and the fine-graining $\mathcal{M}_2^f$ of the CPM $\mathcal{M}_2$ encodes the function $f_B$ (along with a trace on $A_2$). $A_1$ and $A_2$ correspond to qutrits encoding the corresponding variables in the basis $\{|0\rangle, |1\rangle, |\Omega\rangle\}$. Here we have, whenever $\Lambda = 0$, the acyclic model $\mathrm{CM}_0$ but with $A_1$ playing the role of $A$ and $A_2 = \Omega$ and whenever $\Lambda = 1$, the model $\mathrm{CM}_1$ with $A_2$ taking the place of $A$ but $A_1 = \Omega$.

Let us describe explicitly, the encoders and decoders involved in the above example. Consider the encoding-decoding scheme that relates $\mathcal{M}_1: \Lambda \otimes R_A \otimes B \mapsto A$ to $\mathcal{M}_1^f: \Lambda \otimes R_A \otimes B \mapsto A_1 \otimes A_2$. The encoder $\mathrm{Enc}_1$ in this case is a CPTPM from the inputs of $\mathcal{M}_1$ to the inputs of $\mathcal{M}_1^f$, since these are the same systems, the encoder is trivial (identity on all systems). The decoder $\mathrm{Dec}_1$ is a map from $A_1$ and $A_2$ to $A$ which must be CPTP on the image of $\mathcal{M}_1^f \circ \mathrm{Enc}_1$. Notice that this image only has support on the following values of $A_1$ and $A_2$: $\{(0, \Omega), (1, \Omega), (\Omega, 0), (\Omega, 1)\}$. The decoder performs the mapping $\{(i, \Omega), (\Omega, i)\} \mapsto i$ for every $i \in \{0, 1\}$. This can be physically realised through a measurement of $A_1$ and $A_2$ in the basis $\{|0\rangle, |1\rangle, |\Omega\rangle\}$ followed by reporting only the value which lies in $\{0, 1\}$. It is then easy to check that $\mathcal{M}_1 = \mathrm{Dec}_1 \circ \mathcal{M}_1^f \circ \mathrm{Enc}_1$ holds, and also that all signalling relations of $\mathcal{M}_1$ (each of $\{R_A\}$, $\{\Lambda\}$ and $\{B\}$ signal to $\{A\}$) are preserved in $\mathcal{M}_1^f$ (each of $\{R_A\}$, $\{\Lambda\}$ and $\{B\}$ signal to $\{A_1, A_2\}$). Now consider the encoding-decoding scheme that relates $\mathcal{M}_2: \Lambda \otimes R_B \otimes A \mapsto B$ to $\mathcal{M}_2^f: \Lambda \otimes R_B \otimes A_1 \otimes A_2 \mapsto B$. In this case, the decoder $\mathrm{Dec}_2$ is trivial. The encoder $\mathrm{Enc}_2: \Lambda \otimes R_B \otimes A \mapsto \Lambda \otimes R_B \otimes A_1 \otimes A_2$ acts as identity on $R_B$ and whenever $\Lambda = 0$, maps $A$ to $A_1 = A$ and $A_2 = \Omega$ and whenever $\Lambda = 1$, maps $A$ to $A_1 = \Omega$ and $A_2 = A$. It is then easy to verify that $\mathcal{M}_2 = \mathrm{Dec}_2 \circ \mathcal{M}_2^f \circ \mathrm{Enc}_2$ holds and all signalling relations of $\mathcal{M}_2$ are preserved in $\mathcal{M}_2^f$.

To show that the network of Fig. 14b is indeed a fine-graining of the network of Fig. 14a according to our Definition 17, we still need to establish that the above encoding and decoding for the individual maps implies the existence of encoding and decoding schemes between every sub-network in the coarse and fine-grained networks. The remaining sub-networks of our cyclic network are: the network itself (whose induced map is simply a constant number, since it has no in/outputs), the set of sub-networks formed by the sequential composition $\mathcal{M}_2 \circ \mathcal{M}_1$ (through the $A$ system) composed with any subset of the preparations on $\Lambda$, $R_A$ and $R_B$ and another set of sub-networks formed by the sequential composition $\mathcal{M}_1 \circ \mathcal{M}_2$ (through the system $B$) composed with any subset of the preparations. The encoders and decoders for the latter two sets of sub-networks are obtained from those constructed above, by simply applying the appropriate sequential composition to both sides of $\mathcal{M}_1 = \mathrm{Dec}_1 \circ \mathcal{M}_1^f \circ \mathrm{Enc}_1$ and $\mathcal{M}_2 = \mathrm{Dec}_2 \circ \mathcal{M}_2^f \circ \mathrm{Enc}_2$ along with composing the same set of preparations (noting that $\Lambda$, $R_A$ and $R_B$ remain unchanged in the two networks). In particular, the sub-network obtained by applying the sequential composition $\mathcal{M}_2 \circ \mathcal{M}_1$ on all preparations has an induced map with a single in and output system, both isomorphic to $B$. The original cyclic network can be recovered by performing a loop composition on $B$ in this sub-network. The same is true in the fine-grained scenario, the sequential composition $\mathcal{M}_2^f \circ \mathcal{M}_1^f$ acting on the initial preparations yields an induced map having a single in and output isomorphic to $B$ and recovers the fine-grained network through loop composition on this in-output pair. Since the coarse and fine-grained sub-networks act identically in this case (as we have argued above), the loop compositions on $B$ in the two cases will yield identical results, and the same induced map for the original network (Fig. 14a) and fine-grained network (Fig. 14b). This shows that the latter network is indeed a fine-graining of the former relative to this encoding-decoding scheme.

These examples highlight that, even in purely classical scenarios, the causal structure that we associate with a protocol, as well its cyclic/acyclic nature depends on the information captured by the nodes of the causal structure and the level of detail to which we specify this information. Example 3 illustrates a cyclic functional causal model that is inspired by the classical switch process [27] which implements a classical mixture of the orders $A < B$ and $B < A$ in the process framework (cf. Section VI). The quantum generalisation of the process, the quantum switch can also be formalised as a cyclic network in our framework, which admits an acyclic fine-graining that has a similar physical interpretation (see Section IX and Appendix G for details).

## Appendix D: Physical spacetime realisations are causal boxes

In this appendix, we connect spacetime realisations of quantum networks as defined in our top-down formalism to the previously known framework of *causal boxes* [21]. The main result of this section indicates that the most general physical protocols that can be physically realised in a spacetime are those described by causal boxes. Our present formalism and proofs apply to any protocol involving a finite number of finite-dimensional quantum systems, but there is scope for generalising them to the infinite dimensional case in the future (noting that the causal box framework can indeed model infinite dimensional systems). This result provides a top-down justification for the causal box framework, at least for the case of such finite protocols. For such protocols, we can effectively treat the spacetime
poset $\mathcal{T}$ as being finite, without loss of generality. Only a finite set of spacetime locations (or sufficiently localised regions) will be relevant for describing such protocols and we can take $\mathcal{T}$ to only include those locations as all our statements only concern spacetime locations that can embed a system of the protocol. We will not review the causal box framework here but simply outline why it is a powerful approach with several useful features.

Causal boxes describe quantum information processing protocols in an acyclic background spacetime where agents may send/receive messages at a superposition of different spacetime locations, in a manner consistent with relativistic causality. They generalise the standard quantum information formalism to explicitly include spacetime information and also to treat space and time on a more equal footing, in the spirit of special relativity. The standard formalism defines quantum states on fixed time slices (possibly non-localised in space) while causal box formalism allows us to consider states that are not localised in both space and time. Moreover, they preserve the desired mathematical properties of the usual formalism: they are closed under composition, admit a circuit decomposition into a well-defined sequence of isometries that are compatible with the partial order of the spacetime (the sequence representation), as well as admitting Stinespring and Choi matrices and thus being purifiable to unitaries. More generally, the framework is defined on a full infinite-dimensional Fock space and can capture superpositions of different number of messages exchanged between physical devices/agents. The formalism has useful applications for relativistic quantum information processing, in particular for studying composable cryptographic security also for practical relativistic quantum schemes [64].

Causal boxes are defined relative to a fixed partially order set (which models the spacetime), and they satisfy a certain causality condition relative to this poset. The following theorem connects our physical spacetime realisations of networks in our framework to causal boxes (more precisely these realisations map to the subset of finite causal boxes), showing that the causality condition of the causal box formalism can be derived from our relativistic causality condition (or more abstractly, compatibility between signalling structure of CPTPMs and the partial order).

**Theorem 6.** Let $\mathfrak{N}$ be a network of CPTPMs whose signalling structure is compatible with an acyclic causal structure $\mathcal{G}$. Then the induced map of every sub-network of $\mathfrak{N}$ is a causal box relative the partial order induced by $\mathcal{G}$, therefore such a network $\mathfrak{N}$ is itself completely described by the causal box formalism.

*Proof.* Consider any acyclic causal structure $\mathcal{G}$. Being a directed acyclic graph, it induces a partial order, where we say that $N_1 < N_2$ relative to this partial order if and only if there exists a directed path from $N_1$ to $N_2$ in $\mathcal{G}$. Denote the induced poset as $\mathcal{T}^G$. Let $\mathcal{E}(S) \in \text{Nodes}(\mathcal{G})$ define an embedding of the systems $S \in \mathfrak{N}^{sys}$ in $\mathcal{G}$ and equivalently, in the partial order $\mathcal{T}^G$ i.e., $\mathcal{E}(S) \in \mathcal{T}^G$. We can then associate a “region” $\mathcal{R}^S$ or subset of locations in $\mathcal{T}^G$ with every subset of $S \subseteq \mathfrak{N}^{sys}$, with $\mathcal{R}^S := \cup_{S \in \mathcal{S}} \mathcal{E}(S)$. Then define

$$ \text{Past}(\mathcal{R}^S) := \{P \in \mathcal{T}^G : \exists Q \in \mathcal{R}^S, P < Q\}, \qquad (\text{D1}) $$

where $<$ is the partial order relation associated with $\mathcal{T}^G$. Past$(\mathcal{R}^S)$ corresponds to all points that are in the past of the region $\mathcal{R}^S$ relative to this partial order.

Let us now focus on the induced map $\mathcal{N}$ of $\mathfrak{N}$ with In and Out denoting the set of in and output systems of $\mathcal{N}$. Consider $\mathcal{S}_O \subseteq \text{Out}$, and a corresponding $\mathcal{S}_I \subseteq \text{In}$ defined as follows

$$ \mathcal{S}_I := \{S \in \text{In} : \mathcal{E}(S) \in \text{Past}(\mathcal{R}^{\mathcal{S}_O})\}. \qquad (\text{D2}) $$

That is, $\mathcal{S}_I$ consists of all inputs of $\mathcal{N}$ which are embedded at locations in the past of the region $\mathcal{R}^{\mathcal{S}_O}$ associated to $\mathcal{S}_O$. Then, recalling that any poset $\mathcal{T}$ imposes an order relation $\xrightarrow{R}$ on its region $\Sigma(\mathcal{T})$ (cf. Definition 19), $\mathcal{R}^{\text{In}\setminus\mathcal{S}_I} \xrightarrow{\mathcal{R}^{\mathcal{S}_O}} \mathcal{R}^{\mathcal{S}_O}$ follows from the above definition of $\mathcal{S}_I$. Imposing compatibility between the signalling structure of $\mathfrak{N}$ and the causal structure $\mathcal{G}$ implies that we must have $\text{In}\setminus\mathcal{S}_I$ does not signal to $\mathcal{S}_O$ in $\mathfrak{N}$. Applying Definition 10 to the CPTPM $\mathcal{N}$, this non-signalling relation implies that for all states $\rho \in \mathcal{L}(\text{In}\setminus\mathcal{S}_I)$,

$$ \text{Tr}_{\text{Out}\setminus\mathcal{S}_O} \circ \mathcal{N} = \text{Tr}_{\text{Out}\setminus\mathcal{S}_O} \circ \mathcal{N} \circ \text{Tr}_{\text{In}\setminus\mathcal{S}_I}^{\rho}, \qquad (\text{D3}) $$

where $\text{Tr}_{\text{In}\setminus\mathcal{S}_I}^{\rho}$ is a particular choice of local operation on $\text{In}\setminus\mathcal{S}_I$ which discards the incoming state on the subsystems $\text{In}\setminus\mathcal{S}_I$ and replaces it with the fixed state $\rho$. As the embedding $\mathcal{E}$ associates each system to a node in $\mathcal{G}$, we can equivalently represent $\text{Tr}_S$ as $\text{Tr}_{\mathcal{R}^S}$ (where $\mathcal{R}^S$ is the region or subset of nodes in $\mathcal{G}$, associated to the set of systems $S$), to capture the fact that it traces out systems embedded in the region $\mathcal{R}^S$. Then defining a causality function $\chi: \mathcal{R} \to \text{Past}(\mathcal{R})$ for every $\mathcal{R} \subseteq \mathcal{T}^G$, and denoting $\text{Tr}_{\text{Out}\setminus\mathcal{S}_O} \circ \mathcal{N} = \mathcal{N}^{\mathcal{R}^{\mathcal{S}_O}}$ to capture that this map only produces outputs in the region $\mathcal{R}^{\mathcal{S}_O}$ (when embedded in $\mathcal{G}$ according to $\mathcal{E}$) Eq. (D3) is equivalent to
$$ \mathcal{N}^{\mathcal{R}^{SO}} = \mathcal{N}^{\mathcal{R}^{SO}} \circ \mathrm{Tr}_{\mathcal{T}^{\mathcal{G}} \setminus \chi(\mathcal{R}^{SO})}^{\rho} \quad (\mathrm{D4}) $$

Fixing $\rho$ to be the vacuum state $|\Omega\rangle\langle\Omega|$ (0-message space of the causal box Fock space), this is precisely the causality condition for regarding $\mathcal{N}$ as a finite causal box, relative to the causality function $\chi$ [21]. It tells us that the outputs of a causal box in a region $\mathcal{R}$, are independent of the input systems of the causal box that are not in the past of $\mathcal{R}$, relative to the partial order. We can see that $\chi$ is indeed a valid finite causality function. This requires us to show that for every $P \in \mathcal{T}^{\mathcal{G}}$, there exists a finite $n \in \mathbb{N}$ such that $P \notin \chi^n(\mathcal{T}^{\mathcal{G}})$, which ensures that every point in the partial order can be reached in a finite number of steps, through the causality function [21]. To show this, notice that the function $\chi$ defined above has the property that $\chi(\mathcal{R}) \subset \mathcal{R}$ for all $\mathcal{R} \subseteq \mathcal{T}^{\mathcal{G}}$. This is because $\mathcal{T}^{\mathcal{G}}$ being a partial order ensures that every region $\mathcal{R}$ contains at least one “latest” point $L \in \mathcal{R}$ such that $L \notin P$ for all $P \in \mathcal{R}$. Since $\mathcal{T}^{\mathcal{G}}$ is finite, every point $P$ will eventually be excluded from $\chi^n(\mathcal{T}^{\mathcal{G}})$ for some finite number $n$ applications of $\chi$. This establishes that the induced map $\mathcal{N}$ of $\mathfrak{N}$ is a causal box with respect to the partial order induced by $\mathcal{G}$. The same argument as above can be extended to the induced map of every sub-network of $\mathfrak{N}$ to show that they are all causal boxes relative to $\mathcal{G}$ (and using the same embedding $\mathcal{E}(S) \in \mathrm{Nodes}(\mathcal{G})$ for $S \in \mathfrak{N}^{sys}$). $\square$

**Corollary 6.** [Physical spacetime realisations are causal boxes] Consider a spacetime realisation of a network $\mathfrak{N}$ of CPTPMs that satisfies relativistic causality in a spacetime $\mathcal{T}$ with respect to an embedding $\mathcal{E}$. Let $\mathfrak{N}_{\mathcal{T},\mathcal{E}}^f$ be the fine-graining in the specification of the realisation, which is associated with an acyclic region causal structure $\mathcal{G}_{\mathfrak{N}}^{\mathrm{reg},f}$. Then the induced maps of all sub-networks of $\mathfrak{N}_{\mathcal{T},\mathcal{E}}^f$ are causal boxes relative to the order relation defined by $\mathcal{G}_{\mathfrak{N}}^{\mathrm{reg},f}$. Therefore all physical spacetime realisations of networks in our framework can be described within the causal box formalism.

