a quantum channel $\mathcal{E}$ also on A, from the Stinespring dilation theorem and other elementary properties, it follows
that there exists another system B such that for an arbitrary linear operator $\hat{A}$ on A, we have

$$
\begin{align*}
\sum_j \operatorname{tr}_A [\mathcal{E}(\pi_j \hat{A} \pi_j)] &= \sum_j \operatorname{tr}_A [\operatorname{tr}_B [U_{AB}(\pi_j \hat{A} \pi_j) \otimes \rho_B U_{AB}^\dagger]] \\
&= \sum_j \operatorname{tr}_{AB} [U_{AB}(\pi_j \hat{A} \otimes \rho_B \pi_j) U_{AB}^\dagger] \tag{D4} \\
&= \operatorname{tr}_{AB} [\left(\sum_j \pi_j^2\right) \hat{A} \otimes \rho_B] = \operatorname{tr}_A [\hat{A}].
\end{align*}
$$

Applying this equality iteratively to Equation (D3) allows us to simplify it.

$$
\begin{equation}
\begin{split}
P(a_{j_1}, a_{j_2}, \ldots, a_{j_p} | \vec{x}) = & \sum_{\substack{a_1 \text{ if } x_1=1 \ \& \ 1 \notin \text{OUT} \\ a_2 \text{ if } x_2=1 \ \& \ 2 \notin \text{OUT}}} \operatorname{tr}[* *], \\
& \vdots \\
& a_{K-1} \text{ if } x_{K-1}=1 \ \& \ K-1 \notin \text{OUT}
\end{split}
\tag{D5}
\end{equation}
$$

$$
** = \pi_{a_{K-1}, 1}^{A_K} \mathcal{E}^{K-1} \left( \pi_{a_{K-1}, x_{K-1}}^{A_{K-1}} \cdots \mathcal{E}^2 \left( \pi_{a_2, x_2}^{A_2} \mathcal{M}_{\text{unitary}}^{A_2} \left( \mathcal{E}^1 \left( \pi_{a_1, x_1}^{A_1} \mathcal{M}_{\text{unitary}}^{A_1} (\rho_0) \pi_{a_1, x_1}^{A_1} \right) \right) \pi_{a_2, x_2}^{A_2} \right) \cdots \pi_{a_{K-1}, x_{K-1}}^{A_{K-1}} \right).
$$

where $K := \max(\text{OUT})$. We see that this prediction does not depend on any channel $\mathcal{E}^j$, setting $x_j$, nor any other property of an agent $A_j$ for which $j > K$.

Using the definition of conditional probability, we can now derive an expression for setting-conditioned pre-
dictions (Definition III.8), i.e. for the probability of a set of observers {$A_{j_1}, A_{j_2}, \dots, A_{j_p}$} obtaining outcomes
{$a_{j_1}, a_{j_2}, \dots, a_{j_p}$} given {$A_{l_1}, A_{l_2}, \dots, A_{l_q}$} measurement outcomes {$a_{l_1}, a_{l_2}, \dots, a_{l_q}$} and setting $\vec{x}$. When using
the definition of conditional probability, Equation (D3) and simplifying by means of Equation (D5), we obtain

$$
\begin{equation}
\begin{split}
P(a_{j_1}, a_{j_2}, \ldots, a_{j_p} \mid a_{l_1}, \ldots, a_{l_q}, \vec{x}) &:= \sum_{\substack{a_1 \text{ if } x_1=1 \ \& \ 1 \notin \text{OUT} \cup \text{IN} \\ a_2 \text{ if } x_2=1 \ \& \ 2 \notin \text{OUT} \cup \text{IN} \\ \vdots \\ a_{Q-1} \text{ if } x_{Q-1}=1 \ \& \ Q-1 \notin \text{OUT} \cup \text{IN}}} \frac{\text{Numerator}}{\sum_{\substack{a_1 \text{ if } x_1=1 \\ a_2 \text{ if } x_2=1}} \text{Denominator}},
\end{split}
\tag{D6}
\end{equation}
$$

$$
\text{Numerator} := \tag{D7}
$$

$$
\operatorname{tr} \left[ \pi_{a_{Q,1}}^{A_Q} \mathcal{M}_{\text{unitary}}^{A_Q} \left( \mathcal{E}^{Q-1} \left( \pi_{a_{Q-1},x_{Q-1}}^{A_{Q-1}} \cdots \mathcal{E}^2 \left( \pi_{a_2,x_2}^{A_2} \mathcal{M}_{\text{unitary}}^{A_2} \left( \mathcal{E}^1 \left( \pi_{a_1,x_1}^{A_1} \mathcal{M}_{\text{unitary}}^{A_1} (\rho_0) \pi_{a_1,x_1}^{A_1} \right) \right) \pi_{a_2,x_2}^{A_2} \right) \cdots \pi_{a_{Q-1},x_{Q-1}}^{A_{Q-1}} \right) \right) \right] \tag{D8}
$$

$$
\text{Denominator} := \tag{D9}
$$

$$
\operatorname{tr} \left[ \pi_{a_{L,1}}^{A_L} \mathcal{M}_{\text{unitary}}^{A_L} \left( \mathcal{E}^{L-1} \left( \pi_{a_{L-1},x_{L-1}}^{A_{L-1}} \cdots \mathcal{E}^2 \left( \pi_{a_2,x_2}^{A_2} \mathcal{M}_{\text{unitary}}^{A_2} \left( \mathcal{E}^1 \left( \pi_{a_1,x_1}^{A_1} \mathcal{M}_{\text{unitary}}^{A_1} (\rho_0) \pi_{a_1,x_1}^{A_1} \right) \right) \pi_{a_2,x_2}^{A_2} \right) \cdots \pi_{a_{L-1},x_{L-1}}^{A_{L-1}} \right) \right] \quad \text{(D10)}
$$

where $Q := \max(\text{OUT} \cup \text{IN})$, $\text{IN} = \{l_1, l_2, \dots, l_q\}$, $L := \max(\text{IN})$. Furthermore, we have the constraint that $x_{j_1} = x_{j_2} = \dots = x_{j_p} = x_{l_1} = x_{l_2} = \dots = x_{l_q} = 1$, since we are reasoning about these measurement outcomes.
Note that if we now sum $P(a_{j_1}, a_{j_2}, \dots, a_{j_p} | a_{l_1}, \dots, a_{l_q}, \vec{x})$ over the elements in the set $\{a_{j_1}, a_{j_2}, \dots, a_{j_p}\}$ we obtain
one, and thus the distribution is normalised. Notice also that we are not summing over outcomes $a_j$ for which $x_j = 0$ since these correspond to the case said measurement is modelled unitarily. Also, as mentioned previously,
were we have also pre-selected state $\rho_0$. The “post-selected state” is merely the post measurement state of the
last measurement performed by observers $A_j$, $j \in \text{OUT}$. We can easily derive an expression for our predictions
(Definition III.2) using Equation (D6) and our prior $P(\vec{x})$:

$$
P(a_{j_1}, a_{j_2}, \dots, a_{j_p} | a_{l_1}, \dots, a_{l_q}, \vec{x}) := \sum_{\vec{x}} P(a_{j_1}, a_{j_2}, \dots, a_{j_p} | a_{l_1}, \dots, a_{l_q}, \vec{x}) P(\vec{x}), \quad (\text{D11})
$$

where for consistency we have defined $P(a_{j_1}, a_{j_2}, \dots, a_{j_p} | a_{l_1}, \dots, a_{l_q}, \vec{x}) = 0$ if there exists $k \in \{j_1, \dots, j_p, l_1, \dots, l_q\}$ s.t. $a_k \neq \perp$, & $x_k = 0$.
<figure><img src="image_6.png" /><figcaption>FIG. 6: A $C^{sys}$-form standard quantum circuit. Here $\mathcal{O}_i$ is the set of possible non-trivial values of the outcome $a_i$. Each $M^{A_i} := \{\pi_{a_i}^{S_i}\}_{a_i \in \mathcal{O}_i}$ implements a projective measurement of the subset $S_i \subseteq \mathcal{S} := \{S_1, \dots, S_m\}$ of systems.</figcaption></figure>

The simplification coming from the iterative application of Equation (D3) to our conditional probability has important physical consequences, namely that the expressions Equations (D6) and (D11) readily do not depend on the channels, measurement schemes nor settings of agents in the future of when the agents in $\{A_k | k \in \mathcal{OUT} \cup \mathcal{IN}\}$ perform their operations, i.e. in the future of the agents who measurement outcomes are being reasoned about.

## Appendix E: Reduction of the augmented circuit in standard quantum scenarios

In Section VI we defined the subclass of EWFSs which we call standard quantum scenarios (Definition VI.3), and showed that objective, setting-independent predictions emerge in this case. Here we show that in EWFSs corresponding to standard quantum scenarios, the augmented circuit of our framework reduces to a standard form quantum circuit without the “Heisenberg-cut” settings. We define two forms of standard quantum circuit representations below before proving this.

**Definition E.1 (Standard quantum circuit representations).** Consider a quantum protocol involving $N$ agents $A := \{A_1, \dots, A_N\}$ and $m$ systems $\mathcal{S} := \{S_1, \dots, S_m\}$ where each agent $A_i$ performs a projective measurement $M^{A_i} := \{\pi_{a_i}^{S_i}\}_{a_i \in \mathcal{O}_i}$ at time $t_i$ that acts non-trivially on a subset $S_i \subseteq \mathcal{S}$ of the systems, obtaining an outcome $a_i$ that can take values in a set $\mathcal{O}_i$, followed by a channel $\mathcal{E}_i$ that may act on all $\mathcal{S}$. A standard quantum circuit representation of such a protocol corresponds to a circuit of one of the two following types, in one case all measurements are modelled as projectors on the systems $\mathcal{S}$ alone and in another case, all measurements can be equivalently purified to unitaries on the systems and some ancillas.

1. *$C^{sys}$-form quantum circuit (Figure 6)* A quantum circuit acting on $\mathcal{S}$ which is defined through the composition $\mathcal{E}_N \circ M^{A_N} \circ \dots \circ \mathcal{E}_1 \circ M^{A_1}$, where each operation is defined over all of $\mathcal{S}$ but it is given that each $M^{A_i}$ acts non-trivially on some subset $S_i \subseteq \mathcal{S}$. Outcome probabilities are calculated by applying the Born rule to the circuit, with the projective measurements $M^{A_i} := \{\pi_{a_i}^{S_i}\}_{a_i \in \mathcal{O}_i}$ on $S_i$.

2. *$C^{sys+anc}$-form quantum circuit (Figure 7)* A quantum circuit acting on $\mathcal{S} \cup \{M_1, \dots, M_N\}$ where $M_i$ denotes an ancillary quantum system corresponding to the measurement $M^{A_i}$ whose state space is isomorphic to that of the systems $S_i \subseteq \mathcal{S}$ on which the measurement acts non-trivially. It is defined through the composition $\mathcal{E}_N \circ M_{\text{unitary}}^{A_N} \circ \dots \circ \mathcal{E}_1 \circ M_{\text{unitary}}^{A_1}$ where each measurement $M^{A_i}$ is purified to a unitary interaction $M_{\text{unitary}}^{A_i}$ acting on $\mathcal{S} \cup M_i$ (and non-trivially on $S_i \cup M_i$), where $M_{\text{unitary}}^{A_i}$ corresponds to the unitary that implements a coherent copy from $S_i$ to $M_i$ in the orthonormal basis given by the measurement projectors (as defined in Equation (3)). Each $\mathcal{E}_i$ acts only on $\mathcal{S}$. Outcome probabilities are calculated by measuring the ancillas $M_i$ (using isomorphic projectors $\{\pi_{a_i}^{M_i} := |a_i\rangle \langle a_i|_{M_i}\}_{a_i \in \mathcal{O}_i}$) at a time $t_f > t_N$ at the end of the protocol (using the Born rule).

These two forms of circuits are illustrated in Figure 6 and Figure 7.

**Theorem E.1 (Recovering standard quantum circuits).** If an EWFS corresponds to a standard quantum scenario (Definition VI.3), then its augmented circuit can be equivalently reduced to a standard quantum circuit, such that the same (non-trivial) predictions are obtained from the original augmented circuit, the $C^{sys}$-form standard circuit or the $C^{sys+anc}$-form standard circuit. Explicitly, for any disjoint sets $\vec{a}_j = (a_{j1}, \dots, a_{jp})$ and $\vec{a}_l = (a_{l1}, \dots, a_{lq})$ of outcomes, and any choice of settings $\vec{x} = \vec{\xi}$ such that $x_i = 1$ for all $i \in \{j_1, \dots, j_p, l_1, \dots, l_q\}$, we have
<figure><img src="image_7.png" /><figcaption>FIG. 7: A $C^{sys+anc}$-form standard quantum circuit where each measurement in Figure 6 is purified to a unitary using an ancilla. Here $O_i$ is the set of possible non-trivial values of the outcome $a_i$. The measurements $M^{M_i} := \{\pi_{a_i}^{M_i}\}_{a_i}$ applied in the global future, implement the isomorphic projective measurement of the ancilla $M_i$, as do the measurements $M^{A_i}$ (from Figure 6) on the subset $S_i \subseteq S := \{S_1, ..., S_m\}$ of systems. It is immediate to see (and well-known) that the present circuit and that of Figure 6 are operationally equivalent.</figcaption></figure>

$$P_{aug}(\vec{a}_j = \vec{a}_j | \vec{a}_j = \vec{a}_j, \vec{x} = \vec{\xi}) = P_{std}(\vec{a}'_j = \vec{a}_j | \vec{a}'_j = \vec{a}_j), \quad (E1)$$

where the $P_{aug}$ refers to setting-conditioned predictions in the augmented circuit of the EWFS and $P_{std}$ refers to predictions in an equivalent $C^{sys}$-form or $C^{sys+anc}$-form standard quantum circuit (where no settings are involved).

A proof of this theorem can be found in Appendix J.

# Appendix F: Detailed analysis of the FR experiment

## 1. Entanglement version of the FR experiment

In the main text, we provided a brief overview of the entanglement version of the FR scenario as well as a simple explanation of our resolution of the paradox. Having reviewed this scenario in detail in Appendix C2, we now provide a more detailed analysis of the same, showing the explicit calculation for every prediction involved.

We reproduce each of the individual statements used in the reasoning of the FR scenario described in Appendix C2, i.e., those captured by Equation (C4)-Equation (C6). Additionally, this reasoning occurs only in a round where the super-agents observe the outcomes $u = w = \text{ok}$, which is associated with a probability $\frac{1}{12}$ in FR's arguments as shown in Equation (C3). We start by reproducing this probability as an explicit setting conditioning prediction in our framework. The probability of Equation (C3) is equivalent to conventional prediction $P_{conv}^{FR}(u = w = \text{ok})$ (Definition III.6) of the FR scenario, in computing this, FR apply the U assumption implicitly to model the measurements of the agents Alice and Bob purely unitarily (using $M_{unitary}^A$ and $M_{unitary}^B$ as seen in Equation (C3)). This corresponds precisely to the setting choices $x_A = x_B = 0$ (as expected from the general mapping from conventional to setting-conditioned predictions given in Theorem IV.1). Therefore it follows immediately that

$$P(u = w = \text{ok} | |\psi\rangle^{t=2}) := P_{conv}^{FR}(u = w = \text{ok}) = P(u = w = \text{ok} | x_A = x_B = 0) \quad (F1)$$
Note that in our notation for predictions, we don't explicitly condition on the initial state of the scenario as this is taken to be in the common knowledge of all agents. See Section VII A for a discussion on how our framework and arguments can generalise to the case where one relaxes this common knowledge assumption.

Now we proceed to analysing each of the statements that agents make when the above post-selection on $u = w = \text{ok}$ succeeds. Consider the statement obtained in Equation (C4), here Ursula, upon knowing that $u = w = \text{ok}$ reasons about Bob's outcome $b$ using the state $|\psi^{t=2}\rangle_{\text{RASB}}$ and concludes that $u = w = \text{ok} \Rightarrow b = 1$. This logical statement is equivalently expressed in probabilities through the conventional prediction $P_{\text{conv}}(b = 1|u = w = \text{ok}) = 1$.

We can readily extract the setting choices implicit in the calculation of this probability for the FR protocol. Note that $|\psi^{t=2}\rangle$ is obtained by applying $M_{\text{unitary}}^{\text{A}} \otimes M_{\text{unitary}}^{\text{B}}$ to the initial state $|\psi^{t=1}\rangle$ and in calculating the above-mentioned probability for $b = 1$ using the Born rule, FR apply the projector $\pi_{1,1}^{\text{B}} = |11\rangle\langle 11|_{\text{SB}}$ to $|\psi^{t=2}\rangle$. This precisely corresponds to the assigning $x_2 = 1$ for Bob's setting. On the other hand, they model Alice's measurement as a purely unitary evolution ($M_{\text{unitary}}^{\text{A}}$) as seen by Ursula in this reasoning step, therefore the setting choice used for Alice in this reasoning is $x_1 = 0$. Making these setting choices explicit, we see that this probability calculated in the FR reasoning is equivalent to the setting-conditioned prediction $P(b = 1|u = w = \text{ok}, (x_1, x_2) = (0, 1)) = 1$ in our framework.

