For all functions $f: \mathcal{Y} \to \mathcal{X}$, we have

$$ \mathrm{Tr} [E_x^f |y\rangle\langle y|] = \delta_{x,f(y)}. \qquad (133) $$

In addition, $\{E_x^f\}_{x \in \mathcal{X}}$ is a POVM, since $E_x^f$ are positive and it holds

$$ \sum_{x \in \mathcal{X}} E_x^f = \sum_{\substack{x \in \mathcal{X} \\ y \in \mathcal{Y}}} \delta_{x,f(y)} |y\rangle\langle y| = \sum_{\substack{y \in \mathcal{Y} \\ \exists x \in \mathcal{X}: f(y)=x}} |y\rangle\langle y| = \sum_{y \in \mathcal{Y}} |y\rangle\langle y| = 1. \qquad (134) $$

With these results in mind, we proceed to describe functional models within our causal modelling framework.

**Definition 36** (Causal model associated to a functional model). Consider a functional model $f\mathbf{Cm}_G$ (definition 32) on a directed graph $G = (V, E)$. Using the same notation of definition 32, we define a causal model $\mathbf{Cm}(f\mathbf{Cm})_G$ on the causal graph given by definition 34, through:

1. To each edge $e \in E = (v, v')$ associate the finite dimensional Hilbert space $\mathcal{H}_e = \mathcal{H}(\mathcal{X}_v)$, where $\mathcal{H}(\mathcal{X}_v)$ is defined as in equation (1);

2. To each error variable $U_v$ of $v \in V$ associate the finite dimensional Hilbert space $\mathcal{H}(U_v)$, where $\mathcal{H}(U_v)$ is defined as in equation (1), and the state

$$ \sigma^v = \sum_{u_v \in \mathcal{U}_v} p^v(u_v) |u_v\rangle\langle u_v|_{\mathcal{H}(\mathcal{U}_v)}; \qquad (135) $$

3. To each vertex $v \in V$ associate the finite set $\mathcal{X}_v$ as outcome set;

4. To each vertex $v \in V$ associate the set of CP maps

$$ \{\mathcal{M}_x^v : \mathcal{L}(\mathcal{H}_{\mathrm{In}(v)}) \to \mathcal{L}(\mathcal{H}_{\mathrm{Out}(v)})\}_{x \in \mathcal{X}_v} \qquad (136) $$

defined as follows: for all $\rho \in \mathcal{L}(\mathcal{H}_{\mathrm{In}(v)})$,

$$ \mathcal{M}_x^v(\rho) = \mathrm{Tr} \left[ E_x^{f^v}(\rho \otimes \sigma^v) \right] \bigotimes_{e \in \mathrm{Out}(v)} |x\rangle\langle x|_{\mathcal{H}_e}, \qquad (137) $$

where $\{E_x^{f^v}\}_{x \in \mathcal{X}_v}$ is the POVM obtained through applying definition 35 to the function $f^v: \mathcal{X}_{\mathrm{Pa}(v)} \times \mathcal{U}_v \to \mathcal{X}_v$ and $\sigma^v$ is defined in equation (135).

The following lemma proves that given an acyclic functional causal model $f\mathbf{Cm}_G$, the probability distribution evaluated using definition 33 equals the probability distribution of the image causal model (definition 36), $\mathbf{Cm}(f\mathbf{Cm})_G$ evaluated using definition 3.

**Lemma 37** (Equivalence of acyclic probabilities under definition 36). Consider a functional model on a directed and acyclic graph $G = (V, E)$, $f\mathbf{Cm}_G$, and its image under the mapping of definition 36, $\mathbf{Cm}(f\mathbf{Cm})_G$. Let $x := \{x_v \in \mathcal{X}_v\}_{v \in V}$ be a joint observed event, then we have:

$$ \mathrm{Pr}_{\mathrm{f,acyc}}(x)_G = \mathrm{Pr}_{\mathrm{acyc}}(x)_G, \qquad (138) $$

where $\mathrm{Pr}_{\mathrm{f,acyc}_G}$ is the acyclic probability distribution of $f\mathbf{Cm}_G$, evaluated with definition 33, and $\mathrm{Pr}_{\mathrm{acyc}_G}$ the acyclic probability distribution of $\mathbf{Cm}(f\mathbf{Cm})_G$, evaluated using definition 3.
Proof. For all $v \in V$, let us consider the action of $\mathcal{M}_{x_v}^v$ on states of the form

$$ |x_{\mathrm{Pa}(v)}\rangle\langle x_{\mathrm{Pa}(v)}| := \bigotimes_{v' \in \mathrm{Pa}(v)} |x_{v'}\rangle\langle x_{v'}|, \qquad (139) $$

where $x_{v'} \in \mathcal{X}_{v'}$ for all $v' \in \mathrm{Pa}(v)$. Using equation (133) and linearity of the trace, we have

$$ \mathcal{M}_{x_v}^v \left( |x_{\mathrm{Pa}(v)}\rangle\langle x_{\mathrm{Pa}(v)}| \right) = \sum_{u_v \in \mathcal{U}_v} p^v(u_v) \delta_{x_v, f^v(x_{\mathrm{Pa}(v)}, u_v)} \bigotimes_{e \in \mathrm{Out}(v)} |x_v\rangle\langle x_v|_{\mathcal{H}_e}. \qquad (140) $$