*Proof.* This follows from the proof of the above theorem, by taking $\mathfrak{N}_{\mathcal{T},\mathcal{E}}^f$ to play the role of the network $\mathfrak{N}$ used in that proof, and the acyclic causal structure $\mathcal{G}_{\mathfrak{N}}^{\mathrm{reg},f}$ to play the role of $\mathcal{G}$, while noting that relativistic causality (cf. Definition 24) in particular requires the signalling structure of $\mathfrak{N}_{\mathcal{T},\mathcal{E}}^f$ to be compatible with the causal structure $\mathcal{G}_{\mathfrak{N}}^{\mathrm{reg},f}$. $\square$

## Appendix E: Further details of the process matrix framework

In Section VI, we described different classes of processes for the bipartite case. In particular, the set of fixed order processes are relevant for our main results and we review the full multi-partite definition for this class of processes below. This is adapted from definitions originally provided in [31, 32], where the only adaptation is to distinguish the in and output systems $A^I$ and $A^O$ of each agent $A$ rather than grouping them together as a single element.

**Definition 28** (Fixed order processes). An $N$-partite process matrix $W$ is said to be a fixed order process, if there exists a partial order $\mathcal{K}(S_{I/O})$ on the set $S_{I/O} = \{A_1^I, A_1^O, \dots, A_N^I, A_N^O\}$ of the input and output systems of the $N$ agents and associated with the binary relations $<_K$ (first element precedes the second), $>_K$ (first element succeeds the second) and $\prec_K$ (the elements are unordered) such that the following conditions are satisfied

1. For any $i \in \{1, \dots, N\}$, $A_i^I <_K A_i^O$.

2. For any agent $A_i$ and a subset $A^S$ of the remaining agents, such that $A_i^O \prec_K A_S^I$, $\forall A_S \in A_S$ (which is denoted in short as $A_i^O \prec_K A_S^I$) with respect to the partial order $\mathcal{K}(S_{I/O})$, the joint probability distribution $P(x_1, \dots, x_N|a_1, \dots, a_N)$ (cf. Equation (8)) obtained from $W$ for any choice of local measurements of the $N$ agents does not allow the outcome of any of the agents in $A_S$ to depend on the setting of agent $A_i$. That is, taking $x_S$ to denote the set of outcomes $\{x_S\}_{S \in S}$ of the agents in $A_S$, we have the following whenever $A_i^O \prec_K A_S^I$ with respect to $\mathcal{K}(S_{I/O})$

$$ \begin{aligned} P(x_S|a_1, \dots, a_{i-1}, a_i, a_{i+1}, \dots, a_N) &= \sum_{x_j \notin x_S} P(x_1, \dots, x_N|a_1, \dots, a_N) \tag{E1} \\ &= P(x_S|a_1, \dots, a_{i-1}, a_{i+1}, \dots, a_N). \end{aligned} $$
We now review the process matrix description of the quantum switch discussed in Section IX. The quantum switch can be modelled as a 4-partite process involving the agents A, B, C and D. Where the agent C has a trivial (i.e., 1-dimensional) input system and D has a trivial output system, which we can ignore in the description. C's output decomposes as $C^O = C_C^O \otimes C_T^O$ and D's input as $D^I = D_C^I \otimes D_T^I$ corresponding to the control and target systems respectively. Thus C and D act on the control and target, with C acting in the global past of all other agents (i.e., cannot be signalling to by others) and D in the global future of all other agents (i.e., cannot signal to the others). A and B only act on the target qudit. Therefore the dimensions of the non-trivial systems are $d_C^{C_O} = d_D^{D_I} = 2$ and $d_T^{C_O} = d_D^{I} = d_A^I = d_{A^O} = d_B^I = d_{B^O} = d$ where $d \ge 2$. The corresponding process matrix is pure (or equivalently, the process map $W_{QS}$ is unitary) i.e., $W^{QS} = |W^{QS}\rangle\langle W^{QS}|$ where

$$|W^{QS}\rangle = |1\rangle^{C_T^O A^I} |1\rangle^{A^O B^I} |1\rangle^{B^O D_T^I} |00\rangle^{C_C^O D_C^I} + |1\rangle^{C_T^O B^I} |1\rangle^{B^O A^I} |1\rangle^{A^O D_T^I} |11\rangle^{C_C^O D_C^I} \quad (E2)$$

The situation is illustrated in Figure 10. If the lab C prepares an initial state $(\alpha|0\rangle + \beta|1\rangle)^{C_C^O} \otimes |\psi\rangle^{C_T^O}$ (for arbitrary normalised amplitudes $\alpha, \beta$ and qudit state $|\psi\rangle$) and labs A and B perform the respective operations $\mathcal{U}^A : A^I \mapsto A^O$ and $\mathcal{V}^B : B^I \mapsto B^O$ (taken to be unitaries), the final state arriving at lab D is given as follows and recovers the final state of Eq. (15) on the control and target.

$$ \begin{aligned} & \left( (\alpha|0\rangle + \beta|1\rangle)^{C_C^O} \otimes \langle\psi^*|^{C_T^O} \otimes \langle\mathcal{U}^{A^*}|^{A^I A^O} \otimes \langle\mathcal{V}^{B^*}|^{B^I B^O} \right) \cdot |W^{QS}\rangle \\ &= \alpha|0\rangle^{D_C^I} \otimes (\mathcal{V}^B \mathcal{U}^A |\psi\rangle)^{D_T^I} + \beta|1\rangle^{D_C^I} \otimes (\mathcal{U}^A \mathcal{V}^B |\psi\rangle)^{D_T^I}, \end{aligned} \quad (E3) $$

where $\langle\psi^*|$ denotes the complex conjugate of $\langle\psi| = |\psi\rangle^\dagger$ in the computational basis $\{|0\rangle, |1\rangle\}$, such that $\langle\psi^*|i\rangle = \langle i|\psi\rangle$, $i \in \{0, 1\}$. $|\mathcal{U}^{A^*}\rangle^{A^I A^O} = (\mathcal{I} \otimes \mathcal{U}^{A^*})|1\rangle^{A^I A^I}$ and similarly for $\mathcal{V}^B$, where $*$ denotes the complex conjugate in the chosen orthonormal basis.

The process matrix $W^{QS}$ is known to be causally non-separable (i.e., cannot be decomposed as in Equation (9)) but nevertheless causal (i.e., always produces probabilities that decompose as per Equation (10)) [31]. The process map $W_{QS} : C_C^O \otimes C_T^O \otimes A^O \otimes B^O \mapsto D_C^I \otimes D_T^I \otimes A^I \otimes B^I$ is related to $W_{QS}$ through the Choi isomorphism $W_{QS} = \mathcal{I}^{\text{In}} \otimes W_{QS}(|1\rangle\langle 1|^{\text{In}})$, and is obtained from $W_{QS}$ through the inverse Choi isomorphism. Here the superscript In corresponds to the total input system $C_C^O C_T^O A^O B^O$ of the process map. We can also obtain the action of the process map more directly, by considering how it acts on the computational basis $\{|ijkl\rangle^{\text{In}}\}_{ijkl}$ with $i \in \{0, 1\}$ and $j, k, l \in \{0, ..., d-1\}$, which provides a complete orthonormal basis for its inputs. The process map is unitary, and when the control is $|0\rangle^{C_C^O}$, $W_{QS}$ behaves as a fixed order process with A before B. In this case it forwards the target from, $C_T^O$ to $A^I$, then from $A^O$ to $B^I$ and from $B^O$ to $D_T^I$ (as can be read off from the first term of Eq. (E2)). When the control is $|1\rangle^{C_C^O}$, it behaves as a fixed order process with B before A, forwarding $C_T^O$ to $B^I$, $B^O$ to $A^I$ and $A^O$ to $D_T^I$ and in both cases (as can be read off from the first term of Eq. (E2)), and the control being forwarded to $D_C^I$. Therefore we have, for all $j, k, l \in \{0, ..., d-1\}$,

$$ \begin{aligned} W_{QS} : & |0\rangle^{C_C^O} |j\rangle^{C_T^O} |k\rangle^{A^O} |l\rangle^{B^O} \mapsto |0\rangle^{D_C^I} |l\rangle^{D_T^I} |j\rangle^{A^I} |k\rangle^{B^I} \\ W_{QS} : & |1\rangle^{C_C^O} |j\rangle^{C_T^O} |k\rangle^{A^O} |l\rangle^{B^O} \mapsto |1\rangle^{D_C^I} |k\rangle^{D_T^I} |l\rangle^{A^I} |j\rangle^{B^I} \end{aligned} \quad (E4) $$

## Appendix F: Equivalence of notions of signalling in process networks

In this section we define two notions of signalling which are relevant in process networks, one is formulated at the level of the probabilities of the classical settings and outcomes and the other is formulated at the level of the quantum in and output systems of the process map and local operations. We then show an equivalence between these two notions which holds whenever the process map represents a valid process matrix.

**Definition 29** (Probabilistic signalling). Let W be an N-partite process matrix associated with the probability distribution $P(x_1, ..., x_N|a_1, ..., a_N)$. We say that an agent $A_i$ probabilistically signals to a set $A_S$ of the remaining agents if and only if the setting $a_i$ of $A_i$ is correlated with outcomes $x_S := \{x_j\}_{j \in S}$ of the subset of the remaining $N-1$ agents, associated with the indices $S \subseteq \{1, ..., i-1, i+1, ..., N\}$ i.e.,

$$ P(x_S|a_1, ..., a_{i-1}, a_i, a_{i+1}, ..., a_N) \neq P(x_S|a_1, ..., a_{i-1}, a_{i+1}, ..., a_N), \quad (F1) $$

$$ \text{where } P(x_S|a_1, ..., a_{i-1}, a_i, a_{i+1}, ..., a_N) = \sum_{x_j:j \notin S} P(x_1, ..., x_N|a_1, ..., a_N). $$
**Definition 30 (Quantum signalling).** Let $W$ be an $N$-partite process matrix and $\mathfrak{N}_{W,N}$ be the corresponding network in our framework. Let $A_i$ be an agent and $A_S$ be any subset of the remaining agents, and $\mathfrak{N}_{W,\overline{i\cup S}}$ be the sub-network of $\mathfrak{N}_{W,N}$ obtained from composing $W$ with the extended local operations of all agents except those in $A_i \cup A_S$. We say that an agent $A_i$ quantumly signals to a set $A_S$ of the remaining agents if and only if $\{A_i^O\}$ signals to $\{A_j^I\}_{j \in S}$ in the induced map $\mathcal{N}_{W,\overline{i\cup S}}$.

**Theorem 7.** [Equivalence of two notions of signalling] Whenever $W$ is a valid $N$-partite process matrix, $A_i$ probabilistically signals to $A_S$ if and only if $A_i$ quantumly signals to $A_S$.

*Proof.* We first prove the equivalence for bipartite processes $W$ over agents $A$ (Alice) and $B$ (Bob), and later show how this implies the general $N$-partite result.

*Probabilistic signalling implies quantum signalling* Here, we wish to prove that $P(y|ab) \neq P(y|a)$ implies that $\{A^O\}$ signals to $\{B^I\}$ in the process map $W$. We will prove the equivalent contrapositive statement which is that $\{A^O\}$ does not signal to $\{B^I\}$ in $W$ implies $P(y|ab) = P(y|a)$.

First we show that $\{A^O\}$ does not signal to $\{B^I\}$ in $W$ implies $\{A^O\}$ does not signal to $\{A^I, B^I\}$ in $W$. This follows from a result about the decomposition of process matrices proven in [28] which shows that any bipartite process matrix $W \in A^I \otimes A^O \otimes B^I \otimes B^O$ can be expressed in the Hilber-Schmidt basis in the following general form, where a Hilbert-Schmidt basis for linear operators on a Hilbert space $\mathcal{H}^X$ is given by a set of matrices $\{\sigma_{\mu}^X\}_{\mu=0}^{d_X^2-1}$ with $\sigma_0^X = 1^X$, $\text{Tr} \sigma_{\mu}^X \sigma_{\nu}^X = d_X \delta_{\mu\nu}$ and $\text{Tr} \sigma_j^X = 0$ for all $j \in \{1, ..., d_X^2 - 1\}$.

$$ 
\begin{align*}
W &= \frac{1}{d_{A^I} d_{B^I}} (1^{A^I A^O B^I B^O} + \sigma^{B \le A} + \sigma^{A \le B} + \sigma^{A \ne B}), \\
\sigma^{B \le A} &:= \sum_{i,j>0} c_{ij} \sigma_i^{A^I} \otimes 1^{A^O} \otimes 1^{B^I} \otimes \sigma_j^{B^O} + \sum_{i,j,k>0} d_{ijk} \sigma_i^{A^I} \otimes 1^{A^O} \otimes \sigma_j^{B^I} \otimes \sigma_k^{B^O} \\
\sigma^{A \le B} &:= \sum_{i,j>0} e_{ij} 1^{A^I} \otimes \sigma_i^{A^O} \otimes \sigma_j^{B^I} \otimes 1^{B^O} + \sum_{i,j,k>0} f_{ijk} \sigma_i^{A^I} \otimes \sigma_j^{A^O} \otimes \sigma_k^{B^I} \otimes 1^{B^O} \\
\sigma^{A \ne B} &:= \sum_{i>0} \nu_i \sigma_i^{A^I} \otimes 1^{A^O} \otimes 1^{B^I} \otimes 1^{B^O} + \sum_{i>0} x_i 1^{A^I} \otimes 1^{A^O} \otimes \sigma_i^{B^I} \otimes 1^{B^O} \\
&\quad + \sum_{i,j>0} g_{ij} \sigma_i^{A^I} \otimes 1^{A^O} \otimes \sigma_j^{B^I} \otimes 1^{B^O},
\end{align*}
\tag{F2} $$

where $c_{ij}, d_{ijk}, e_{ij}, f_{ijk}, \nu_i, x_i, g_{ij} \in \mathbb{R}$. We then use the fact that the non-signalling relations $\{A^O\} \leftrightarrow \{B^I\}$ and $\{A^O\} \leftrightarrow \{A^I, B^I\}$ in the process map $W$ can be equivalently expressed as the following conditions on its Choi matrix (which is by construction, the process matrix $W$),

$$ 
\begin{align}
\mathrm{Tr}_{A^I} W &= \mathrm{Tr}_{A^I A^O} W \otimes \frac{1^{A^O}}{d_{A^O}} \tag{F3} \\
W &= \mathrm{Tr}_{A^O} W \otimes \frac{1^{A^O}}{d_{A^O}}. \nonumber
\end{align} $$

Then we can immediately see for each term of $W$, except the term $\sigma^{A \le B}$, both conditions of the above equation are satisfied when replacing $W$ with the term under consideration. Thus a violation of the second condition by a given $W$ could only come from the presence of terms of the form of $\sigma^{A \le B}$. We see that the term $\sigma^{A \le B}$ always violates both conditions, whenever there are any non-zero $e_{ij}$ or $f_{ijk}$ coefficients. This is because $\sigma_j^{A^O} \neq 1^{A^O}$ by construction (since the $j$ values start from 1, and for all such $\sigma_j^{A^O}$, the trace is 0 which is distinct from the trace of the identity). Thus, tracing and replacing the $A^O$ system will change the term, and violate the second condition, and by the same argument we can see that it will also change the term even if we have already traced out $A^I$, hence the first condition will also be violated.