Indeed one can calculate this prediction from the augmented circuit of Figure 4 for the FR protocol (using the Born rule and the well-known rule for conditional probabilities) and would obtain the same. We demonstrate this below, for further details on how the probability rule for setting-conditioned predictions in augmented circuits is derived, see Appendix D.

$$P(b = 1|u = w = \text{ok}, (x_1, x_2) = (0, 1)) = \frac{P(b = 1, u = w = \text{ok}|(x_1, x_2) = (0, 1))}{P(u = w = \text{ok}|(x_1, x_2) = (0, 1))} \quad (\text{F2})$$

That this expression evaluates to unit probability is evident from the following calculation of the numerator and denominator of this expression for the FR protocol.

$$ \begin{aligned} P(b = 1, u = w = \text{ok}|(x_1, x_2) = (0, 1)) &= \frac{1}{12} \\ &= |\langle \text{ok}|_{\text{RA}} \otimes \langle \text{ok}|_{\text{SB}} (1_{\text{RA}} \otimes \pi_{x_2=1,b=1}^{\text{B}}) |\psi^{t=2}\rangle|^2 \quad (\text{F3}) \\ &= |\langle \text{ok}|_{\text{RA}} \otimes \langle \text{ok}|_{\text{SB}} (1_{\text{RA}} \otimes |11\rangle\langle 11|_{\text{SB}}) |\psi^{t=2}\rangle|^2. \end{aligned} $$

$$ \begin{aligned} P(u = w = \text{ok}|(x_1, x_2) = (0, 1)) &= \frac{1}{12} \\ &= \sum_{b \in \{0,1\}} |\langle \text{ok}|_{\text{RA}} \otimes \langle \text{ok}|_{\text{SB}} (1_{\text{RA}} \otimes \pi_{x_2=1,b}^{\text{B}}) |\psi^{t=2}\rangle|^2 \quad (\text{F4}) \\ &= \sum_{b \in \{0,1\}} |\langle \text{ok}|_{\text{RA}} \otimes \langle \text{ok}|_{\text{SB}} (1_{\text{RA}} \otimes |bb\rangle\langle bb|_{\text{SB}}) |\psi^{t=2}\rangle|^2. \end{aligned} $$

Having formalised FR's logical statement $u = w = \text{ok} \Rightarrow b = 1$ as the setting-conditioned prediction $P(b = 1|u = w = \text{ok}, (x_1, x_2) = (0, 1)) = 1$ in our framework, we obtain the corresponding explicit version of the statement:

$$u = w = \text{ok} \wedge (x_1, x_2) = (0, 1) \Rightarrow b = 1. \quad (\text{F5})$$

We now proceed to the next statement of the FR reasoning, given by Equation (C5), where Ursula reasons about Bob's reasoning of Alice through the statement $b = 1 \Rightarrow a = 1$. This is equivalently expressed in terms of probabilities through the conventional prediction $P_{\text{conv}}(a = 1|b = 1) = 1$.

To evaluate this probability using the Born rule as FR do, we must apply the projector $\pi_{1,1}^{\text{A}} \otimes \pi_{1,1}^{\text{B}} = |11\rangle\langle 11|_{\text{RA}} \otimes |11\rangle\langle 11|_{\text{SB}}$ to $|\psi^{t=2}\rangle_{\text{RASB}}$ and it is evident that the implicit setting choices for Alice and Bob used here are $(x_1, x_2) = (1, 1)$. Therefore, making this explicit, we have $P(a = 1|b = 1, (x_1, x_2) = (1, 1)) = 1$. We can again verify this from the probability rule for our augmented circuit (which is simply the Born rule and standard conditional probability rule).

$$P(a = 1|b = 1, (x_1, x_2) = (1, 1)) = \frac{P(a = 1, b = 1|(x_1, x_2) = (1, 1))}{P(b = 1|(x_1, x_2) = (1, 1))} \quad (\text{F6})$$
That this evaluates to unit probability is immediate from the following expressions for the numerator and denominator.

$$ 
\begin{aligned}
P(a = 1, b = 1 | (x_1, x_2) = (1, 1)) &= \frac{1}{3} \\
&= |(\pi_{x_1=1, a=1}^A \otimes \pi_{x_2=1, b=1}^B) |\psi^{t=2}\rangle|^2 \qquad (F7) \\
&= |(1|11\rangle \langle 11|_{RA} \otimes |11\rangle \langle 11|_{SB}) |\psi^{t=2}\rangle|^2.
\end{aligned}
$$

$$ 
\begin{aligned}
P(b = 1 | (x_1, x_2) = (1, 1)) &= \frac{1}{3} \\
&= |(1_{RA} \otimes \pi_{x_2=1, b=1}^B) |\psi^{t=2}\rangle|^2 \qquad (F8) \\
&= |(1_{RA} \otimes |11\rangle \langle 11|_{SB}) |\psi^{t=2}\rangle|^2.
\end{aligned}
$$

From this, as before, we can extract the explicit version of the logical statement.

$$ b = 1 \wedge (x_1, x_2) = (1, 1) \Rightarrow a = 1. \qquad (F9) $$

We now turn to the third statement of the FR reasoning given in Equation (C6), where Ursula reasons about Bob's reasoning about Alice's reasoning about Wigner, through the statement $a = 1 \Rightarrow w = \text{fail}$. This equivalent probabilistic version is given by the conventional prediction $P_{conv}(w = \text{fail}|a = 1) = 1$.

Analysing how FR calculate this probability using the Born rule, we see that this involves applying the projector $\pi_{1,1}^A = |11\rangle \langle 11|_{RA}$ to the state $|\psi^{t=2}\rangle$ which gives us Alice's setting $x_1 = 1$. Moreover, Bob is modelled purely unitarily here, through $M_{unitary}^B$ and we have $x_2 = 0$. Therefore, the explicit setting-conditioned prediction corresponding to this reasoning step of FR is $P(w = \text{fail}|a = 1, (x_1, x_2) = (1, 0)) = 1$. This can be verified within our framework as follows.

$$ P(w = \text{fail}|a = 1, (x_1, x_2) = (1, 0)) = \frac{P(w = \text{fail}, a = 1|(x_1, x_2) = (1, 0))}{P(a = 1|(x_1, x_2) = (1, 0))} \qquad (F10) $$

The numerator and denominator are evaluated below, which makes it evident that the expression above evaluates to unity.

$$ 
\begin{aligned}
P(w = \text{fail}, a = 1|(x_1, x_2) = (1, 0)) &= \frac{2}{3} \\
&= |1_{RA} \otimes \langle \text{fail}|_{SB} (\pi_{x_1=1, a=1}^A \otimes 1_{SB}) |\psi^{t=2}\rangle|^2 \qquad (F11) \\
&= |1_{RA} \otimes \langle \text{fail}|_{SB} (|11\rangle \langle 11|_{RA} \otimes 1_{SB}) |\psi^{t=2}\rangle|^2
\end{aligned}
$$

$$ 
\begin{aligned}
P(a = 1|(x_1, x_2) = (1, 0)) &= \frac{2}{3} \\
&= |(\pi_{x_1=1, a=1}^A \otimes 1_{SB}) |\psi^{t=2}\rangle|^2 \qquad (F12) \\
&= |(1|11\rangle \langle 11|_{RA} \otimes 1_{SB}) |\psi^{t=2}\rangle|^2
\end{aligned}
$$

Then the corresponding, explicit version of the logical statement is,

$$ a = 1 \wedge (x_1, x_2) = (1, 0) \Rightarrow w = \text{fail}. \qquad (F13) $$

Therefore, we have explicitly derived all the statements in Table I while highlighting the setting choices implicit in each of FR's statements. As we have seen, making explicit these setting choices is sufficient to resolve the apparent paradox.
<figure><img src="image_8.png" /><figcaption>FIG. 9: Equivalent version of the augmented circuit (Figure 8) of the prepare and measure FR protocol. Note that the blue box corresponding to $\Pi_{x_1}^{\overline{F}}$ commutes with the controlled Hadamard gate since the control is on the same basis as the measurements associated with the box. Then, it is easy to verify that the joint state of $R\overline{L}\text{SL}$ just before the blue boxes (i.e., just after F's measurement of S) is precisely the same state as $|\psi^{t=2}\rangle_{\text{RASB}}$ (Equation (C2)) of the entanglement version of the FR scenario, with $\overline{L}\setminus$ and $L\setminus$ playing the role of A and B.</figcaption></figure>

subscript indicating the $(x_2, x_1)$ setting that said observer is using when making the statement in the corresponding augmented circuit: $G_{(x_1,x_2)}^t$: “k”. The setting $(0,0)$ corresponds to the case where the agent is not reasoning about $F$ nor $\overline{F}$'s measurement outcome. Note that the reasoning of a particular observer only depends on their choice of $(x_1, x_2)$ settings and not on those of the other observers. Furthermore, we will say $G_{(x_1,x_2)}^t$ = correct if $G^t$ holds under settings $(x_1, x_2)$, $G_{(x_1,x_2)}^t$ = false if statement $G^t$ does not hold under settings $(x_1, x_2)$, and $G_{(x_1,x_2)}^t = \emptyset$ if statement $G^t$ involves reasoning about an observer's measurement outcome when the choice $(x_1, x_2)$ does not allow observer $G$ to reason about said observer.

The first statement is made by $\overline{F}$ at time $n$: 00: $\overline{F}^{n:00}$: “The value $w$ is obtained by a measurement of $L$ w.r.t. basis $\{\pi_{w=\text{ok}}^{n:10}, \pi_{w=\text{fail}}^{n:10}\}$, which is completed at time $n$: 31”. Here $\pi_{w=\text{ok}}^{n:10}, \pi_{w=\text{fail}}^{n:10}$ are the projectors onto the $|\text{ok}\rangle_L, |\text{fail}\rangle_L$ basis. This statement is correct according to the framework of this paper for all $(x_1, x_2)$ i.e. $\overline{F}_{(0,0)}^{n:00} = \overline{F}_{(0,1)}^{n:00} = \overline{F}_{(1,0)}^{n:00} = \overline{F}_{(1,1)}^{n:00} = \text{correct}$.

Then, if $\overline{F}$ got $r = \text{tails in her measurement of the coin flip, she would make the statement } \overline{F}^{n:01}$: “The spin $S$ is in state $|\rightarrow\rangle_S$ at time $n$: 10”. In our formalism, this statement would also hold since $\overline{F}$ is only reasoning about the measurement she made, we thus have $\overline{F}_{(1,0)}^{n:01} = \overline{F}_{(1,1)}^{n:01} = \text{correct}$ and $\overline{F}_{(0,0)}^{n:01} = \overline{F}_{(0,1)}^{n:01} = \emptyset$. Any further statements made by agent $\overline{F}$ would have to be pre-selected on the spin being in state $|\rightarrow\rangle_S$. In Figure 5 this would correspond to the selection of branch 1 and multiplying it by $\sqrt{3}/2$ to re-normalise the branch. The authors then go on to make the following claim: $\overline{F}^{n:00}$ and $\overline{F}^{n:01}$ inserted into $Q$ imply $w = \text{fail}$ (this is claim $\overline{F}^{n:02}$ in table 3). This statement only holds if one does not take into account $F$'s measurement at time $n$: 10. In other words, $\overline{F}_{(1,0)}^{n:02} = \text{correct}, \overline{F}_{(1,1)}^{n:02} = \text{false}$. To see this, note from Figure 5 that their conclusion follows from noting the cancellation of the two blue $|\text{ok}\rangle_L$ kets when we pre-select on branch 2. However, branch 2 is further split into sub-branches 2.1.1. and 2.1.2. via $F$'s measurement. This splitting causes the blue coloured kets to not cancel each other out from $\overline{F}$'s perspective when reasoning under settings $(x_1, x_2) = (1, 1)$. In terms of equations, under settings $(x_1, x_2) = (1, 1)$ we would conclude that $\overline{F}^{n:00}$ and $\overline{F}^{n:01}$ imply that the probability that $w = \text{fail}$ is
$$
\begin{align}
& P(w = \text{fail} | r = \text{tails}, (x_1, x_2) = (1, 1)) \nonumber \\
&= \sum_{z \in \{-1/2, 1/2\}} P(w = \text{fail}, z | r = \text{tails}, (x_1, x_2) = (1, 1)) \nonumber \\
&= \frac{\sum_{z \in \{-1/2, 1/2\}} \text{tr} \left[ \pi_{\text{fail}}^{\text{W}} \pi_z^{\text{F}} \pi_{\text{tail}}^{\overline{\text{F}}} \rho_{\text{Fi}} \pi_{\text{tail}}^{\overline{\text{F}}} \pi_z^{\text{F}} \right]}{\sum_{w \in \{\text{ok, fail}\}} \text{tr} \left[ \pi_w^{\text{W}} \pi_z^{\text{F}} \pi_{\text{tail}}^{\overline{\text{F}}} \rho_{\text{Fi}} \pi_{\text{tail}}^{\overline{\text{F}}} \pi_z^{\text{F}} \right]} \tag{F14} \\
&= \sum_{z \in \{-1/2, 1/2\}} \frac{\text{tr} \left[ \pi_{\text{fail}}^{\text{W}} \pi_z^{\text{F}} \pi_{\text{tail}}^{\overline{\text{F}}} \rho_{\text{Fi}} \pi_{\text{tail}}^{\overline{\text{F}}} \pi_z^{\text{F}} \right]}{\text{tr} \left[ \pi_{\text{tail}}^{\overline{\text{F}}} \rho_{\text{Fi}} \right]} = \frac{1}{2}, \nonumber
\end{align}
$$

where $\rho_{\text{Fi}} := U_F U_{\text{spin}} U_{\overline{\text{F}}} \rho_0 (U_F U_{\text{spin}} U_{\overline{\text{F}}})^{\dagger}$ with $\rho_0 := |\psi_0\rangle\langle\psi_0|$, $|\psi_0\rangle$ being the initial state (i.e. l.h.s. of Equation (C11)). The unitaries $U_F$, $U_{\text{spin}}$, $U_{\overline{\text{F}}}$ are those of the protocol (see Figure 5) and give rise to the final state $\rho_{\text{Fi}} = |\psi_{\text{Fi}}\rangle\langle\psi_{\text{Fi}}|$ with $|\psi_{\text{Fi}}\rangle$ given by the r.h.s. of Equation (C11). F's measurement outcome at time $n$: 00 is denoted by $r$ (with $a$ denoting the corresponding random variable) and $\{\pi_z^F\}_{z \in \{-1/2, 1/2\}}$ is the PVM of F's measurement at time $n$: 10. We note that in the notation of the general framework of Section III, these projectors would be explicitly written as $\{\pi_{1,z}^F\}_{z \in \{-1/2, 1/2\}}$ as they correspond to the case of choosing the setting to be 1 for that measurement. In order to avoid clutter, here and in the following, we drop the setting subscript “1” in all such projectors as the meaning is evident from the context of the protocol at hand. In Equation (F14) we have used the fact that the unitary transformations taking us from the initial state to the final state commute with the measurement projectors, i.e. we have used the equivalence between the circuits of Figures 8 and 9.

Since this probability in Equation (F14) is less than one, we can conclude that $\overline{F}^{n:02}_{(1,1)} = \text{false}$. Now it is F's turn to make a statement. $F^{n:10}$: “The value $z$ is obtained by a measurement of spin $S$ with respect to $\{\pi_{z=-1/2}^{n:10}, \pi_{z=1/2}^{n:10}\}$, which is completed at time $n$: 11”. This statement clearly holds for all setting choices for F in which F can reason about her outcome, since it is merely stating one of the rules of the protocol: $F^{n:10}_{(0,1)} = F^{n:10}_{(1,1)} = \text{correct}$. They now further go on to state: Suppose now that F observed $z = 1/2$ in round $n$. Since $\langle\downarrow|\pi_{z=-1/2}^{n:10}|\downarrow\rangle = 1$, it follows from $Q$ that $S$ was not in state $|\downarrow\rangle_S$, and hence that the random value $r$ was not heads. Therefore $F^{n:12}$: “I am certain that $\overline{F}$ knows that $r = \text{tails at time } n$: 11”. Here F is reasoning about both her measurement outcome and that of $\overline{F}$. Therefore, by definition, this statement only makes sense when F chooses setting $(x_1, x_2) = (1, 1)$ since in the case $(x_1, x_2) = (0, 1)$, F cannot reason about $\overline{F}$'s measurement outcome since there is no classical outcome to assign to it. We therefore have $F^{n:12}_{(0,1)} = F^{n:12}_{(1,0)} = \emptyset$. Meanwhile, the following equation verifies that $F^{n:12}_{(1,1)} = \text{correct}$. Using Equation (D6), if we post-select on F getting $z = 1/2$ in round $n$, then we find that the probability that $\overline{F}$ got tails is one:

$$
P(r = \text{tails} | z = 1/2, (x_1, x_2) = (1, 1)) = \frac{\text{tr}[\pi_{1/2}^{\overline{F}} \pi_{\text{tail}}^{\overline{F}} \rho_{\text{Fi}} \pi_{\text{tail}}^{\overline{F}}]}{\sum_{r \in \{\text{tails, heads}\}} \text{tr}[\pi_{1/2}^{\overline{F}} \pi_r^{\overline{F}} \rho_{\text{Fi}} \pi_r^{\overline{F}}]} = 1, \quad (\text{F15})
$$

where $\pi_{\text{tails}}^{\overline{F}} = |\overline{t}\rangle\langle\overline{t}|_{\overline{L}}$, $\pi_{\text{heads}}^{\overline{F}} = |\overline{h}\rangle\langle\overline{h}|_{\overline{L}}$ are the projectors onto the lab of $\overline{F}$, corresponding to the two outcomes of the coin toss. The last equality follows from noting that $\text{tr}[\pi_{1/2}^{\overline{F}}\pi_{\text{heads}}^{\overline{F}}\rho_{\text{Fi}}\pi_{\text{heads}}^{\overline{F}}] = 0$.

