange of $M_i$ and $S_i$ for both setting choices $x_i = 0$ and $x_i = 1$, as shown in Equation (3) and Equation (7). This symmetry exists because the memory is perfectly correlated with the system in the measurement basis, obtained by coherently or incoherently copying (depending on the setting) the system state in that basis. Therefore, any scenario where an agent prepares a new state for another agent based on the state stored in their memory can be perfectly mimicked by an equivalent operation that prepares the new state based on the state of the system, acting trivially (as an identity) on the memory.

<footnote>[^8]: Keep in mind that each agent in our formalism is associated with a single time step, and a physical agent acting at many time steps would be modelled as multiple agents acting here, thus one agent has to act later in time than another in order to act on the latter's memory.</footnote>
However, a scenario where Wigner performs a Hadamard operation on a Friend's brain, or undoes a Friend's measurement, involves a non-trivial joint operation on the memory and system that cannot be emulated by an operation acting on the system alone. Therefore, when $A_i$ acts before $A_j$ in the causal order, the key property distinguishing whether or not $A_j$ acts as a superagent to $A_i$ is whether $A_j$'s operations necessarily act jointly on $A_i$'s lab (system and memory). With these physical intuitions in mind, we provide the following technical definitions.

**Definition VI.1 (Operationally equivalent EWFSs).** We say that two N-agent EWFSs involving sets $\{A'_1, ..., A'_N\}$ and $\{A_1, ..., A_N\}$ of agents are operational equivalent if and only if the following hold

* There is one-to-one identification between the systems $S' = \{S'_1, ..., S'_m\}$ and $S = \{S_1, ..., S_m\}$, agents $\{A'_1, ..., A'_N\}$ and $\{A_1, ..., A_N\}$, memories $\{M'_1, ..., M'_N\}$ and $\{M_1, ..., M_N\}$, measurements $\{M^{A'_1}, ..., M^{A'_N}\}$ and $\{M^{A_1}, ..., M^{A_N}\}$, subsets $S'_i \subseteq S' \cup M'\{M'_i\}$ and $S_i \subseteq S' \cup M\{M_i\}$ of systems on which each measurement acts non-trivially, and sets of operations $\{\mathcal{E}'_1, ..., \mathcal{E}'_N\}$ and $\{\mathcal{E}_1, ..., \mathcal{E}_N\}$, with outcome sets $\mathcal{O}'_i$ and $\mathcal{O}_i$ being equivalent.

* The augmented circuits of the two EWFS yield the same setting-conditioned predictions, i.e., for all disjoint subsets $\vec{a}_j$ and $\vec{a}_l$ of outcomes in one EWFS and corresponding subsets $\vec{a}'_j$ and $\vec{a}'_l$ of outcomes in the other, as well as all settings $\vec{x}$ in one and corresponding settings $\vec{x}'$ in the other, we have

$$ \begin{aligned} P(\vec{a}_j = \vec{a}'_j | \vec{a}_l = \vec{a}'_l, \vec{x} = \vec{\xi}) \\ = P(\vec{a}'_j = \vec{a}'_j | \vec{a}_l = \vec{a}_l, \vec{x}' = \vec{\xi}). \end{aligned} \quad (21) $$

**Definition VI.2 (Non-superagent structure).** We say that an EWFS involving a set $\{A_1, ..., A_N\}$ of agents respects a non-superagent structure $nSA$ whose elements are pairs $(A_i, A_j)$ of agents, if the given EWFS is operationally equivalent to another EWFS involving a set $\{A'_1, ..., A'_N\}$ of agents, such that for all $(A_i, A_j) \in nSA$ the following conditions hold.

1. If $i = j$, then $\mathcal{E}'_i$ acts trivially on the memory $M'_i$.

2. If $i \neq j$, then one of the following holds

* $A'_i \not\subset A'_j$

* If $A'_i \subsetneq A'_j$, then $\mathcal{E}'_i$ acts trivially on the memory $M'_i$ and $A'_i \not\subsetneq M'_i$

If $(A_i, A_j) \in nSA$, we say that $A_j$ does not act on the memory of $A_i$ or does not act as a superagent to $A_i$.

Note that the $nSA$ of an EWFS is also a physical, objective and perspective-independent property of the

protocol. Although the notion of operational equivalence refers to settings, the property is in fact independent of settings as it must hold for all settings.

For instance, in Wigner's original experiment, where the agent Wigner W can measure the lab of the Friend F in the Bell basis while the Friend only acts on some quantum system (that does not include Wigner's memory/lab), applying this definition, we find that $(F, W) \notin nSA$ and $(W, F) \in nSA$. This indicates that W can act as a superagent to F but not vice-versa. More generally, if another agent Ursula U can "ask" Wigner about his observed measurement outcomes but does not have non-trivial quantum control over Wigner's full lab, we can simulate Ursula's operation of "asking Wigner about his outcome" as an operation that acts directly on Wigner's system (Friend's lab) and not on his memory (as discussed earlier in this section). This simulation would produce the same setting-conditioned predictions in they augmented circuit as the original scenario.

Therefore $(W, U), (U, W) \in nSA$ while $(F, U) \notin nSA$ and $(W, U) \in nSA$, and of course $(F, F), (W, W), (U, U) \in nSA$ since no agent acts on their own memory through the channel $\mathcal{E}_i$ that they implement after their measurement.

In the (entanglement version of the) FR protocol reviewed in Appendix C 2, we have $nSA = \{(A, B), (B, A), (U, W), (W, U), (U, B), (B, U), (W, A), (A, W), (U, A), (W, B), (A, A), (B, B), (U, U), (W, W)\}$, but this is not a standard quantum experiment as the pairs $(A, U)$ and $(B, W)$ don't appear in $nSA$, the latter agent acts as a superagent to the former. The $nSA$ of the prepare and measure version of the FR protocol is identical.

We are now ready to formally define what we mean by a standard quantum experiment.

**Definition VI.3 (Standard quantum scenario).** An EWFS with N agents $\{A_1, ..., A_N\}$ is said to correspond to a standard quantum scenario if $(A_i, A_j) \in nSA$ for all $A_i, A_j \in \{A_1, ..., A_N\}$. In case this holds for a subset of the first k agents $\{A_1, ..., A_k\}$ (who act at time steps $t_1 < ... < t_k$) in the context of a larger N-agent EWFS we call $\{A_1, ..., A_k\}$ a standard quantum sector of the EWFS. Otherwise we call it a non-standard scenario or sector accordingly.

For instance, in FR's protocol, the agents Alice and Bob who act first form a standard quantum sector, but once we include the superagents Ursula and Wigner, the sector is no longer standard.

**Non-standardness as a signature of genuine Wigner's Friend-ness?** We have provided a concrete definition of standard quantum scenarios among a general class of EWFSs, based on the operational causal structure of the scenario. An interesting question is whether scenarios that are non-standard according to this definition can be considered as exhibiting a genuinely Wigner's Friend-type aspect. One intuition in favour of this is that by definition, non-standardness captures that there is at least one pair of agents such
that one (say $A_j$) acts on the memory of another (say $A_i$) in a non-trivial manner that cannot be regarded as $A_j$ simply “asks” $A_i$ their outcome (for the latter can be simulated in an operationally equivalent scenario with trivial action on memory, as shown before). However, this point needs to be further investigated both at a conceptual and technical level to make conclusive statements, after all there can be different equally well-motivated criteria for “genuineness” of a non-classical resource, as the vast literature on quantum non-locality and entanglement highlights.

Our formalism provides a first consistent, general and fully formal platform for formulating and investigating such questions for EWFS. This gives the potential to understand from causal principles, the quantum resource associated with Wigner's Friend type no-go results that fundamentally distinguish them from existing quantum no-go results where agents are not treated quantum mechanically. We leave this for future work.

## B. Recovering Heisenberg cut independence in standard quantum experiments

In the previous sections of the paper, we have shown that the choice of settings (which formalise Heisenberg cuts in our framework) do affect the empirical predictions in Wigner's original scenario as well as its extensions such as FR and LF. That is, the I assumption (setting-independence) is violated here. This differs from our standard intuitions and usage of quantum theory to describe realistic experiments, where we do not have to consider any such settings or Heisenberg cuts, and where observed outcomes appear to be objective records independent of any such concepts.

In order to show that our framework correctly reproduces the known predictions and observations of standard quantum experiments conducted so far (i.e., where agents do not have full quantum control over each other's labs/memories), we must show that the setting-variables can be safely dropped from the predictions and the augmented circuit when analysing such experiments. The following results formalise this intuition concretely using our definition of standard quantum scenarios. At a foundational level, this will shed light on how the perceived objectivity or non-relationalism of observed measurement events emerges within standard quantum sectors.

**Theorem VI.1** (Non-action on memory and setting-independence). Consider an EWFS and a subset $A_K$ of agents therein. Suppose that $A_i \notin A_K$ is another agent in the EWFS such that no agent in $A_K$ acts as a superagent to $A_i$ i.e., $(A_i, A_k) \in nSA \forall A_k \in A_K$. Then for every partition $A_K = \{A_{j_1}, \dots, A_{j_p}\} \cup \{A_{l_1}, \dots, A_{l_q}\}$ of $A_K$, the setting-conditioned prediction $P(\vec{a}_j = \vec{a}_j | \vec{a}_l = \vec{a}_l, \vec{x} = \vec{\xi})$ is independent of the setting $x_i$ that is,

$$ 
\begin{aligned}
P(\vec{a}_j | \vec{a}_l, (\xi_1, \dots, \xi_N)) &= \\
P(\vec{a}_j | \vec{a}_l, (\xi_1, \dots, \xi_{i-1}, \xi_{i+1}, \dots, \xi_N)).
\end{aligned} 
\qquad (22) 
$$

Recall that this expression is equivalent to the conditional independence given in Equation (10).

A proof of the above theorem can be found in Appendix J.

Then, as a corollary of Theorem VI.1, we can immediately recover the full setting-independence of predictions in standard quantum experiments.