Thus we have shown that in each term of a valid $W$, if there is a violation of the second condition of Eq. (F3), then there will also be a violation of the first condition. This implies (by linearity) that for all valid $W$'s this implication holds, and consequently that $\{A^O\}$ signals to $\{A^I, B^I\}$ implies $\{A^O\}$ signals to $\{B^I\}$. Moreover, $\{A^O\}$ signals to $\{A^I, B^I\}$ which is equivalent to the second condition of Eq. (F3) is also shown to be equivalent to $P(y|ab) = P(y)$ in [32]. Therefore, $P(y|ab) \neq P(y|b)$ implies that $\{A^O\}$ signals to $\{A^I, B^I\}$ in $W$, which in turn implies that $\{A^O\}$ signals to $\{B^I\}$ in $W$ and altogether, this establishes that $A$ probabilistically signals to $B$ implies $A$ quantumly signals to $B$.
Quantum signalling implies probabilistic signalling Suppose that we have the quantum signalling relation {$A^O$} signals to {$B^I$} in W. For any CPTP map W, this is equivalent [67] to the condition that there exists a state $\sigma^{B^O}$ on $B^O$ and distinct states $\rho^{A^O}$ and $\tilde{\rho}^{A^O}$ on $A^O$ such that

$$ \mathrm{Tr}_{A^I} \circ W(\rho^{A^O} \otimes \sigma^{B^O}) \neq \mathrm{Tr}_{A^I} \circ W(\tilde{\rho}^{A^O} \otimes \sigma^{B^O}). $$

In other words, the state on $B^I$ when W acts on $\rho^{A^O} \otimes \sigma^{B^O}$ is distinct from the state on $B^I$ when it acts on $\tilde{\rho}^{A^O} \otimes \sigma^{B^O}$ and hence there exists a measurement {$M_{y|b}^B$}$_y$ that B can perform to distinguish these states with a non-zero probability. Then, we can define A's local operation $M^A$ to be such that whenever the setting $a = 0$ is input on $A^S$, it discards the input on $A^I$ and prepares the state $\rho^{A^O}$ to send to W and whenever the setting $a = 1$ is input on $A^S$, it discards the input on $A^I$ and prepares the state $\tilde{\rho}^{A^O}$ to send to W. This in turn implies that $P(y|a = 0, b) \neq P(y|a = 1, b)$ or $P(y|ab) \neq P(y|b)$, and establishes the claim.

**Generalisation to the multi-partite case** For the multi-partite case, we consider the signalling relation $A_i$ signals to the set $A_S$ in terms of the two notions of signalling. We can map this to the proof for the bipartite case which we have shown above, by treating $A_i$ as A in the above proof, and the set $A_S$ of agents together as B. In this case, instead of signalling in W, we would consider signalling in the induced map $N_{W,\overline{i \cup S}}$ of the sub-network $\mathfrak{N}_{W,\overline{i \cup S}}$ (note that in the bipartite case, this sub-network reduces to the network $(W, \emptyset)$ whose induced map is W). Then the proof of the statement that probabilistic implies quantum signalling carries forth unchanged, we can again prove the equivalent contrapositive statement using the same arguments, showing that {$A_i^O$} does not signal to {$A_j^I$}$_{j \in S}$ in $N_{W,\overline{i \cup S}}$ (i.e., no quantum signalling) implies that {$A_i^O$} does not signal to {$A_j^I$}$_{j \in i \cup S}$ in $N_{W,\overline{i \cup S}}$ which implies that $A_i$ does not probabilistically signal to $A_S$.

For the statement that quantum signalling implies probabilistic signalling, the only aspect that needs additional consideration is that, when we treat Bob as a collection of more than one agent $A_S$ in the multi-partite case, then he can only apply extended local maps that act independently on each input $A_j^I$ for $j \in S$ and not arbitrary joint operations. Bob's role, as the receiver of the signal is to be able to distinguish any two distinct states that arise on the subsystems {$A_j^I$}$_{j \in S}$ which are the outputs of the CPTP map $N_{W,\overline{i \cup S}}$, which arise as a result of different local operations that Alice may perform on the input $A_i^O$ of the same map. Quantum theory satisfies the property that any two states of a multi-partite system can be distinguished through local measurements performed on the subsystems i.e., there will exist a local measurement on each subsystem such that the outcome probabilities for the two states differ when performing those local measurements simultaneously on all the systems. This is known as the *local distinguishability* or *local discriminability* property of quantum theory [93, 94]. This property ensures that all the arguments used above for proving the equivalence in the bipartite case also carry over to the general multi-partite case. $\square$

# Appendix G: Fine-grained acyclic network describing a spacetime realisation of QS

Here we describe a spacetime realisation of the QS network $\mathfrak{N}_{W_{QS,U,V}}$ that satisfies relativistic causality, and the corresponding fine-grained network which will be associated with a definite and acyclic (information-theoretic) causal structure (as shown in Theorem 5 and Corollary 3). The realisation is associated with the spacetime embedding given in Eq. (16) and Eq. (17) of the main text and the fine-grained description will correspond to a *causal box* [21], as expected from our results of Appendix D. We first detail the fine-grained causal box description, and then show that this is indeed a fine-graining of the original network $\mathfrak{N}_{W_{QS,U,V}}$ as per Definition 17.

## 1. Causal box description of the fine-grained network

Consider the quantum switch process map $W_{QS}$ described in Appendix E, which is known to be a unitary, along with local unitaries $U^A$ and $V^B$ of Alice and Bob. Recall from the main text that the corresponding fine-grained process map $W_{QS}^f$ associated with the spacetime embedding of Eq. (16) and Eq. (17) has the in and output systems, In = {$C_C^O, C_T^O, A_1^O, A_2^O, B_1^O, B_2^O$} and Out = {$D_C^I, D_T^I, A_1^I, A_2^I, B_1^I, B_2^I$}, where $d_{C_C^O} = d_{D_C^I} = 2$, $d_{C_T^O} = d_{D_T^I} = d$ ($d \ge 2$) and $d_{A_1^I} = d_{A_2^I} = d_{A_1^O} = d_{A_2^O} = d_{B_1^I} = d_{B_2^I} = d_{B_1^O} = d_{B_2^O} = d + 1$. The $d + 1$ dimensional systems are associated with the Hilbert space $\mathbb{C}^d \oplus |\Omega\rangle$, where $|\Omega\rangle$ is the vacuum state. The fine-grained local operations $U_{1,2}^{A,f} : A_1^I \otimes A_2^I \mapsto A_1^O \otimes A_2^O$
<figure><img src="image_17.png" /><figcaption>FIG. 15: An equivalent representation of the fine-grained network of Fig. 12: This representation makes explicit that $W_{QS}^f$ itself is a CPTPM which is composed with the fin-grained local operations through feedback loops. The blue path and orange wires represent the paths of the non-vacuum target when the control is $|0\rangle$ and $|1\rangle$ respectively and correspond to the blue and orange paths in Fig. 12. Although there are feedback loops involved, $W_{QS}^f$ is a causal box and decomposes into a sequence of CPTPMs as shown in Fig. 16. This means that every causal influence between the systems involved in this network, is that such whenever $S$ is a cause of $S'$, then $S$ is embedded in the past of $S'$ according to the spacetime embedding associated with this realisation (Eq. (16) and Eq. (17)). As the spacetime forms a partial order, this implies that the causal structure of this network is acyclic as shown in Fig. 12.</figcaption></figure>

and $V_{1,2}^{B,f} : B_1^I \otimes B_2^I \to B_1^O \to B_2^O$ are given as

$$ U_{1,2}^{A,f} = U_1^{A,f} \otimes U_2^{A,f}, \quad V_{1,2}^{B,f} = V_1^{B,f} \otimes V_2^{B,f}, \qquad (G1) $$

where $U_i^{A,f}$ and $V_i^{B,f}$ act on vacuum and non-vacuum states as in Eq. (18) (leaving the vacuum $|\Omega\rangle$ invariant and applying a qudit unitary $U_i^A$ or $V_i^B$ on a non-vacuum input). We repeat this action here for convenience.

$$ \begin{aligned} U_i^{A,f} |\Omega\rangle^{A_i'} &= |\Omega\rangle^{A_i^O} \\ U_i^{A,f} |\psi\rangle^{A_i'} &= U_i^A |\psi\rangle^{A_i^O} \end{aligned} \qquad (G2) $$

The resultant fine-grained network is identical to that of Fig. 12 at the information-theoretic level (maps and compositions) and can be equivalently represented as shown in Fig. 15.

We now proceed to describe the internal decomposition of $W_{QS}^f$. This is identical to the description of the quantum switch originally given in the causal box framework [21] and we repeat it here in our notation. $W_{QS}^f$ can be decomposed
into three internal CPTP maps, $QS_1^f$, $QS_2^f$ and $QS_3^f$, which are illustrated in Fig. 16. When $W_{QS}^f$ is composed with the fine-grained local operations $U_{1,2}^{A,f}$ and $V_{1,2}^{B,f}$ described above, we have the following well-defined sequence of operations in the network.

1. $QS_1^f$ Receives the initial state of control and target $(\alpha|0\rangle + \beta|1\rangle)^{C_C^O} \otimes |\psi\rangle^{C_T^O}$ from $C$, stores the control in an internal memory $Q$, and forwards $|\psi\rangle$ to $A_1^I$ and $|\Omega\rangle$ to $B_1^I$ when the control is in state $|0\rangle$, and $|\Omega\rangle$ to $A_1^I$ and $|\psi\rangle$ to $B_1^I$ when the control is in the state $|1\rangle$.

2. $U_1^{A,f}$ and $V_1^{B,f}$ are applied on the inputs $A_1^I$ and $B_1^I$ and the corresponding outputs $A_1^O$ and $B_1^O$ are fed back to $W_{QS}^f$.

3. $QS_2^f$ Forwards the state in $A_2^O$ to $B_2^I$ and the state in $B_1^O$ to $A_2^I$.

4. $U_2^{A,f}$ and $V_2^{B,f}$ are applied on the inputs $A_2^I$ and $B_2^I$ and the corresponding outputs $A_2^O$ and $B_2^O$ are fed back to $W_{QS}^f$.

5. $QS_3^f$ consults the control qubit stored in $Q$, if it is in the state $|0\rangle$, forwards the state in the output $B_2^O$ to $D_T^I$ and it the control is in the state $|1\rangle$, forwards the output $A_2^O$ to $D_T^I$, and in both cases forwards the control in $Q$ to $D_C^I$.

One can easily verify (see [21] for details) that this implements the transformation of Eq. (19) from $C_C^O \otimes C_T^O$ to $D_C^I \otimes D_T^O$ whenever the local maps act on the vacuum as given in Eq. (G2) and in the case where $U_1^{A,f} = U_2^{A,f}$ and $V_1^{B,f} = V_2^{B,f}$ this implements the quantum switch transformation of Eq. (15). Moreover, as shown in [21], one can additionally include counters with the local operations, initialised to $|0\rangle$ and which increment by 1 whenever the local operation is applied on a non-vacuum state. At the end of the protocol, both Alice and Bob's counters would then record 1, which verifies that in this network, each of Alice and Bob's operations act exactly once on a non-vacuum $d$-dimensional state.

Moreover, this sequence decomposition ensures that the fine-grained causal structure of the network is acyclic as shown in Fig. 12, with a causal influence $S \rightsquigarrow S'$ only if the fine-grained systems $S$ and $S'$ are associated with spacetime locations $P^S$ and $P^{S'}$ (through the embedding $\mathcal{E}$ of Eq. (16)) where $P^S < P^{S'}$. Consequently, the signalling relations of the fine-grained network are compatible with the partial order of the spacetime. We can view the fine-grained map $W_{QS}^f$ as a process map associated with the agents $A_1, A_2, B_1, B_2, C$ and $D$. Notice that the in and output systems of these 6 agents are localised in the spacetime according to the embedding Eq. (16), and satisfy relativistic causality with respect to this embedding. This implies, by Corollary 1 that $W_{QS}^f$ must correspond to a fixed order process. The fine-grained causal structure of Fig. 12 along with the embedding Eq. (16) make it clear that the order involves $C$ and $D$ being in the global past and future respectively of other agents, $A_1$ before $B_2$ and $B_1$ before $A_2$ (with $A_1$ and $A_2, B_1$ and $B_2$ not necessarily ordered among themselves).

Here, we have only considered the 0-message (vacuum) and 1-message spaces of $d$ dimensions, for Alice and Bob. More generally, all fine-grained systems can be modelled as Fock spaces which allow for multiple messages of a given dimension and the above fine-grained operations can be consistently extended to the full Fock space as shown in [21]. In follow up work based on the master's thesis [77], we show that such causal box descriptions associated with an acyclic causal structure, and their extensions to the full Fock space can also be obtained for a much general class of processes called QCQCs [41], which realise arbitrary quantum controlled superpositions of orders.

## 2. Encoders and decoders relating the fine and coarse grained networks

We now describe the encoder and decoder that allow us to recover the coarse-grained network $\mathfrak{W}_{QS,U,V}$ from the fine-grained causal box description of the previous sub-section, and show that these respect the conditions of Definition 17. Having described the coarse and fine-grained systems of the network, we know that the encoder and decoder relating $W_{QS}$ and $W_{QS}^f$ must be maps of the following form.

$$ \begin{aligned} \text{Enc}_{QS}: \quad & C_C^O \otimes C_T^O \otimes A^O \otimes B^O \mapsto C_C^O \otimes C_T^O \otimes A_1^I \otimes A_2^I \otimes B_1^O \otimes B_2^O, \\ \text{Dec}_{QS}: \quad & D_C^I \otimes D_T^I \otimes A_1^I \otimes A_2^I \otimes B_2^I \otimes B_2^I \mapsto D_C^I \otimes D_T^I \otimes A^I \otimes B^I. \end{aligned} \qquad (G3) $$
<figure><img src="image_18.png" /><figcaption>FIG. 16: Sequence representation of the causal box $W_{QS}^f$ of Figure 15</figcaption></figure>

A first requirement of Definition 17 is that $Enc_{QS}$ must be CPTP on the whole input space while $Dec_{QS}$ must be CPTP on the image of $W_{QS}^f \circ Enc_{QS}$. We completely specify the action of $Enc_{QS}$ by its action on a complete orthonormal basis of $C_C^O \otimes C_T^O \otimes A^O \otimes B^O$, which is given by the computational basis $\{|ijkl\rangle\}_{i \in \{0,1\}, j,k,l \in \{0,...,d-1\}}$. This is given as follows, for any $j,k,l \in \{0,...,d-1\}$

$$ \begin{aligned} \text{Enc}_{QS}: & \quad |0\rangle^{C_C^O}|j\rangle^{C_T^O}|k\rangle^{A^O}|l\rangle^{B^O} \mapsto |0\rangle^{C_C^O}|j\rangle^{C_T^O}|k\rangle^{A_1^O}|\Omega\rangle^{A_2^O}|\Omega\rangle^{B_1^O}|l\rangle^{B_2^O}, \\ \text{Enc}_{QS}: & \quad |1\rangle^{C_C^O}|j\rangle^{C_T^O}|k\rangle^{A^O}|l\rangle^{B^O} \mapsto |0\rangle^{C_C^O}|j\rangle^{C_T^O}|\Omega\rangle^{A_1^O}|k\rangle^{A_2^O}|l\rangle^{B_1^O}|\Omega\rangle^{B_2^O}. \end{aligned} \quad (G4) $$

We can see that such an encoder is in fact unitarity (and therefore a CPTPM). We could physically realise it as follows, including an ancilla $E_A$ for Alice and ancilla $E_B$ for Bob, both initialised in the state $|\Omega\rangle$, then performing a controlled swap which sends $A^O$ to $A_1^O$ and $E_A$ to $A_2^O$ when $|0\rangle^{C_C^O}$ and $A^O$ to $A_2^O$ and $E_A$ to $A_1^O$ when $|1\rangle^{C_C^O}$, and another analogous controlled swap for Bob. Then we can see that the image of $W_{QS}^f \circ Enc_{QS}$ (using that $W_{QS}^f$ can be decomposed as in Fig. 16) is always such that it lies within the subspace defined by the set of projectors $\{\Pi_{0jkl}^f, \Pi_{1jkl}^f\}_{jkl \in \{0,...,d-1\}}$ associated with the following states

$$ \begin{aligned} |\Psi_{0jkl}^f\rangle^{\text{Out}} := & |0\rangle^{D_C^I}|j\rangle^{D_T^I}|k\rangle^{A_1^I}|\Omega\rangle^{A_2^I}|\Omega\rangle^{B_1^I}|l\rangle^{B_2^I}, \\ |\Psi_{1jkl}^f\rangle^{\text{Out}} := & |1\rangle^{D_C^I}|j\rangle^{D_T^I}|\Omega\rangle^{A_1^I}|k\rangle^{A_2^I}|l\rangle^{B_1^I}|\Omega\rangle^{B_2^I}. \end{aligned} \quad (G5) $$

Then consider a decoder that acts as follows for all $i,j,k$

$$ \begin{aligned} \text{Dec}_{QS}: & \quad |\Psi_{0jkl}^f\rangle^{\text{Out}} \mapsto |0\rangle^{D_C^I}|j\rangle^{D_T^I}|k\rangle^{A^I}|l\rangle^{B^I}, \\ \text{Dec}_{QS}: & \quad |\Psi_{1jkl}^f\rangle^{\text{Out}} \mapsto |1\rangle^{D_C^I}|j\rangle^{D_T^I}|k\rangle^{A^I}|l\rangle^{B^I}. \end{aligned} \quad (G6) $$

Such a decoder will be unitary (and therefore CPTP) on the image of $W_{QS}^f \circ Enc_{QS}$. We could physically realise this by sending the state in $A_1^I$ or $A_2^I$ to $A^I$ depending on $D_C^I$ and tracing out the other state, and similarly for Bob. Note that when applied on the image of $W_{QS}^f \circ Enc_{QS}$, the state being traced out will always be the vacuum $|\Omega\rangle$ and will decouple from the rest such that the decoder is unitary on the subspace and maps pure states to pure states as described.