Similarly, the above equation can also be concluded from Figure 5 by noting that if we are in branch 2.1.2 (this corresponds to F getting outcome $z = 1/2$), then the only measurement outcome of $\overline{F}$ which leads to this branch is $r = \text{tails}$.

The authors of the FR paradox then claim: *Therefore from $F^{n:12}$ and invoking $Q$, we conclude $F^{n:13}$: “I am certain that $\overline{F}$ is certain that $W$ will observe $w = \text{fail at time } n$: 31.”. Statements $\{F^{n:13}_{(x_1,x_2)}\}_{x_1,x_2}$ do not correspond to a single statement in our framework since $F^{n:13}$ is a concatenation of two statements “upon observing $z = 1/2$, I know with certainty that $r = \text{tails}$” made by F and “upon observing $r = \text{tails}$, I know with certainty that $w = \text{fail}$” made by $\overline{F}$. Note that these two statements are precisely FR's $F^{n:12}$ and $\overline{F}^{n:02}$ respectively and they are combined using the assumptions C and D to give $F^{n:13}$. Moreover, as we have seen that $\overline{F}^{n:02}_{(1,0)} = \text{correct}$, $\overline{F}^{n:02}_{(1,1)} = \text{false}$ and $F^{n:12}_{(0,1)} = \emptyset$ and $F^{n:12}_{(1,1)} = \text{correct}$, hence there is no common setting $(x_1, x_2)$ for which both $\overline{F}^{n:02}$ and $F^{n:12}$ are correct. Hence there are no setting choice under which $F^{n:13}$ can be derived from $F^{n:12}$ and $\overline{F}^{n:02}$ as FR do. Agent F can always inherit both statements via the knowledge operator Equation (12), in a similar way to how you, the reader, is “inheriting” all the statements in this article when you read them. This, however, by itself poses little value due to the setting mismatch.
Alternatively, one can attempt a more direct derivation of the prediction associated with $F^{n:13}$, which tells us something about the outcome $w$=fail based on the observation of the outcome $z = 1/2$. This requires the setting $x_2 = 1$. Then, from F's perspective under settings $(x_1, x_2) = (1, 1)$ and after obtaining measurement outcome $z = 1/2$, she would conclude that the probability of $w$= fail is only

$$ 
\begin{aligned}
& P(w = \text{fail} | z = 1/2, (x_1, x_2) = (1, 1)) \\
&= \sum_{r \in \{\text{heads}, \text{tails}\}} P(r | z = 1/2, (x_1, x_2) = (1, 1)) P(w = \text{fail} | r, z = 1/2, (x_1, x_2) = (1, 1)) \\
&= P(w = \text{fail} | r = \text{tails} \ \& \ z = 1/2, (x_1, x_2) = (1, 1)) \\
&= \frac{\text{tr}[\pi_{\text{fail}}^{\text{W}} \pi_{1/2}^{\text{F}} \pi_{\text{tails}}^{\overline{\text{F}}} \rho_{\text{Fi}} \pi_{\text{tails}}^{\overline{\text{F}}} \pi_{1/2}^{\text{F}}]}{\sum_{w \in \{\text{ok, fail}\}} \text{tr}[\pi_{w}^{\text{W}} \pi_{1/2}^{\text{F}} \pi_{\text{tails}}^{\overline{\text{F}}} \rho_{\text{Fi}} \pi_{\text{tails}}^{\overline{\text{F}}} \pi_{1/2}^{\text{F}}]} \\
&= \frac{\text{tr}[\pi_{\text{fail}}^{\text{W}} \pi_{1/2}^{\text{F}} \pi_{\text{tails}}^{\overline{\text{F}}} \rho_{\text{Fi}} \pi_{\text{tails}}^{\overline{\text{F}}} \pi_{1/2}^{\text{F}}]}{\text{tr}[\pi_{1/2}^{\text{F}} \pi_{\text{tails}}^{\overline{\text{F}}} \rho_{\text{Fi}} \pi_{\text{tails}}^{\overline{\text{F}}}]}
\end{aligned}
\quad (\text{F16})
$$

where we have used Equation (F15) and the last inequality follows from noting that if we are on branch 2.1.2., then the possibility of W measuring fail cannot be one since he can get outcome ok too (since due to F's measurement, the blue cancellation does not take place). Note also that the same probability is obtained under the settings $(x_1, x_2) = (0, 1)$.

The authors now proceed to reason from the perspective of the super-observers. The first statement is $\overline{W}^{n:21}$: "System R is initialised to $|\text{init}\rangle_R$ at time $n: 00$. This statement is true for all settings, since it is a statement about the protocol, which all agents are assumed to know, so $\overline{W}_{(0,1)}^{n:21} = \overline{W}_{(1,0)}^{n:21} = \overline{W}_{(1,1)}^{n:21} = \text{correct}$. The authors then point out that the state $U_{\overline{F}} |\text{init}, \phi_0, S_0\rangle$ (r.h.s of Equation (C10)) is orthogonal to $|\overline{\text{ok}}\rangle_{\overline{L}} |\downarrow\rangle_S$. This is indeed correct, as can be readily seen from Figure 5 by observing that $|\downarrow\rangle\langle\downarrow|_S$ projects onto the superposition of branches 1.1.1. and 2.1.1. and that for these branches, the purple $|\overline{\text{ok}}\rangle_{\overline{L}}$ terms cancel each other out. The authors then state this in the form of an expectation value of a projector, namely $\langle\text{init}|\pi_{(\overline{w},z)\neq(\overline{\text{ok}},-1/2)}^{n:00}|\text{init}\rangle = 1$. Here $\pi_{(\overline{w},z)\neq(\overline{\text{ok}},-1/2)}^{n:00} = 1 - \pi_{(\overline{w},z)=(\overline{\text{ok}},-1/2)}^{n:00}$, where $\pi_{(\overline{w},z)=(\overline{\text{ok}},-1/2)}^{n:00}$ is a Heisenberg picture projector that would first transform $|\text{init}\rangle_R$ to $U_{\overline{F}} |\text{init}, \phi_0, S_0\rangle$ (through the appropriate isometry that appends $|\phi_0, S_0\rangle_{\overline{L}\setminus S}$ and performs $U_{\overline{F}}$) and then projects onto the outcomes $(\overline{w}, z) = (\overline{\text{ok}}, -1/2)$, as $|\overline{\text{ok}}\rangle\langle\overline{\text{ok}}|_{\overline{L}} \otimes |\downarrow\rangle\langle\downarrow|_S \otimes 1_{\overline{L}}$ i.e, $\pi_{(\overline{w},z)\neq(\overline{\text{ok}},-1/2)}^{n:00}$ is the Heisenberg projector onto the complement of outcomes $(\overline{w}, z) = (\overline{\text{ok}}, -1/2)$. They then claim Agent $\overline{W}$, who uses $Q$, can hence be certain that $(\overline{w}, z) \neq (\overline{\text{ok}}, -1/2)$ and that this implies (when $\overline{w}=\overline{\text{ok}}$) the statement $\overline{W}^{n:22}$: "I am certain that F knows that $z = 1/2$ at time $n: 11$". Now the authors are allowing $\overline{W}$ to take into account the measurement outcome of F in their reasoning but not the measurement of $\overline{F}$. In other words, they are using settings $(0, 1)$ and thus assigning

$$ 
\begin{aligned}
& P(z = 1/2 | \overline{w} = \overline{\text{ok}}, (x_1, x_2) = (0, 1)) \\
&= \sum_{w \in \{\text{ok, fail}\}} P(z = 1/2, w | \overline{w} = \overline{\text{ok}}, (x_1, x_2) = (0, 1)) \\
&= \sum_{w \in \{\text{ok, fail}\}} \frac{\text{tr}[\pi_w^W \pi_{\overline{\text{ok}}}^W \pi_{1/2}^F \rho_{\text{Fi}} \pi_{1/2}^F]}{\sum_{w' \in \{\text{ok, fail}\}} \text{tr}[\pi_{w'}^W \pi_{\overline{\text{ok}}}^W \pi_z^F \rho_{\text{Fi}} \pi_z^F]} \\
&= \frac{\text{tr}[\pi_{\overline{\text{ok}}}^W \pi_{1/2}^F \rho_{\text{Fi}} \pi_{1/2}^F]}{\sum_{z \in \{-1/2, 1/2\}} \text{tr}[\pi_{\overline{\text{ok}}}^W \pi_z^F \rho_{\text{Fi}} \pi_z^F]}
\end{aligned}
\quad (\text{F17})
$$

where the last line follows from observing that in Figure 5 we have that $\text{tr}[\pi_{\overline{\text{ok}}}^W \pi_{-1/2}^F \rho_{\text{Fi}} \pi_{-1/2}^F] = 0$ and thus the r.h.s. of above is one, in accordance with what the authors claim. We thus have $\overline{W}_{(0,1)}^{n:22} = \text{correct}$.

While they are taking into account F's measurement while reasoning, they are not taking into account $\overline{F}$'s measurement when reasoning. We can check that the statement $\overline{W}_{(1,1)}^{n:22} = \text{false}$ since the following probability,
which takes into account both $\overline{\text{F}}$'s and $\text{F}$'s measurements, is strictly less than one:

$$
\begin{align}
& P(z = 1/2 | \overline{w} = \overline{\text{ok}}, (x_1, x_2) = (1, 1)) \nonumber \\
&= \sum_{\substack{w \in \{\text{ok,fail}\} \\ r \in \{\text{heads,tails}\}}} P(z = 1/2, w, r | \overline{w} = \overline{\text{ok}}, (x_1, x_2) = (1, 1)) \nonumber \\
&= \frac{\sum_{\substack{w \in \{\text{ok,fail}\} \\ r \in \{\text{heads,tails}\}}} \text{tr}[\pi_w^\text{W} \pi_{\overline{\text{ok}}}^\text{W} \pi_{1/2}^\text{F} \pi_r^\text{F} \rho_{\text{Fi}} \pi_r^\text{F} \pi_{1/2}^\text{F}]}{\sum_{\substack{z \in \{-1/2, 1/2\} \\ w \in \{\text{ok,fail}\} \\ r \in \{\text{heads,tails}\}}} \text{tr}[\pi_w^\text{W} \pi_{\overline{\text{ok}}}^\text{W} \pi_z^\text{F} \pi_r^\text{F} \rho_{\text{Fi}} \pi_r^\text{F} \pi_z^\text{F}]} \tag{F18} \\
&= \frac{\sum_{r \in \{\text{heads,tails}\}} \text{tr}[\pi_{\overline{\text{ok}}}^\text{W} \pi_{1/2}^\text{F} \pi_r^\text{F} \rho_{\text{Fi}} \pi_r^\text{F} \pi_{1/2}^\text{F}]}{\sum_{z \in \{-1/2, 1/2\}} \text{tr}[\pi_{\overline{\text{ok}}}^\text{W} \pi_z^\text{F} \pi_r^\text{F} \rho_{\text{Fi}} \pi_r^\text{F} \pi_z^\text{F}]} = \frac{1}{3} < 1, \nonumber
\end{align}
$$

Furthermore $\overline{W}^{n:22}_{(1,0)} = \emptyset$ since the statement $\overline{W}^{n:22}$ is about $\text{F}$'s measurement outcome. Before we move on, observe that Equations (F17) and (F18) take on different values and provides another example of collider bias. In particular, these two equations show that the probability of $\text{F}$'s outcome does depend on $\overline{\text{F}}$'s setting $x_1$ given the knowledge that the post-selection on $\overline{w}$ succeeded. This means that the probability $P(z = 1/2|\overline{w} = \overline{\text{ok}})$ that FR consider, is not well-defined when the setting or the prior over the settings is not specified.

The authors then go on to claim ...because agent $\overline{W}$ announces $\overline{w}$, agent $W$ can be certain about $\overline{W}$'s knowledge, which justifies statement $W^{n:26}$, where $W^{n:26}$ is (assuming $\overline{W}$ announces $\overline{w} = \overline{\text{ok}}$ at time $n: 21$.) "I am certain that $\overline{W}$ knows that $\overline{w} = \overline{\text{ok}}$ at time $n: 21$." We agree that $W$ can be sure of any correct announcement made by $\overline{W}$, irrespective of their choice of $(x_1, x_2)$ settings. Thus $W^{n:26}_{(0,1)} = W^{n:26}_{(1,0)} = W^{n:26}_{(1,1)} = \text{correct}$.

Next the authors make the claim ...according to quantum mechanics, agent $\overline{W}$ can be certain that the outcome $(\overline{w}, w) = (\overline{\text{ok}}, \text{ok})$ occurs after finitely many rounds. They make this based on the fact that

$$ \langle \psi_{\text{Fi}} | \pi_{(\overline{w},w)=(\overline{\text{ok}},\text{ok})} | \psi_{\text{Fi}} \rangle = \frac{1}{12}, \qquad (\text{F19}) $$

where $|\psi_{\text{Fi}}\rangle$ is the final state given on the r.h.s. of Equation (C11), $\pi_{(\overline{w},w)=(\overline{\text{ok}},\text{ok})} = |\overline{\text{ok}}\rangle \langle \overline{\text{ok}}|_{\overline{\text{L}}} \otimes |\text{ok}\rangle \langle \text{ok}|_{\overline{\text{L}}}$. Note that this can also be seen as the case with $(x_1, x_2) = (0, 0)$, as we will see later in Equation (F23). The authors then claim $W^{n:00}$: "I am certain that there exists a round $n$ in which the halting condition at time $n: 40$ is satisfied." Noting that

$$
\begin{align}
& P(\overline{w} = \overline{\text{ok}}, w = \text{ok} | (x_1, x_2) = (1, 1)) \nonumber \\
&= \sum_{\substack{z \in \{-1/2, 1/2\} \\ r \in \{\text{heads,tails}\}}} \text{tr}[\pi_{\overline{\text{ok}}}^{\text{W}} \pi_{\overline{\text{ok}}}^{\text{W}} \pi_z^{\text{F}} \pi_r^{\text{F}} \rho_{\text{Fi}} \pi_r^{\text{F}} \pi_z^{\text{F}}] \tag{F20} \\
&= \sum_{\substack{z \in \{-1/2, 1/2\} \\ r \in \{\text{heads,tails}\}}} |\langle \overline{\text{ok}} | \langle \text{ok} | \pi_z^{\text{F}} U_F U_{\text{Spin}} \pi_r^{\text{F}} U_{\overline{\text{F}}} | \text{init}, \phi_0, S_0 \rangle_{\overline{\text{LS}}} | \phi_0 \rangle_{\text{L}}| = \frac{5}{6} \nonumber
\end{align}
$$

is positive, we conclude that $W^{n:00}_{(1,1)} = \text{correct}$. Similarly, we observe that

$$ P(\overline{w} = \overline{\text{ok}}, w = \text{ok} | (x_1, x_2) = (0, 1)) = \sum_{z \in \{-1/2, 1/2\}} \text{tr}[\pi_{\overline{\text{ok}}}^{\text{W}} \pi_{\overline{\text{ok}}}^{\text{W}} \pi_z^{\text{F}} \rho_{\text{Fi}} \pi_z^{\text{F}}] = \frac{1}{2} \qquad (\text{F21}) $$

in positive, thus $W^{n:00}_{(0,1)} = \text{correct}$. Likewise, there exists $r \in \{\text{tails, heads}\}$ such that

$$ P(\overline{w} = \overline{\text{ok}}, w = \text{ok} | (x_1, x_2) = (1, 0)) = \sum_{r \in \{\text{heads,tails}\}} \text{tr}[\pi_{\overline{\text{ok}}}^{\text{W}} \pi_{\overline{\text{ok}}}^{\text{W}} \pi_r^{\text{F}} \rho_{\text{Fi}} \pi_r^{\text{F}}] = \frac{1}{2} \qquad (\text{F22}) $$

is positive thus $W^{n:00}_{(1,0)} = \text{correct}$. Finally,

$$ P(\overline{w} = \overline{\text{ok}}, w = \text{ok} | (x_1, x_2) = (0, 0)) = \text{tr}[\pi_{\overline{\text{ok}}}^{\text{W}} \pi_{\overline{\text{ok}}}^{\text{W}} \rho_{\text{Fi}}] = \frac{1}{12}, \qquad (\text{F23}) $$
hence $\overline{W}_{(0,0)}^{n:00} = \text{correct}$. Therefore we conclude from Equations (F20) to (F23) that while the statement $\overline{W}_{(x_1,x_2)}^{n:00}$ is correct for all settings, the value of the corresponding probability which FR calculate is not correct for all settings.