**Corollary VI.1** (Full setting-independence in standard quantum theory). In any EWFS corresponding to a standard quantum scenario, every non-trivial setting-conditioned prediction, is setting independent. Formally, for any disjoint sets $\vec{a}_j$ and $\vec{a}_l$ of outcomes in the EWFS, $P(\vec{a}_j = \vec{a}_j | \vec{a}_l = \vec{a}_l, \vec{x} = \vec{\xi})$ is independent of settings $x_i$ for all $i \notin \{j_1, \dots, j_p, l_1, \dots, l_q\} := \mathcal{JL}$ i.e., Equation (22) holds for all such $i$. Specifically, in such standard scenarios, all non-trivial predictions i.e., those where $a_i \neq \perp$ for all $i \in \mathcal{JL}$, can equivalently be expressed in a fully setting-independent manner as given below.

$$ 
\begin{aligned}
& P(\vec{a}_j = \vec{a}_j | \vec{a}_l = \vec{a}_l, \vec{x} = \vec{\xi}) \\
& = P(\vec{a}_j = \vec{a}_j | \vec{a}_l = \vec{a}_l, x_i = 1, \forall i \in \mathcal{JL}) \\
& := P(\vec{a}_j = \vec{a}_j | \vec{a}_l = \vec{a}_l).
\end{aligned} 
\qquad (23) 
$$

Going beyond predictions, and to the underlying circuit representation, we expect that standard quantum experiments involving some measurements and channels, can be represented in terms of quantum circuits with no ambiguity in how a measurement is modelled (i.e., circuits with no setting variables for measurements). Here, two equivalent types of circuit representations are possible, which are both commonly used within the standard quantum computing paradigm: we can either model each measurement as acting only on the measured system and yielding a (non-trivial) classical outcome at the time of the measurement, or we can model each measurement as a unitary interaction between a system and ancilla at the time of measurement and probabilities can be extracted by measuring all the ancillas at a later time at the end of the experiment.

We leave a formal definition of these to Appendix E, where we prove that for EWFSs corresponding to standard quantum scenarios, the augmented circuit can equivalently be reduced to either of these expected forms. These two types of standard quantum circuits are illustrated in Figure 6 and Figure 7 in the same appendix.

# VII. DISCUSSIONS

## A. Sound scientific reasoning in EWFS and analogies to classical multi-agent reasoning

The results of Section III guarantee that our formalism allows quantum agents to make predictions and rea-
son consistently about physical experiments and each other's knowledge, even when unitary quantum theory is universally valid.

Here, we illustrate a general paradigm for scientific reasoning indicated by our results and discuss their wider scope, contrasting the quantum and classical aspects. Specifically, while there are potentially genuine quantum aspects related to agents' reasoning in EWFS, by formalising EWFSs as done here, all considerations regarding the consistency of agents' reasoning can be reduced to analogous issues arising in classical multi-agent reasoning. This allows to extend the scope of our proposal to more general multi-agent scenarios by considering how analogous generalisations would work in the classical case.

**Genuinely quantum and relational aspects** We have seen a concrete rule for choosing settings to make predictions in universal quantum theory (Theorem IV.1), equivalent to conventional quantum predictions in the EWFS literature. This default rule models the maximum number of measurements as pure unitaries. It is implied by the choice of prediction one wishes to compute: if we wish to compute the probability of measurement outcomes $a_1$ and $a_2$, the settings $x_1$ and $x_2$ of the corresponding measurements are $x_1 = x_2 = 1$, while the settings for all other measurements are set to 0. This does not allude to agents or depend on them. However, when agents incorporate this rule while reasoning about each other's knowledge, the setting choices depend on the agent who is reasoning and the agent being reasoned about.

For example, if Alice reasons about Bob's outcome $b$ based on her outcome $a$, she will use a prediction $P(b|a, \vec{x})$ choosing $x_A = x_B = 1$ and $x_C = 0$ for the measurement of Charlie. If Alice reasons about Charlie instead of Bob then the corresponding prediction $P(c|a, \vec{x})$ will have $x_A = x_C = 1$ and $x_B = 0$. Within the quantum formalism, we can interpret our settings as choices of Heisenberg cuts (this interpretation need not hold in hidden variable models for reproducing quantum predictions, see Appendix I). Then, the default rule captures subjective choices of Heisenberg cuts: each agent places themselves and the agent whose classical outcome they are reasoning about on the “classical side” of the cut, and everyone else on the “quantum side” of the cut. This allows the notion of an observed event (classical outcome) to be subjective and relative to a choice of cut.

This type of relationalism, non-absoluteness of observed events and the general setting-dependence (or Heisenberg-cut dependence) of predictions in EWFS are arguably non-classical aspects. This is because classical theories lack a non-trivial concept of Heisenberg cuts, generally have no ambiguities in how a measurement has to be fundamentally modelled.

Moreover, for one agent $A_j$ to act as a super-agent to another agent $A_i$ (i.e., $(A_i, A_j) \notin nSA$), $A_j$ must perform a non-trivial joint operation on the lab of $A_i$. If both agents measure in the same basis, we have already seen in section VI that $(A_i, A_j) \in nSA$ as the opera-

tion can be simulated by acting on part of the lab, but this is not the case when $A_j$ measures $A_i$'s lab in a complementary superposition basis. This suggests that some notion of measurement complementarity (not typically a classical feature) may be necessary for having a non-trivial WF-like scenario. There is scope for future work on identifying and characterising non-classical resources in EWFS as discussed in Section VI A and Section VIII, which would be needed for making these observations fully rigorous.

**Aspects reducible to classical issues** Our formalisation of Heisenberg cuts in quantum theory as different choices of channels in a circuit implies, based on classical probability theory and logic, that one must generally condition on these choices in their reasoning unless (1) it is known that all agents in the scenario employ the same fixed choices, or (2) it has been established that these choices do not matter.

Suppose Alice and Bob who are reasoning about the output of a physical classical channel acting on an input state $\rho$ that they previously agreed upon, but they assume different noise models $N_A$ and $N_B$ for the channel. Then they will generally arrive at distinct output states/probabilities, and their conclusions can seem inconsistent if they do not communicate the conditioning on the assumed noise model. One can obtain logical paradoxes akin to FR in such classical scenarios (see Appendix G for an explicit example). Note that in this example, the agents need not communicate the initial state $\rho$ as it is common knowledge.

If Alice and Bob perform their analysis with the same noise model but one of them before and the other after lunch, there would be no inconsistencies in their conclusions even if they forget to mention whether they had eaten, as the predictions are independent of this parameter. Finally, if Alice observes that the channel's output differs from her prediction, this can falsify her assumption regarding the noise model $N_A$. If she believes the experiment was performed correctly, she would update her knowledge of the noise model based on a closer examination of the experimental results.

**General reasoning paradigm** These classical examples highlight a general paradigm for sound scientific reasoning that ensures agents do not arrive at inconsistencies:

1. Identify fixed common knowledge vs variable parameters

2. Drop redundant parameters

3. Fix a choice of remaining parameters and condition on them

4. Check if choices are falsified by observed data and update them if needed

Such careful conditioning on all relevant variable parameters ensures that the set of statements is consistent according to Definition III.5, ensuring logical and probabilistic consistency.

All these aspects are incorporated in our formalism and results. We take the protocol description of an
EWFS (Definition III.1) to be the common knowledge of all agents and construct the augmented circuit from this knowledge alone. The only variable parameters are the settings $\vec{x}$. Our results (Theorem IV.1 and Theorem VI.1) provide concrete criteria for identifying redundant settings based on the operational causal structure and non-super agent structure, both of which are objective properties of the protocol. Our default rule then fixes a choice of the remaining settings for every prediction/statement and explicitly conditions on them.

Finally, the discussions throughout our paper (see also Section VII C and Appendix I) highlight how one can falsify setting conditioned predictions through experimental data. Our default rule is based on the premise of universal validity of quantum theory, but a yet-undiscovered physical mechanism for objective collapse may falsify these predictions (obtained through the default rule) in a future experiment and one would then have to update the setting choices given by this rule (changing $x_i = 0$ to $x_i = 1$) for certain settings, if the experimental demonstration of the falsification is deemed “loophole free” and sufficiently convincing.

**Efficiency of reasoning** Our results guarantee that the reasoning rules we propose for quantum agents remain consistent and respect causality principles in general EWFSs. We now discuss the computational efficiency of this reasoning process, aiming to illustrate that the complexity of applying our reasoning rules is comparable to standard quantum or classical reasoning under similar assumptions.

First, computing setting-conditioned predictions in any EWFS involves applying the Born rule to a circuit with the same number of gates as the physical operations in the protocol, making it as complex as standard quantum reasoning in standard quantum scenarios. The additional rules for processing and assigning settings do not introduce any inefficiencies, as they simply involve reading off independences from the protocol's causal structure and non-super agent structure, which can be largely pre-computed.<fnref n="9" />

The only potential extra resource cost is in communicating and storing non-redundant settings in EWFS. However, this cost is minimal since the vector of non-redundant settings has binary entries and dimensions typically smaller than $N$, the number of agents. Moreover, as illustrated before, even in classical examples, under similar ambiguities in how a (classical) channel is modelled, consistency necessitates that agents communicate and keep track of the information that removes this ambiguity. Further, we have also shown that the no additional rules are needed for combining predictions/statements about measurement outcomes in quantum EWFS once the settings are accounted for,

just classical probability theory and axioms of classical logic.

Finally, we have shown that reasoning in our framework respects the C assumption which formalised FR's C rule for setting-conditioned predictions. This means that when a setting-conditioned prediction or statement is communicated to agent $A_i$ by another agent $A_j$, $A_i$ does not need to recalculate the prediction by simulating $A_j$'s reasoning process, and can directly inherit this knowledge. Notice that despite this lack of agent-labels on predictions, the agent-dependence and relationalism are incorporated through the fact that our default rule allows different agents to choose different settings in their reasoning.

**Scope of our resolution** In this work, we have taken the protocol description to be in the common knowledge of all the agents for simplicity and to illustrate the core features of Wigner's Friend scenarios that extend beyond more standard experiments. However, situating our framework within the broader reasoning paradigm described above, generalizing our consistency results to partial knowledge scenarios is entirely analogous to generalizing a classical circuit framework (for reasoning in a classical world) from full knowledge to partial knowledge scenarios.

In both classical and quantum cases, allowing agents only partial knowledge about the protocol increases the number of variable parameters that need to be conditioned on to avoid inconsistencies, as common knowledge is reduced. In the earlier classical example, if Alice and Bob had not agreed on the input state $\rho$ to the channel, or if they are unsure whether the other person knows this state, they would need to condition on their choice of state in their communicated statements in order to avoid paradoxes.