The next step is to show that $W_{QS} = \text{Dec}_{QS} \circ W_{QS}^f \circ \text{Enc}_{QS}$ holds for the encoder and decoder defined above. $W_{QS}$ is unitary and its action is as given in Eq. (E4). From the causal box description of $W_{QS}^f$ given in the previous sub-section, we can see that it is unitary on the image of $Enc_{QS}$ and acts as follows (noting from Eq. (G4) that the image of the encoder is a subspace of $C_C^O \otimes C_T^O \otimes A_1^O \otimes A_2^O \otimes B_1^O \otimes B_2^O$ that is spanned by states of the form given
below)

$$
\begin{align}
W_{QS}^{f}: \quad & |0\rangle^{C_C^O}|j\rangle^{C_T^O}|k\rangle^{A_1^O}|\Omega\rangle^{A_2^O}|\Omega\rangle^{B_1^O}|l\rangle^{B_2^O} \mapsto |0\rangle^{D_C^I}|l\rangle^{D_T^I}|j\rangle^{A_1^I}|\Omega\rangle^{A_2^I}|\Omega\rangle^{B_1^I}|k\rangle^{B_2^I}, \nonumber \\
W_{QS}^{f}: \quad & |1\rangle^{C_C^O}|j\rangle^{C_T^O}|\Omega\rangle^{A_1^O}|k\rangle^{A_2^O}|l\rangle^{B_1^O}|\Omega\rangle^{B_2^O} \mapsto |1\rangle^{D_C^I}|k\rangle^{D_T^I}|\Omega\rangle^{A_1^I}|l\rangle^{A_2^I}|j\rangle^{B_1^I}|\Omega\rangle^{B_2^I}. \tag{G7}
\end{align}
$$

Notice that the final states above correspond to the states $|\Psi_{0ljk}^f\rangle$ and $|\Psi_{1klj}^f\rangle$ of Eq. (G5). Then it is immediate from combining the description of the encoder (Eq. (G4)), the description of the fine-grained process map (Eq. (G7)) and that of the decoder (Eq. (G6)), that we recover the action of the coarse-grained process (Eq. (E4)) i.e., $W_{QS} = \text{Dec}_{QS} \circ W_{QS}^f \circ \text{Enc}_{QS}$ holds. Moreover, it is easy to see that all the signalling relations are preserved, in particular the signalling relations $\{A^O\} \to \{B^I\}$ and $\{B^O\} \to \{A^I\}$ between the main agents in $W_{QS}$ is preserved in the corresponding signalling relations $\{A_1^O, A_2^O\} \to \{B_1^I, B_2^I\}$ and $\{B_1^O, B_2^O\} \to \{A_1^I, A_2^I\}$ in $W_{QS}^f$.

Recall that in order to regard the causal box realisation of the process network $\mathfrak{N}_{W_{QS,U,V}}$ as its fine-graining, we must compare all corresponding sub-networks of the two descriptions and find corresponding encoders and decoders to translate between their induced maps. $W_{QS}$ that we have analysed above is the induced map of one such sub-network of $\mathfrak{N}_{W_{QS,U,V}}$. Other sub-networks of $\mathfrak{N}_{W_{QS,U,V}}$ are associated with the individual local maps $\mathcal{U}^A$, $\mathcal{V}^B$ (without any composition) as well as the networks formed by composing $W_{QS}$ with one or both of $\mathcal{U}^A$ and $\mathcal{V}^B$. The latter type of networks either correspond to fully connecting a local operation to $W_{QS}$ by composing the inputs and outputs of the operation with $W_{QS}$, or sequentially composing it to $W_{QS}$ (in any order) which would correspond to connecting only its input system or only its output system to the corresponding system of $W_{QS}$ with the same label. In all of these cases, we can construct the corresponding encoders and decoders in an entirely analogous manner as we did for $W_{QS}$ in the above paragraphs, and we therefore do not repeat the construction for each case. For instance, consider the sub-network of $\mathfrak{N}_{W_{QS,U,V}}$ obtained by composing $\mathcal{V}^B$ but not $\mathcal{U}^A$ with $W_{QS}$. The induced map then maps $C_C^O \otimes C_T^O \otimes A^O$ to $D_C^I \otimes D_T^I \otimes A^I$ and the corresponding induced map in the fine-grained network maps $C_C^O \otimes C_T^O \otimes A_1^O \otimes A_2^O$ to $D_C^I \otimes D_T^I \otimes A_1^O \otimes A_2^O$. The encoder and decoder in the case would be constructed in the same manner as Eq. (G4) and Eq. (G6) to ensure that the non-vacuum and vacuum states end up in the right fine-grained system for Alice depending on the control. They would also satisfy all the required properties by the same arguments as before.

The only remaining sub-networks are those corresponding to the individual local operations $\mathcal{U}^A$ and $\mathcal{V}^B$. Consider the fine-grained local maps of Eq. (G1), which act on the vacuum state according to Eq. (18). Further suppose that one the fine-grained operations $\mathcal{U}_i^A$ acting on the non-vacuum subspace, say for $i=1$, is identical to the coarse-grained operations $\mathcal{U}^A$, and similarly for Bob i.e., $\mathcal{U}_1^A = \mathcal{U}^A$ and $\mathcal{V}_1^B = \mathcal{V}^B$. Here, the encoders and decoders are maps of the following form.

$$
\begin{align}
\text{Enc}_A: A^I &\mapsto A_1^I \otimes A_2^I, \quad \text{Enc}_B: B^I &\mapsto B_1^I \otimes B_2^I, \nonumber \\
\text{Dec}_A: A_1^O \otimes A_2^O &\mapsto A^O, \quad \text{Dec}_B: B_1^O \otimes B_2^O &\mapsto B^O. \tag{G8}
\end{align}
$$

Then it is sufficient to take encoders are simply identity channels from the coarse-grained system (e.g., $A^I$) to the first fine-grained system (e.g., $A_1^I$) and prepare an arbitrary, independent state on the second fine-grained system. Similarly, it is sufficient to consider decoders that act as identity channels from the first fine-grained system to the coarse-grained system and trace out the second fine-grained system. Then it is clear that the necessary properties to regard the map $\mathcal{U}_{1,2}^{A,f}$ as a fine-graining of $\mathcal{U}^A$ and the map $\mathcal{V}_{1,2}^{B,f}$ as a fine-graining of $\mathcal{V}^B$ are trivially satisfied.

From the above, it follows that the causal box description of the previous sub-section is indeed a fine-graining of the process network (Definition 17) and preserves all of it signalling relations, even though it admits an acyclic causal structure (unlike the original coarse-grained process network which has a cyclic causal structure).

**Remark 6 (Local distinguishability of the order).** The fixed spacetime realisation of QS described in Figure 12 and in this section is such that the spacetime location at which Alice or Bob’s operation is applied (on a non-vacuum state) is perfectly correlated with the control. Therefore, Alice and Bob can perfectly and locally distinguish the two orders by measuring the time of arrival (or more generally, spacetime location of arrival) of a non-vacuum state to their lab. This would of course collapse the superposition of orders. Alternatively, they can use the location of arrival as a control parameter based on which they apply different unitary operations depending on the order (cf. Eq. (19)). In [45, 49], a different type of experimental realisation of QS in Minkowski spacetime has been proposed, which has the interesting property that local measurements by Alice and Bob of the (space)time location at which they receive a non-vacuum state would not reveal significant information about the control since there is a large uncertainty in this spacetime
location even when the operations are applied in a fixed order (i.e., where the control is in one of the computational basis states). Our results are general and apply to both these type of realisations, and independently of the agents' ability to perfectly distinguish the orders locally. For the realisations of [46, 49], the fine-grained description would not correspond to that of Figure 12 but one where Alice and Bob act on a larger number of fine-grained systems (associated with more than two spacetime locations) such that the location at which they receive a non-vacuum state would not tell them much about the order. This means that we would need to consider a larger number of agents than 6 in order to describe the experiment using a fixed order processes (cf. Theorem 5), and this also means that we may not be able to perfectly implement the transformation of Eq. (19) where the unitary applied by each agent is perfectly correlated with the order in which they act. However, this does not affect the conclusions of Theorem 5 that the fine-grained description of such an experiment would correspond to a fixed order process that is compatible with a definite and acyclic causal structure.

## Appendix H: Comparing fixed spacetime and quantum gravitational realisations

In [53], a protocol has been proposed for distinguishing between the physically realised optical realisations and theoretically proposed gravitational realisations of QS in a manner that does not disturb the coherence between the different branches of the superposition. For the definite spacetime case, [53] focuses on a QS implementation in Minkowski spacetime described with respect to a single global frame where the in/output events of Alice and Bob are spatially localised but not temporally localised. Here we show that their core argument can be generalised to arbitrary fixed spacetime realisations.

The main property that the protocol of [53] seeks to distinguish is whether the realisation involves a superposition of orders between two spacetime events (one for Alice and one for Bob) or whether at least one agent is associated with more than one spacetime event. The protocol introduces an additional agent $F$, hereby known as “Friend” to whom Alice and Bob send out photons in addition to performing their usual operations in the QS scenario. $F$ is assumed to be spatially localised<fnref n="14" /> and can measure information regarding the times of arrival of the photons arriving from Alice and Bob to decide whether or not the local operation in each lab was a single spacetime event. In the optical implementation of QS, at least one of Alice and Bob must act (on a non-vacuum state) at an earlier time or a distinct later time depending coherently on the control qubit (while the other agent may act at a fixed spacetime location). In this case, at least one agent is associated with two spacetime events which translates into $F$ receiving at least one of the agents’ photons arriving to them at a coherent superposition of different times, in the optical realisation in Minkowski spacetime. On the other hand, [53] proposes (theoretical) quantum gravitational realisations where the superposition of spacetime metrics can used to ensure that any photon from Alice always arrives to $F$ at the same time $t^A$ and any photon from Bob always arrives to $F$ at the same time $t^B$. A non-demolition measurement is then performed by $F$ to distinguish these two scenarios without collapsing the superposition of orders (see [53] for details).

We can model this protocol by considering a new 5-partite process map $W_{QS}^{+F}$ obtained from the 4-partite quantum switch map $W_{QS}$ by including the agent $F$ with input systems $F_A^I$ and $F_B^I$ and a trivial output space. We can give additional outputs $A_F^O$ and $B_F^O$ to Alice and Bob for the photons being sent to $F$ and the process vector $|W_{QS}^+\rangle$ is then identical to $|W_{QS}\rangle$ of Equation (E2), but with the additional factor $|1\rangle^A F_A^I |1\rangle^B F_B^I$ in both terms of the superposition, representing the identity channels from $A_F^O$ to $F_A^I$ and $B_F^O$ to $F_B^I$. [53] argue that in the Minkowski spacetime realisation, assuming that Alice and Bob communicate to $F$ using photons (i.e., light speed communication), then it cannot be the case that $F$ (assumed to be spatially localised) receives Alices’s photon at time $t^A$ and Bob’s at time $t^B$ independently of the order in which they act. This is because such a realisation of QS involves at least one agent (say Alice) acting at multiple spacetime events, and the light-like surfaces emanating from the two events would intersect $F$’s spatially localised worldline at two distinct times. That is, it is impossible for the systems $F_A^I$ and $F_B^I$ to both be localised in the spacetime if we assume light speed communication from $A$ and $B$ to $F$, in a realisation satisfying relativistic causality. Notice that without this assumption, it is in-principle possible to adjust the communication speed according to the order in which Alice acts such that her communication reaches $F$ at the same time (e.g., if Alice acts earlier, she uses free space communication and if she act later, she uses a fibre optic cable

<footnote n="14">¹⁴ In a classical background spacetime, which is what we consider here, the notion of spatial localisation is well defined. To model a quantum spacetime, [53] considers a region of the manifold within which there can be a superposition of metrics while there is a well defined classical geometry outside this region where localisation is well-defined as in classical spacetimes. The friend $F$ is assumed to be spatially localised in such a classical region of the spacetime.</footnote>
with a suitable refractive index). Here, we suggest a way to generalise the arguments of [53] to arbitrary spacetime realisations, without assuming light speed communication.

For this, consider an arbitrary but definite and acyclic spacetime $\mathcal{T}$, and that $F_A^I$ is embedded at a single spacetime location $P^F_A$ and $F_B^I$ at the location $P^F_B$, taking $P^F_B < P^F_A$ (the argument for $P^F_A < P^F_B$ is analogous). Denoting $\text{Past}(P) := \{Q \in \mathcal{T} | Q < P\}$ to be the past of a spacetime point $P$, we have $\text{Past}(P^F_B) \subset \text{Past}(P^F_A)$. Then, relativistic causality, along with the fact that Alice and Bob must in general, signal to $F$ in the protocol implies that $\mathcal{R}^{A^O} \xrightarrow{R} P^F_A$ and $\mathcal{R}^{B^O} \xrightarrow{R} P^F_B$. Now if we can ensure that $P^F_B$ is outside the lightcone of all points in $\mathcal{R}^{A^O}$ i.e., that it is impossible for Alice to have communicated to $F$ such that Alice's message could arrive to $F$ at the location $P^F_B$ in any relativistically causal protocol, then we would have $\mathcal{R}^{A^O} \subseteq \text{Past}(P^F_A) \setminus \text{Past}(P^F_B)$. Together with $\mathcal{R}^{B^O} \subseteq \text{Past}(P^F_B)$, this would imply that $\mathcal{R}^{A^O} \not\subset \mathcal{R}^{B^O}$. However, since we have $\{A^O\} \to \{B^O\}$ in the QS protocol, this would be a violation of relativistic causality.