Next, the authors state Agent F may insert agent's $\overline{F}$'s statement $\overline{F}^{n:02}$ into $F^{n:12}$, obtaining statement $F^{n:13}$. As we pointed out above, said statement only holds under certain settings. The authors then go on to say By virtue of [assumption] C, she [agent F] may then conclude that statement $F^{n:14}$ holds, too. This statement is $F^{n:14}$: "I am certain that W will observe $w$= fail at time $n$:31". This statement is merely the same as $F^{n:13}$ but now with the difference that F herself is certain that W will observe $w$= fail at time $n$:31 rather that F being merely certain that $\overline{F}$ is certain. There are no settings F can select for which this statement is true since

$$P(w = \text{fail} | z = 1/2, (x_1, x_2) = (x_1, 1)) < 1 \qquad (\text{F24})$$

for all settings $x_1 \in \{0,1\}$. Moreover, F cannot inherit the statement "I am certain that W will observe $w$= fail at time $n$:31" via the transfer of knowledge operator since this statement does not make reference to agent $\overline{F}$ and thus it would require F himself to use settings $(x_1, x_2) = (1,0)$ just after obtaining the measurement outcome $z = \frac{1}{2}$; yet the latter requires setting $x_2 = 1$ (This is distinct to the case of $F^{n:13}$).

The remaining statements made by W and $\overline{W}$ are derived from the previous statements under the assumptions Q, U, C, D, and S. However, they do so disregarding the setting parameters and hence reaching their apparent contradiction.

In Table II, we compare the original statements of the FR paper used in deriving the apparent contradiction and the explicit version of those statements obtained within our framework. This is analogous to Table I of the entanglement case and it can be immediately seen that while the original statements (which ignore setting information) can be combined to yield the apparent contradiction, the explicit statements (which specify the setting choice) cannot be combined in this manner even using the standard rules of classical logic. These explicit statements can also be derived directly within the augmented circuit of the EWFS at hand, which is the prepare and measure version of the FR scenario. This augmented circuit is illustrated in Figure 8 and an equivalent version of this circuit (that makes the mapping to the entanglement formulation of the scenario more evident) is given in Figure 9.

In summary, in this section we have shown that each of the individual statements that FR use to derive a contradiction (see Table II) hold under some choice of settings in our framework (note this also follows from Theorem IV.1 of the general framework). However, each of these statements requires a different setting choice and can no longer be combined using the FR assumptions to yield a paradox.

### 3. Setting-dependence in FR's experiment

It Appendix F 1, we mapped each FR probability ($P_{conv}(b = 1|u = w = \text{ok}) = 1$, $P_{conv}(a = 1|b = 1) = 1$ and $P_{conv}(w = \text{fail}|a = 1) = 1$) to a unique setting choice $(x_1, x_2) \in \{(1,0), (1,1), (0,1)\}$ that reproduces the same probability in our framework. It is illustrative to go beyond this analysis and consider other possible setting choices we can assign to each statement. Then we find that in contrast to the above-mentioned settings which do reproduce the exact probabilities of the FR paper, alternate setting choices not longer give the same probabilities, and therefore do not yield the desired logical statements needed in FR's reasoning. For instance, we showed in Appendix F 1 that $P_{conv}(b = 1|u = w = \text{ok}) = P(b = 1|u = w = \text{ok}, (x_1, x_2) = (0,1)) = 1$. We can alternatively consider $P(b = 1|u = w = \text{ok}, (x_1, x_2) = (1,1))$. Note that we cannot consider any setting choice with $x_2 = 0$ here because the prediction involves Bob's outcome $b = 1$ and therefore must involve a non-trivial projector on Bob's side in its evaluation.

$$ \begin{align} P(b = 1, u = w = \text{ok}|(x_1, x_2) = (1, 1)) &= \frac{1}{12} \nonumber \\ &= \sum_{a \in \{0,1\}} |\langle \text{ok} |_{\text{RA}} \otimes \langle \text{ok} |_{\text{SB}} (\pi_{x_1=1,a}^{\text{A}} \otimes \pi_{x_2=1,b=1}^{\text{B}}) | \psi^{t=2} \rangle|^2 \tag{F25} \\ &= \sum_{a \in \{0,1\}} |\langle \text{ok} |_{\text{RA}} \otimes \langle \text{ok} |_{\text{SB}} (|aa\rangle\langle aa|_{\text{RA}} \otimes |11\rangle\langle 11|_{\text{SB}}) | \psi^{t=2} \rangle|^2 \nonumber \end{align} $$
<table><caption>TABLE II: Table 3 of [2] with the additional implicit assumptions needed to make each statement. Without the implicit assumptions, the implications drawn from Q in the original FR paper are summarised in the modal logic language in column 4. The corresponding explicit version of the same implications that take the additional implicit assumptions into account are given in the last column. While original implications can be combined using C and the distributive axiom to yield the further implications listed in Table 3 of the FR paper, the explicit version of these implications cannot be combined even using the standard rules of classical logic such as C and the distributive axiom. We therefore see that the paradox never arises even when using Q, C and S and modelling agents unitarily as long as we are careful to use the explicit version of the implications. The variables x and y appearing in the last column are the settings appearing in the circuit of Figure 8, these encode the additional implicit assumptions.</caption><thead><tr><th>Agent</th><th>Assumed observation</th><th>Statement inferred via Q</th><th>Original implication obtained from Q</th><th>Additional implicit assumption</th><th>Explicit implication obtained from Q</th></tr></thead><tbody><tr><td>F&#773;</td><td>r = tails at n : 01</td><td>F&#773;<sup>n:02</sup>: I am certain that W will observe w = fail at n : 31</td><td>K<sub>F&#773;</sub>(r = tails ⇒ w = fail)</td><td>F&#773;'s outcome is reasoned about and F&#773;'s lab is modelled as a closed quantum system</td><td>K<sub>F&#773;</sub>((x, y) = (1, 0) ∧ r = tails ⇒ w = fail)</td></tr><tr><td>F</td><td>z = +<sup>1</sup>&frasl;<sub>2</sub> at n : 11</td><td>F&#773;<sup>n:12</sup>: I am certain that F&#773; knows that r = tails at n : 01</td><td>K<sub>F</sub>(z = +<sup>1</sup>&frasl;<sub>2</sub> ⇒ K<sub>F&#773;</sub>(r = tails))</td><td>F&#773;'s and F&#773;'s outcomes are both reasoned about</td><td>K<sub>F</sub>((x, y) = (1, 1) ∧ z = +<sup>1</sup>&frasl;<sub>2</sub> ⇒ K<sub>F&#773;</sub>(r = tails))</td></tr><tr><td>W&#773;</td><td>w&#773; = ok at n : 21</td><td>W&#773;<sup>n:22</sup>: I am certain that F&#773; knows z = +<sup>1</sup>&frasl;<sub>2</sub> at n : 11</td><td>K<sub>W&#773;</sub>(w&#773; = ok ⇒ K<sub>F</sub>(z = +<sup>1</sup>&frasl;<sub>2</sub>))</td><td>F&#773;'s outcome is reasoned about and F&#773;'s lab is modelled as a closed quantum system</td><td>K<sub>W&#773;</sub>((x, y) = (0, 1) ∧ w&#773; = ok ⇒ K<sub>F</sub>(z = +<sup>1</sup>&frasl;<sub>2</sub>))</td></tr><tr><td>W</td><td>Announcement by W&#773; that w&#773; = ok at n : 21</td><td>W&#773;<sup>n:26</sup>: I am certain that W&#773; knows that w&#773; = ok at n : 21</td><td>K<sub>W</sub>K<sub>W&#773;</sub>(w&#773; = ok)</td><td>(none)</td><td>K<sub>W</sub>K<sub>W&#773;</sub>(w&#773; = ok)</td></tr></tbody></table>

$$
\begin{align}
P(u = w = \text{ok}|(x_1, x_2) = (1, 1)) &= \frac{1}{4} \nonumber \\
&= \sum_{a,b \in \{0,1\}} |\langle \text{ok} |_{\text{RA}} \otimes \langle \text{ok} |_{\text{SB}} (\pi_{x_1=1,a}^{\text{A}} \otimes \pi_{x_2=1,b}^{\text{B}}) | \psi^{t=2} \rangle|^2 \tag{F26} \\
&= \sum_{a,b \in \{0,1\}} |\langle \text{ok} |_{\text{RA}} \otimes \langle \text{ok} |_{\text{SB}} (|aa\rangle \langle aa|_{\text{RA}} \otimes |bb\rangle \langle bb|_{\text{SB}}) | \psi^{t=2} \rangle|^2 \nonumber
\end{align}
$$

Putting this together using the rule for conditional probabilities, we have

$$P(b=1|u=w=\text{ok}, (x_1, x_2) = (1, 1)) = \frac{1}{3} \neq 1 \qquad (\text{F27})$$

The FR probability $P_{conv}(a = 1|b = 1) = 1$ corresponds to $P(a = 1|b = 1, (x_1, x_2) = (1, 1)) = 1$ in our framework, but no alternative setting choices are possible here as the prediction refers to both Alice and Bob's classical outcomes. Finally, the FR probability $P_{conv}(w = \text{fail}|a = 1) = 1$ corresponds to $P(w = \text{fail}|a = 1, (x_1, x_2) = (1, 0)) = 1$ in our framework and we consider the alternative setting choice $(x_1, x_2) = (1, 1)$ in this case (again $x_1 = 0$ is not a possible setting choice here as the prediction involves Alice's classical outcome). The corresponding prediction $P(w = \text{fail}|a = 1, (x_1, x_2) = (1, 1))$ is calculated below, and is also not equal to 1.

$$
\begin{align}
P(a = 1, w = \text{fail}|(x_1, x_2) = (1, 1)) &= \frac{1}{3} \nonumber \\
&= \sum_{b \in \{0,1\}} |1_{\text{RA}} \otimes \langle \text{fail} |_{\text{SB}} (\pi_{x_1=1,a=1}^{\text{A}} \otimes \pi_{x_2=1,b}^{\text{B}}) | \psi^{t=2} \rangle|^2 \tag{F28} \\
&= \sum_{b \in \{0,1\}} |1_{\text{RA}} \otimes \langle \text{fail} |_{\text{SB}} (|11\rangle \langle 11|_{\text{RA}} \otimes |bb\rangle \langle bb|_{\text{SB}}) | \psi^{t=2} \rangle|^2 \nonumber
\end{align}
$$
$$ 
\begin{align}
P(a = 1 | (x_1, x_2) = (1, 1)) &= \frac{2}{3} \nonumber \\
&= \sum_{b \in \{0,1\}} | (\pi_{x_1=1,a=1}^A \otimes \pi_{x_2=1,b}^B) |\psi^{t=2}\rangle|^2 \tag{F29} \\
&= \sum_{b \in \{0,1\}} | (|11\rangle \langle 11|_{RA} \otimes |bb\rangle \langle bb|_{SB}) |\psi^{t=2}\rangle|^2 \nonumber
\end{align}
$$

This gives us

$$ P(w = \text{fail}|a = 1, (x_1, x_2) = (1, 1)) = \frac{1}{2} \neq 1. \tag{F30} $$

In summary, we have found that $P(b = 1|u = w = \text{ok}, (x_1, x_2) = (0, 1)) \neq P(b = 1|u = w = \text{ok}, (x_1, x_2) = (1, 1))$ and $P(w = \text{fail}|a = 1, (x_1, x_2) = (1, 0)) \neq P(w = \text{fail}|a = 1, (x_1, x_2) = (1, 1))$. That is the predictions do depend on the setting. This highlights that a core reason for the FR paradox in this scenario is that the FR reasoning involves ignoring the conditioning on setting values in a scenario where the predictions do depend on these value. This illustrates our general result of Corollary IV.2, which shows that in any EWFS, any potential inconsistencies can only arise in this manner: ignoring the conditioning on settings, in predictions which are setting-dependent.

Reformulating this insight in a more framework-independent manner, this means that the choice of Heisenberg cuts under which we evaluate the predictions for different sets of outcomes does matter, the probabilities do depend on this choice (captured by the settings in our framework). Our framework which explicitly takes into account the Heisenberg cut, provides a natural resolution to any such paradox without forsaking unitary quantum theory or classical logic.

## Appendix G: Classical example reproducing certain features of the FR correlations

We noted earlier that the FR correlations are an example of collider bias, or in this case, they exhibit signalling under post-selection. Further, we have also seen that the assumption **I** is necessary for reproducing the apparent paradox. In this section, we provide an example of a classical protocol that reproduces these features of the FR correlations. We note however that the settings in our example do not have the same physical meaning as the settings of the FR augmented circuit (which relate to the choice of Heisenberg cut). Nevertheless, the example gives an intuition for our resolution. We will thus refer to the **I** applied to the settings of this classical example as **I**<sub>C</sub> to distinguish it from the version of the assumption applied to the settings of our augmented circuit that model Heisenberg cuts.

The augmented circuit Figure 4 of the entanglement version of the FR protocol can be represented as shown in Figure 10(a). Consider now the circuit of Figure 10(b), which has the same configuration as the circuit of Figure 10(a) for the FR protocol and consider the following classical operations in place of the boxes **A**, **B**, **U** and **W**. The initial state is simply a uniformly distributed binary random variable $\Lambda$. The operation **A** of Alice takes the given value $\lambda \in \{0, 1\}$ of $\Lambda$ together with the binary setting $x_1$, and generates the outcome $a \in \{0, 1\}$ by taking the XOR of the two $a = \lambda \oplus x_1$, and forwards $a$ to Ursula's operation **U**. **U** takes $a$ from Alice's operation, internally generates a uniformly distributed bit $k_U \in \{0, 1\}$ and outputs $u = a \cdot k_U$ where $\cdot$ denotes the logical AND. Bob's side is identical, his operation **B** takes a value of $\Lambda$ along with $x_2$ and outputs $b = \lambda \oplus x_2$ while forwarding a copy of the classical bit $b$ to Wigner's operation **W**. **W** takes $b$ from Bob, generates a uniform bit $k_W$ internally and outputs $w = b \cdot k_W$.

Given this simple model, it is easy to verify that $P(a, b|x_1, x_2)$ is non-signalling i.e., $P(a|x_1, x_2) = P(a|x_1)$ and $P(b|x_1, x_2) = P(b|x_2)$. However, $P(a, b|x_1, x_2, u)$ allows signalling from Alice to Bob i.e., $P(b|x_1, x_2, u) \neq P(b|x_2, u)$ and $P(a, b|x_1, x_2, w)$ allows signalling from Bob to Alice i.e., $P(a|x_1, x_2, w) \neq P(a|x_1, w)$. This is because, for instance, conditioned on the knowledge that $u = 1$ (say), we know that since $u = a \cdot k_U$, $a = k_U = 1$. Further, since $a = \lambda \oplus x_1 = 1$ and $b = \lambda \oplus x_2$, we know that $b = x_1 \oplus x_2 \oplus 1$ and Bob's outcome $b$ clearly depends on Alice's setting $x_1$ now. This means that when reasoning given the knowledge about some post-selections on $u$ and $w$, even in classical probability theory, the agents must be careful to account for any new correlations that this may introduce between their settings, else they may end up making wrong conclusions.

Interestingly, in [14] Healey pointed out that whether or not Bob's state is collapsed influences the probabilities of Alice's outcome in the FR scenario due to the post-selection. Denying this dependence is what [14] refers to as the assumption of *intervention insensitivity*. Healey argues that FR implicitly assumes intervention insensitivity in order to obtain the logical contradiction. While Healey's intervention sensitivity may seem similar in spirit to the setting-dependence in our framework, [14] appears to regard intervention sensitivity as a special non-local feature of quantum theory. In the FR case, the setting-dependence takes the particular form of enabling signalling
<figure><img src="image_9.png" /><figcaption>FIG. 10: Left: A concise representation of the augmented circuit (Figure 4) corresponding to the entanglement version of the FR protocol. Here the boxes *A* and *B* correspond to Alice and Bob's measurements modelled as unitaries $M_{\text{unitary}}^{\mathcal{A}}$ and $M_{\text{unitary}}^{\mathcal{B}}$ followed by the setting-dependent projectors $\{\pi_{x_1}^{\mathcal{A}}\}$ and $\{\pi_{x_2}^{\mathcal{B}}\}$. The boxes *U* and *W* model to ok, fail basis measurements of Ursula and Wigner on the joint systems RA and SB producing outcomes *u*, *w*. Right: A classical circuit with the same configuration as the left hand side circuit of the FR protocol. Here $\Lambda$, *a*, *b*, *u*, *w*, $x_1$ and $x_2$ are all associated with classical, binary variables and the operations $\mathcal{A}$, $\mathcal{B}$, $\mathcal{U}$ and $\mathcal{W}$ and initial distribution over $\Lambda$ are described in the text. Here as well, as in the FR scenario, $x_1$ and *b* although initially independent (due to no-signalling) become correlated when conditioning on the future outcome *w*, and similarly $x_2$ and *a* become correlated under post-selection on *u*.</figcaption></figure>

under post-selection that we have explained here. However, the present classical example shows that this is not a
special feature of the quantum correlations of the FR scenario but rather a common feature that arises in classical
probability theory and causal inference [29] where it often goes by the name of *collider bias*.