Our quantum reasoning rules for the complete knowledge case can be straightforwardly generalised to partial knowledge scenarios using the ideas described here. We consider this generalisation to be entirely analogous to how it would be implemented in a purely classical theory of multi-agent reasoning, and therefore do not detail it here.

**Status of proposed reasoning challenges** With these contributions, we believe that challenges related to the logical reasoning of quantum agents in EWFS (including partial knowledge scenarios) are fully addressed to the same extent that analogous challenges are considered resolved in purely classical theories.

For example, Renner and del Rio [25] have recently proposed a challenge for agents' reasoning in quantum theory, based on the FR paradox. This challenge includes scenarios where agents have partial knowledge about a protocol. However, due to ambiguities—similar to those in the original FR paper—regarding assumptions about the Heisenberg cut, the formal modeling of measurement channels, and conditioning on agents' knowledge, we believe it is currently unclear whether the challenge is well-defined. Inconsistencies in agents' predictions can also arise in purely classical theories when similar assumptions are not carefully accounted

<footnote n="9">⁹ The irrelevance of certain settings for specific predictions is derived from the protocol's structural properties alone and can naturally be included in the protocol description initially given to agents.</footnote>
for, especially in scenarios where agents may have in-
complete information about the protocol. As a result,
we believe it remains an open question whether such a
challenge admits solutions even in classical theories.

Therefore, we consider the challenges for quantum
theory raised by FR's original paper [2] and the re-
cent article [25] as effectively addressed by the frame-
work and results proposed here, in the sense explained
above (while noting the subtleties surrounding classical
multi-agent inconsistencies). A further point support-
ing our conclusion is that we have demonstrated in de-
tail, the consistent resolution of the FR paradox—on
which such challenges are based—through our formal-
ism (Section V for the entanglement version and Ap-
pendix F 2 for the original prepare-and-measure ver-
sion). Moreover, to our knowledge, no example of
a quantum EWFS (including those involving partial
knowledge) exists in which our approach fails to en-
sure consistent reasoning or violates any fundamental
physical principle, such as causality.

While these consistency issues for reasoning quan-
tum agents appear to be resolved, several intriguing
open questions remain regarding EWFSs, including
both agents’ reasoning and meta-physical aspects such
as the absoluteness of observed events. Our work offers
a solid foundation with which these questions can be
formally explored. We discuss in Section VIII.

B. Interpretations of quantum theory

Our results are presented in three levels of general-
ity that allow to distinguish different conceptual points.
Firstly there is a general formalism for defining EWFSs,
which allows for different rules that one may prescribe
to compute probabilities P(ā_j|ā_l, k) of some outcomes
ā_j, ā_l given certain parameters k describing the as-
sumptions made about the scenario (such as its states,
channels etc). Even when specialising these k param-
eters to the settings x̄ of our augmented circuit in the
second step, there is freedom is choosing the values
of these settings. Different interpretations of quan-
tum theory can suggest different sets of parameters k
to be considered (e.g., certain interpretations such as
Bohmian mechanics may require the description of ad-
ditional hidden variables λ to be included in k), and can
also assign different values to the settings (e.g., collapse
theories would suggest x_i = 1 modelling the projection
postulate, for all measurements, while a many-worlds
type interpretation would suggest x_i = 0 or unitary
evolution for measurements).

Our consistency results of Theorem IV.1 apply to all
choices of settings in the augmented circuit, and there-
fore show that logical and probabilistic inconsistencies
can be avoided across interpretations by applying our
formalism. In Appendix I we discuss in detail how dif-
ferent interpretations of quantum mechanics (such as
many-worlds, collapse theories, hidden variable theo-
ries, relational quantum mechanics and QBism) could
apply our framework consistently to resolve all appar-

ent FR-type paradoxes. In this sense, our general for-
malism is interpretation independent.

In the third step, we prescribe reasoning rules for
universal quantum theory, without assuming absolute
events and while maintaining causality principles. The
assumption of universal quantum theory as well as ad-
herence to causality principles excludes certain inter-
pretations such as collapse theories or retrocausal in-
terpretations, this part of our results is still applica-
ble with different interpretations such as many-worlds,
QBism or relational quantum mechanics as the applica-
bility of the reasoning rules do not depend on whether
the states and channels of the augmented circuit are
interpreted in an ontological or epistemic manner.

Finally, we note that there are certain previous works
that, at a rather high level, may appear similar to some
of our results, especially with regards to our resolu-
tion of the FR paradox. This includes the consistent
histories interpretation of quantum mechanics [26, 27],
works on quantum decoherence [28] and another rea-
soning rule proposed for quantum theory to avoid FR
paradoxes [10]. We discuss the relation to these previ-
ous proposals in Appendix I.

C. Physical interpretation of settings

Before discussing interpretations of settings, it is im-
portant to remind the reader of the role they already
play in the existing literature of EWFSs.

A setting choice is required for a prediction to be
made in EWFS experiments. Settings specify how a
measurement is modelled (as purely quantum unitary
evolution or as being associated with classical records
identified by projectors), and this specification is neces-
sary for computing any prediction in quantum theory
using the Born rule<fnref n="10" />. We have shown that conven-
tional predictions considered in the literature when re-
ferring to “predictions of quantum theory” in an EWFS
are in fact equivalent to specific setting-conditioned
predictions (Theorem IV.1). This highlights that par-
ticular setting choices are already present in EWFS ar-
guments. This applies to FR’s arguments related to
consistent reasoning as well as arguments related to
the absoluteness of events, as discussed in Section V.<fnref n="11" />
Our framework makes the assumptions about the set-
tings explicit and highlights that this information can-
not generally be neglected in EWFSs.

This serves as an important prelude to the discus-
sion of the question about physical interpretations of
the settings. Due to the interpretation-independence
aspect of our formalism, there isn’t a unique answer

<footnote n="10">¹⁰ Note that this is the case even in scenarios where one
does not apply the projection postulate to specify the post-
measurement state, see Remark V.1.</footnote>

<footnote n="11">¹¹ Although the setting-dependence can have different consequences for the conclusions drawn from these types of arguments.</footnote>
to this. The interpretation of the settings depend on
the assumptions we make about the completeness of
quantum theory and our beliefs about how far quan-
tum theory might extend to the macroscopic domain.

a. **Heisenberg cuts and classical records** Let us assume that quantum theory is complete (i.e., that measurement outcomes are not described by hidden variables; see Section VIIIB for when this does not hold). Then, from a more physical perspective, the settings can also be associated with the Heisenberg cut. Setting $x_i = 0$ for a measurement $M^{A_i}$ implies that the memory $M_i$ of the agent $A_i$ (or more generally their lab) is inside the Heisenberg cut and is treated as a quantum system, while setting $x_i = 1$ implies that the agent's memory $M_i$ can be treated as a classical database (in the measurement basis) that is outside the cut.

This choice of Heisenberg cut or settings can gener-
ally be dependent on the prediction being considered,
or the perspective of an agent. For instance, when con-
sidering a prediction about Alice's outcome *a*, the mea-
surement producing the outcome *a* is modelled with
setting 1 as we refer to its classical record. This would
also be consistent with the perspective of Alice who
would perceive her own memory (which stores the clas-
sical outcome *a* that she observes) as being outside the
Heisenberg cut and would thus assign setting 1 to her
own measurement in her reasoning process. This per-
spective and setting choice is still consistent with the
universal validity of unitary quantum theory because
the perceived classicality is only relative to the basis
in which the agent (here, Alice) accesses their memory.
However, one agent can still model the measurements of
other agents as unitary evolutions of the respective labs
and therefore assign setting 0 to these, as they may not
have access to these classical records and may be able
to perform arbitrary quantum operations on those labs
that can destroy the associated classical measurement
records.

Crucially, we note that if we assume the universal validity of unitary quantum theory, the setting choice $x_i = 1$ of our framework does not correspond to a projector that was “actually performed” or an objective “wave function collapse”, as no such objective account may exist. Rather, the projectors associated with $x_i = 1$ are only required for calculating the probability of the outcome of the agent $A_i$ through the Born rule, which is necessary when one wishes to reason about said outcome.

b. **Falsifying setting choices through experiments** We have shown that every EWFS can be equivalently formulated in terms of a unique quantum circuit, which we called an augmented EWFS. It is important to note that these settings do not correspond to different choices of operations that are actually performed in the protocol. Rather, they can capture choices made by agents when scientifically reasoning about the protocol (as discussed in the previous paragraph), or the fundamental dynamics imposed by different extensions of quantum theory to the macroscopic domain. The latter becomes relevant when we

do not a priori assume that unitary quantum theory is
universally valid.

Different interpretations of quantum theory can gen-
erally predict different types of fundamental dynam-
ics for macroscopic quantum systems such as agents'
labs, and can thus assign different settings. Con-
sider Wigner's original experiment, with Alice being
the friend and Wigner the super-agent. Objective col-
lapse models defy unitary quantum theory beyond a
certain scale; if agents and measurement devices in-
volved are larger than this scale, Alice's measurement
would fundamentally correspond to $x_A = 1$ evolution
in this case, such that the associated projectors are
“actually implemented” if the physical world followed
such a theory. The prediction for Wigner would then
be $P(w|x_A = 1)$. In many-worlds type interpretations,
one would believe that fundamental dynamics is gener-
ally unitary and assign $x_A = 0$ to Alice's measurement,
computing the prediction $P(w|x_A = 0)$ for Wigner's
outcome.

As it is still an open question whether or not quan-
tum theory is universally applicable, we do not know
which of these predictions will be confirmed by a hy-
pothetical future experiment of this Wigner’s Friend
Scenario. If there is a collapse mechanism that breaks
unitary quantum theory, then the data of the experi-
ment could falsify the prediction $P(w|x_A = 0)$, and if
unitary quantum theory prevails, then the prediction
$P(w|x_A = 1)$ could be falsified by experimental data.

Different setting choices lead to different predictions,
and as such, one may be inclined to believe that there
is one setting choice for each measurement which is ac-
tually the “correct” one. This would be the case for
models that objectively fix the Heisenberg cut for all
measurements, such as the objective collapse case. This
is definitely a valid interpretation of the settings, but it
is not the only one. As we have seen, our formulation
does not impose the absoluteness of events and per-
mits a relational interpretation where there is no “one
correct” or “one absolute” setting assignment.