This shows that the following properties cannot be simultaneously achieved in any fixed spacetime realisation of [53]'s modified QS protocol that satisfies relativistic causality: (1) the spatially localised agent $F$ receives the communication from Alice at a single spacetime location $P^F_A = (\vec{r}^F, t^F_A)$ and from Bob at a distinct, single location $P^F_B = (\vec{r}^F, t^F_B)$ and (2) for at least one agent, say $A$, the spacetime location associated with the other agent, $P^F_B$ is such that $P^F_B$ is outside the future light-cone of $A$'s output locations i.e, $\mathcal{R}^{A^O} \not\subset P^F_B$. Notice that (2) is a weaker condition that requiring $A$ and $B$ to communicate to $F$ only at light speed. If consider Minkowski spacetime and we restrict communication from $A$ and $B$ to $F$ to be light speed communication, then we can recover the argument of [53] as a special case. In this case, assuming (1) and relativistic causality, would imply that every point $Q \in \mathcal{R}^{A^O}$ must lie on the past light-like surface of $P^F_A$. Taking $P^F_B < P^F_A$ w.l.o.g, and recalling that $P^F_B$ and $P^F_A$ have the same spatial co-ordinate $\vec{r}^F$, it is clear that the past light-like surface of $P^F_A$ cannot have any overlap with the past light-cone of $P^F_B$. Therefore $\mathcal{R}^{A^O} \subseteq \text{Past}(P^F_A) \setminus \text{Past}(P^F_B)$ and consequently $\mathcal{R}^{A^O} \not\subset P^F_B$ holds. However, in a gravitational implementation where the spacetime locations are described with respect to local quantum reference frames [60], it might nevertheless be possible to satisfy all these conditions in a physical implementation of QS, as suggested by the gravitational realisations proposed in [53]. It would be interesting to study, in the future, how (2) can be imposed and justified in quantum gravitational realisations.

# Appendix I: Relation to causal loops without superluminal signalling

In recent works [2, 3] involving one of the authors, a formalism for describing cyclic and non-classical information-theoretic causal models as well as their compatibility with spacetime has been developed. While the present paper is inspired by the spirit of this work, namely to disentangle the two causality notions and relate them through compatibility conditions (which capture relativistic causality constraints), there are some important differences between the two formalisms as they were developed for describing different types of scenarios. For instance, [2, 3] can model certain beyond-quantum theories and correlations known as jamming non-local theories/correlations [95], however it focused on interventions on observed classical nodes in information-theoretic causal models rather than interventions directly on the underlying non-classical systems. On the other hand, the present formalism focuses on quantum information-theoretic networks where agents can intervene on quantum systems, which enables us to explicitly model indefinite causal structures and process matrices. In the following, we discuss the relation between the two formalisms, and in particular, clarify the link between our results about the acyclic fine-grained causal structure of spacetime realised networks and the surprising result of [3] regarding the possibility of causal loops without superluminal signalling in Minkowski spacetime.

As highlighted in Section IIIB and in previous works [2, 3, 7, 59], information-theoretic causation and signalling are distinct notions. It is possible to have causal influence (~~>) without signalling (—), which implies that no superluminal causation and no superluminal signalling correspond to two distinct relativistic causality principles. Let us illustrate this more concretely with the CPTPM $\mathcal{M}: I \mapsto O_1 \otimes O_2$ from Fig. 5, where $\{I\} \to \{O_1, O_2\}$ but $\{I\} \not\to \{O_1\}$ and $\{I\} \not\to \{O_2\}$. Suppose that the in/output systems are embedded at distinct spacetime locations $P^S$, $S \in \{I, O_1, O_2\}$. Then in any decomposition of $\mathcal{M}$, $I$ must be a cause of at least one of $O_1$ or $O_2$, the decompositions
<figure><img src="image_19.png" /><figcaption>FIG. 17: (a) The directed cyclic graph over the nodes A, B, C, Λ represents the information-theoretic causal structure of the causal loop of [3]. The loop corresponds to a classical functional causal model (reviewed in Appendix C) on this causal structure where all nodes are binary variables, the parentless node Λ is associated with the uniform distribution $P(\Lambda)$, and the remaining nodes depend on their parent nodes through the functional relations: $A = \Lambda \oplus B$, $C = \Lambda$ and $B = A \oplus C$. Here only A, B and C are accessible to agents, Λ is a latent node and cannot be intervened upon. The signalling relations of the causal model are such that B signals jointly to A and C but not individually to A or C. The causal model is embedded in (1+1)-Minkowski spacetime by assigning a single spacetime event to each of its nodes as shown, the 45 degree straight lines (blue) denote light-like surfaces. In the given embedding, it is shown in [3] that this causal loop does not lead to superluminal signalling, since the signalling from B to {A,C} can only be verified in the blue shaded region which is contained in the future of B. Notice however that the causal influence from B to A is superluminal in this embedding. (b) Equivalent representation of the causal loop of [3] as a cyclic network in our framework. Notice that the map $M_1$ acts identically to the map $M_1$ of Fig. 5b, with B, A and C playing the role of I, $O_1$ and $O_2$ respectively. If this network is embedded in spacetime, our relativistic causality condition (Definition 23) would require at least one of A or C to be embedded in the future light-cone of B (since {$B$} signals to {$A,C$}), the spacetime embedding of (a) for this network therefore violates the relativistic causality condition of the present paper, although it respects the strictly weaker relativistic causality condition of [2, 3] and does not lead to superluminal signalling as discussed in the text (and shown in [3].</figcaption></figure>

illustrated in Fig. 5 have this property.<fnref n="15" /> Therefore, a necessary condition for ensuring that information-theoretic causal influences flow from past to future in the spacetime (i.e, there is no superluminal causation) is that at least one of $P^I < P^{O_1}$ or $P^I < P^{O_2}$ must hold. This is what our relativistic causality condition of Definition 23 imposes. On the other hand, noting that the signalling {$I$} — {$O_1, O_2$} can only be verified when $O_1$ and $O_2$ are jointly accessed, imposing that the joint future (intersection of the future light-cones) of $P^{O_1}$ and $P^{O_2}$ is contained within the future of I would be sufficient for ensuring no superluminal signalling in this scenario [2].<fnref n="16" /> This condition is strictly weaker than requiring $P^I < P^{O_1}$ or $P^I < P^{O_2}$ to hold, as either of these would imply that the joint future of the outputs would be contained in the inputs but the converse is not true: the joint future condition can be satisfied even when $P^I, P^{O_1}$ and $P^{O_2}$ are space-like separated [95]. The compatibility condition of [2, 3] which is necessary and sufficient for ensuring no superluminal signalling in that framework, imposes these weaker conditions.

<footnote n="15">¹⁵ For if it were not (i.e., we had a decomposition where there is no internal wire connection I to either of $O_1$ or $O_2$, then {$I$} signals to {$O_1, O_2$} would be impossible.</footnote>

<footnote n="16">¹⁶ Provided that the systems I, $O_1$ and $O_2$ are the only ones that are accessible to agents (systems involved in the internal decomposition of $M$ being outside their access).</footnote>
The difference between the two compatibility conditions, which correspond to two distinct (but related) relativistic causality principles as outlined above, leads to different characterisations of the set of information processing protocols which can be realised in a fixed spacetime. Interestingly, only demanding no superluminal signalling (the weaker condition) does not rule out the possibility of genuine causal loops being embedded in Minkowski spacetime as shown in [3]. In the language of our formalism, such genuine loops would correspond to those where the maximally fine-grained description is also cyclic i.e., the causal loop of [3] is a counter-example which shows that Theorem 5 would not hold under the weaker compatibility condition. [3] does not suggest that such loops are physical in any way, rather, the main message of [3] is that the principle of no superluminal signalling is not sufficient for ruling out causal loops in Minkowski spacetime, as commonly believed. In light of this, it remains important to understand the physical principles that can rule them out. Our work suggests that the stronger relativistic causality constraint (which is necessary for ruling out superluminal causation) is sufficient for ruling out such genuine causal loops in a fixed spacetime, at least in quantum theory. Whether this is also a necessary condition for ruling out such loops (even in quantum and classical theories) in a fixed spacetime, and whether the sufficiency result generalises to post-quantum theories remain open questions. In Fig. 17 we illustrate the information-theoretic causal model of the causal loop proposed in [3], the cyclic network by which we can model it within the present framework and the spacetime embedding proposed in [3] whereby the causal loop does not lead to superluminal signalling. While we have discussed some of the relationships and distinctions between the two formalisms, characterising the precise relationship(s) between the classes of causal loops proposed in [2, 3] and the cyclic quantum networks proposed here, specifically the subset that describes indefinite causal structure processes, remains an interesting open question.

We stress that although our compatibility condition is stronger than the condition of [2, 3] and is necessary for ensuring the absence of superluminal causation, it is formulated only at the level of the operational signalling structure of the network and without referring to the underlying causal decompositions of the maps involved. This is because the signalling, although distinct from causation, does allow us to operationally infer the presence of certain causal influences. For the questions of interest in the present work, namely for understanding the physical and experimental realisations of indefinite causal structures in spacetime, the stronger compatibility condition that we have used is the more relevant one, as we are not yet aware of the existence of any physical mechanisms for superluminal causation in Minkowski spacetime. However, one way to evade the consequences of our main no-go theorems would be through a theoretical model which incorporates such superluminal causal mechanisms (which, could in-principle be possible without violating the principle of superluminal signalling). However, such causal explanations are typically disfavoured as they would need to be fine-tuned [4, 7].

## Appendix J: Proofs of all results from the main text

**Theorem 1.** [Embedding arbitrary, cyclic signalling relations in spacetime] For every signalling structure $\mathcal{G}^{\text{sig}}$, there exists a fixed acyclic spacetime $\mathcal{T}$ and an embedding $\mathcal{E}$ of $\mathcal{G}^{\text{sig}}$ in a region causal structure $\mathcal{G}_{\mathcal{T}}^{\text{reg}}$ of $\mathcal{T}$ that respects relativistic causality.

*Proof.* In our framework, a signalling structure $\mathcal{G}_{\mathfrak{N}}^{\text{sig}}$ of some network $\mathfrak{N}$ (which will simply be denoted as $\mathcal{G}^{\text{sig}}$ in the rest of the proof) is in general a directed graph where the nodes are subsets of in/output systems in the network, i.e., Nodes($\mathcal{G}^{\text{sig}}$) $\subseteq \Sigma(\mathfrak{N}^{\text{sys}})$. An embedding $\mathcal{E}$ of $\mathcal{G}^{\text{sig}}$ in a spacetime $\mathcal{T}$ corresponds to an assignment of spacetime regions to each system in $\mathfrak{N}^{\text{sys}}$. This immediately implies an embedding for all systems in $\Sigma(\mathfrak{N}^{\text{sys}})$: for any subset $\mathcal{S}$ of $\mathfrak{N}^{\text{sys}}$ the spacetime region assigned to $\mathcal{S}$ through the embedding $\mathcal{E}$ is simply the union $\mathcal{R}^{\mathcal{S}} = \bigcup_{S \in \mathcal{S}} \mathcal{R}^{\mathcal{S}}$ of all the spacetime regions assigned to the individual elements $S \in \mathcal{S}$ under the embedding $\mathcal{E}$. Thus, in order to establish the theorem statement, we first find an embedding for systems in $\mathfrak{N}^{\text{sys}}$, such that the signalling relations in $\mathcal{G}^{\text{sig}}$ over the nodes corresponding to systems in $\mathfrak{N}^{\text{sys}}$ respects relativistic causality. We will later see that this immediately implies an embedding of all the nodes of $\mathcal{G}^{\text{sig}}$ such that relativistic causality is still preserved. Therefore, for the purpose of the next few paragraphs, we will treat $\mathcal{G}^{\text{sig}}$ as a directed graph $\mathcal{G}$ over the nodes $\mathfrak{N}^{\text{sys}}$ with directed edges $\to$, and generalise the result to the powerset $\Sigma(\mathfrak{N}^{\text{sys}})$ at the end.

We first need to set out some nomenclature. For every node $N$ of a directed graph $\mathcal{G}$, let the set $\text{Par}(N) := \{N' \in \text{Nodes}(\mathcal{G}) : N' \to N \in \mathcal{G}\}$ denote the set of all parents of the node $N$, the set $\text{Ch}(N) := \{N' \in \text{Nodes}(\mathcal{G}) : N \to N' \in \mathcal{G}\}$ denote the set of all children of the node $N$, the set $\text{Anc}(N) := \{N' \in \text{Nodes}(\mathcal{G}) : \exists \text{ directed path } N' \to \dots \to N \in \mathcal{G}\}$ denote the set of all ancestors of $N$ in $\mathcal{G}$ and the set $\text{Desc}(N) := \{N' \in \text{Nodes}(\mathcal{G}) : \exists \text{ directed path } N \to \dots \to N' \in \mathcal{G}\}$ denote the set of all descendants of $N$ in $\mathcal{G}$. Then the set of all nodes in $\mathcal{G}$ that are involved in at least one cycle is defined as

$$ \text{Cycle}(\mathcal{G}) = \{N \in \text{Nodes}(\mathcal{G}) : N \in \text{Anc}(N)\}. $$
That is, every node $N$ that is its own ancestor belongs to the set Cycle($G$). Note that since $N_1 \in \text{Anc}(N_2)$ is equivalent to $N_2 \in \text{Desc}(N_1)$, we could have equivalently defined this set as Cycle($G$) = {$N \in \text{Nodes}(G) : N \in \text{Desc}(N)$}. In particular this implies that $N \in \text{Anc}(N) \cap \text{Desc}(N)$ (or $N$ is contained in its own strongly connected component [62]), for every $N \in \text{Cycle}(G)$. Moreover, $G$ is a directed acyclic graph if and only if Cycle($G$) = $\emptyset$.

The rest of the proof proceeds as follow. We will show that we can fine-grain (cf. Definition 2) any signalling structure $\mathcal{G}$ into a directed acyclic graph $\mathcal{G}'$ by “splitting” nodes in Cycle($\mathcal{G}$), such that when the split nodes in $\mathcal{G}'$ are recombined, we get back the original graph $\mathcal{G}$. Once we have a directed acyclic graph $\mathcal{G}'$, we can always embed it in a partially ordered spacetime $\mathcal{T}$ though an embedding $\mathcal{E}'$ that respects relativistic causality. We can then coarse-grain the embedding $\mathcal{E}'$ of $\mathcal{G}'$ to an embedding $\mathcal{E}$ of the original structure $\mathcal{G}$ that also respects relativistic causality, which will establish the theorem statement.

Construct a new directed graph $\mathcal{G}'$ from the original directed graph $\mathcal{G}$ as follows. If Cycle($\mathcal{G}$) = $\emptyset$, set $\mathcal{G}'$ = $\mathcal{G}$. If Cycle($\mathcal{G}$) $\neq$ $\emptyset$, then split every node $N \in \text{Cycle}(\mathcal{G})$ into two nodes $N_1$ and $N_2$ such that $\text{Par}(N_1) = \text{Par}(N)\setminus\text{Cycle}(\mathcal{G})$ and $\text{Ch}(N_1) = \text{Ch}(N)$ and $\text{Par}(N_2) = \text{Par}(N)$ and $\text{Ch}(N_2) = \text{Ch}(N)\setminus\text{Cycle}(\mathcal{G})$ i.e., $N_1$ contains no incoming arrows from nodes in Cycle($\mathcal{G}$) but all the same outgoing arrows as $N$ does in $\mathcal{G}$ while $N_2$ contains no outgoing arrows to nodes in Cycle($\mathcal{G}$) but all the same incoming arrows as $N$ does in $\mathcal{G}$. Nodes that do not belong to Cycle($\mathcal{G}$), and other edges not featuring in the above construction remain unaffected. This fully defines $\mathcal{G}'$. $\mathcal{G}'$ constructed in this way is a fine-graining of $\mathcal{G}$ (cf. Definition 2). More precisely, when we recombine $N_1$ and $N_2$ back into a single node $N$, for each pair of split nodes and without altering the edge structure, we recover the original graph $\mathcal{G}$ since $\text{Par}(N_1 \cup N_2) = \text{Par}(N)$ and $\text{Ch}(N_1 \cup N_2) = \text{Ch}(N)$ for every $N \in \text{Cycle}(\mathcal{G})$, and the nodes $N \notin \text{Cycle}(\mathcal{G})$ were not split or altered in going from $\mathcal{G}$ to $\mathcal{G}'$. This implies that we can define a graph fine-graining $\mathcal{F}^{\text{graph}}$: $\text{Nodes}(\mathcal{G}) \mapsto \Sigma(\text{Nodes}(\mathcal{G}'))$ such that $\mathcal{F}^{\text{graph}}(N) = \{N_1, N_2\}$ for all $N \in \text{Cycle}(\mathcal{G})$ and $\mathcal{F}^{\text{graph}}(N) = N$ otherwise, with respect to which $\mathcal{G}'$ is a fine-graining of $\mathcal{G}$.