In the FR protocol, this feature is not immediately apparent as there are no settings in the actual protocol.
However, as we have argued in the main text, these settings are implicit in the reasoning steps, and once made
explicit, we see that there is indeed signalling under post-selection also in the FR scenario. Note that our general
resolution and results do not depend on this aspect of signalling under post-selection or collider bias, although
this is a feature of FR correlations. The general results show that imposing the **I** assumption in a scenario where
it fails is necessary for recovering FR type apparent paradoxes (independently of this feature), and we
have seen that **I** does fail in the FR scenario.

This is also the case in our classical example, we can show that imposing $I_C$ also leads to an apparent paradox
here, as the predictions do depend on the settings (the scenario violates $I_C$). For instance, take $u = 1$ and
$x_2 = 1$, then we have $a = k_u = 1$ and hence $\Lambda = x_1 \oplus 1$, which gives $b = x_1 \oplus x_2 \oplus 1 = x_1$. Therefore we have
$P(b = 0 | x_1 = 0, u = 1, x_2 = 1) = 1$ and $P(b = 1 | x_1 = 1, u = 1, x_2 = 1) = 1$. If we ignore the conditioning on the
settings, as allowed by $I_C$, we apparently have the paradoxical probability assignments $P(b = 0 | u = 1) = 1$ and
$P(b = 1 | u = 1) = 1$ that correspond to $b$ deterministically being 0 and 1 (both with certainty) whenever $u$ is 1,
which is an apparent violation of S as in the FR scenario. However we see that there is no real paradox once we
correctly take into account all the settings and account for the correlations of the outcomes with the settings.

While the property of signalling under postselection holds in this simple classical circuit as it does in the
quantum FR scenario, the exact correlations of the FR scenario are Bell non-local (or more generally, contextual)
and cannot be reproduced using classical resources alone, once we fix the above circuit configuration where there is
no information exchange between Alice and Bob's sides (except the shared initial state). In order to reproduce the
FR correlations classically, one must necessarily modify the circuit configuration to allow for additional connections
such as allow the settings $x_1$ and $x_2$ to depend on the state preparation, or allow for causal connections between
the $\mathcal{A}$, $\mathcal{U}$ and $\mathcal{B}$, $\mathcal{W}$ operations, analogous to superdeterministic or non-local hidden variable explanations of Bell
correlations (see for instance [42]). The correspondence between the FR argument and Hardy's logical argument
for contextuality of quantum correlations is discussed in the next section.

**Appendix H: Relation to Hardy's logical proof of contextuality**

In [43] Lucien Hardy proves the (Bell) non-locality, and hence the contextuality of the correlations arising from a
set of bipartite quantum states and measurements, through a logical argument that does not rely on the violation
of Bell-type inequalities. The extremal state and measurements of Hardy's argument, as well as the chain of logical
reasoning are in direct correspondence with those of the entanglement version of the FR scenario presented here. We explain the relationship here and also clarify how FR's construction can be seen as an alternative proof of Hardy's theorem, thereby establishing the contextuality of this scenario.

However FR's claimed no-go theorem regarding a logical reasoning paradox between agents is a stronger statement, and we have shown in the main-text that such a paradox can always be avoided within our framework. In other words, while the FR chain of reasoning cannot lead to a logical paradox using quantum theory and classical logic (for observed outcomes) once implicit assumptions about settings are accounted for, the same chain of reasoning can be used in the FR setup to prove the contextuality of the scenario as Hardy did. The relationship between the FR scenario and Hardy's proof has been noted several times before in the literature (for instance, [5, 13, 36, 44]), this section is to be considered as a concrete overview of this relationship along with additional insights provided by our framework.

Consider the bipartite Hardy state

$$|\psi_{Hardy}\rangle_{AB} := \frac{1}{\sqrt{3}}(|00\rangle + |10\rangle + |11\rangle)_{AB}$$

shared between Alice and Bob and suppose that they perform a Bell type experiment on this state with the setting choices $x_1 \in \{0, 1\}$ for Alice and $x_2 \in \{0, 1\}$ for Bob and corresponding outcomes $a, b \in \{0, 1\}$. Suppose the settings $x_1 = 0, x_2 = 0$ correspond to Hadamard basis ($|+\rangle, |-\rangle$) measurements on the A and B subsystems respectively, in which case we take $a, b = 0$ corresponding to the + outcome and $a, b = 1$ corresponding to the – outcome. And let the settings $x_1 = 1, x_2 = 1$ correspond to computational basis ($|0\rangle, |1\rangle$) measurements on the A and B subsystems. Hardy's argument establishes that the resulting distribution $P(a, b|x_1, x_2)$ is (Bell) non-local (and consequently, contextual), through a logical argument that does not involve a consideration of (Bell-like) inequalities.

This is established as follows. If the correlations were Bell local, then we could simultaneously assign values to the outcomes of all the measurements. Explicitly, let $a_{x_1=0}$ and $a_{x_1=1}$ denote the outcome $a$ of Alice corresponding to the setting choice $x_1 = 0$ and $x_1 = 1$ respectively, and similarly let $b_{x_2=0}$ and $b_{x_2=1}$ denote the outcomes of Bob when his setting $x_2 = 0$ and $x_2 = 1$ respectively. Consider a run of the experiment where the settings $x_1 = x_2 = 0$ are chosen and the outcomes $a_{x_1=0} = b_{x_2=0} = 1$ are obtained. Now using the state $|\psi_{Hardy}\rangle_{AB}$, one can argue that whenever $a_{x_1=0} = 1$, we must have $b_{x_2=1} = 1$. This is because $a_{x_1=0} = 1$ implies that the post-measurement state on A is $|-\rangle_A$, and $\langle-|_A \langle0|_B |\psi_{Hardy}\rangle_{AB} = 0$. We can also see that whenever $b_{x_2=1} = 1$, we must have $a_{x_1=1} = 1$ since $\langle0|_A \langle1|_B |\psi_{Hardy}\rangle_{AB} = 0$. Finally, whenever $a_{x_1=1} = 1$, we must have $b_{x_2=0} = 0$ since $\langle1|_A \langle-|_B |\psi_{Hardy}\rangle_{AB} = 0$. This contradicts the fact that $a_{x_1=0} = b_{x_2=0} = 1$ was obtained in the said experimental run, establishing that in such an experimental run one cannot jointly assign values to the outcomes of all measurements. Such correlations are said to exhibit logical contextuality, and we refer the reader to [45] for further details on the same.

We now return to the FR scenario and explain how these directly correspond to the above correlations. For this, note that the $\{|00\rangle, |11\rangle\}$ basis measurements on the RA and SB subsystems of $|\psi^{t=2}\rangle_{RASB} = \frac{1}{\sqrt{3}}(|0000\rangle + |1100\rangle + |1111\rangle)_{RASB}$ are operationally equivalent to the computational basis $\{|0\rangle, |1\rangle\}$ measurements on the A and B subsystems of the bipartite (Hardy) state $|\psi_{Hardy}\rangle_{AB} := \frac{1}{\sqrt{3}}(|00\rangle + |10\rangle + |11\rangle)_{AB}$. Further, the $\{|\text{ok}\rangle, |\text{fail}\rangle\}$ basis measurements on the RA and SB subsystems of $|\psi^{t=2}\rangle_{RASB}$ are operationally equivalent to the Hadamard basis $\{|+\rangle, |-\rangle\}$ measurements on the A and B subsystems of $|\psi_{Hardy}\rangle_{AB}$. In the FR scenario, $x_1 = 0$ ensures that the $\{|\text{ok}\rangle_{RA}, |\text{fail}\rangle_{RA}\}$ basis measurement is performed directly on $|\psi^{t=2}\rangle_{RASB}$ giving the outcome $u$, while $x_1 = 1$ ensures that the $\{|00\rangle_{RA}, |11\rangle_{RA}\}$ basis measurement is performed on $|\psi^{t=2}\rangle_{RASB}$, giving the outcome $a$, similarly on Bob's side. Thus if we generate a new outcome $a'$ locally on Alice's side such that $a' = 0$ when $x_1 = 0$ and $u = \text{fail}$, $a' = 1$ when $x_1 = 0$ and $u = \text{ok}$, and $a' = a$ when $x_1 = 1$, and similarly the outcome $b'$ on Bob's side such that $b' = 0$ when $x_2 = 0$ and $w = \text{fail}$, $b' = 1$ when $x_2 = 0$ and $w = \text{ok}$ and $b' = b$ when $x_2 = 1$, the distribution $P(a', b'|x_1, x_2)$ is the same as that of the Hardy construction. This distribution is non-local and therefore cannot be explained by a local hidden variable model.

From the above construction, we can see that the choice of setting $x_1$ (or $x_2$) determines which of the two measurements (the one corresponding to the computational basis or that corresponding to the Hadamard basis) is performed directly on the subsystem RA (or SB) of the state $|\psi^{t=2}\rangle_{RASB}$ isomorphic to $|\psi_{Hardy}\rangle_{AB}$. These measurements are complementary, and $(x_1, x_2) \in \{(0, 0), (0, 1), (1, 0), (1, 1)\}$ specify the four possible measurement contexts in this setup. The Hardy model, and therefore the FR model is logically contextual which means that we cannot jointly assign values to the outcomes of all these measurements [45]. The arguments for the logical proof of contextuality are also in direct correspondence with the logical reasoning steps leading to the paradoxical chain of Equation (C7), with $a_{x_1=0} = 0/1$, $b_{x_2=0} = 0/1$ in the Hardy case corresponding to $u = \text{fail/ok}$, $w = \text{fail/ok}$ in the FR case and $a_{x_1=1} = 0/1$, $b_{x_2=1} = 0/1$ corresponding to $a = 0/1$, $b = 0/1$.

In fact, the FR set-up can also be used to prove Hardy's theorem i.e., to provide a logical proof of contextuality of the underlying quantum states and measurements. The FR argument for the entanglement scenario establishes precisely this (by mirroring the Hardy argument as we have explained above). Therefore, even though the FR
and Hardy theorems have a one-to-one mapping between the states, measurements and statements involved in the proofs, Hardy's conclusion regarding the non-locality/contextuality of the scenario holds true while the validity of FR's claim that "Quantum theory cannot consistently justify the use of itself" (which is a stronger statement than Hardy's) does not generally hold true for quantum theory as our results show, but can be understood as holding only for a specific version of quantum theory that additionally assumes independence of statements from choices of Heisenberg cuts (i.e., when additional assuming I), as we have discussed in Section V.

# Appendix I: Relation to previous works: a more unified picture

There are several previous works discussing and analyzing the FR apparent paradox, which can be categorised into three broad groups. The first category includes works (eg. [28, 46-49]) which provide fundamental reasons for rejecting one of the assumptions of the FR no-go theorem. This is a natural response to any no-go theorem, as it necessitates identifying which assumptions to discard.

The second category consists of papers such as [5, 10, 17] which (possibly after identifying implicit assumptions beyond Q, C, and S) conclude that there is indeed a paradox in the FR thought experiment. Some of these papers suggest possible resolutions by proposing additional reasoning rules.

The third category includes works such as [11, 13-16, 44, 50] that question the correctness of the theorem. They argue that the reasoning in the FR paper includes implicit assumptions, one or more of which are invalid in quantum theory, and conclude that there is no real paradox between quantum theory and logic as FR claim.<fnref n="16" />

These responses are often discussed independently and can be interpretation-specific. Here, we provide a more unified picture of these arguments within our framework, also discussing how different interpretations of quantum mechanics could apply our framework to resolve the apparent paradox in any EWFS.

## 1. Previous works rejecting one of FR's assumptions

Here we consider previous arguments which fundamentally reject one of the assumptions Q, U, C, D, or S in the FR no-go theorem [2, 5]. We discuss these cases in detail below, relating them to different interpretations of quantum mechanics and demonstrating how these arguments play out within our framework.

**Rejecting Q:** In FR's original paper, the authors interpret the violation of Q as a violation of unitary quantum theory. Following [5], we have separated these two assumptions Q and U explicitly, such that the Q assumption refers to the validity of the Born rule (or a weaker possibilistic version thereof), which can be independently violated without giving up U.

Rejecting Q means at least one prediction of the FR scenario (e.g., Equation (16) for the entanglement version) does not comply with the Born rule. Not all predictions can be simultaneously tested in a single experiment, leaving room for observable compliance with the Born rule without satisfying Q. As noted in [51], certain versions of Bohmian mechanics, a non-local hidden variable interpretation, violate Q in this manner, by deviating from the Born rule for inaccessible predictions.

In our framework, predictions in an EWFS are given by $P(\vec{a}_j|\vec{a}_l, k)$ (Definition III.2), where $\vec{a}_j$ and $\vec{a}_l$ are sets of measurement outcomes and $k$ are a set of parameters describing the scenario. In hidden variable interpretations, in addition to quantum states, channels, and measurements (Definition III.1), $k$ can include descriptions of the hidden variables. $P(\vec{a}_j|\vec{a}_l, k)$ may be computed using this information, not necessarily following the Born rule.

Such predictions can violate assumption Q formalised in our work, and they would not correspond to the setting-conditioned predictions of the augmented circuit (Definition III.8) which are derived using the Born rule. However, the settings are still meaningful in such interpretations, they provide different descriptions of the quantum measurement channel that the hidden variables must “emulate”. It is to be noted that within such theories, settings cannot be interpreted as choices of Heisenberg cuts since classical theories do not have a non-trivial notion of such a cut. But such a fully classical theory could “emulate” a unitary measurement channel or setting $x_i = 0$ through non-local hidden mechanisms.

Moreover, by applying the general premise of our reasoning rules discussed in Section VII A, FR type paradoxes can also avoided in any EWFS within such interpretations as well. This would involve using settings to clearly specify the measurement channels being assumed, as well as potentially additional parameters in $k$ and conditioning on these variables in the reasoning (even when the probabilities do not arise through the Born rule).

**Rejecting U:** The U assumption, first noted in [5], involves rejecting the idea that agents' labs evolve as unitary closed quantum systems. Interpretations involving objective mechanisms for “wavefunction collapse” reject

<footnote n="16">¹⁶ The cited works are only representatives of the extensive research generated by the FR paper. For further references on previous responses to FR, see [9].</footnote>
this, where unitary quantum theory breaks down at certain macroscopic scales. Such interpretations also violate the **U** assumption in our work. In this view, one would assign setting $x_i = 1$ for all measurements, assuming they involve systems more macroscopic than the scale at which the collapse occurs. Whether such models always avoid FR-type paradoxes depends on whether the predicted objective collapse scale is smaller than the scale at which a quantum system can be regarded as an “agent”, which remains an open question.

In our formalisation of the **U** assumption as **U**, we have made explicit an additional aspect that is often implicit in previous works, relating to quantum control over labs of other agents. Interpretations compatible with unitary quantum theory can still violate **U** by arguing that full quantum control over an agent’s lab is impractical. For instance, in decoherence-based interpretations, measurements are always associated with an inaccessible environment that decoheres the quantum superposition. To regard each $M^{A_i}$ in our EWFS definition (Definition III.1) as a “measurement” according to this view point, one must restrict to standard quantum scenarios (Definition VI.3), where one agent cannot have complete quantum control over the labs of other agents. As shown by Corollary VI.1, in these scenarios, a joint distribution for all (non-trivial) measurement outcomes can be assigned, which would be equivalent to the distribution obtained by choosing $x_i = 1$ for all settings.

Notably, a recent work [28] makes an argument for objective decoherence based on the concept of pre-measurements. The notion of a pre-measurement coincides with the case of setting 0 in our framework i.e., the modelling of the measurement purely as a unitary evolution. They argue that pre-measurements cannot produce outcomes consistently in quantum theory, and to produce an outcome one requires an irreversible evolution. This aspect is captured within our framework in the fact that no non-trivial measurement outcome can be assigned to a measurement modelled with setting 0, non-trivial outcomes require setting 1 (applying the projection postulate). The authors of [28] then suggest that this observation resolves the FR apparent paradox as FR’s reasoning assigns outcome values to pre-measurements. However, [28] imposes an objective distinction between pre-measurements and measurements, considering the operations of the agents Alice and Bob only as pre measurements (setting 0) and those of the super-agents Ursula and Wigner as measurements (setting 1). Moreover, the criterion for objectively fixing this choice in general EWFSs is not explicitly discussed in prior works.