This discussion highlights that even though setting
choices are linked to agents’ reasoning and beliefs rather
than their choices of different physical operations, they
do still have physical and empirical consequences in
EWFSs.

c. **Time dependence of setting choices and knowledge update** The falsification of predictions goes hand-in-hand with the updating of knowledge. If one makes a prediction that is falsified by experimental data, one is forced to question the assumptions under which said prediction was made.

Given an EWFS, if a prediction made under a cer-
tain setting choice is not consistent with observed ex-
perimental outcomes in a physical realisation of the
EWFS, one can update one's knowledge about the set-
tings. Such a situation can only arise in a scenario that
has setting-dependence, as setting-independent predic-
tions would not allow us to infer anything about the
setting choices.

For example, in FR, predictions concerning Alice's
outcome are independent of Bob's setting and vice-versa<sup>12</sup>, and hence Alice's outcomes cannot be used to falsify Bob's setting choice and vice-versa. However, we have seen that at later times, when super-agents Ursula and Wigner perform their measurements, the joint predictions for the outcomes of the two super-observers are dependent on Alice's and Bob's settings. In particular, there are certain experimental outcomes which are inconsistent with Alice's and Bob's settings being 1, observing which can falsify this setting choice.

Note that updating knowledge in light of new data is also common in classical theories. However, the difference is that in such classical theories, one updates their knowledge about something that exists and to which there is an “absolute fact of the matter” regardless of whether it is measured. On the other hand, in our formalism for EWFSs, the knowledge update represents our belief about the existence/nonexistence of classical records of measurements, which may no longer be an absolute, fact of the matter and which can be updated in time in light of new data.

For example, we may safely treat all the measurements performed by experimentalists around the world as having setting 1, as we do not believe that anyone currently has access to a device that could potentially Hadamard the memories of other agents (or quantum computers that can act as measuring reasoning agents), even if some of us may believe that the world is ultimately quantum mechanical.<fnref n="13" /> However, once we are sufficiently convinced in the future that such devices do exist, we would need to consider assigning setting 0 to some measurements in order to consistently explain the results of future experiments where such devices can “undo” or erase classical records of measurements.

This suggests yet another operational interpretation of the settings. The setting 1 case could also be interpreted as a result of unitary evolution: to do so, one simply uses the model where these incoherent measurement channels are purified via an ancillary system. Then the setting being 1 can be understood as reflecting one's belief that a superobserver does not have access to the ancillary system and thus does not have the ability to destroy the classical measurement records through a non-trivial joint operation on Alice's lab (which now includes the ancilla). Thus, the classical records of what we presently observe persist for as long as this belief holds true, but may no longer be a matter of the fact if a powerful future super-agent gains control over sufficient quantum degrees of freedom. This also complies with a decoherence type interpretation, where the world is

fundamentally unitary but classical records emerge due to decohering interactions with an environment (here, ancilla). However, such interpretations typically do not consider the premise that the environment could be accessed by a future super-agent, and the resulting time-dependence with regard to the persistency of classical records.

d. **Analogy to the Maxwell's demon paradox**
The broader message delivered by our resolution of the apparent FR paradox is analogous to the resolution of Maxwell's demon paradox in thermodynamics. In short, the latter is a thermodynamical thought experiment where a microscopic demon with access to knowledge of the microstates of gas molecules in a box could exploit this knowledge to apparently extract work from the gas and violate the second law of thermodynamics. However, the paradox is resolved once we are consistent in the perspective (of the microscopic demon with knowledge of microstates or a macroscopic observer without this knowledge) that is taken while calculating thermodynamic quantities such as entropies, we then find that no such violation of the second law ensues.

From the perspective of the macroscopic observer, the problem never arises in the first place and from the perspective of the demon, work must be performed in order to erase the information gained in the process which in turn ensures that the second law is not violated through the work extraction. It is only when we argue from both perspectives while ignoring the perspective that was used, that we run into an apparent paradox—the work is extracted from taking the demon's perspective and there is apparently no work performed in the process when taking the macroscopic observer's perspective.

This is analogous to the message of Corollary IV.3. The settings of our framework are related to the perspective of agents and their Heisenberg cuts as discussed above. Taking this into account allows us to make predictions that are consistent with a given perspective. We have shown that this ensures that no logical contradictions arise. It is only when we use different settings to derive a set of predictions and then ignore this setting choice through the assumption *I* that we obtain an apparent paradox.

VIII. CONCLUSIONS AND OUTLOOK

Wigner's thought experiment [1] exposes fundamental challenges in applying quantum theory to observers or agents. Recent no-go arguments (e.g., [2-4]) extend this experiment to multiple agents, suggesting that the universal validity of unitary quantum theory radically challenges our understanding of logic, scientific reasoning, causality, and the absoluteness of observed events.

In this work, we have developed a comprehensive theoretical framework for Extended Wigner's Friend Scenarios (EWFSs), enabling sound scientific reasoning without assuming absolute measurement events. The main theorems establish the framework's consistency

<footnote n="12">¹² In the entanglement version this follows from the causal structure, and in the prepare and measure version, this follows from the non-super agent structure, even though the causal structure permits a dependence due to communication.</footnote>

<footnote n="13">¹³ Notice that even if measurements are modelled as unitary evolutions of systems and memories/labs, as long as there are no “super-agents” who perform non-trivial operations on another agent’s whole lab, we may safely treat this as the setting 1 case as we have setting-independence (c.f. Corollary VI.1).</footnote>
and preservation of causality, distinguishing objective (e.g., causal structure) and subjective (e.g., predictions, agents’ knowledge) aspects of EWFSs. These results can be applied to ensure global consistency of scientific reasoning in EWFSs, in a manner independent of the particular interpretation of quantum theory that one subscribes to. Further, we have discussed in Remark V.1 how our solution ensures consistent reasoning in FR-like EWFSs even when measurements are modelled as unitary evolutions of agents’ labs and where the Born rule is applied to reason, without invoking the projection postulate or state update rule of quantum theory.

As a key application, our framework fully resolves all FR-type logical paradoxes in quantum theory, and more generally ensures both logical and probabilistic consistency in all EWFSs. We provided a physically motivated set of reasoning rules for quantum agents that are simultaneously consistent with the universal validity of quantum theory and classical logic applied to observed measurement outcomes, and also the fundamental relationalism of agents’ perspectives in EWFSs. This demonstrates in a constructive manner that quantum theory is perfectly consistent in all EWFSs that one can construct within the theory, and that there is no threat to the ability to consistently program future quantum computers that play the role of agents. This is contrary to FR’s broader claim [2] that “Quantum theory cannot consistently justify the use of itself”, and we have discussed a refined interpretation of FR’s arguments in light of our results, such that FR’s statement about the apparent paradox does not conflict with our statements about consistently.

The key insight is to make explicit how measurements are modelled in each reasoning step, as physical predictions in EWFSs do depend on whether a quantum measurement is regarded as producing a classical outcome or as a purely quantum unitary evolution (capturing the choice of Heisenberg cut associated with the measurement). This sheds light on the core reason for apparent FR paradoxes, as arising from ignoring the choices of Heisenberg cuts used in the reasoning.

Extending beyond agent reasoning, our framework explains the emergence of objective, Heisenberg-cut-independent predictions in real-world quantum experiments. This provides a concrete view on how quantum theory can naturally accommodate the non-absoluteness of events and fundamental relationalism in general EWFSs (where agents can have arbitrary quantum control over each others’ labs), while remaining consistent with objective scientific observations made so far. We outline future research directions and some broader implications of these findings for the field of EWFSs below.

a. **Local-Friendliness and non absolute events** Building on Brukner’s no-go theorem for the absoluteness of observed events (AoE) [3], the Local-Friendliness (LF) no-go theorem [4] imposes strong constraints on any physical theory satisfying AoE (and other reasonable physical assumptions

relating to causality and free choice), demonstrating that such theories cannot explain quantum predictions in a specific EWFS. This EWFS is similar to the entanglement version of FR, but importantly, allows super-agents to choose different measurements to perform on the agents’ labs. Although we have not discussed scenarios with physical measurement choices here due to space constraints, we apply our framework to model the LF scenario and analyse their no-go theorem in forthcoming work [18], highlighting the rather distinct yet relevant role played by setting or Heisenberg-cut dependence (violation of I) in the LF theorem compared to FR’s analysis.

In future work, it would be interesting to further explore the implications of these links between the I and AoE assumptions, for characterizing novel scenarios that yield no-go theorems for AoE in combination with other fundamental assumptions and quantum phenomena. These include measurement complementarity, contextuality, indefinite causal order and relativistic causality principles. We discuss some of these possible directions below.

b. **Quantum and relativistic causality in EWFSs** Causal models offer a rigorous framework to connect observed data with causal explanations, widely used in classical data-driven fields [29]. Bell’s theorem exposes fundamental challenges for classical causal models in explaining quantum correlations consistently with relativity, driving the development of quantum causal modelling frameworks [30, 31] within an information-theoretic paradigm.

The LF theorem [4] is suggested to present more radical challenges for causality [6], even to existing quantum causal models as these assume AoE. AoE violations suggest that the causal structure may become subjective, affecting the notion of spacetime events and relativistic principles [6]. This raises the need for a framework for quantum causal modelling and relativistic causality that does not assume AoE.

Our framework shows that all predictions in an EWFS, possibly subjective, can be recovered within a single objective causal structure of the protocol. The augmented circuit respects this causal structure, forming an acyclic circuit that can be appropriately embedded in spacetime to preserve relativistic causality. The settings which model the Heisenberg cut choices are explicit inputs in this circuit, and the only part that can be subjective are the priors which specify the agents’ choices of settings or Heisenberg cuts.

This paves a concrete pathway for developing a quantum causal modelling framework for EWFSs that does not assume AoE, which is relational, perspectival, and operational (considering agents’ interventions and knowledge) and fully consistent with free choice and relativistic causality principles in space-time. This is the subject of a follow-up work.