Now, we can show that $\mathcal{G}'$ is a directed acyclic graph. For this, observe that a node of a graph can belong to the cycle of the graph only if it has a parent and a child that belong to the cycle of the graph. This is because $N \in \text{Cycle}(\mathcal{G})$ implies that there is a directed path from $N$ to itself in $\mathcal{G}$ which implies a directed path from $N$ to at least one $N' \in \text{Par}(N)$ and this directed path together with $N' \to N$ tells us that $N' \in \text{Cycle}(\mathcal{G})$. Similarly, $N \in \text{Cycle}(\mathcal{G})$ implies a directed path from at least one $N'' \in \text{Ch}(N)$ to $N$ and this directed path together with $N \to N''$ implies that $N'' \in \text{Cycle}(\mathcal{G})$. Next, notice that if $N \notin \text{Cycle}(\mathcal{G})$, then by construction, we have $N \notin \text{Cycle}(\mathcal{G}')$. Moreover, for each $N \in \text{Cycle}(\mathcal{G})$, we have a pair of nodes $N_1$ and $N_2$ in $\mathcal{G}'$ where all parents of $N_1$ and all children of $N_2$ are nodes of the form $N \notin \text{Cycle}(\mathcal{G})$ (and hence $N \notin \text{Cycle}(\mathcal{G}')$). Then, it follows that no node $N$ in $\mathcal{G}'$ can have both a parent and a child in Cycle($\mathcal{G}'$), and therefore that $N$ itself cannot belong to Cycle($\mathcal{G}'$). In other words, this establishes that Cycle($\mathcal{G}$) = $\emptyset$ or that $\mathcal{G}'$ is acyclic.

Since $\mathcal{G}'$ is a directed acyclic graph, there exists an embedding $\mathcal{E}'$: $\text{Nodes}(\mathcal{G}') \mapsto \mathcal{T}$ of $\mathcal{G}'$ in a partially ordered set $\mathcal{T}$ (associated with the order relation $<$) such that $N_i \to N_j$ in $\mathcal{G} \Rightarrow \mathcal{E}'(N_i) < \mathcal{E}'(N_j)$. By virtue of being a partial order, $\mathcal{T}$ satisfies our minimal definition of spacetime structure, according to Definition 18. Then the required embedding $\mathcal{E}$ of $\mathcal{G}$ in the spacetime $\mathcal{T}$ simply associates two spacetime locations with each node $N \in \text{Cycle}(\mathcal{G})$, the two locations being precisely those assigned by $\mathcal{E}'$ to each of the split nodes i.e., $\mathcal{E}(N) := \{\mathcal{E}'(N_1), \mathcal{E}'(N_2)\}$. For all nodes $N \notin \text{Cycle}(\mathcal{G})$, $\mathcal{E}(N) = \mathcal{E}'(N)$ noting that these nodes never got split. Then it is clear that the embedding $\mathcal{E}$ of $\mathcal{G}$ respects relativistic causality whenever the embedding $\mathcal{E}'$ of $\mathcal{G}'$ respects relativistic causality (which it does by construction).

We now describe how the proof generalises to case where $\text{Nodes}(\mathcal{G}) = \Sigma(\mathfrak{N}^{\text{sys}})$. For this, note that our above proof covers all cases where $\mathcal{G}$ has the property that $S_1 \to S_2$ for two subsets $S_1, S_2$ of $\mathfrak{N}^{\text{sys}}$, then there exists $S_1 \in S_1$ and $S_2 \in S_2$ such that $S_1 \to S_2$. However, suppose that we have a signalling relation $S_1 \to S_2$ in $\mathcal{G}$ such that there is no signalling relation between individual elements of these two sets (this is indeed possible, see for instance Fig. 5). The relativistic causality condition implied by this signalling relation on the corresponding spacetime embedding is that $\mathcal{R}^{S_1} \xrightarrow{R} \mathcal{R}^{S_2}$ (cf. Definition 23). Since $\mathcal{R}^{S_1} = \bigcup_{S_1 \in \mathcal{R}^{S_1}} \mathcal{R}^{S_1}$ (and similarly for $S_2$), this is equivalent to saying that there exists $S_1 \in S_1$ and $S_2 \in S_2$ such that the corresponding spacetime regions satisfy $\mathcal{R}^{S_1} \xrightarrow{R} \mathcal{R}^{S_2}$. In other words, the relativistic causality constraints on the spacetime embedding of $\mathcal{G}$ are the same irrespective of whether or not $\mathcal{G}$ satisfies the aforementioned property. Thus the above proof also applies to establish the theorem statement for signalling structures $\mathcal{G}$ not satisfying this property and hence applies to all signalling structures.

Finally, we note that according to Definition 18 any partially ordered set corresponds to a spacetime. This rather minimal definition allows us to derive general results that only depend on the order relation between spacetime
points and does not require the spacetime to have any further symmetries, or a smooth differentiable structure. However, under this minimal definition, one might regard two different partially ordered sets $\mathcal{T}$ and $\mathcal{T}'$ as two different “spacetimes”. On the other hand, if we consider the more standard method of modelling spacetime as a differentiable manifold $\mathcal{M}$, as done in relativistic physics, we could sample different sets of points on the same manifold to generate different partially ordered sets<fnref n="17" /> $\mathcal{T}$ and $\mathcal{T}'$ from the same spacetime. If we model spacetime as a globally hyperbolic manifold that ensures the absence of closed timelike curves, then the statement of the present theorem would instead become “For every signalling structure $\mathcal{G}^{\text{sig}}$ and every globally hyperbolic manifold $\mathcal{M}$, there exists an embedding $\mathcal{E}$ of $\mathcal{G}^{\text{sig}}$ in a region causal structure $\mathcal{G}_{\mathcal{M}}^{\text{R}}$ of $\mathcal{M}$ that respects relativistic causality, where each node of $\mathcal{G}_{\mathcal{M}}^{\text{R}}$ is a finite set of points in $\mathcal{M}$.” This can be shown as follows. If $\mathcal{G}$ is a directed acyclic graph, then it can be embedded in any globally hyperbolic manifold $\mathcal{M}$ through an embedding $\mathcal{E}: \text{Nodes}(\mathcal{G}) \to \mathcal{M}$ that assigns a point in $\mathcal{M}$ to each node of $\mathcal{G}$ (see [53] for an explicit construction of such an embedding for the acyclic case). This is because the graph has a finite number of nodes and we can always sample a suitable set of points in the manifold having the required order relations. One can apply this embedding to the acyclic graph $\mathcal{G}'$ constructed in the proof above, this would define the embedding $\mathcal{E}'$: $\text{Nodes}(\mathcal{G}) \to \mathcal{M}$. The rest of the proof will be the same as the above case for partially ordered sets $\mathcal{T}$.

**Lemma 2.** [Probabilities from loop composition] For every process map $\mathcal{W}$, the joint probability distribution obtained through the induced map $\mathcal{P}_{\mathcal{W}}$ of the network $\mathfrak{N}_{\mathcal{W},N}$ as in Equation (13) is equivalent to that obtained in the process matrix framework through Equation (8). Moreover, $\mathcal{W}$ being a valid process matrix guarantees that the denominator of Equation (13) equals unity.

*Proof.* Recall from Eq. (12) that the induced map of the network $\mathfrak{N}_{\mathcal{W},N}$ which we denote by $\mathcal{P}_{\mathcal{W}}$ is given as follows

$$ \mathcal{P}_{\mathcal{W}} := \left( \mathcal{W} \otimes_{k=1}^{N} \mathcal{M}^{A_k} \right)^{\{A_k^I \to A_k'^I, A_k^O \to A_k'^O\}_{k=1}^{N}}, \quad (J1) $$

where $\mathcal{W}$ is an $N$-partite process map with inputs $\{A_k^{IO}\}_k$ and outputs $\{A_k^I\}_k$, while $\mathcal{M}^{A_k}$ is the extended local map of the agent $A_k$, associated with a quantum input $A_k'^I$ and classical input $A_k^S$ (for the setting $a_k$) and quantum output $A_k^O$ and a classical output $A_k^o$ (for the outcome $x_k$). Here the primed and corresponding unprimed systems are isomorphic to each other, which allows them to be loop composed together. Then the map $\mathcal{P}_{\mathcal{W}}$ obtained this way has $N$ classical inputs $\{A_k^S\}_k$ and $N$ classical outputs $\{A_k^o\}_k$. The above equation can be read as a parallel composition of the process map with the $N$ local maps followed by $2N$ loop compositions.

In the following, for brevity, we detail the proof for the bipartite case. However, the proof readily generalises to the $N$ agent case. In the bipartite case, taking the agents to be $A$ and $B$ with local settings associated with input systems $A^S, B^S$ and outcomes associated with output systems $A^o, B^o$, the parallel composition yields the map $\mathcal{M}^A \otimes \mathcal{M}^B \otimes \mathcal{W}$ with input systems $\{A'^I, A^S, B'^I, B^S, A'^O, B'^O\}$ and output systems $\{A^O, A^o, B^O, B^o, A^I, B^I\}$. Applying the loop formula of Eq. (1) to compose the corresponding prime and unprimed systems, we obtain the following expression for the induced map $\mathcal{P}_{\mathcal{W}}$ acting on an input state $|a\rangle\langle a|^{A^S} \otimes |b\rangle\langle b|^{B^S}$.

$$ \begin{aligned} \mathcal{P}_{\mathcal{W}}(|a\rangle\langle a|^{A^S} \otimes |b\rangle\langle b|^{B^S}) &= \sum_{k \dots r} \langle kmoq|^{A^I A^O B^I B^O} \left( \mathcal{M}_a^A (|k\rangle\langle l|^{A'^I}) \otimes \mathcal{M}_b^B (|o\rangle\langle p|^{B'^I}) \otimes \mathcal{W} (|mq\rangle\langle nr|^{A'^O B'^O}) \right) |lnpr\rangle^{A^I A^O B^I B^O} \\ &= \sum_{k \dots r} \langle knor|^{A^I A^O B^I B^O} \left( [\mathcal{M}_a^A (|k\rangle\langle l|)]^T \otimes [\mathcal{M}_b^B (|o\rangle\langle p|)]^T \otimes \mathcal{W} (|mq\rangle\langle nr|) \right) |lmpq\rangle^{A^I A^O B^I B^O}, \end{aligned} $$

where we have used the notation $\mathcal{M}_a^A(|k\rangle\langle l|^{A'^I}) := \mathcal{M}^A(|a\rangle\langle a|^{A^S} \otimes |k\rangle\langle l|^{A'^I})$ (and similarly for $B$'s operation) and suppressed the system labels where they are evident from context. Denoting the factor $([\mathcal{M}_a^A(|k\rangle\langle l|^{A'^I})]^T \otimes$

<footnote n="17">¹⁷ That is, if the manifold is globally hyperbolic, in more exotic spacetimes with closed timelike curves, we can also obtain pre-ordered sets from sampling suitable points.</footnote>
$[\mathcal{M}_b^B(|o\rangle\langle p|^{B'})]^T \otimes \mathcal{W}(|m\rangle\langle n|^{A'} \otimes |q\rangle\langle r|^{B'})$ by (...), introducing factors of the identity $\mathcal{I} = \sum_j |j\rangle\langle j|$, and then rearranging the resulting inner products we have

$$
\begin{align*}
& \mathcal{P}_W(|a\rangle\langle a|^{A^s} \otimes |b\rangle\langle b|^{B^s}) \\
&= \sum_{ijst k...r} \langle k|^{A^I} \langle n|j\rangle \langle j|^{A^O} \langle o|^{B^I} \langle r|s\rangle \langle s|^{B^O} \left( \dots \right) |i\rangle^{A^I} \langle i|l\rangle^{A^I} \langle m|^{A^O} \langle t|^{B^I} \langle t|p\rangle^{B^I} \langle q|^{B^O} \\
&= \sum_{ijst k...r} \langle i|^{A^I} |l\rangle \langle k|^{A^I} \langle j|^{A^O} \langle t|^{B^I} |p\rangle \langle o|^{B^I} \langle s|^{B^O} \left( \dots \right) |i\rangle^{A^I} \langle m|^{A^I} \langle n|^{A^O} \langle j|^{A^O} \langle t|^{B^I} \langle q|^{B^O} \langle r|^{B^O} |s|^{B^O} \\
&= \operatorname{Tr}_{A^I A^O B^I B^O} \left[ \sum_{k...r} |l\rangle\langle k|^{A^I} \otimes |p\rangle\langle o|^{B^I} \otimes \left( \dots \right) \otimes |m\rangle\langle n|^{A^O} \otimes |q\rangle\langle r|^{B^O} \right] \\
&= \operatorname{Tr}_{A^I A^O B^I B^O} \left[ \left( \sum_{kl} |l\rangle\langle k|^{A^I} \otimes [\mathcal{M}_a^A (|k\rangle\langle l|^{A^I})]^T \right) \otimes \left( \sum_{op} |p\rangle\langle o|^{B^I} \otimes [\mathcal{M}_b^B (|o\rangle\langle p|^{B^I})]^T \right) \right. \\
&\qquad \left. \otimes \left( \sum_{mnqr} |mq\rangle\langle nr|^{A^O B^O} \otimes \mathcal{W}(|mq\rangle\langle nr|^{A^O B^O}) \right) \right]
\end{align*}
$$

Now, we wish to calculate the probability that the output of $\mathcal{P}_W(|a\rangle\langle a|^{A^s} \otimes |b\rangle\langle b|^{B^s})$ is $|x\rangle\langle x|^{A^o} \otimes |y\rangle\langle y|^{B^o}$ i.e., the outcomes $x$ and $y$ are obtained by Alice and Bob upon measuring the settings $a$ and $b$. This is given by Eq. (13). We now show that the numerator of this expression i.e, $\operatorname{Tr}\left[ (|x\rangle\langle x|^{A^o} \otimes |y\rangle\langle y|^{B^o})(\mathcal{P}_W(|a\rangle\langle a|^{A^s} \otimes |b\rangle\langle b|^{B^s})) \right]$ equals the process matrix probability rule of Eq. (8). Using the expression for $\mathcal{P}_W(|a\rangle\langle a|^{A^s} \otimes |b\rangle\langle b|^{B^s})$ derived above, along with Eq. (11) to absorb the outcome projectors into the definition of the maps $\mathcal{M}_{x|a}^A$ and $\mathcal{M}_{y|b}^B$, we immediately obtain the following which coincides with Eq. (8).

$$ P(xy|ab) = \operatorname{Tr}\left[ (\mathcal{M}_{x|a}^{A^I A^O} \otimes \mathcal{M}_{y|b}^{B^I B^O}) W \right], \qquad (J2) $$

where $W = \mathcal{I} \otimes \mathcal{W}|1\rangle\langle 1|$ is the process matrix and $\mathcal{M}_{x|a}^{A^I A^O} = [\mathcal{I} \otimes \mathcal{M}_{x|a}^A (|1\rangle\langle 1|)]^T$ is the Choi matrix of the local map $\mathcal{M}_{x|a}^A$ (and similarly for $B$'s operation) as defined in Section VI. Whenever $W$ is valid process matrix, this distribution is normalised, which implies that $\sum_x \operatorname{Tr}[(\mathcal{M}_{x|a}^{A^I A^O} \otimes \mathcal{M}_{y|b}^{B^I B^O})W] = 1$, which in turn implies that the denominator of Eq. (13) equals unity. $\square$

**Lemma 3.** [Sub-networks and reduced processes] For any $N$-partite process map $W$, the sub-network $\mathfrak{N}_{W,l<N}$ of the process network formed by composing $W$ with some set of fixed local operations $\{\mathcal{M}_{a_k}^{A_k}\}_{k=1}^l$ for the first $l < N$ agents is a CPTP map whose Choi matrix is the reduced process matrix $\overline{W}(\mathcal{M}_{a_1}^{A_1 A_1^O}, \dots, \mathcal{M}_{a_l}^{A_l A_l^O})$.