The resolution proposed here is significantly more general. It applies to all EWFSs, allowing for generally subjective setting assignments (given by an explicit rule) while consistently accounting for relational interpretations as well.

**Rejecting C or D:** The assumption **C** allows agents to inherit each other’s knowledge. FR suggested that relational approaches, such as relational quantum mechanics [52] and QBism [53, 54], reject this assumption by allowing different subjective perspectives. However, since the original **C** assumption left unclear the formal modeling of agents’ knowledge and choices of Heisenberg cuts, we have proposed a formalisation of this assumption in our framework, as **C**.

In subjective/relational approaches to EWFSs, agents can have different choices of Heisenberg cuts. Super-agents model agents as “inside the cut,” assuming unitary evolution of their labs, while agents model themselves as “outside the cut,” associating classical outcomes to their measurements. This is captured in our framework through settings, our reasoning rule (see Theorem IV.1 and Section VII A) allows the choices of these settings to be subjective and agent-dependent in a precise manner.

Despite this relational aspect and allowing dynamic updates of agents’ knowledge (based on new observations), our framework still allows agents to freely inherit each other’s knowledge as it satisfies the assumption **C**. Even when agents use different settings, if assumptions about setting choices are explicitly stated, **C** and other logical rules, such as the distributive axiom **D** and transitivity, are always satisfied as shown in Corollary IV.2. Ignoring these setting choices leads to apparent breakdowns of these axioms, but, as argued in Section VII A, similar logical breakdowns can occur in classical scenarios if assumptions about channels used in reasoning are ignored.

There also exist proposals, such as [55], which reject classical logical axioms in light of FR’s work. Our results highlight that such rejection is unnecessary for preserving the consistency of quantum theory or relational interpretations.

**Rejecting S:** Previous discussions [2, 5, 9] suggest that many-worlds interpretations [56] would tend to reject **S** because an outcome $a_i$ could be 0 in one “branch” of the wavefunction and 1 in another. Treating $a_i$ as a quantum object associated with a quantum state $|a_i\rangle$, the violation of **S** seems less paradoxical since the quantum state not being $|0\rangle$ does not imply it is $|1\rangle$, but could be another non-orthogonal state [16]. Violation of **S** is paradoxical only when $a_i$ is regarded as a classical random variable.

In our framework, there is no ambiguity between classical and quantum objects. Even with subjective points of view, agents’ measurement outcomes are classical variables, with the trivial value $a_i = \perp$ when the measurement is unitary. Therefore, rejecting **S** within our framework would indeed constitute a paradox, leading to an invalid probability distribution $P(a_i)$. However, we have shown this does not happen in our framework.

Thus, many-worlds interpretations would not reject **S** in our framework. We do not believe any previous proposal or interpretation of quantum theory are pathological enough to fail this assumption. Moreover, it is important to note that selecting setting $x_i = 1$ when computing probabilities of $a_i$ does not conflict with many-worlds interpretations where the universe undergoes unitary evolution. The projectors applied for the $x_i = 1$ case
need not be interpreted as a physical projection or wavefunction collapse, but can be seen as a mathematical tool
necessary for computing probabilities through the Born rule. Alternatively, this can also be understood akin to
classical probability theory, as conditioning on an agents' knowledge of seeing a particular outcome $a_i = a_i$ in
a particular "branch" of a many-worlds wavefunction, even when other branches can exist, from the perspective
of other agents. It would be interesting to extend such ideas to formally define a perspectival version of the
many-worlds interpretation, from an operational approach.

## 2. Previous suggestions for consistent reasoning

Another category of previous works is those that propose ways around FR's apparent paradox, either through
conceptual discussions or by suggesting additional reasoning rules.

We have shown that if the FR scenario and its conventional quantum predictions are equivalently described
using our augmented circuit, there is no paradox, eliminating the need for additional reasoning rules.<fnref n="17" /> Our
formal version of each FR assumption is satisfied without contradictions. Nevertheless, it is insightful to interpret
previous works' proposed reasoning rules.

**Adding context labels** At a more pedagogical level, [5] suggests an additional reasoning rule that could
incorporated to avoid the paradox, which involves tagging all statements made by agents with certain "context
labels" and only combining statements with matching labels. However no particular model for these labels or a
rigorous formalisation of the rule was proposed there, and the question of efficiency and causal consistency (namely
regarding the number of contexts to be checked at each step, and whether one needs to account for the contexts of
future measurements) was left open. The settings of our framework can be interpreted as a concrete instantiation
of these abstract context labels, but we have seen that no additional reasoning rules beyond quantum theory and
classical logic are required once the settings are explicitly specified.

Importantly, our framework does not impose that only statements derived from the same setting labels can be
combined. In fact, we have shown that in certain situations the statements become independent of the certain
setting labels and the choice of those setting labels no longer affects the validity of the statement (see Theorem IV.1
and Theorem VI.1). We have also addressed the efficiency and causality issues.

**Parsing rule for measurements** Another recent work [10] proposes a parsing rule for quantum theory to
determine when a unitary/isometry can be considered a measurement with a classical outcome. This rule deems an
operation a measurement if no future non-commuting operations act on the memory system, ensuring consistency
in the FR scenario by limiting agents' reasoning to such operations. This idea also aligns with the principle
of superpositional solipsism in [17]. Unlike our framework, which uses settings to describe measurements, [10]
maintains ambiguity in measurement modeling, but the additional parsing rule clarifies when the ambiguity can
be safely ignored in agents' reasoning.

If an operation $M^{A_i}$ in the EWFS description is deemed a measurement according to this parsing rule, then it
would imply that there are no super-agents to the agent $A_i$ in the scenario (also according to our definition of the
non-superagent structure). As shown in Theorem E.1, $A_i$'s setting can then be safely ignored in the predictions,
allowing all agents to reason about $A_i$'s outcome without inconsistencies in that scenario. However, we note that
our non-superagent structure is formalised without reference to commutativity of operations. In particular, a
non-commuting operation acting solely on the memory $M_i$ might fail the parsing rule of [10] but would still allow
$A_i$'s setting to be safely ignored in our framework.

Moreover our framework does not restrict the ability to reason about $A_i$'s outcome based on whether there
are super-agents to $A_i$ (i.e., someone "Hadamarding" $A_i$'s brain) in the scenario. As we have seen, each of FR's
statements can be reproduced in our formalism. This is in contrast to [10] and another work [12] which suggest
that certain statements of FR that refer to outcomes of agents' whose brains will later be Hadamarded, should not
be allowed. While such additional rules or restrictions are sufficient to ensure logical consistency, the adherence
to causality principles and efficiency of programming remain open question. Our work shows that such additional
rules are not necessary for the purpose of ensuring consistency in any EWFS and that logical, causal consistency
and efficiency of reasoning are possible with weaker restrictions on the reasoning, though it may be of interest to
impose such rules based on other physical considerations or interpretations.

**Consistent histories interpretation** Another notable approach is the consistent histories (CH) interpretation
of quantum theory [26, 27]. In this approach, one specifies a set of possible histories for a given scenario, and a
consistency criterion that tells us when a set of histories is consistent, allowing probabilities to be assigned to such
consistent sets. The approach has been applied to explain a number of quantum paradoxes arising in standard
(non-Wigner's Friend like) quantum scenarios, such as contextuality paradoxes, pre and post-selection paradoxes

<footnote n="17">¹⁷ Although additional steps can be employed to simplify the reasoning by dropping redundant settings and parameters, as is also the case in classical multi-agent scenarios (cf. Section VII A).</footnote>
etc. In the context of Wigner's Friend scenarios, [11] shows that the reasoning used in FR's derivation of the paradox requires computing probabilities in an inconsistent family of histories, and the authors then argue that such reasoning is therefore not valid in quantum theory.

At a high level, this bears resemblance to the general result of Corollary IV.3, where we demonstrated that any EWF paradox is rooted in computing predictions across distinct setting choices and combining such statements while ignoring the setting labels (even though the predictions depend on these labels). However, our approach is distinct from the CH interpretation in key ways, and arguably offers a simpler and more minimal resolution to EWFS paradoxes.

The CH approach requires considering commutation relations between families of projectors to determine consistency, while our settings are formalised without reference to commutation relations and do not involve such additional rules to ensure consistency. We have seen that conditioning on settings that model a measurement corresponds to conditioning on the choice of channel used in computing a probability, no additional rule is required to forbid combinations of statements made under different settings (as in needed in the CH approach to avoid combining inconsistent histories).

Moreover, our setting independence results, highlight that in certain cases, classical probability theory and classical logic ensure that even statements made under different settings can be consistently combined, as those statements are independent of the setting choice. While non-commuting projectors central to the CH approach are a non-classical aspect, we have discussed how our resolution of the paradox shares strong similarities with how analogous multi-agent inconsistencies are resolved in purely classical theories (see also the previous paragraph on parsing rules, for further discussion on the link between non-commutativity and our resolution).

Moreover, while CH's solution is sufficient to avoid logical inconsistencies, it does not provide an explicit reasoning rule for how to select the set of histories to be used when reasoning about predictions or agents' knowledge in an EWFS [9]. In particular, [9] noted that, if unitary quantum theory were universally valid, the perspectives and predictions of different agents would indeed correspond to different histories and even in a single experiment such as FR's, there is no single objective history of events that is realised.

A natural question that arises is whether there is a unified framework with a concrete set of rules to construct it, where all these perspectives and predictions of an EWFS can be consistently incorporated, while recovering the predictions of real-world quantum experiments performed so far. Here, we have demonstrated that all EWFSs in quantum theory can be completely described within a single consistent quantum circuit framework that is capable of resolving general EWF paradoxes. We also provided an explicit rule for selecting the settings (modelling the Heisenberg cuts) in accordance with universal validity of unitary quantum theory and without assuming the existence of objective notion of observed measurement events, the observations and predictions in our formalism are relative to a choice of such settings.

### 3. Previous works discussing the validity of FR's claim

A third category of papers are those which question the validity of FR's theorem, due to additional implicit assumptions (other than Q, U, C, D and S discussed in [2, 5]) which are violated [11, 13–16, 50]. Some specific examples include Scott Aaronson's blog post that refers to an additional “unformalised” assumption of FR, Healey's assumption of intervention insensitivity [14], and the assumption regarding collapse/no-collapse that Araujo points out [15]. To quote Aaronson,

> But I reject an assumption that Frauchiger and Renner never formalise. That assumption is, basically: “it makes sense to chain together statements that involve superposed agents measuring each other's brains in different incompatible bases, as if the statements still referred to a world where these measurements weren't being done.”

In our work, we have concretely shown that an additional assumption I (setting-independence) is violated in the FR scenario, but must be necessarily imposed for reproducing the apparent paradox. We have seen that the assumptions Q, U, C, D and S about the validity of quantum theory and classical logic are always consistent in any EWFS in our formalism. In our understanding, the assumption I formally embodies the spirit of the additional implicit assumptions noted in the above examples of previous works. All these previous works argue that the respective assumption is necessary to reproduce the apparent paradox of FR, but suggest that the assumption fails in the FR scenario, due to the incompatible measurements, and/or (Bell) non-locality of the correlations involved.

While the assumption I of our framework is formulated in a much more general manner, and is a priori independent of quantum features such as incompatible measurements and non-locality, when applied to the FR scenario, it captures the features of these previously noted assumptions. Aaronson's assumption is reflected in our framework by noting that the four possible settings $(x_1, x_2) \in \{(0,0), (0,1), (1,0), (1,1)\}$ for the FR protocol are in one-to-one correspondence with the four possible measurement contexts (i.e., sets of compatible measurements) of
a bipartite Bell experiment with binary choice of measurements on each side, this correspondence is explained in Appendix H where we discuss the relation between FR's protocol and Hardy's proof regarding (Bell) non-locality without inequalities.

The settings also tell us whether or not we have “collapsed” the state of an agents’ system and memory (or rest of the lab) by applying a projector corresponding to their outcome, and provide a way to formalise Araujo’s assumption. Araujo also noted issues with FR’s treatment of post-selection, which are accounted for in our analysis by explicitly computing probabilities conditioned on the post-selection and avoiding collider bias. We have also seen that in the FR scenario (Appendix F 2), the prediction $P(w = \text{fail}|r = \text{tails}, (x_1, x_2))$ does depend on the setting $x_2$ of F. This is analogous to the property that Healey calls intervention sensitivity.

While our results which establish the general consistency of quantum theory are certainly contrary to FR’s popular summary that quantum theory cannot consistently justify the use of itself, the validity of FR’s claimed theorem depends on how the assumptions are interpreted. In Section V C, we discussed a refined interpretation of FR’s theorem in which it is correct, this would be to say that a version of quantum theory that additionally allows Heisenberg cuts to be freely ignored leads to contradictions with classical logic. However, in this interpretation, the result is not as surprising as the popular summary claims it to be, although it has undoubtedly fuelled an intriguing research program on EWFSs in quantum foundations inspiring other no-go theorems (such as [3, 4]) based on a similar set-up where the underlying assumptions are formalised more rigorously.

# Appendix J: Proofs of all results

## 1. Proofs of results from the main text

**Lemma III.1.** If $\Sigma$ is a set of consistent predictive statements, then

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

where $\neg S$ denotes the negation of the statement S.

*Proof.* Let $S \in \Sigma$ be an arbitrary statement, by construction this of the form: “If the outcomes $\vec{a}_l$ take values $\vec{a}_l$ and the additional parameters of the scenario take the value $k = \bar{k}$, then the outcomes $\vec{a}_l$ take values $\vec{a}_l$ with a probability $P(\vec{a}_j = \vec{a}_j|\vec{a}_l = \vec{a}_l, k = \bar{k})$.” Then the negation of S is “If the outcomes $\vec{a}_l$ take values $\vec{a}_l$ and the additional parameters of the scenario take the value $k = \bar{k}$, then the outcomes $\vec{a}_l$ do not take values $\vec{a}_l$ with a probability $P(\vec{a}_j = \vec{a}_j|\vec{a}_l = \vec{a}_l, k = \bar{k})$.” More precisely, this is a set of statements:

$\neg S := \{ \text{If the outcomes } \vec{a}_l \text{ take values } \vec{a}_l \text{ and the additional parameters of the scenario take the value } k = \bar{k}, \text{ then the outcomes } \vec{a}_l \text{ take values } \vec{a}_l \text{ with a probability } P'(\vec{a}_j = \vec{a}_j|\vec{a}_l = \vec{a}_l, k = \bar{k}). \} \}_{P \neq P'}$.

It is then immediate from Definition III.5 that if $S \in \Sigma$ then no $S' \in \neg S$ can be such that $S' \in \Sigma$. This completes the proof. $\square$

**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.

*Proof.* The statement immediately follows from noting that fixing the setting choice for each measurement fully specifies the circuit of Figure 1 and therefore the joint state of all systems and memories $S_1, \dots, S_m, M_1, \dots, M_N$ at each time-step. If A and B make the same choice of settings for all measurements, then they fully agree on the circuit (initial states and all channels) and therefore assign the same joint state to all systems at each time step.

Since all agents apply the Born rule to calculate the probabilities (Definition III.8), and agree on the states and measurements for which the probabilities are calculated, they obtain the same probabilities and therefore make the same predictions for all measurements.

More explicitly, Appendix D shows that in any EWFS, given a choice of settings, every setting-conditioned prediction can be uniquely computed by applying the Born rule to our augmented circuit. Therefore if all agents in an EWFS use the augmented quantum circuit to reason, picking the same setting choice or prior distribution $P(\vec{x})$, then they agree on all the predictions made in that scenario. $\square$

## 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 $i \in \{j_1, \dots, j_p, l_1, \dots, l_q\}$ and $x_i = 0$ for all $i \notin \{j_1, \dots, j_p, l_1, \dots, l_q\}$.
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 $A_i \not\prec A_k$ 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 $\vec{x} = (x_1, ..., x_N)$.

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

*Proof.* 1. Completeness: Recall that a conventional prediction $P_{conv}(\vec{a}_j = \vec{a}_j|\vec{a}_l = \vec{a}_l)$ in an EWFS (Definition III.6) is computed by modelling the measurements $M^{A_i}$ for all $i \in \{j_1, ..., j_p, l_1, ..., l_q\}$ as purely unitary evolutions of the agents' labs (Equation (3)). By construction of the augmented circuit for the EWFS, this corresponds to the case where $x_i = 0$ for all $i \in \{j_1, ..., j_p, l_1, ..., l_q\}$.

For $i \notin \{j_1, ..., j_p, l_1, ..., l_q\}$, in a conventional prediction, a projective measurement of $\{\Pi_{a_i}^S = |a_i\rangle\langle a_i|_S\}_{a_i \in \mathcal{O}_i}$ is performed followed by a CNOT in the same basis with the system $S_i$ as control and memory $M_i$ as target (capturing the memory update after measurement). This CNOT is precisely the unitary $M_{unitary}^{A_i}$ (Equation (3)).