The EWFSs in our framework correspond to protocols where agents’ operations occur in a fixed, acyclic order, consistent with the time direction. Quantum theory permits so-called indefinite causal order pro-
cesses [32–34] involving quantum superpositions of the order of agents’ operations, and cyclic generalisations of quantum causal models enable a description of these [23, 24, 35]. A consistent formalism for reconciling such cyclic and indefinite causal structures (which are defined through an information-theoretic notion of causality) with spacetime (which has a definite causal structure according to a relativistic definition of causality) was developed in [23, 24]. Combining techniques from our present formalism with [23, 24] offers scope for generalisation towards a unified framework for quantum agents, quantum causality and spacetime structure, providing a platform for exploring phenomena at the intersection of EWFSs, quantum processes without a definite order of operations, relativity and quantum correlations in space and time.

c. **EWFSs beyond quantum theory** Both arguments related to agents’ reasoning (such as FR) and those related to AoE (such as LF) have been studied in broader theoretical contexts beyond quantum theory.

EWFS beyond quantum theory were first considered in [36], where agents memories are modelled as physical systems of a given theory. A theory-independent analogue of the projective and unitary perspectives on quantum measurements was formalised, with the latter represented by the concept of an information-preserving memory update. Using this, it was shown that agents sharing a PR-box (a post-quantum resource) [37] can encounter an apparent contradiction similar to the FR scenario. This shows that FR-type apparent paradoxes are not exclusive to quantum theory. Moreover, it was shown in [38] that any physical theory allowing for information-preserving memory updates, Bell non-classical correlations, and satisfying a locality principle would lead to violations of AoE akin to those witnessed in the quantum LF scenario [4]. Thus, the measurement problem is also not unique to quantum theory.

Future work can explore generalizing the current framework beyond quantum theory to resolve multi-agent paradoxes in post-quantum theories and illuminate the nature of AoE violations and the measurement problem therein. Many concepts and tools developed here are amenable to such generalisation.

d. **Resource-theoretic characterisation of EWF results** What are the information-theoretic resources in EWFSs responsible for FR and LF type results, and which distinguish Wigner's Friend setups from standard quantum experiments?

While we have shown that apparent Wigner's Friend paradoxes can always be resolved in quantum theory with careful specification of Heisenberg cuts, it is intriguing to characterise scenarios where this choice can be safely ignored without leading to inconsistencies. Our results Theorem IV.1 and Theorem VI.1 provide sufficient conditions for safely ignoring settings based on general structural properties (the causal and non-superagent structures), but there is scope to study the role of state and measurement dependence and derive tight necessary and sufficient conditions for scenarios

that violate I (have setting-dependence).

Specifically, in the FR scenario, the correlations are known to be identical to those in Hardy's proof of quantum contextuality (see Appendix H for a discussion). Is contextuality a necessary feature for apparent Wigner's Friend paradoxes? In an upcoming work [39], it is shown that for a large class of multi-agent paradoxes within EWFSs in general physical theories (including FR's quantum paradox and [36]'s PR-box based paradox, but excluding Wigner's original 2-agent quantum scenario), a logical form of contextuality is a necessary property. Studying the necessary and sufficient conditions for different classes of multi-agent reasoning paradoxes in a theory-independent manner, comparing the structure of such apparent paradoxes in quantum vs more general theories, and relating them to physical principles and informational resources of the theory remain interesting future directions.

While we have provided a general resolution to all EWF quantum paradoxes, we have also highlighted that there remain interesting and less-explored questions relating to the apparent paradoxes initiated by the FR paper [2]. Our work provides a formal and consistent toolkit for exploring these other promising avenues towards understanding the structure of quantum correlations and measurements, through the study of EWFSs and Heisenberg-cut dependence of predictions.

Similar questions can also be posed for understanding the limits of AoE. In [39, 40], it is shown that Bell non-locality is not necessary for FR type paradoxes, as contextuality without Bell non-locality suffices. Is it possible to construct no-go theorems for AoE using contextuality as a resource and is this a necessary feature? More broadly, the non-super agent structure introduced here to distinguish standard quantum scenarios from genuine Wigner's Friend scenarios could be relevant for developing a resource theory of EWFSs (as discussed in Section VI).

## ACKNOWLEDGMENTS

We thank Renato Renner, Victor Gitton, Yilè Yng, Marina Maciel Ansanelli, Joe Renes, Eric Cavalcanti and Nuriya Nurgalieva for insightful discussions. V.V. acknowledges support from an ETH Postdoctoral Fellowship. M.P.W. acknowledges funding from the Swiss National Science Foundation (AMBIZIONE Fellowship, No. PZ00P2_179914). Both authors acknowledge support from NCCR QSIT.
# APPENDIX

## Appendix A: Generality of the definition of EWFSs

Here we discuss the justification for the generality of Definition III.1 of an Extended Wigner's Friend Scenario that we have proposed in this paper. The idea is that Definition III.1 encompasses all finite multi-agent quantum protocols where agents' memories (in which they store the measurement outcome) are modelled as quantum systems, and where one agent can have full quantum control over the labs (measured system and memory) of other agents in the scenario. Here the finiteness applies both to the Hilbert space dimensions and the number of information-processing steps.

Generically, we can model such scenarios by considering a set of $N$ agents $A = \{A_1, \dots, A_N\}$ and a set of $m$ systems $S = \{S_1, \dots, S_m\}$ under study and a set $M := \{M_1, \dots, M_N\}$ of systems, one for each agent $A_i$ which models their memory where they store the outcomes of measurements that they perform.

For simplicity but without loss of generality, we take the lab of each agent $A_i$ to consist of the system $S_i$ that they measure, along with their memory $M_i$. Each agent $A_i \in A$ performs a measurement $M^{A_i}$ on some subset $S_i \subseteq S \cup M \setminus \{M_i\}$ of systems, which can include the memories of other agents, and stores the outcome of the measurement in their memory $M_i$.<fnref n="14" /> Note that $S_i$ is a subset of systems and memories of other agents, which allows each agent to possibly act as a superagent to any subset of the other agents by measuring their memories. We have no loss of generality in assuming that each agent performs one measurement, because any scenario where one agent performs multiple measurements can be bought to this form by modelling them as multiple agents, each performing one measurement.

Next, also without loss of generality, we can assume that each agent $A_i$ acts at a distinct time step $t^i$ with $t^i < t^{i+1}$ for $i < j$, since any physical scenario where the same agent acts at different times can equivalently be modelled in terms of multiple agents each acting at distinct times (since our definition allows for communication channels that carry the relevant information between the time steps). Further, we can model each measurement $M^{A_i}$ as acting on the whole set $S$ of systems and the set of all memories $M_1, \dots, M_{i-1}$ of previous agents since any operation on a subset $S_i$ of systems can be trivially enlarged into an operation on all systems by appending the identity on the complementary set of systems. Agent $A_i$ may also perform certain fixed transformations (corresponding to quantum channel $\mathcal{E}^i$ on $S \cup M$) in between time steps $t_i$ and $t_{i+1}$, for instance, agent $A_1$ may measure the system $S_1 \in S$ and depending on their measurement outcome, could perform a different transformation on some initial state of $S_2$ which they send to agent $A_2$ who could then measure $S_2$.

The circuit of Figure 1 illustrates this general form of an EWFS that we consider, where $\{\mathcal{E}^i\}_{i=1}^N$ are fixed transformations that all agents agree on as part of the protocol while $\{M^{A_i}\}_{i=1}^N$ denote the measurements, one for each agent.

Furthermore, consider that an agent $A_i$ measures a subset $S_i$ of the systems through a measurement $M^{A_i}$. Let $a_i$ be the random variable associated with the measurement outcome, which takes values $a_i$ in the set $\{0, 1, \dots, d_{S_i} - 1\}$, where $d_{S_i}$ is the Hilbert space dimension (assumed to be finite) of the system $S_i$ measured by $A_i$. Without loss of generality, we can model this as a projective measurement, since any measurement on finite dimensional systems can be purified to a projective measurement involving rank 1 projectors on a larger set of systems through Neumark dilation; see e.g. [41, Sec. 9-6]. Explicitly, we can use the projective measurement $\{\pi_{\vec{a}_i}^{S_i} = |a_i\rangle\langle a_i|_{S_i}\}_{a_i \in \mathbb{O}_i}$, where $\{|a_i\rangle_{S_i}\}_{a_i \in \mathbb{O}_i}$ forms an orthonormal basis of $S_i$. While this looks like a computational basis due to the choice of outcome labels $\{0, 1, \dots, d_{S_i} - 1\}$, the choice of which basis to associate with these labels is arbitrary and therefore the measurement may correspond to an arbitrary orthonormal basis.

Together with all these simplifications, we arrive at the general form of Definition III.1.

## Appendix B: Distinguishing predictive and observational statements: refining consistency

In the main text, we used $\Sigma$ to refer to the set of all statements obtained from predictions in an EWFS, in Definition III.3. We motivated the need to distinguish such predictive statements from observational statements through examples of simple classical scenarios. Here we define observational statements and discuss how our results immediately generalise to ensure consistency of predictive and observational statements together. To make the distinction clear, in this section, we will explicitly write $\Sigma_{pred} := \Sigma$.

**Definition B.1 (Observational statements).** Observational statements are statements in the set $\Sigma_{obs} := \{ \text{"Based on observation, I am certain that the outcomes } \vec{a}_j \text{ takes values } \vec{a}_j. \} \}_{\vec{a}_j, \vec{a}_j}$. In logical notation, we will denote elements of the set as $\vec{a}_j = \vec{a}_j|_{obs}$.

<footnote n="14">¹⁴ The memory $M_i$ of an agent $A_i$ is chosen to be a system of at least the same dimension as the system $S_i$ that the agent $A_i$ measures.</footnote>
**Definition B.2 (Consistency of observational statements).** A set $\Sigma_{obs}$ of observational statements is consistent iff $\vec{a}_j = \vec{a}_j|_{obs} \in \Sigma_{obs}$ implies that $\vec{a}_j = \neg \vec{a}_j|_{obs} \notin \Sigma_{obs}$.

**Definition B.3 (Global consistency).** A set $\Sigma_{pred} \cup \Sigma_{obs}$ of predictive and observational statements obtained in an EWFS, is said to be globally consistent if $\Sigma_{pred}$ and $\Sigma_{obs}$ are consistent according to Definition III.5 and Definition B.2 respectively, and additionally, $S := \vec{a}_j = \vec{a}_j|_{obs} \in \Sigma_{obs}$, then $S' \in \Sigma_{pred}$ where $S'$ is a statement associated with a prediction $P(\vec{a}_j = \vec{a}_j|k = \bar{k}) > 0$.