*Proof.* The proof method is very similar to that of Lemma 2 but we provide it here for completeness and follow the same notation as the previous proof. Again, we restrict to the bipartite case for simplicity but the proof easily generalises to the multi-partite case. Consider a bipartite process map $W$ associated with the agents $A_1 \equiv A$ and $A_2 \equiv B$, with input systems $A'^O, B^O$ and output systems $A^I, B^I$. Let $\mathcal{M}_a^A : A'^I \to A^O$ be a local operation of the agent $A$ associated with a particular setting $a$, and where all the primed systems are isomorphic to the corresponding unprimed ones (we only add the primes on $A$'s systems as we will consider only their composition in the next steps). The sub-network $\mathfrak{N}_{W,1<2}$ (which we denote here as $\mathfrak{N}_{W,A}$) formed by composing $W$ only with the operation of $A$, through $A^I \to A'^I, A^O \to A'^O$ will have an induced map $\mathcal{N}_{W,A}$ which is associated with the input $B^O$ and output $B^I$. We now show that for every choice of setting $a$, the Choi matrix of the induced sub-network map $\mathcal{N}_{W,A}$ is the reduced process matrix $\overline{W}(\mathcal{M}_a^{A^I A^O})$ over $B$ (cf. Equation (14)). This will complete the proof as the same arguments, under exchange of $A$ and $B$ apply for showing that the Choi matrix of the subnetwork $\mathcal{N}_{W,B}$ is the reduced process matrix $\overline{W}(\mathcal{M}_b^{B^I B^O})$ on $A$, and these two are the only non-trivial reduced processes here. The Choi matrix of $\mathcal{N}_{W,A}$ is given as

$$ \sum_{m,n} |m\rangle\langle n|^{B^O} \otimes \mathcal{N}_{W,A}(|m\rangle\langle n|^{B^O}). $$
Using the composition operation, we can write $\mathcal{N}_{W,A}(|m\rangle\langle n|^{BO})$ as

$$ 
\begin{aligned}
\mathcal{N}_{W,A}(|m\rangle\langle n|^{BO}) &= \sum_{i,j,k,l} \langle i|^{A^I} \langle k|^{A^O} (M_a^A \otimes W)(|i\rangle\langle j|^{A'^I} \otimes |km\rangle\langle ln|^{A'^O B^O})|j\rangle^{A^I} |l\rangle^{A^O} \\
&= \sum_{i,j,k,l} \langle k|^{A^O} (M_a^A(|i\rangle\langle j|^{A'^I})|l\rangle^{A^O} \otimes \langle i|^{A^I} W(|km\rangle\langle ln|^{A'^O B^O})|j\rangle^{A^I}
\end{aligned}
$$

Plugging this back into the Choi matrix, inserting factors of the identity and rearranging, we can write the Choi matrix of $\mathcal{N}_{W,A}$ as

$$ 
\begin{aligned}
& \sum_{i,j,k,l,m,n} |m\rangle\langle n|^{BO} \otimes \langle k|^{A^O} M_a^A(|i\rangle\langle j|^{A'^I}) |l\rangle^{A^O} \otimes \langle i|^{A^I} W(|km\rangle\langle ln|^{A'^O B^O}) |j\rangle^{A^I} \\
&= \sum_{i,j,k,l,m,n} |m\rangle\langle n|^{BO} \otimes \langle l|^{A^O} [M_a^A(|i\rangle\langle j|^{A'^I})]^T |k\rangle^{A^O} \otimes \langle i|^{A^I} W(|km\rangle\langle ln|^{A'^O B^O}) |j\rangle^{A^I} \\
&= \sum_{i,j,k,l,m,n,p,q} |m\rangle\langle n|^{BO} \otimes \langle l|^{A^O} |p\rangle\langle p|^{A^O} [M_a^A(|i\rangle\langle j|^{A'^I})]^T |k\rangle^{A^O} \otimes \langle i|^{A^I} |q\rangle\langle q|^{A^I} W(|km\rangle\langle ln|^{A'^O B^O}) |j\rangle^{A^I} \\
&= \sum_{i,j,k,l,m,n,p,q} |m\rangle\langle n|^{BO} \otimes \langle p|^{A^O} [M_a^A(|i\rangle\langle j|^{A'^I})]^T |k\rangle^{A^O} \langle l|^{A^I} |p\rangle^{A^O} \otimes \langle q|^{A^I} W(|km\rangle\langle ln|^{A'^O B^O}) |j\rangle^{A^I} \langle i|^{A^I} |q\rangle^{A^I} \\
&= \sum_{p,q} \langle pq|^{A^O A^I} \left( 1^{B^I B^O} \otimes \sum_{i,j} |j\rangle\langle i| \otimes [M_a^A(|i\rangle\langle j|^{A'^I})]^T \right) \left( \sum_{k,l,m,n} |km\rangle\langle ln| \otimes W(|km\rangle\langle ln|^{A'^O B^O}) \right) |pq|^{A^O A^I} \\
&= \mathrm{Tr}_{A^I A^O} \left( (1^{B^I B^O} \otimes M_a^{A^I A^O}) \cdot W \right) = \overline{W}(M_a^{A^I A^O}),
\end{aligned}
$$

where we have used the definition of the Choi matrices of the local operations and the process map (see Section VI) in the last line, along with that of the reduced process matrix (Equation (14)). Moreover, we have identified the primed and unprimed versions of the systems $A^I$ and $A^O$ as they get composed to form effectively a single system. Moreover the reduced process matrix $\overline{W}(M_a^{A^I A^O})$ being a valid process matrix implies that the map $\mathcal{N}_{W,A}$ of which it is the Choi matrix is a CPTP map [31]. This completes the proof. $\square$

**Theorem 2.** The network $\mathfrak{N}_{W,N}$ formed the composition of any valid process map $W$ with a set of extended local operations $\{M^{A_k}\}_{k=1}^N$ is a network of CPTP maps.

*Proof.* The sub-networks of $\mathcal{N}_{W,N}$ are formed by taking subsets of its maps and compositions (cf. Eq. (12)). Subsets of $\mathcal{N}_{W,N}^{maps}$ are of two types: either the subset includes $W$ or it does not. In the latter case, none of the compositions of $\mathcal{N}_{W,N}^{comp}$ can be applied, as all compositions involve $W$. Thus such sub-networks are always guaranteed to be CPTP since they correspond to the parallel composition (i.e., tensor product) of CPTP maps, and we only need to consider sub-networks formed by composing $W$ with some subset of the extended local operations through some subset of the compositions in $\mathcal{N}_{W,N}^{comp}$. W.l.o.g, consider the subset of the first $l < N$ extended local operations. The compositions associated with these operations and $W$ are $\{A_k^I \to A_k^{I'}, A_k^O \to A_k^{O'}\}_{k=1}^l$. If we were to perform all these compositions, we obtain the sub-network $\mathfrak{N}_{W,l<N}$, whose induced map is CPTP as it corresponds to a reduced process matrix associated with the remaining $N-l$ agents, as we have shown in Lemma 3.

Suppose we perform only a subset of the compositions in $\{A_k^I \to A_k^{I'}, A_k^O \to A_k^{O'}\}_{k=1}^l$. Consider all the agents $A_k$ for which both compositions $\{A_k^I \to A_k^{I'}, A_k^O \to A_k^{O'}\}$ are included in this subset. Again, w.l.o.g, we can take these to be the first $p < l$ agents. As the order in which we perform the compositions does not matter (cf. Lemma 6), we can perform all the operations in the subset under consideration by first performing the compositions $\{A_k^I \to A_k^{I'}, A_k^O \to A_k^{O'}\}_{k=1}^p$ and then the rest. By Lemma 3 if again follows that performing the compositions $\{A_k^I \to A_k^{I'}, A_k^O \to A_k^{O'}\}_{k=1}^p$ results in a CPTP map $\mathcal{N}_{W,p<N}$. Now by construction, the remaining compositions, include at most one of $A_k^I \to A_k^{I'}$ or $A_k^O \to A_k^{O'}$ for each agent in $\{A_{p+1}, ..., A_l\}$. It is easy to see that each of these compositions would correspond to a regular sequential composition of the CPTP map $\mathcal{N}_{W,p<N}$ with a CPTP map $M^{A_k}$ for an agent $A_k$ in $\{A_{p+1}, ..., A_l\}$. The sequential composition of two CPTP maps (in any given order) is always a CPTP map. This shows that the remaining compositions also yield a CPTP map. We have covered all possible sub-networks in the above and
shown their induced maps to be CPTP. This establishes that our original network is a network of CPTP maps, as required. □

**Theorem 3.** A process matrix $W$ is not a fixed order process if and only if the signalling structure of the corresponding network $\mathfrak{N}_{W,N}$ certifies the cyclicity of any causal structure $\mathcal{G}$ that it is compatible with.

*Proof.* Observe that in the network $\mathfrak{N}_{W,N}$, we have $\{A_k^I\}$ signals to $\{A_k^O\}$ for every agent $A_k$ since the extended local map for each agent encodes all possible operations between their these in and output systems and in particular the identity channel which clearly signals from the input to output. This implies that when we compatibly embed (Definition 14) the systems of $\mathfrak{N}_{W,N}$ in a causal structure $\mathcal{G}$ with directed edges $\xrightarrow{\mathcal{G}}$ (thereby allowing us to treat each system as a node of a directed graph with these edges), then we must have $A_k^I \xrightarrow{\mathcal{G}} A_k^O$ for each $k \in \{1, ..., N\}$. By Definition 28 of fixed ordered processes and Definition 29, we can see that a process $W$ is a fixed order process if any only if there exists a partial order on the in/output systems of all agents where $A_i^O$ precedes $A_j^I$ in the partial order for some $j \in S$ whenever an agent $A_i$ probabilistically signals to a set of agents $A_S$, and $A_k^I$ precedes $A_k^O$ in this partial order for every agent $A_k$. By Theorem 7, $A_i$ probabilistically signals to $A_S$ is equivalent to $\{A_i^O\}$ signals to $\{A_j^I\}_{j \in S}$ in the network. Recall that compatibility of a signalling relation $\{A_i^O\}$ signals to $\{A_j^I\}_{j \in S}$ with an embedding in a causal structure $\mathcal{G}$ implies that there exists $j \in S$ such that there is a directed path $A_i^O \xrightarrow{\mathcal{G}} ... \xrightarrow{\mathcal{G}} A_j^I$ with respect to the order induced by $\mathcal{G}$ on the embedded systems (cf. Definition 14). Then the above arguments, taken together imply that $W$ is a fixed order process if and only if there exists a directed acyclic graph $\mathcal{G}$ with which the signalling relations of the process network $\mathfrak{N}_{W,N}$ are compatible. Equivalently $W$ is not a fixed order process if and only if any directed graph $\mathcal{G}$ with which the signalling relations of $\mathfrak{N}_{W,N}$ are compatible is a directed cyclic graph i.e., the signalling relations of $\mathfrak{N}_{W,N}$ being compatible with an embedding in some $\mathcal{G}$ certify the cyclicity of $\mathcal{G}$ whenever $W$ is not a fixed order process. □

**Theorem 4.** [*No-go theorem for spacetime realisations of processes*] No realisation (Definition 22) of an $N$-partite process network $\mathfrak{N}_{W,N}$ in a fixed spacetime structure $\mathcal{T}$ (Definition 18) with respect to a spacetime embedding $\mathcal{E}$ can simultaneously satisfy the following three assumptions.

1. *W is not a fixed order process (Definition 28).*

2. *The spacetime realisation satisfies relativistic causality (cf. Definition 24).*

3. *The region causal structure $\mathcal{G}_{\mathcal{T}}^{\text{reg}}$ induced by the embedding $\mathcal{E}$ (Definition 21) is acyclic.*

*Proof.* Recall that a spacetime realisation of a network $\mathfrak{N}_{W,N}$ is specified by a spacetime embedding $\mathfrak{N}_{W,N,\mathcal{T},\mathcal{E}}$ of the network (obtained by associating a spacetime region $\mathcal{E}(S) \subseteq \mathcal{T}$ for each system $S$ in the network) along with a fine-graining $\mathfrak{N}_{W,N,\mathcal{T},\mathcal{E}}^f$. For this theorem, the fine-graining and its properties will be irrelevant, we only need to focus on the spacetime embedded network $\mathfrak{N}_{W,N,\mathcal{T},\mathcal{E}}$ to prove the result. Observe that the relativistic causality condition of Definition 24 implies, in particular, that the signalling structure of $\mathfrak{N}_{W,N,\mathcal{T},\mathcal{E}}$ must be compatible with the region causal structure $\mathcal{G}_{\mathcal{T}}^{\text{reg}}$ induced by the embedding $\mathcal{E}$. Moreover, the signalling structure of $\mathfrak{N}_{W,N,\mathcal{T},\mathcal{E}}$ is, by construction, the same as the signalling structure of $\mathfrak{N}_{W,N}$, since the only difference in the two networks is in the relabelling of each system $S$ to the corresponding spacetime embedded system $S^{\mathcal{E}(S)}$. This implies that the signalling structure of $\mathfrak{N}_{W,N}$ must be compatible with the causal structure $\mathcal{G}_{\mathcal{T}}^{\text{reg}}$ if condition 2 of the theorem must hold. Now if we impose condition 1, that the associated process $W$ is not a fixed order process, it immediately follows from Theorem 3 that $\mathcal{G}_{\mathcal{T}}^{\text{reg}}$ must contain a directed cycle and therefore that it is not an acyclic graph, which would violate condition 3. This proves the theorem. □

**Corollary 2.** [*Time localisation in a global frame*] No realisation of an $N$-partite process network $\mathfrak{N}_{W,N}$ in a fixed spacetime structure $\mathcal{T}$ with respect to a spacetime embedding $\mathcal{E}$ can simultaneously satisfy the following three assumptions.

1. *W is not a fixed order process (cf. Definition 28).*

2. *The spacetime realisation satisfies relativistic causality (cf. Definition 24).*

3. *The region causal structure $\mathcal{G}_{\mathcal{T}}^{\text{reg}}$ induced by $\mathcal{E}$ is such that there exists an agent $A$ from whose perspective every region $\mathcal{R} \in \text{Nodes}(\mathcal{G}_{\mathcal{T}}^{\text{reg}})$ is time-localised (cf. Definition 27).*
**Proof.** This corollary follows from noting that the first two assumptions are identical to the first two assumptions of Theorem 4, while the third assumption here implies the third assumption of the theorem. More explicitly, assumption 3 here requires that for each system $S$, the corresponding spacetime region $\mathcal{R}^S \subseteq \mathcal{T}$ assigned to $S$ by the embedding $\mathcal{E}$ is such that all spacetime points in $\mathcal{R}^S$ have the same time coordinate, say $t^S$ in a global reference frame. Then, Definitions 18 and 19 tell us that $\mathcal{R}^{S_1} \xrightarrow{R} \mathcal{R}^{S_2}$ implies $t^{S_1} < t^{S_2}$. Since we can never have a sequence of times in some global reference frame such that $t^{S_1} < t^{S_2} < \dots < t^{S_n} < t^{S_1}$, it follows that the set of spacetime regions satisfying assumption 3 of this corollary can never contain a sequence of regions such that $\mathcal{R}^{S_1} \xrightarrow{R} \mathcal{R}^{S_2} \xrightarrow{R} \dots \xrightarrow{R} \mathcal{R}^{S_n} \xrightarrow{R} \mathcal{R}^{S_1}$, i.e., the regions satisfy assumption 3 of Theorem 4. $\square$

**Theorem 5.** Any spacetime realisation of a process network $\mathfrak{N}_{W,N}$ in a spacetime $\mathcal{T}$ relative to an embedding $\mathcal{E}$ that satisfies relativistic causality will admit an explanation in terms of a fine-grained network $\mathfrak{N}_{W^f,M}$ which will be described by a fixed order and therefore causally separable process $W^f$ associated with $M \ge N$ (possibly communicating) agents. Here $M$ is given in terms of the cardinality of the associated input spacetime regions $M = \sum_{k=1}^N |\mathcal{R}^{A_k^I}|$.