Now, following Section III B of the main text, consider an initial state $|\psi\rangle_{S_i} \otimes |0\rangle_{M_i}$ of the system and memory (on which the measurement $M^{A_i}$ acts) where $|\psi\rangle_{S_i} = \sum_{a_i \in \mathcal{O}_i} c_{a_i} |a_i\rangle_{S_i}$. That section of the main text shows that the following two procedures are operationally equivalent: (1) a projective measurement $\{\Pi_{a_i}^S = |a_i\rangle\langle a_i|_S\}_{a_i \in \mathcal{O}_i}$ is applied on the system and then the unitary channel $M_{unitary}^{A_i}$ is applied on the system and memory (2) the unitary channel $M_{unitary}^{A_i}$ is applied on the system and memory, and then the projective measurement $\{\Pi_{a_i}^{S,M_i} = |a_i a_i\rangle\langle a_i a_i|_{S,M_i}\}_{a_i \in \mathcal{O}_i}$ is performed on the system and memory. The operational equivalence of (1) and (2) entails that they yield the same transformation on the initial state, and also that the measurements in both cases yield the same probabilities for an outcome $a_i$.

Generally by linearity, the argument about the equivalence of (1) and (2) extends to all initial states $\rho_{S_i} \otimes |0\rangle\langle 0|_{M_i}$ of the system and memory. (1) is the procedure used for dealing with a measurement $M^{A_i}$ when computing conventional predictions involving the outcome $a_i$ (Definition III.6) while (2) is the procedure used in the augmented EWFS for calculating the setting conditioned predictions (Definition III.8) involving the outcome $a_i$ where by default we will have $x_i = 1$.

This shows that the conventional prediction for the probability of $\vec{a}_j = \vec{a}_j$ given $\vec{a}_l = \vec{a}_l$ and the augmented circuit prediction for the same yield the same answer when using the setting assignment $\vec{x} = \vec{\xi}^*$ defined as: $x_i = 1$ for $i \in \{j_1, ..., j_p, l_1, ..., l_q\}$ and $x_i = 0$ otherwise. The only difference being that the corresponding prediction in the augmented circuit explicitly conditions on the setting choice $\vec{x} = \vec{\xi}^*$, and we have $P_{conv}(\vec{a}_j = \vec{a}_j|\vec{a}_l = \vec{a}_l, \vec{x} = \vec{\xi}^*) = P(\vec{a}_j = \vec{a}_j|\vec{a}_l = \vec{a}_l, \vec{x} = \vec{\xi}^*)$.

This establishes the claim that all the conventional predictions of any given EWFS can be derived as particular cases of setting-conditioned predictions in the single augmented circuit of the EWFS, showing the completeness of our formalism.

2. Consistency: The fact that the set $\Sigma^{aug}$ of statements associated with an EWFS in our framework satisfies the consistency condition of Definition III.5 immediately follows from the procedure through which these statements are derived.

As defined in Definition III.10, each statement $S \in \Sigma^{aug}$ is associated with a setting-conditioned prediction $P(\vec{a}_j = \vec{a}_j|\vec{a}_l = \vec{a}_l, \vec{x} = \vec{\xi})$ of the given EWFS. Such predictions are computed by applying the Born rule to a single, well-defined quantum circuit (explicitly detailed in Appendix D), which implies that for any given setting choice $\vec{x} = \vec{\xi}$ and sets of outcome values $\vec{a}_j = \vec{a}_j, \vec{a}_l = \vec{a}_l$ in a given EWFS, a unique setting-conditioned prediction $P(\vec{a}_j = \vec{a}_j|\vec{a}_l = \vec{a}_l, \vec{x} = \vec{\xi})$ can be computed, which corresponds to a valid, well-defined and normalised conditional probability distribution.

Since $\Sigma^{aug}$ only contains statements associated with setting-conditioned predictions in the augmented circuit, and it is impossible to have $P(\vec{a}_j = \vec{a}_j|\vec{a}_l = \vec{a}_l, \vec{x} = \vec{\xi})$ and $P'(\vec{a}_j = \vec{a}_j|\vec{a}_l = \vec{a}_l, \vec{x} = \vec{\xi})$ in the same augmented EWFS for $P \neq P$, the consistency of $\Sigma^{aug}$ according to Definition III.5 follows.

3. Causality: Our framework provides a single well-defined quantum circuit (the augmented circuit) from which all setting-conditioned predictions in an EWFS can be derived, and in which all the operations are applied in an acyclic order (given by the DAG $G$). From this, the desired result follows immediately by applying well-known results on quantum causal networks or causal models, such as the d-separation theorem
[29–31, 57]. However, in the interest of not introducing new concepts, we describe the proof in terms of concepts and results introduced in this paper.

Consider the setting-conditioned prediction $P(\vec{a}_j = \vec{a}'_j|\vec{a}_l = \vec{a}'_l, \vec{x} = \vec{\xi})$, and the condition (C) $A_i \not\subset A_k$ for all $k \in \{j_1, ..., j_p, l_1, ..., l_q\}$. Notice that an $A_i$ satisfies this condition, in particular when $t_i > \max(t_{j_1}, ..., t_{j_p}, t_{l_1}, ..., t_{l_q})$. In Appendix D, we have established that any setting-conditioned prediction $P(\vec{a}_j = \vec{a}'_j|\vec{a}_l = \vec{a}'_l, \vec{x} = \vec{\xi})$ in an augmented EWFS can be simplified to the form of Equation (D6), where the components $x_i$ of the setting vector $\vec{x}$ which are associated with a time $t_i > \max(j_1, ..., j_p, l_1, ..., l_q)$ do not feature in the probability expression, which implies the required independence Equation (10) for all such settings $x_i$.

For cases where we have $t_i < t_k$ for some $k \in \{j_1, ..., j_p, l_1, ..., l_q\}$ but $A_i \not\subset A_k$, by construction, this would only happen when the circuit contains no directed path of wires from the measurement $M^{A_i}$ at $t_i$ to the measurement $M^{A_k}$ at $t_k > t_i$ (absence of directed paths between corresponding nodes in $G$). In this case, the two measurements act on disjoint sets of systems and the non-signalling property of quantum theory would then guarantee the required independence of the outcome of one measurement from the setting of the other.

Alternatively, in such cases, we can always transform to an equivalent circuit (with the same channels, systems, states and same connectivity between channels) where $A_i$ and $A_k$ are assigned time $t'_i$ and $t'_k$ with the opposite time order, $t'_i > t_k$, allowing us to apply the argument from the previous paragraph to establish the required independence.<fnref n="18" />

□

**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**.

*Proof.* The proof proceeds by showing that all 5 assumptions are satisfied by statements $\Sigma^{aug}$ derived in our augmented circuit framework, and the consistency of our framework shown in Theorem IV.1 guarantees that all 5 can be consistently applied to reason without any contradictions.

**Q**, **U** are by construction satisfied for all the statements $\Sigma^{aug}$, since these are associated with predictions computed using the Born rule and consider all possible setting choices (including the unitary modelling). Moreover, all our results apply to general EWFS (Definition III.1) which allows agents to have full quantum control over the labs of others, in the precise sense described in the formalisation of **U**.

**S** holds for all statements $\Sigma^{aug}$ in our framework because the statements are obtained from setting-conditioned predictions, which are well-defined normalised probabilities (see Appendix D) and the set of statements $\Sigma^{aug}$ is consistent as proven in Theorem IV.1. No single valid normalised probability distribution can assign $P(\vec{a}_j = \vec{a}'_j) = 1$ and $P(\vec{a}_j = \vec{a}'_j) = 1$ and the consistency result forbids the possibility of two distributions in the same scenario with $P(\vec{a}_j = \vec{a}'_j) = 1$ and $P'(\vec{a}_j = \vec{a}'_j) = 1$.

**C** holds for all statements in our framework because if an agent $A_i$ knows that an agent $A_j$ knows a statement $S \in \Sigma^{aug}$, then $A_i$ can directly compute the setting-conditioned prediction $P(\vec{a}_j = \vec{a}'_j|\vec{a}_l = \vec{a}'_l, \vec{x} = \vec{\xi})$ that defines the statement $S$ and thereby inherit the knowledge. Consistency as shown in Theorem IV.1 guarantees that there is only one such probability assignment any user of our framework can arrive at for a given choice $\vec{x} = \vec{\xi}$ and outcome values $\vec{a}_j = \vec{a}'_j, \vec{a}_l = \vec{a}'_l$.<fnref n="19" />

Finally the distributive axiom **D** is a rather basic axiom of logical inference, that it holds in our framework can be seen as follows. Firstly, since $\Sigma_L^{aug} \subseteq \Sigma^{aug}$, consistency of the superset as shown in Theorem IV.1 implies consistency of the subset. Then consider a logical statement $S_1 \in \Sigma_L^{aug}$ associated with a logical setting-conditioned prediction $P(\vec{a}_j = \vec{a}'_j|\vec{a}_l = \vec{a}'_l, \vec{x} = \vec{\xi}) \in \{0, 1\}$, it can be of the following form where the set $\vec{a}_l$ of outcomes can be empty

$$ \begin{aligned} \vec{a}_l &= \vec{a}_l \wedge \vec{x} = \vec{\xi} \Rightarrow \neg(\vec{a}_j = \vec{a}'_j), \\ \vec{a}_l &= \vec{a}_l \wedge \vec{x} = \vec{\xi} \Rightarrow \vec{a}_j = \vec{a}'_j. \end{aligned} \qquad (J1) $$

<footnote n="18">¹⁸ Physically if we regard the circuit as embedded in space-time such that the absence of directed paths between $A_i$ and $A_k$ correspond to space-like separation, then the above transformation can be seen as transforming to another reference frame where the time order relative to the co-ordinate time is reversed.</footnote>

<footnote n="19">¹⁹ This fact is independent of whether or not different agents agree on the setting choices to model their perspective of the experiment, as for instance, even if Wigner models the Friend's lab as a unitarily evolving closed quantum system ($x_F = 0$), both Wigner and the Friend can still use our augmented circuit to compute predictions for the case where $x_F = 1$ and will arrive at the same answer.</footnote>
Let $S_2$ be another statement of the same form, but relative to a potentially different set of outcomes $\vec{a}_m$ and $\vec{a}_n$ and setting values $\vec{x} = \vec{\xi}'$ i.e.,

$$ 
\begin{aligned}
\vec{a}_n &= \vec{a}_n \wedge \vec{x} = \vec{\xi}' \Rightarrow \neg(\vec{a}_m = \vec{a}_m), \\
\vec{a}_n &= \vec{a}_n \wedge \vec{x} = \vec{\xi}' \Rightarrow \vec{a}_m = \vec{a}_m.
\end{aligned} 
\qquad (J2) $$

This corresponds to $P(\vec{a}_m = \vec{a}_m | \vec{a}_n = \vec{a}_n, \vec{x} = \vec{\xi}') \in \{0, 1\}$. Denote the probabilities associated with $S_1$ and $S_2$ as $P_1$ and $P_2$ in short. If $S_1 \Rightarrow S_2$, then $P_2$ can be derived from $P_1$ through the rules of classical probability theory and usual manipulation of quantum circuits (in this case for the augmented circuit).

This implies that if one were to directly compute the probabilities for $\vec{a}_m, \vec{a}_n$ under the setting choice $\vec{x} = \vec{\xi}'$ in the augmented circuit, one must arrive at $P_2$ (otherwise there can be two distinct probability assignments to the same outcomes given these settings, which is not possible due to consistency, Definition III.5 and Theorem IV.1). This shows that $S_2 \in \Sigma_L^{aug}$. As this holds for all agents using our framework to reason, it follows that Equation (13) holds. □

**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.

*Proof.* The existence claim follows from the main results of our framework Section IV and Corollary IV.2. The consistent description satisfying **Q**, **U**, **C**, **D** and **S** is given by the augmented circuit. The violation of **I** for certain logical setting-conditioned prediction follows from the analysis of Appendix F 3, where we have shown that the logical predictions $P(b = 1 | u = w = \text{ok}, (x_1, x_2) = (0, 1)) = 1$ and $P(w = \text{fail} | a = 1, (x_1, x_2) = (1, 0)) = 1$ are setting-dependent (thus violating **I**), since

$$ 
\begin{aligned}
P(b = 1 | u = w = \text{ok}, (x_1, x_2) = (0, 1)) &\neq P(b = 1 | u = w = \text{ok}, (x_1, x_2) = (1, 1)), \\
P(w = \text{fail} | a = 1, (x_1, x_2) = (1, 0)) &\neq P(w = \text{fail} | a = 1, (x_1, x_2) = (1, 1))
\end{aligned} 
\qquad (J3) $$

In the prepare and measure version analysed in Appendix F 2, the logical predictions $P(w = \text{fail} | r = \text{tails}, (x_1, x_2) = (1, 0)) = 1$ and $P(z = +\frac{1}{2} | \overline{w} = \overline{\text{ok}}, (x_1, x_2) = (0, 1)) = 1$ are setting-dependent and violate **I**, since

$$ 
\begin{aligned}
P(w = \text{fail} | r = \text{tails}, (x_1, x_2) = (1, 0)) &\neq P(w = \text{fail} | r = \text{tails}, (x_1, x_2) = (1, 1)), \\
P(z = +\frac{1}{2} | \overline{w} = \overline{\text{ok}}, (x_1, x_2) = (0, 1)) &\neq P(z = +\frac{1}{2} | \overline{w} = \overline{\text{ok}}, (x_1, x_2) = (1, 1)).
\end{aligned} 
\qquad (J4) $$

These precisely correspond to the three statements $\overline{F}^{n:02}$ and $\overline{W}^{n:22}$ of FR that are combined to yield the apparent paradox (as shown in Table II).

The fact that assuming **I** is necessary to reproduce the apparent paradox in both versions of the protocol, follows from Corollary IV.3 and the sufficiency follows from noting that assuming **I** for all logical setting-conditioned predictions allows us to ignore the setting information on all such predictions and consequently on all associated logical statements. It is clear from Table I and Table II that this recovers the original statements of FR (in both versions) and therefore the apparent paradox. □

**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).
Proof. Given any set $A_K$ of agents and another agent $A_i$ such that $(A_i, A_k) \in nSA \forall A_k \in A_K$, we will first show that the joint probability of the outcomes of all agents in $A_K$ (these outcomes will be denoted using the vector $\vec{a}_K$) is independent of the setting $x_i$ i.e.,

$$P(\vec{a}_K|(x_1, ..., x_N)) = P(\vec{a}_K|(x_1, ..., x_{i-1}, x_{i+1}, ..., x_N)). \qquad (J5)$$

Here we have shortened the usual notion using value assignments $a = a$ in probabilities to just the variable $a$, since the meaning of the equation is clear from context. From this, the desired result will follow easily through the rules of conditional probability.

To establish Equation (J5), we consider the operationally equivalent augmented circuit (involving agents $\{A'_1, ..., A'_N\}$) to the original augmented circuit (involving agents $\{A_1, ..., A_N\}$), whose causal structure more explicitly reflects non-superagent structure $nSA$ of the original EWFS. Recall that such an equivalent circuit is guaranteed to exist by Definition VI.2. The equivalence implies in particular, the equivalence of predictions in the two circuits, that is, for all disjoint sets of outcomes $\vec{a}_j$ and $\vec{a}_l$ in the original EWFS,

$$P(\vec{a}_j = \vec{a}_j | \vec{a}_l = \vec{a}_l, \vec{x} = \vec{\xi}) = P(\vec{a}'_j = \vec{a}'_j | \vec{a}'_l = \vec{a}'_l, \vec{x}' = \vec{\xi}). \qquad (J6)$$

We establish Equation (J5) for the primed EWFS, and by the above equivalence of predictions, obtain the same for the original EWFS. We do so by dividing the problem into different cases, depending on the causal structure of the (equivalent) augmented circuit. According to Definition VI.2, $(A_i, A_k) \in nSA$ (along with the fact that we have $i \neq k$) implies that in the equivalent EWFS, one of the following cases must hold

* Case 1: $A'_i \not\subset A'_k$

* Case 2: $A'_i \subsetneq A'_k$, and $\mathcal{E}'_i$ acts trivially on $M'_i$ and $A_i \not\subsetneq M'_i A_k$

Case 1 In this case Equation (J5) immediately follows from the causality result of Theorem IV.1.

Case 2 In this case we must have the following property: for any agent $A'_j$ with $A'_i \subsetneq M'_i A'_j$ we must have $A'_j \not\subset A'_k$ for all $k \in K$. Otherwise $A'_i \subsetneq M'_i A'_j$ along with the directed path $A'_j \subsetneq A'_k$ would immediately imply by Definition IV.1 that $A'_i \subsetneq M'_i A'_k$, which would contradict the condition of Case 2 and hence the fact that $(A_i, A_k) \in nSA$ Definition VI.2.