Recall that within our framework the scenario parameters $k$ are instantiated by the setting vector $\vec{x}$, which give us setting-conditioned predictions e.g., $P(\vec{a}_j = \vec{a}_j|\vec{x} = \vec{\xi})$.

In any EWFS, a given choice of settings $\vec{x} = \vec{\xi}$ allows to fix all the channels involved the scenario. Generically, the choice of how these setting values must be chosen is given by some set of reasoning rules $\mathcal{R}$. This rule may either provide an absolute choice of setting values $\vec{x} = \vec{\xi}$ that must be applied to every prediction in the scenario, or it may provide a different choice of setting values relative to each prediction one wishes to compute. An example of the former would be collapse theories or any interpretation that rejects universal validity of unitary quantum theory, which would require all settings to be 1. An example of the latter is our reasoning rule given by the setting choices illustrated in the completeness result of Theorem IV.1: in every prediction, e.g., $P(\vec{a}_j = \vec{a}_j|\vec{x} = \vec{\xi})$, the settings for all outcomes that appear in the probability are set to 1, and the settings for the remaining measurements are set to 0.

Now, given an EWFS together with a set of rules $\mathcal{R}$ for choosing the settings in the augmented circuit, we can consider what happens when we design an experiment to observe the outcome referred to in a prediction. Let us do so with a simple example, referring back to Wigner’s original thought experiment, where Alice was the agent and Bob the superagent. When considering the probability of Bob’s outcome $b$, we have a choice for Alice’s setting $x_A$, and can compute for instance $P(b|x_A = 0)$. If it was possible to physically perform a Wigner’s Friend type experiment involving these agents, and if indeed unitary quantum theory were universally valid, then Bob’s observations regarding $b$ would be consistent with this prediction in the sense of Definition B.3: if these premises are satisfied, then Bob can observe $b = \delta$ only if $P(b = \delta|x_A = 0) > 0$. On the other hand, if unitary quantum theory was not universally valid, but there was an additional (yet undiscovered) physical mechanism for objective collapse, then Bob’s physical observations would be consistent with $P(b = \delta|x_A = 1)$ (according to Definition B.3), and need not be globally consistent relative to $P(b = \delta|x_A = 0) > 0$. Note that in these discussions, we have omitted $x_B$, which will by default be $x_B = 1$ here since these predictions refer to a non-trivial outcome of Bob.

We have proven in our framework that (setting-conditioned) predictions obtained under any possible rule $\mathcal{R}$ for choosing the settings are mutually consistent. Moreover, as the above example illustrates, when considering observational statements, it is important to consider the physical dynamics leading to the said observations.

From the consistency of the predictions, and the definition of global consistency, it is straightforward to see that if the physical dynamics leading to the observations respects the channel choices specified by the reasoning rules $\mathcal{R}$ (and assuming that the physical probabilities also respect the Born rule relative to those channel choices), then the predictions and observations will be globally consistent. On the other hand, if the physical dynamics leading to the observations differs from the $\mathcal{R}$ used to compute the predictions (e.g., when there is physical collapse for all measurements but the rules model certain measurements as pure unitaries), then it is possible to violate global consistency. This provides a way to operationally falsify the rule $\mathcal{R}$ (see also Section VII A), assuming that the experiment was performed in a faithful way, in the sense that indeed Bob’s operation acted on Alice’s whole lab and there was no unexpected information leakage which is not accounted for in the scenario description.

Considering conventional predictions used in FR and LF type arguments, we have seen that these imply a particular default rule $\mathcal{R}^{\text{def}}$ for selecting the settings Theorem IV.1 (as also discussed in Section VII A). In this rule, only the measurements whose outcomes appear in a given prediction are assigned projectors (in order to compute said probability through the Born rule), while all other measurements are modelled as pure unitary evolutions. Therefore, if unitary quantum theory were indeed valid universally, our consistency result of Theorem IV.1 (which only refers to predictive statements) is sufficient to guarantee global consistency of predictive statements obtained through this rule for a given EWFS together with any observations made within a hypothetical experimental realisation of that EWFS. Therefore, we have no loss of generality in restricting only to predictive statements in the main text.

**Consistency and agents’ knowledge** This distinction between predictive and observational statements emphasised here, provides a precise understanding of what does not constitute an inconsistency. The consistency for predictive statements requires that one should not be able to obtain two different probability assignments $P$ and $P'$ for the same outcomes in a scenario, conditioned on the same information. This only refers to predictions i.e., probabilities associated with running the protocol for several rounds.

It is important to note that even in a consistent theory (such as purely classical physics) agents can nevertheless assign different probabilities to events due to different knowledge based on observations made in a given round. If Alice and Bob have a fair coin at hand, where Alice knows that the outcome of the coin flip is $c = \text{heads}$ in one round, she would assign probability 1 to $c = \text{heads}$ in that round while Bob would assign a uniform probability
to $c = \text{heads}$ if he does not know the outcome. This is not a contradiction because Alice's probability relates to an observation in a particular round, her certainty in this case would be captured by an observational statement $c = \text{heads}|_{\text{obs}}$. This example leads to consistent predictions since both agents would predict a uniform probability for $c = \text{heads}$ if asked what the probability of heads will be over many coin flips.

Furthermore, Alice may wish to make a prediction about the outcome $b$ of a bet given that she observed $c = \text{heads}$ (and possibly other information $k = \bar{k}$ that she may know about the scenario). This would then correspond to a prediction $P(b = \text{win}|c = \text{heads}, k = \bar{k})$. On the other hand, Bob who does not know the outcome of the coin flip but has the same background information $k = \bar{k}$ about the scenario, would make a prediction $P(b = \text{win}|k = \bar{k})$ which can be different from $P(b = \text{win}|c = \text{heads}, k = \bar{k})$. This is also not a contradiction, since the two predictions are conditioned on different knowledge, and this conditioning is important to ensure consistency. It is immediate to see that even in these simple classical examples, ignoring the conditioning on agents' knowledge and/or the background assumptions they make about the scenario at hand, one can obtain apparent inconsistencies quite easily.

# Appendix C: Overview of the Frauchiger-Renner apparent paradox

## 1. The FR no-go theorem

Here we review the assumptions Q, C and S of FR's claimed no-go theorem, as well as the additional assumptions U and D that are also relevant to the FR analysis, as pointed out in [5]. The FR protocol and formal statement of their no-go theorem will be reviewed in Appendix C 3.

**Assumption (Q):** it asserts that an agent can be certain that a given proposition holds whenever the quantum-mechanical Born rule assigns probability 1 to it. Specifically:

> Suppose that agent A has established that Statement A<sup>(i)</sup>: “System S is in state $|\psi\rangle_S$ at time $t_0$.” Suppose furthermore that agent A knows that Statement A<sup>(ii)</sup>: “The value $x$ is obtained by a measurement of S w.r.t. the family {$\pi_x^{t_0}$}$_{x \in \chi}$ of Heisenberg operators relative to time $t_0$, which is completed at time $t$.” If $\langle\psi|\pi_\xi^{t_0}|\psi\rangle = 1$ for some $\xi \in \chi$, then agent A can conclude that Statement A<sup>(iii)</sup>: “I am certain that $x = \xi$ at time $t$.”

**Assumption (C):** It asserts that one agent can inherit the knowledge of another agent who uses the same theory as them to arrive at their conclusions.

> Suppose that agent A has established that Statement A<sup>(i)</sup>: “I am certain that agent A’, upon reasoning within the same theory as the one I am using, is certain that $x = \xi$ at time $t$”. Then agent A can conclude that Statement A<sup>(ii)</sup>: “I am certain that $x = \xi$ at time $t$”.

**Assumption (S):** from the viewpoint of an agent who carries out a particular measurement, this measurement has one single outcome. Specifically:

> Suppose that agent A has established that Statement A<sup>(i)</sup>: “I am certain that $x = \xi$ at time $t$” The agent A must necessarily deny that Statement A<sup>(ii)</sup>: “I am certain that $x \neq \xi$ at time $t$”.

Note that a violation of S can itself be interpreted as a logical paradox, as it would imply that the outcome $x$ is both $\xi$ and not $\xi$ with certainty.

In [5], the authors also identify additional assumption U (unitarity) that FR use (as part of Q) but did not explicitly state in their set of assumptions.

**Assumption (U):**

> An agent A can model measurements performed by any other agent B as reversible unitary evolutions in B's lab.

Furthermore they also note that the FR reasoning involves another basic rule of classical logical inference namely the distributive axiom.

**Assumption (D):**

> If an agent A knows a statement $s_1$ and also knows that $s_1$ implies another statement $s_2$ then agent A can conclude that they know $s_2$.

They formulate this in terms of a knowledge operator which formally keeps track of which agent knows which statement [5]. We will also introduce said notation later. Assumption D is implicitly used in FR when the statements $s_1$ and $s_2$ correspond to measurement outcomes, e.g. $s_1$ could be “I am certain that $x = \xi$ at time $t$”,
$s_2$ could be “I am certain that $x' = \xi'$ at time $t'$”. Then if agent A who knows $s_1$ also knows “If I am certain that $x = \xi$ at time $t$, then I am certain that $x' = \xi'$ at time $t'$”, they would use D to conclude that agent A is certain of the statement $s_2$, “I know that $x' = \xi'$ at time $t'$”.

In [5], the FR argument is refined by making explicit the additional assumptions U and D. Thus, they suggest the implication that the assumptions Q, U, C, D and S cannot be simultaneously satisfied in the protocol proposed by FR. We note that this result is proven in [5] by formalising these assumptions using the Kripke structure of epistemic modal logic (which is the branch of logic that refers to knowledge of agents) and we refer the reader to [5] for the formal statement of this theorem in this mathematical language. We will not review the full modal logic framework here but will refer to aspects of it wherever necessary.

At a broad level, the proof proceeds by considering the FR thought experiment where agents reason about each other's knowledge using assumptions Q, U, C and D and claims to show that such agents would always arrive at a violation of S, which as we have explained above can be interpreted as a paradox.

## 2. Entanglement version of the FR experiment

In Section V and Figure 3 of the main text, we provided an overview of the entanglement version of the FR scenario and the main arguments. Here, we review in more detail the entanglement-based version of the FR thought-experiment and apparent paradox that highlights the proof method typically employed to prove the no-go claim regarding a contradiction between the assumptions Q, U, C and D (reviewed in the subsection above).