**Proof.** We prove the bipartite case here for simplicity, but the proof method readily generalises to the $N$-partite case. Let $W$ be a bipartite process matrix associated with the agents $A$ and $B$ and $\mathfrak{N}_{W,N}$ be the associated network with $N=2$. Consider a spacetime embedding $\mathfrak{N}_{W,N,\mathcal{T},\mathcal{E}}$ of $\mathfrak{N}_{W,N}$ in a fixed acyclic spacetime $\mathcal{T}$. If the region causal structure $\mathcal{G}_{\mathcal{T}}^{\text{reg}}$ induced by the embedding $\mathcal{E}$ is acyclic, then this spacetime embedding fully specifies a spacetime realisation of the network (Definition 22). In this case, if we impose the first condition of the theorem, namely that relativistic causality condition of Definition 24 is satisfied, then it follows that the signalling structure of $\mathfrak{N}_{W,N,\mathcal{T},\mathcal{E}}$ is compatible with the causal structure $\mathcal{G}_{\mathcal{T}}^{\text{reg}}$ (which is acyclic in the case under consideration). This signalling structure is the same as the signalling structure of $\mathfrak{N}_{W,N}$ (since the spacetime embedded network only differs from the original network in the fact that each system $S$ is additionally labelled by the corresponding spacetime region $\mathcal{E}(S)$), which implies that the signalling structure of $\mathfrak{N}_{W,N}$ is compatible with an acyclic causal structure, which is only possible if $W$ is a fixed order process (cf. Theorem 3).

Therefore we consider the case where $\mathcal{G}_{\mathcal{T}}^{\text{reg}}$ is cyclic. In this case, we require a fine-graining $\mathfrak{N}_{W,N,\mathcal{T},\mathcal{E}}^f$ of $\mathfrak{N}_{W,N}$, associated with an acyclic region causal structure $\mathcal{G}_{\mathcal{T}}^{\text{reg},f}$ in order to fully specify the spacetime realisation. Moreover, $\mathcal{G}_{\mathcal{T}}^{\text{reg},f}$ would be a fine-graining of the region causal structure $\mathcal{G}_{\mathcal{T}}^{\text{reg}}$ induced by the original embedding $\mathcal{E}$. Invoking the input-output correspondence between the regions in the embedding, we can denote by $\mathcal{R}^{A^I} \in \text{Nodes}(\mathcal{G}_{\mathcal{T}}^{\text{reg}})$ the region assigned to $A$'s input systems $A^s$ and $A^I$ by $\mathcal{E}$ and by $\mathcal{R}^{A^O} \in \text{Nodes}(\mathcal{G}_{\mathcal{T}}^{\text{reg}})$ the region assigned to $A$'s output systems $A^o$ and $A^O$, and similarly for $B$. Further, since $\mathcal{G}_{\mathcal{T}}^{\text{reg},f}$ is a fine-graining of $\mathcal{G}_{\mathcal{T}}^{\text{reg}}$, there is a fine-graining map that maps $\mathcal{R}^{A^I} \in \text{Nodes}(\mathcal{G}_{\mathcal{T}}^{\text{reg}})$ to a set of nodes $\{\mathcal{P}_1^I, \dots, \mathcal{P}_n^I\} \subseteq \text{Nodes}(\mathcal{G}_{\mathcal{T}}^{\text{reg},f})$ which partition it, $\mathcal{R}^{A^I} = \bigcup_{j=1}^n \mathcal{P}_j^I$. The input-output correspondence ensures that $\mathcal{R}^{A^O} \in \text{Nodes}(\mathcal{G}_{\mathcal{T}}^{\text{reg}})$ is mapped to a corresponding set of nodes $\{\mathcal{P}_1^O, \dots, \mathcal{P}_n^O\} \subseteq \text{Nodes}(\mathcal{G}_{\mathcal{T}}^{\text{reg},f})$ under the fine-graining, which partition it as $\mathcal{R}^{A^O} = \bigcup_{j=1}^n \mathcal{P}_j^O$ and where $\mathcal{P}_i^I \xrightarrow{R} \mathcal{P}_i^O$ for each $i \in \{1, \dots, n\}$. Similarly, we have the set of regions $\{\mathcal{Q}_1^I, \dots, \mathcal{Q}_m^I\} \subseteq \text{Nodes}(\mathcal{G}_{\mathcal{T}}^{\text{reg},f})$ associated with Bob's input $B^I$ under the fine-graining, and a corresponding set of regions $\{\mathcal{Q}_1^O, \dots, \mathcal{Q}_m^O\} \subseteq \text{Nodes}(\mathcal{G}_{\mathcal{T}}^{\text{reg},f})$ associated with $B^O$, where $\mathcal{Q}_j^I \xrightarrow{R} \mathcal{Q}_j^O$ for each $j \in \{1, \dots, m\}$, with $\mathcal{R}^{B^I} = \bigcup_{j=1}^m \mathcal{Q}_j^I$ and $\mathcal{R}^{B^O} = \bigcup_{j=1}^m \mathcal{Q}_j^O$.

Then in the fine-grained network $\mathfrak{N}_{W,N,\mathcal{T},\mathcal{E}}^f$, we have $n$ input and $n$ output systems associated with Alice (an input system for each $\mathcal{P}_i^I$ and an output system for each $\mathcal{P}_i^O$) and $m$ input and $m$ output systems associated with Bob (an input system for each $\mathcal{Q}_j^I$ and an output system for each $\mathcal{Q}_j^O$). Moreover, since $\mathfrak{N}_{W,N,\mathcal{T},\mathcal{E}}^f$ is a fine-graining of $\mathfrak{N}_{W,N}$, each sub-network of the former must be a fine-graining of a corresponding sub-network of the latter (Definition 17). As the process map $W$ itself corresponds to a sub-network of $\mathfrak{N}_{W,N}$, the fine-grained network $\mathfrak{N}_{W,N,\mathcal{T},\mathcal{E}}^f$ must be associated with a fine-grained process map $W_{\mathcal{T},\mathcal{E}}^f$ whose inputs are the $n$ output systems of Alice, along with the $m$ output systems of Bob, and whose outputs are the $n$ input systems of Alice and the $m$ input systems of Bob. We then split Alice into $n$ and Bob into $m$ agents, associated with the corresponding in/output systems in order to model $W_{\mathcal{T},\mathcal{E}}^f$ as an $n+m$-partite process map. In the following we show that whenever the spacetime realisation satisfies the relativistic causality condition, $W_{\mathcal{T},\mathcal{E}}^f$ is indeed a valid $n+m$-partite process map which corresponds to a fixed order process.

In Corollary 6 we have shown that spacetime realisations of any network that respect the relativistic causality condition correspond to so-called *causal boxes* which are CPTPMs that are closed under arbitrary compositions.
Imposing relativistic causality for the spacetime realisation of the original network $\mathfrak{N}_{W,N}$, this result implies that the induced map of each sub-network of $\mathfrak{N}_{W,N,\mathcal{T},\mathcal{E}}^f$ must be a causal box and in particular that $W_{\mathcal{T},\mathcal{E}}^f$ is a causal box. This implies that composing $W_{\mathcal{T},\mathcal{E}}^f$ together with any choice of extended local operations (which are themselves CPTP maps) between each input-output pair from the $n+m$ inputs associated with $\{\mathcal{P}_1^I, \dots, \mathcal{P}_n^I, \mathcal{Q}_1^I, \dots, \mathcal{Q}_m^I\}$ and $n+m$ outputs associated with $\{\mathcal{P}_1^{O}, \dots, \mathcal{P}_n^{O}, \mathcal{Q}_1^{O}, \dots, \mathcal{Q}_m^{O}\}$ can only lead to valid normalised probabilities. This composition is also a valid composition of causal boxes as the input regions are ordered before the corresponding output regions relative to the partial order induced by the acyclic region causal structure $G_{\mathcal{T}}^{\text{reg},f}$. From this argument, we conclude that $W_{\mathcal{T},\mathcal{E}}^f$ is a valid $n+m$-partite process map whenever the given spacetime realisation of $\mathfrak{N}_{W,N}$ satisfies the relativistic causality condition. Then it immediately follows from Theorem 3, just as in the first paragraph of this proof, that $W_{\mathcal{T},\mathcal{E}}^f$ must be a fixed-order process as its signalling structure is compatible with an acyclic causal structure $G_{\mathcal{T}}^{\text{reg},f}$ (which is implied by the relativistic causality condition).

Finally, we note that $\mathfrak{N}_{W,N,\mathcal{T},\mathcal{E}}^f$ being a fine-graining of $\mathfrak{N}_{W,N}$ also implies that the extended local maps $\mathcal{M}^A$ and $\mathcal{M}^B$ of Alice and Bob in the original network get fine-grained to extended local maps $\mathcal{M}^{A,f}$ and $\mathcal{M}^{B,f}$ acting on $n$ and $m$ in/output systems respectively. As we have not placed any constraints on the fine-grained local maps, these need not factorise as $\mathcal{M}^{A,f} = \bigotimes_{i=1}^n \mathcal{M}_i^{A,f}$ with each $\mathcal{M}_i^{A,f}$ acting between the in/output systems associated with $\mathcal{P}_i^I$ and $\mathcal{P}_i^O \stackrel{R}{\leftarrow} \mathcal{P}_i^I$. Thus the $n+m$ agents in the fine-grained process can perform non-product local operations by communicating to each other. The fact that the spacetime realisation satisfying relativistic causality can be described as a causal box guarantees that the process $W_{\mathcal{T},\mathcal{E}}^f$ will lead to valid normalised probabilities even when composed with such non-product local operations. This completes the proof.

As an additional remark, we note that above argument using causal boxes also guarantees that each fine-grained local operation will admit a decomposition into a sequential composition of CPTP maps (the sequence representation), whenever relativistic causality is imposed. This means that the local operations correspond to quantum combs (these are defined, for instance in [29]) and allows us to recover the product structure of the local operations through a simple trick originally proposed in [73]. One can always map a process and set of non-product local operations modelled as quantum combs (on which the action of the process leads to normalised probabilities) to a new process and a set of product local operations such that the composition is identical in the two cases, yielding the same correlations. This can be done by absorbing all communication channels in the non-product local operations into the definition of the new process. $\square$

**Lemma 4.** [No-go result for the quantum switch] Let $\mathfrak{N}_{W_{QS},U,V}$ be the network describing the action of $W_{QS}$ on any two non-trivial local operations $U^A: A^I \mapsto A^O$ and $V^B: B^I \mapsto B^O$, where $W_{QS}$ is the process map associated with the process matrix $W_{QS}$ of the quantum switch. Then any realisation of this network in a fixed spacetime $T$ with respect to an embedding $\mathcal{E}$ cannot simultaneously satisfy both of the following assumptions

1. The spacetime realisation satisfies relativistic causality (cf. Definition 24).

2. The region causal structure over the nodes $\mathcal{E}(S)$ for $S \in \{A^I, A^O, B^I, B^O\}$ is acyclic.

*Proof.* We first show that in $W_{QS}$, when the initial state of the control and target systems is $|\psi_C\rangle := \alpha|0\rangle + \beta|1\rangle$ and $|\psi_T\rangle$ respectively where $\alpha$ and $\beta$ are both non-zero amplitudes and $|\psi_T\rangle$ is an arbitrary qudit state, then we have $\{A^O\} \to \{B^I\}$ irrespective of the state input on $B^O$ and $\{B^O\} \to \{A^I\}$ irrespective of the state input on $A^O$, in $W$. This implies that both signalling relations can be realised in the network $\mathfrak{N}_{W_{QS},U,V}$ for any fixed choice of non-trivial operations $U^A$ and $V^B$ without needing to consider the extended local operations. This is in contrast to the general case of Theorem 4 where the signalling relation being realised may depend on the choice of local operations, such that not all signalling relations allowed by $W$ are realised when it is composed with a given fixed set of local operations.

Consider the action of the process map $W_{QS} : C_C^O \otimes C_T^O \otimes A^O \otimes B^O \mapsto D_C^I \otimes D_T^I \otimes A^I \otimes B^I$ on the input state $|\Psi\rangle_{C_C^O C_T^O A^O B^O} := (\alpha|0\rangle + \beta|1\rangle)_{C_C^O} \otimes |\psi_T\rangle_{C_T^O} \otimes |\psi_A\rangle_{A^O} \otimes |\psi_B\rangle_{B^O}$, where $|\psi_A\rangle$ and $|\psi_B\rangle$ are arbitrary qudit states. We see that $W_{QS}$ maps the initial state $|\Psi\rangle_{C_C^O C_T^O A^O B^O}$, to the final state

$$ \alpha|0\rangle^{D_C^I} |\psi_B\rangle^{D_T^I} |\psi_T\rangle^{A^I} |\psi_A\rangle^{B^I} + \beta|1\rangle^{D_C^I} |\psi_A\rangle^{D_T^I} |\psi_B\rangle^{A^I} |\psi_T\rangle^{B^I} := |\Phi\rangle^{D_C^I D_T^I A^I B^I}. $$

This gives,

$$ \begin{aligned} \operatorname{Tr}_{D_C^I D_T^I A^I} [W_{QS}(|\Psi\rangle\langle\Psi|^{C_C^O C_T^O A^O B^O})] &= |\alpha|^2 |\psi_A\rangle\langle\psi_A|^{B^I} + |\beta|^2 |\psi_T\rangle\langle\psi_T|^{B^I} \\ \operatorname{Tr}_{D_C^I D_T^I B^I} [W_{QS}(|\Psi\rangle\langle\Psi|^{C_C^O C_T^O A^O B^O})] &= |\alpha|^2 |\psi_T\rangle\langle\psi_T|^{A^I} + |\beta|^2 |\psi_B\rangle\langle\psi_B|^{A^I} \end{aligned} \quad (J3) $$
Notice that $\text{Tr}_{D_C^I D_T^I A^I} [\mathcal{W}_{QS}(|\Psi\rangle\langle\Psi|)]$ which is the output on $B^I$ does depend on $\psi_A$, the input on $A^O$ but not depend on $\psi_B$, the input on $B^O$. Similarly $\text{Tr}_{D_C^I D_T^I B^I} [\mathcal{W}_{QS}(|\Psi\rangle\langle\Psi|)]$ depends on $\psi_B$ but does not depend on $\psi_A$. This implies that given the knowledge of the initial control and target states, $\alpha|0\rangle + \beta|1\rangle$ and $|\psi_T\rangle$, Alice and Bob can signal to each other by suitable choices of $|\psi_A\rangle$ and $|\psi_B\rangle$ on their respective output systems $A^O$ and $B^O$, irrespective of the local operation of the other agent. The above proof easily generalises to arbitrary input states $\rho^{C_C^O C_T^O A^O B^O} := |\psi_C\rangle\langle\psi_C|^{C_C^O} \otimes |\psi_T\rangle\langle\psi_T|^{C_T^O} \otimes \rho^{A^O B^O}$, where the input state $\rho^{A^O B^O}$ on $A^O$ and $B^O$ may be an entangled state, $\text{Tr}_{D_C^I D_T^I A^I} [\mathcal{W}_{QS}(|\Psi\rangle\langle\Psi|)]$ depends only on the marginal of the initial state over $A^O$ which is unaffected by local operations on $B^O$.

This establishes that the network $\mathfrak{N}_{\mathcal{W}_{QS},U,V}$ gives rise to a directed cycle of signalling relations $\{A^O\} \to \{B^I\} \to \{B^O\} \to \{A^I\} \to \{A^O\}$ for any non-trivial $U^A$ and $V^B$ (where $\{A^I\} \to \{A^O\}$ and $\{B^I\} \to \{B^O\}$ come from the non-triviality of these operations). Then implies (by relativistic causality) that the spacetime regions must satisfy $\mathcal{P}^{A^O} \xrightarrow{R} \mathcal{P}^{B^I} \xrightarrow{R} \mathcal{P}^{B^O} \xrightarrow{R} \mathcal{P}^{A^I} \xrightarrow{R} \mathcal{P}^{A^O}$, which violates assumption 3. This completes the proof. $\square$

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