Now, the prediction of interest from Equation (J5) can be calculated explicitly by applying Equation (D3) as detailed in Appendix D. The above property guarantees that for all $A'_j$ such that $A'_i \subsetneq M'_i A'_j$, the operations of $A'_j$ and those of any agent $A'_k \in A'_K$ must act of disjoint sets of systems Definition IV.1. Note that $A'_i \subsetneq M'_i A'_j$ implies that $A'_j$ acts after $A'_i$ in the augmented circuit. Then by the non-signalling property of quantum theory, and as we can explicitly see from Equation (D3), the trace in the probability calculation will act on all the outputs of $A'_j$'s operation. Since $A'_j$ is not in the set $A'_K$, we sum over their outcomes in the probability, therefore all operations of such agents are completely positive and trace preserving maps (CPTPMs). For CPTPMs tracing the output after applying the map is equivalent to tracing the inputs directly $\mathrm{tr} \circ \mathcal{E} = \mathrm{tr}$.

The above implies that all the operations of the agents $A'_j$ such that $A'_i \xrightarrow{M'_i} A'_j$ will drop out of the probability expression for Equation (J5). This probability expression will then have the trace over $M'_i$ acting directly on the operation $\mathcal{E}'_i \circ \mathcal{M}^{A'_i}$ of the agent $A'_i$. From the definition of Case 2, we also know that $\mathcal{E}'_i$ acts trivially on $M'_i$, which implies that the trace over $M'_i$ will act directly on the measurement $\mathcal{M}^{A'_i}$.

It is easy to see that $\mathrm{tr}_{M'_i} \circ \mathcal{M}^{A'_i}$ is the same, independently of the setting $x_i \in \{0, 1\}$, because the setting dictates whether we coherently ($x_i = 0$) or incoherently ($x_i = 1$) copy the system state onto the memory in the basis of the measurement, and the post-measurement state on the system alone (obtained by tracing out the memory) is the same in both cases. A more explicit proof of this fact can be found in the proof of Theorem E.1. This is sufficient to establish the required Equation (J5) for this case.

Having established Equation (J5) for all sets $A_K$ of agents of the form required by the theorem statement, it is immediate that the same setting independence of Equation (J5) also holds for all subsets of agents $A_K$ (this can be seen by computing the relevant marginals of Equation (J5)). Now, for any partition $A_K = \{A_{j_1}, ..., A_{j_p}\} \cup \{A_{l_1}, ..., A_{l_q}\}$ of $A_K$, we can compute prediction $P(\vec{a}_j|\vec{a}_l, \vec{x})$ by applying the conditional probability rule

$$P(\vec{a}_j|\vec{a}_l, \vec{x}) = \frac{P(\vec{a}_j, \vec{a}_l|\vec{x})}{P(\vec{a}_l|\vec{x})}. \qquad (J7)$$

The fact that both the numerator and denominator of this expression are independent of the component $x_i$ of the setting vector $\vec{x}$ are then immediate from Equation (J5). This establishes the theorem. □
## 2. Proofs of results from the Appendix

**Theorem E.1 (Recovering standard quantum circuits).** If an EWFS corresponds to a standard quantum scenario (Definition VI.3), then its augmented circuit can be equivalently reduced to a standard quantum circuit, such that the same (non-trivial) predictions are obtained from the original augmented circuit, the $C^{sys}$-form standard circuit or the $C^{sys+anc}$-form standard circuit. Explicitly, for any disjoint sets $\bar{a}_j = (a_{j1}, ..., a_{jp})$ and $\bar{a}_i = (a_{i1}, ..., a_{iq})$ of outcomes, and any choice of settings $\vec{x} = \vec{\xi}$ such that $x_i = 1$ for all $i \in \{j_1, ..., j_p, l_1, ..., l_q\}$, we have

$$P_{\text{aug}}(\bar{a}_j = \bar{a}_j | \bar{a}_j = \bar{a}_j, \vec{x} = \vec{\xi}) = P_{\text{std}}(\bar{a}'_j = \bar{a}_j | \bar{a}'_j = \bar{a}_j), \qquad (\text{E1})$$

where the $P_{\text{aug}}$ refers to setting-conditioned predictions in the augmented circuit of the EWFS and $P_{\text{std}}$ refers to predictions in an equivalent $C^{sys}$-form or $C^{sys+anc}$-form standard quantum circuit (where no settings are involved).

*Proof.* In a standard quantum scenario, by Definition VI.3, we must have $(A_i, A_j) \in nSA$ for all $i, j \in \{1, ..., N\}$. This means that the augmented circuit of the given EWFS over agents $\{A_1, ..., A_N\}$ can be reduced to an operationally equivalent augmented circuit over corresponding agents $\{A'_1, ..., A'_N\}$ such that for any $A'_i$ and $A'_j$ we have: either $A'_j$ acts before $A'_i$ in time $t'_i > t'_j$ or in the alternative cases where $i = j$ or $t'_j > t'_i$, no operation that acts after the measurement $M^{A'_i}$ in the circuit (including the operation $\mathcal{E}'_i$) acts non-trivially on the memory $M'_i$. Here we have used the fact that $A'_i < A'_j$ according to the operational causal structure (Definition IV.1) implies $t'_i < t'_j$.

Applying this argument to every pair of agents in the scenario, this implies that for each measurement $M^{A'_i}$, the corresponding set of systems $S'_i$ of systems (excluding the memory $M'_i$) on which it acts non-trivially is a subset of the systems $S'$ and does not include any of the memories $M'_j \in M'$ of other agents (in contrast to a general EWFS of Definition III.1 where the “system” for one agent’s measurement may include the “memories” of other agents). All the following arguments will refer to the equivalent augmented circuit over the primed agents.

Let the measurement outcomes $a_i$ of $A'_i$’s measurement take values in the same set $a_i \in O_i$ as that of the original scenario w.l.o.g.<fnref n="20" /> Then $A'_i$’s measurement on the systems $S'_i$ is associated with the projectors $\{|a_i\rangle \langle a_i|_{S'_i}\}_{a_i \in O_i}$. Let $\rho_{S'_i}$ be an arbitrary state, which corresponds to the state of the system $S'_i$ just before $A'_i$’s measurement. We can express this state is the basis of the measurement as follows, for some coefficients $c_{a_i, \bar{a}_i}$.

$$\rho_{S'_i} = \sum_{a_i, \bar{a}_i} c_{a_i, \bar{a}_i} |a_i\rangle \langle \bar{a}_i|_{S'_i},$$

where $\{|a\rangle_i\}$ and $\{|\bar{a}\rangle_i\}$ both correspond to the same measurement basis. Recall that the memory $M'_i$ is initialised to $|0\rangle \langle 0|_{M'_i}$. Now consider the case where the setting $x'_i = 0$. Then we model $A'_i$’s measurement as the unitary evolution $M^{A'_i}_{\text{unitary}}$ which as we have seen before, is simply a unitary implementing a coherent copy in the measurement basis with $S'_i$ being the control and $M'_i$ being the target. The post-measurement state of $S'_i M'_i$ in this case is given as

$$M^{A'_i}_{\text{unitary}}(\rho_{S'_i} \otimes |0\rangle \langle 0|_{M'_i}) M^{A'_i, \dagger}_{\text{unitary}} = \sum_{a_i, \bar{a}_i} c_{a_i, \bar{a}_i} |a_i a_i\rangle \langle \bar{a}_i \bar{a}_i|_{S'_i M'_i}. \qquad (\text{J8})$$

Then it is easy to see that the following holds, where we use the notation $\pi_{\bar{a}_i}^{S'_i} := |a_i\rangle \langle a_i|_{S'_i}$, $\pi_{a_i}^{M'_i} := |a_i\rangle \langle a_i|_{M'_i}$, $\pi_{a_i}^{S'_i M'_i} = |a_i a_i\rangle \langle a_i a_i|_{S'_i M'_i}$.

$$(1_{S'_i} \otimes \pi_{\bar{a}_i}^{M'_i}) (M_{\text{unitary}}^{A'_i} (\rho_{S'_i} \otimes |0\rangle \langle 0|_{M'_i}) M_{\text{unitary}}^{A'_i, \dagger}) (1_{S'_i} \otimes \pi_{\bar{a}_i}^{M'_i}) \\ = \pi_{a_i}^{S'_i M'_i} M_{\text{unitary}}^{A'_i} (\rho_{S'_i} \otimes |0\rangle \langle 0|_{M'_i}) M_{\text{unitary}}^{A'_i, \dagger} \pi_{a_i}^{S'_i M'_i} \qquad (\text{J9})$$

We now show that the post-measurement on the system $S'_i$ alone is independent of the setting $x'_i$, and this indeed the post-measurement state one would get by directly applying the projective measurement of $C^{sys}$ on $S'_i$. This will allows us to reduce our augmented circuit to a $C^{sys}$-form standard quantum circuit as required. For this,

<footnote n="20">²⁰ Since the scenarios are operationally equivalent with a one-one-one correspondence between the outcome sets, we can use the same labels without loss of generality.</footnote>
first consider this state under the setting $x'_i = 0$. This is given as $\text{tr}_{M'_i}[\mathcal{M}_{\text{unitary}}^{A'_i}(\rho_{S'_i} \otimes |0\rangle\langle 0|_{M'_i})\mathcal{M}_{\text{unitary}}^{A'_i,\dagger}]$. Noting that $\sum_{a_i \in \mathcal{O}_i} \pi_{a_i}^{M'_i} = 1_{M'_i}$, we can expand this as follows

$$
\begin{align*}
& \text{tr}_{M'_i} [\mathcal{M}_{\text{unitary}}^{A'_i}(\rho_{S'_i} \otimes |0\rangle\langle 0|_{M'_i}) \mathcal{M}_{\text{unitary}}^{A'_i, \dagger}] \\
&= \sum_{\bar{a}_i \in \mathcal{O}_i} (1_{S'_i} \otimes \langle \bar{a}_i |_{M'_i}) (\mathcal{M}_{\text{unitary}}^{A'_i}(\rho_{S'_i} \otimes |0\rangle\langle 0|_{M'_i}) \mathcal{M}_{\text{unitary}}^{A'_i, \dagger}) (1_{S'_i} \otimes |\bar{a}_i\rangle_{M'_i}) \\
&= \sum_{\bar{a}_i \in \mathcal{O}_i} (1_{S'_i} \otimes \langle \bar{a}_i |_{M'_i}) (1_{S'_i} \otimes \sum_{a_i \in \mathcal{O}_i} \pi_{\bar{a}_i}^{M'_i}) (\mathcal{M}_{\text{unitary}}^{A'_i}(\rho_{S'_i} \otimes |0\rangle\langle 0|_{M'_i}) \mathcal{M}_{\text{unitary}}^{A'_i, \dagger}) (1_{S'_i} \otimes \sum_{a_i \in \mathcal{O}_i} \pi_{a_i}^{M'_i}) (1_{S'_i} \otimes |\bar{a}_i\rangle_{M'_i}) \tag{J10} \\
&= \sum_{\bar{a}_i \in \mathcal{O}_i} \sum_{a_i \in \mathcal{O}_i} (1_{S'_i} \otimes \langle \bar{a}_i |_{M'_i}) (1_{S'_i} \otimes \pi_{\bar{a}_i}^{M'_i}) (\mathcal{M}_{\text{unitary}}^{A'_i}(\rho_{S'_i} \otimes |0\rangle\langle 0|_{M'_i}) \mathcal{M}_{\text{unitary}}^{A'_i, \dagger}) (1_{S'_i} \otimes \pi_{\bar{a}_i}^{M'_i}) (1_{S'_i} \otimes |\bar{a}_i\rangle_{M'_i}) \\
&= \text{tr}_{M'_i} \left[ \sum_{a_i \in \mathcal{O}_i} (1_{S'_i} \otimes \pi_{a_i}^{M'_i}) (\mathcal{M}_{\text{unitary}}^{A'_i}(\rho_{S'_i} \otimes |0\rangle\langle 0|_{M'_i}) \mathcal{M}_{\text{unitary}}^{A'_i, \dagger}) (1_{S'_i} \otimes \pi_{a_i}^{M'_i}) \right].
\end{align*}
$$

In the second equality above, we have summed over the same indices $\sum_{a_i \in \mathcal{O}_i}$ in both occurrences of the projectors $\pi_{a_i}^{M'_i}$ as the cross terms disappear due to the trace over $M'_i$. Using this and Equation (J9), we immediately obtain the following

$$
\begin{align}
& \text{tr}_{M'_i} [\mathcal{M}_{\text{unitary}}^{A'_i}(\rho_{S'_i} \otimes |0\rangle\langle 0|_{M'_i}) \mathcal{M}_{\text{unitary}}^{A'_i, \dagger}] \nonumber \\
& = \text{tr}_{M'_i} \left[ \sum_{a_i \in \mathcal{O}_i} \pi_{a_i}^{S'_i M'_i} \mathcal{M}_{\text{unitary}}^{A'_i}(\rho_{S'_i} \otimes |0\rangle\langle 0|_{M'_i}) \mathcal{M}_{\text{unitary}}^{A'_i, \dagger} \pi_{a_i}^{S'_i M'_i} \right] \tag{J11} \\
& = \sum_{a_i \in \mathcal{O}_i} c_{a_i, a_i} |\bar{a}_i\rangle \langle \bar{a}_i|_{S'_i}. \nonumber
\end{align}
$$

The right hand side is precisely the expression for the post-measurement state of $S'_i$ under the setting $x_i = 1$ when using the trace preserving form of the evolution for this setting (c.f. Equation (9)). Thus we have shown that $A'_i$'s measurement implements the following map on an arbitrary input state $\rho_{S'_i}$ of the system $S'_i$ irrespective of the setting $x'_i$.

$$ \rho_{S'_i} = \sum_{a_i, \bar{a}_i \in \mathcal{O}_i} c_{a_i, \bar{a}_i} |a_i\rangle \langle \bar{a}_i|_{S'_i} \mapsto \sum_{a_i \in \mathcal{O}_i} c_{a_i, a_i} |a_i\rangle \langle a_i|_{S'_i} \qquad (J12) $$

This map can equivalently be described as

$$ \rho_{S'_i} \mapsto \sum_{a_i \in \mathcal{O}_i} \pi_{a_i}^{S'_i}(\rho_{S'_i}) \pi_{a_i}^{S'_i}. \qquad (J13) $$

Or, for the case of a particular outcome, we have the following trace non-increasing map from pre to post measurement state

$$ \rho_{S'_i} \mapsto \frac{\pi_{a_i}^{S'_i}(\rho_{S'_i}) \pi_{a_i}^{S'_i}}{\text{tr}[\pi_{a_i}^{S'_i}(\rho_{S'_i}) \pi_{a_i}^{S'_i}]} \qquad (J14) $$

These correspond precisely to the how the measurements $\mathcal{M}^{A'_i} = \{\pi_{a_i}^{S'_i}\}_{a_i \in \mathcal{O}_i'}$ of a $\mathcal{C}^{\text{sys}}$-form circuit (Definition E.1) over $\{A'_1, ..., A'_N\}$ act.

To show the full reduction of the augmented circuit to a $\mathcal{C}^{\text{sys}}$-form circuit, let $S'_i := S' \setminus S'_i$ denote the complement of $S_i \subseteq S'$ (the systems that $\mathcal{M}^{A'_i}$ acts non-trivially on). The above argument, immediately generalises to the case where we consider $\rho_{S'_i}$ to be a reduced state of a larger state $\rho_{S'_i}^{cS'_i}$ over all systems $S'$ in the EWFS. Since under both settings, the measurement operation of $A'_i$ only acts locally on $S'_i$ by construction, the above argument implies that the joint state on $S'_i^{cS'_i}$ is also independent of the settings. It follows that for all agents $A'_i$, we can replace their measurement in the (primed) augmented circuit with the setting independent projective measurement $\{|a_i\rangle\langle a_i|_{S'_i}\}_{a_i \in \mathcal{O}_i'}$ on the system $S'_i \subseteq S'$ alone while preserving all the predictions i.e., the augmented circuit reduces to an equivalent $\mathcal{C}^{\text{sys}}$-form standard circuit. Since the primed circuit is operationally equivalent to the original augmented EWFS, it follows that the $\mathcal{C}^{\text{sys}}$-form standard circuit obtained here is also operationally equivalent to the original augmented circuit.
To obtain an equivalent $C^{sys+anc}$-form circuit, we can keep the memories (even if they are not acted upon after $A'_i$'s measurement) and model all measurements with $x'_i = 1$ (recalling that we have established full setting independence of predictions, therefore an arbitrary fixing of settings will not change the predictions). Then, it follows from Equation (J9) that this is equivalent to modelling each measurement in its unitary form $M_{unitary}^{A'_i}$ at the time $t'_i$ of the measurement followed by a projective measurement $\{\pi_{a'_i}^{M'_i}\}_{a'_i \in \mathcal{O}_i}$ on the memory $M'_i$ alone. Since the memory is not subsequently acted upon until the time $t'_N$ at which the protocol ends, the measurement $\{\pi_{a'_i}^{M'_i}\}_{a'_i \in \mathcal{O}_i}$ acting on the memory alone, can be equivalently performed at any time $t_f$ after $t'_N$. This immediately yields $C^{sys+anc}$-form standard circuit which is equivalent to the original augmented circuit, where the memories $\{M'_1, ..., M'_N\}$ act as the ancillas. This completes the proof of the current theorem. □

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