The entanglement version of the FR protocol is originally attributed to Lluis Masanes (based on a talk), and was also mentioned by Matthew Pusey in [8]. It is much simpler than FR's original (prepare and measure based) protocol, but makes the important/salient features of the FR protocol more readily accessible. The resolution to EWFS paradoxes given by our work is however fully general and applies in particular to both the entanglement and to the original prepare and measure version of FR's arguments. For the interested reader, we provide a review of the original FR thought-experiment in Appendix C 3 and apply our framework to resolve it, in Appendix F 2.

In this section we follow the notation and agent naming conventions of [5, 36]. Here we have two agents Alice and Bob who measure individual subsystems of a bipartite system while two superagents Ursula and Wigner measure the labs (system and memory) of Alice and Bob respectively. The protocol can be broken down into three steps: a bipartite state preparation (pre-selection) at an initial time $t = 1$, intermediate local measurements by Alice and Bob on the state at time $t = 2$, a final measurement and post-selection by two superagents Ursula and Wigner at time $t = 3$.

* **Pre-selection at time $t = 1$:** An initial state $|\psi^{t=1}\rangle_{\text{RS}} := \frac{1}{\sqrt{3}}(|00\rangle_{\text{RS}} + |10\rangle_{\text{RS}} + |11\rangle_{\text{RS}})$ is prepared and shared between Alice and Bob where R and S label the subsystems belonging to Alice and Bob respectively. Alice and Bob's memories A and B are initialised to $|0\rangle_A$ and $|0\rangle_B$. Hence the initial preparation (i.e., pre-selected state) on RASB is given by

$$|\psi^{t=1}\rangle_{\text{RASB}} := \frac{1}{\sqrt{3}}(|0000\rangle + |1000\rangle + |1010\rangle)_{\text{RASB}} \quad (C1)$$

* **Intermediate operations at time $t = 2$:** Alice and Bob measure their respective systems in the computational basis $\{|0\rangle, |1\rangle\}$ and store the outcomes a and b of their respective measurements in their memory systems. From the outside perspective, Alice's measurement is modelled as a unitary $M_{\text{unitary}}^A := (\text{CNOT})_{\text{RA}}$ on the joint system RA which performs a CNOT operation with R as control and A as target, while Bob's measurement is similarly modelled as a unitary $M_{\text{unitary}}^B := (\text{CNOT})_{\text{SB}}$.

* **Post-selection at time $t = 3$:** The super-observers Ursula and Wigner post-select on the following final state $|\phi^{t=3}\rangle_{\text{RASB}} := |\text{ok}\rangle_{\text{RA}}|\text{ok}\rangle_{\text{SB}}$, which they achieve by measuring RA and SB respectively in the basis $\{|\text{ok}\rangle := \frac{1}{\sqrt{2}}(|00\rangle - |11\rangle), |\text{fail}\rangle := \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)\}$ and halting when both of them obtain the outcome $u = \text{ok}, w = \text{ok}$ (where $u$ and $w$ denote Ursula's and Wigner's outcomes, the halting condition is checked by announcing these measurement outcomes in each run).

When modelling both measurements as unitaries, the joint state of RASB just after time $t = 2$ is given by

$$|\psi^{t=2}\rangle_{\text{RASB}} = (M_{\text{unitary}}^{\text{A}} \otimes M_{\text{unitary}}^{\text{B}}) |\psi^{t=1}\rangle_{\text{RASB}} = \frac{1}{\sqrt{3}}(|0000\rangle + |1100\rangle + |1111\rangle)_{\text{RASB}} \quad (C2)$$

The super-observers can then calculate the probability of success of the post-selection given the pre-selection and intermediate unitary evolution as

$$P(u = w = \text{ok}|\psi^{t=2}) = |\langle\phi^{t=3}|\psi^{t=2}\rangle|^2 = |\langle\phi^{t=3}|(M_{\text{unitary}}^{\text{A}} \otimes M_{\text{unitary}}^{\text{B}})|\psi^{t=1}\rangle|^2 = \frac{1}{12} \quad (C3)$$
By this, they establish that they have a non-zero probability of obtaining $u = w = \text{ok}$ and can thus repeat the protocol until they succeed. Upon successfully obtaining the desired outcomes, the protocol is halted and the agents reason about each others' knowledge as follows, where we recall that $K_A(S)$ denotes that Agent A knows the statement $S$.

* Upon obtaining $u = \text{ok}$ on measuring RA, Ursula reasons using the joint state $|\psi^{t=2}\rangle$ (Equation (C2)) that Bob must have certainly obtained the outcome $b = 1$ upon measuring S, since $\langle\text{ok}|_{\text{RA}}\langle 00|_{\text{SB}}|\psi^{t=2}\rangle_{\text{RASB}} = 0$. This gives

$$K_U(u = w = \text{ok} \Rightarrow b = 1) \qquad (C4)$$

* Using the same state, Ursula knows that if Bob obtained $b = 1$ on measuring S, he would have concluded with certainty that Alice obtained $a = 1$ on measuring R since $\langle 00|_{\text{RA}}\langle 11|_{\text{SB}}|\psi^{t=2}\rangle_{\text{RASB}} = 0$.

$$K_U K_B(b = 1 \Rightarrow a = 1) \qquad (C5)$$

* Again using the same state, Ursula further reasons that Bob knows that if Alice had obtained $a = 1$, she would have concluded with certainty that Wigner would obtain $w = \text{fail}$, since $\langle 11|_{\text{RA}}\langle \text{ok}|_{\text{SB}}|\psi^{t=2}\rangle_{\text{RASB}} = 0$. This gives

$$K_U K_B K_A(a = 1 \Rightarrow w = \text{fail}). \qquad (C6)$$

As shown in [5], the above three statements can be combined using the assumption C of the FR paper (in the form of Equation (12)) and the distributive axiom D to yield the following paradoxical chain of statements. To obtain this result, the assumption C in the form of Equation (12) only needs to be used between the following pairs of agents: Alice and Bob, Alice and Wigner, Ursula and Bob, Ursula and Wigner, as other pairs of agents need not trust each other [5, 36].<fnref n="15" />

$$K_U(u = w = \text{ok} \Rightarrow b = 1 \Rightarrow a = 1 \Rightarrow w = \text{fail}), \qquad (C7)$$

or in short $K_U(u = w = \text{ok} \Rightarrow w = \text{fail})$. This argument aims to establish that agents reasoning using Q, U, C and D will arrive at a contradiction with S as they conclude through such a reasoning that $w = \text{ok}$ and $w = \text{fail}$ must both hold with certainty, and is therefore regarded as a proof of the FR no-go theorem (and its refinement as given in [5]) regarding the incompatibility of Q, U, C, D and S.

### 3. Prepare and measure version of the FR experiment

The scenario in [2] describes a protocol realised by agents F, $\overline{F}$, W and $\overline{W}$. Agents F and $\overline{F}$ have their own individual labs while W and $\overline{W}$ are so-called super-observers, i.e. W can perform arbitrary measurements on F and the lab of F, while $\overline{W}$ can perform arbitrary measurements on $\overline{F}$ and the lab of $\overline{F}$. The labs of F and $\overline{F}$ are completely isolated from W and $\overline{W}$ until W and $\overline{W}$ measure at the end of the protocol. It is also assumed that the labs F and $\overline{F}$ are initially in pure states. The lab systems of F and $\overline{F}$ are denoted L and $\overline{L}$ respectively. $\overline{L}$ includes everything in $\overline{F}$'s lab such as the agent $\overline{F}$ and a random generator R that they use, but excludes a spin qubit S which will start off in the lab of $\overline{F}$ and move to the lab of F during the protocol. The lab system L of F will include the spin S that arrives to F, as well as the agent F and other devices in their lab which are not explicitly specified.

The protocol is repeated $n$ times. Between each implementation, it is reset to the initial state. There is a halting condition which is examined at the end of each round. When the condition is satisfied, the protocol is stopped and the last round of the experiment is analysed. The $n^{th}$ round of the protocol from the perspective of W and $\overline{W}$ is as follows:

Before time $n:00$, $\overline{F}$ tosses a coin in her lab which gives heads with probability $1/3$ and tails with probability $2/3$. This coin toss is a random variable which is obtained by measuring the following quantum state in the same basis in which it is expressed.

$$|\text{init}\rangle_R := \sqrt{\frac{1}{3}}|\text{heads}\rangle_R + \sqrt{\frac{2}{3}}|\text{tails}\rangle_R. \qquad (C8)$$

<footnote n="15">¹⁵ In [5], the assumption C is replaced by what they call the trust axiom which asserts that an agent $A^i$ can inherit the knowledge of another agent $A^j$ as per Equation (12) only if $A^i$ trusts $A^j$. In the FR setup, it is precisely these pairs of agents who can be said to trust each other. Other pairs such as Alice and Ursula need not trust each other as one agent Hadamard's the memory of the other.</footnote>
<figure><img src="image_5.png" /><figcaption>FIG. 5: The FR thought experiment expressed using branching notation. The quantum state as per a unitary description from the viewpoint of W and $\overline{W}$ at any time, is the sum of the states of each branched at said time. Arrows indicate the splitting of the amplitudes due to the actions of individual observers. We refer to these as branches and each branch has its own number above the corresponding arrow. The two purple kets on branches 1.1.1. and 2.1.1 cancel each other out when said branches are added together, while the two blue kets on branches 2.1.1. and 2.1.2. cancel when said branches are added together.</figcaption></figure>

The lab of $\overline{F}$ consists of the random generator (or coin) R, a spin S and $\overline{L}$ which represents the rest of the lab. In the following, $\overline{L} := R \otimes \overline{L}$ is the lab of $\overline{F}$, excluding the spin S. Then the unitary $U_{\overline{F}}$ (from the perspective of F, W and $\overline{W}$) that describes F's coin toss (measurement of R) implements the evolution

$$ U_{\overline{F}} : |\text{init}, \phi_0, S_0\rangle_{\overline{L}S} \rightarrow \left( \sqrt{\frac{1}{3}} |\overline{h}\rangle_{\overline{L}} + \sqrt{\frac{2}{3}} |\overline{t}\rangle_{\overline{L}} \right) |S_0\rangle_S, \qquad (C9) $$

where $|\text{init}, \phi_0, S_0\rangle_{\overline{L}S} := |\text{init}_R\rangle_{\overline{L}} |\phi_0\rangle_{\overline{L}} |S_0\rangle_S$ with $|\phi_0\rangle_{\overline{L}}$ the initial state of $\overline{F}$'s lab excluding the spin system and the random generator; the latter two being $|S_0\rangle_S$ and $|\text{init}_R\rangle_{\overline{L}}$ respectively, where $|S_0\rangle_S$ is some initial state of S. The kets $|\overline{h}\rangle, |\overline{t}\rangle$ represent the state of $\overline{F}$'s lab (excluding the spin qubit) after her measurement of the random variable ($|\overline{h}\rangle$ in the case her measurement revealed heads while $|\overline{t}\rangle$ when she obtained tails).

Following the coin toss, $\overline{F}$ prepares a spin qubit in her lab in state $|\downarrow\rangle_S$ if she gets heads, while $|\rightarrow\rangle_S$ if she gets tails. From the perspective of F, W and $\overline{W}$, this is a unitary process since her lab is an isolated system. The unitary that describes this preparation of S is denoted as $U_{\text{spin}}$ and together with $U_{\overline{F}}$, this implements the evolution

$$ U_{\text{spin}} U_{\overline{F}} : |\text{init}, \phi_0, S_0\rangle_{\overline{L}S} \rightarrow \sqrt{\frac{1}{3}} |\overline{h}, \downarrow\rangle_{\overline{L}S} + \sqrt{\frac{2}{3}} |\overline{t}, \rightarrow\rangle_{\overline{L}S}, \qquad (C10) $$

Meanwhile, F waits patiently in her lab, which we denote $|\phi_0\rangle_{\overline{L}}$ initially, where $\overline{L}$ denotes F's lab without the spin qubit S and we will use $\overline{L} := S \otimes \overline{L}$. Between times $n:10$ and $n:20$, $\overline{F}$ sends her qubit system to F's lab and F subsequently measures the qubit in the $|\uparrow\rangle_S, |\downarrow\rangle_S$ basis, denoting her outcome +1/2 and -1/2 respectively. From the perspective of W and $\overline{W}$, the labs of F and $\overline{F}$ are subsequently modelled through the series of evolutions $U_{\overline{F}}$, followed by $U_{\text{spin}}$ followed by $U_{\overline{F}}$ where $U_{\overline{F}}$ is the unitary evolution corresponding to F's measurement of S.

$$ U_{\overline{F}} U_{\text{spin}} U_{\overline{F}} : |\text{init}, \phi_0, S_0\rangle_{\overline{L}S} |\phi_0\rangle_{\overline{L}} \rightarrow \sqrt{\frac{1}{3}} \left( |\overline{h}\rangle_{\overline{L}} |\downarrow, -1/2\rangle_{\overline{L}} + |\overline{t}\rangle_{\overline{L}} |\downarrow, -1/2\rangle_{\overline{L}} + |\overline{t}\rangle_{\overline{L}} |\uparrow, 1/2\rangle_{\overline{L}} \right), \qquad (C11) $$

where $|\downarrow, -1/2\rangle_{\overline{L}}$ denotes the spin in the down state $|\downarrow\rangle_S$ and the state of the rest of lab L when F obtains measurement outcome -1/2; similarly for $|\downarrow, -1/2\rangle_{\overline{L}}$. Between times $n:20$ and $n:30$, $\overline{W}$ measures lab $\overline{L}$ in the basis $\{|\overline{ok}\rangle_{\overline{L}}, |\overline{fail}\rangle_{\overline{L}}\}$, where $|\overline{ok}\rangle_{\overline{L}} := \sqrt{\frac{1}{2}}(|\overline{h}\rangle_{\overline{L}} - |\overline{t}\rangle_{\overline{L}})$, $|\overline{fail}\rangle_{\overline{L}} := \sqrt{\frac{1}{2}}(|\overline{h}\rangle_{\overline{L}} + |\overline{t}\rangle_{\overline{L}})$. After time $n=30$, W measures lab L in the basis $\{|\overline{ok}\rangle_{\overline{L}}, |\overline{fail}\rangle_{\overline{L}}\}$, where $|\overline{ok}\rangle_{\overline{L}} := \sqrt{\frac{1}{2}}(|\downarrow, -1/2\rangle_{\overline{L}} - |\uparrow, 1/2\rangle_{\overline{L}})$, $|\overline{fail}\rangle_{\overline{L}} := \sqrt{\frac{1}{2}}(|\downarrow, -1/2\rangle_{\overline{L}} + |\uparrow, 1/2\rangle_{\overline{L}})$.
The final state of the labs (right hand side of Equation (C11)) in this measurement basis takes on the form

$$ \sqrt{\frac{1}{12} (|\overline{\text{fail}}\rangle_L + |\overline{\text{ok}}\rangle_L) (|\text{fail}\rangle_L + |\text{ok}\rangle_L)} + \sqrt{\frac{1}{6} (|\overline{\text{fail}}\rangle_L - |\overline{\text{ok}}\rangle_L) |\text{fail}\rangle_L} \quad (C12) $$

In Figure 5, we summarise this thought experiment using branching notation. If the outcomes of W and $\overline{W}$ are not ok, ok respectively, then the protocol is re-set and repeated. If their outcomes are ok, ok, then the experiment is finished and F, $\overline{F}$, W and $\overline{W}$ reason about their ok, ok measurement outcomes and what they should be able to predict about the measurement outcomes of the other agents in the last round of the protocol. The obtention of ok, ok by W and $\overline{W}$ respectively is the halting condition mentioned previously, and from the above form of the final state, it is clear that this outcome occurs with probability $\frac{1}{12}$.

While reasoning, they assume that all agents are aware of the entire experimental procedure as described above, and that they all employ the same theory. As we will see, all agents, when reasoning from *their* perspective, assume unitary dynamics for all agent's measurements other than those they are reasoning about. In other words, their protocol is a special case of an EWFS. In Figure 8 we present the augmented circuit of the original FR protocol as a simplified version that makes the mapping back to the entanglement version more apparent.

In FR's reasoning, they make the three assumptions Q, C, S explicitly, in addition to the assumptions U and D implicitly, as described in Appendix C 1.

In particular, the authors then claim to prove the following theorem:

**Theorem 1: Any theory that satisfies assumptions (Q), (C), and (S) yields contradictory statements when applied to their thought experiment of Box 1.** By “Box 1” the authors are referring to the above protocol.

## Appendix D: Derivation of predictions in the augmented EWFS

This section serves the purpose of deriving explicit expressions for predictions and setting-conditioned predictions in the augmented circuit of an EWFS from the Born rule and probability theory.

Our sample space $\Omega$ is chosen to be the set of all measurement outcomes under all settings, namely

$$ \Omega = \{a_1, x_1, a_2, x_2, \dots, a_N, x_N\}_{\{a_j, x_j\}_j}, \quad (D1) $$

where $a_j \in \mathbb{P}$ for $x_j = 0$ and $a_j \in \mathbb{P}$ for $x_j = 1$. The set of events is the power set of $\Omega$. It follows from applying the Born rule to our augmented circuit (see also Figure 2) that if agents $A_1, A_2, \dots, A_N$ were to perform measurements under settings $\vec{x}$ with outcomes corresponding to projectors $\pi_{a_1,x_1}^{A_1}, \pi_{a_2,x_2}^{A_2}, \dots, \pi_{a_N,x_N}^{A_N}$ respectively, then the probability of these elementary events is

$$ \begin{aligned} P(a_1, a_2, \dots, a_N, \vec{x}) &= P(a_1, a_2, \dots, a_N | \vec{x}) P(\vec{x}) \\ &= \text{tr} \left[ \mathcal{E}^N \left( \pi_{a_N, x_N}^{A_N} \mathcal{M}_{\text{unitary}} \left( \dots \mathcal{E}^2 \left( \pi_{a_2, x_2}^{A_2} \mathcal{M}_{\text{unitary}} \left( \mathcal{E}^1 \left( \pi_{a_1, x_1}^{A_1} \mathcal{M}_{\text{unitary}}(\rho_0) \pi_{a_1, x_1}^{A_1} \right) \right) \pi_{a_2, x_2}^{A_2} \right) \dots \right) \pi_{a_N, x_N}^{A_N} \right) \right] P(\vec{x}), \end{aligned} \quad (D2) $$

where $\rho_0$ is the initial “pre-selected” state, namely $\rho_{S_1, \dots, S_m} \otimes |0\rangle\langle 0|^{\otimes N}$ and $P(\vec{x})$ the unconditional probability distribution over settings. The other term in Equation (D2) is the probability of outcomes $\{a_1, a_2, \dots, a_N\}$ conditioned on setting $\vec{x}$.

The probability of the other events can be derived by marginalising over this distribution. In particular, we will be interested in the probability of events corresponding to agents $\{A_{j_1}, A_{j_2}, \dots, A_{j_p}\}$ obtaining outcomes $\{a_{j_1}, a_{j_2}, \dots, a_{j_p}\}$, conditioned of the setting being $\vec{x}$. This is given by

$$ P(a_{j_1}, a_{j_2}, \dots, a_{j_p} | \vec{x}) = \sum_{\substack{a_1 \text{ if } x_1 = 1 \ \& \ 1 \notin \text{OUT} \\ a_2 \text{ if } x_2 = 1 \ \& \ 2 \notin \text{OUT} \\ \vdots \\ a_N \text{ if } x_N = 1 \ \& \ N \notin \text{OUT}}} \text{tr} \left[ \mathcal{E}^N \left( \pi_{a_N, x_N}^{A_N} \dots \mathcal{E}^2 \left( \pi_{a_2, x_2}^{A_2} \mathcal{M}_{\text{unitary}} \left( \mathcal{E}^1 \left( \pi_{a_1, x_1}^{A_1} \mathcal{M}_{\text{unitary}}(\rho_0) \pi_{a_1, x_1}^{A_1} \right) \right) \pi_{a_2, x_2}^{A_2} \right) \dots \pi_{a_N, x_N}^{A_N} \right) \right], \quad (D3) $$

$\text{OUT} := \{j_1, j_2, \dots, j_p\}$. We have not summed over settings $x_j = 0$ since the corresponding observer is modelled unitarily with a deterministic outcome, $a_i = \mathbb{P}$. Note that if we have a set of projectors $\{\pi_j\}_j$ on a system A and