The maps associated to vertices are composed according to the graph connectivity (see definition 3 for more details) taking as input $1 \in \mathbb{C}$. In addition, the output of any CP map $\mathcal{M}_x^v$ is always decohered in the basis specified by the values of $x_v$ (this is always the case for maps associated to observed vertices). Thus, the input of all maps is always of the form considered in equation (140), up to a constant factor. Since the graph is finite and acyclic, the output of the maps composition will eventually be in $\mathbb{C}$. By linearity and because of trace-preservation, such composition will equal the product of the proportionality factors mentioned above, i.e., $\sum_{u_v} p^v(u_v) \delta_{x_v, f^v(x_{\mathrm{Pa}(v)}, u_v)}$ for each $v \in V$.

Thus, applying definition 3 gives

$$ \begin{aligned} \mathrm{Pr}(x)_G &= \bigcirc_{v \in V} \mathcal{M}_{x_v}^v = \prod_{v \in V} \sum_{u_v \in \mathcal{U}_v} p^v(u_v) \delta_{x_v, f^v(x_{\mathrm{Pa}(v)}, u_v)} \\ &= \sum_{u_v} \prod_{v \in V} p^v(u_v) \delta_{x_v, f^v(x_{\mathrm{Pa}(v)}, u_v)} = \mathrm{Pr}_{\mathrm{f,acyc}}(x)_G, \end{aligned} \qquad (141) $$

where the sum $\sum_{u_v}$ runs over $u_v = \{u_v \in \mathcal{U}_v\}_{v \in V}$ and $x = \{x_v\}_{v \in V}$. Thus proving the equivalence of this definition and definition 33. $\square$

**Remark.** We described functional models within our framework as causal models on a causal graph having only observed vertices. In the functional modelling literature, if a vertex is unobserved the probability distribution over the remaining ones is obtained through marginalising the joint distribution. Within our framework, unobserved variables can be naturally represented as unobserved vertices of the causal graph. Consider a functional model as in definition 32, and assume that a subset of vertices are unobserved, $V_u \subseteq V$. Then, we associate to this graph a causal graph where all edges are classical, $E = E_{\mathrm{cl}}$, the unobserved vertices are $V_u$ and the observed are $V_o = V \setminus V_u$. We define a causal model on this causal graph by preserving the same associations of definition 36 to edges, observed vertices $v \in V_o$ and all error variables. However, to each unobserved vertex $v \in V_u$, we associate the CPTP map such that, for all $\rho \in \mathcal{L}(\mathcal{H}_{\mathrm{In}(v)})$,

$$ \mathcal{E}^v(\rho) = \sum_{x \in \mathcal{X}_v} \mathcal{M}_x^v(\rho) = \sum_{x \in \mathcal{X}_v} \mathrm{Tr} \left[ E_x^{f^v}(\rho \otimes \sigma^v) \right] \bigotimes_{e \in \mathrm{Out}(v)} |x\rangle\langle x|_{\mathcal{H}_e}, \qquad (142) $$

where $\rho_x^v$ and $\mathcal{M}_x^v$ are as in definition 36.

The probability distribution of this causal model, evaluated through definition 3, gives for all $x = \{x_v \in \mathcal{X}_v\}_{v \in V}$

$$ \begin{aligned} \mathrm{Pr}_{\mathrm{acyc}}(\{x_v\}_{v \in V_o})_G &= \bigcirc_{v \in V_o} \mathcal{M}_{x_v}^v \bigcirc_{v' \in V_u} \mathcal{E}^{v'} = \sum_{\{x_v\}_{v \in V_u}} \bigcirc_{v \in V} \mathcal{M}_{x_v}^v \\ &= \sum_{\{x_v\}_{v \in V_u}} \mathrm{Pr}_{\mathrm{f,acyc}}(x)_G = \mathrm{Pr}_{\mathrm{f,acyc}}(\{x_v\}_{v \in V_o})_G. \end{aligned} \qquad (143) $$

where the sum runs over $\{x_v \in \mathcal{X}_v\}_{v \in V_u}$. Therefore, the distribution of the causal model which accounts for unobserved vertices in the causal graph is equal to the distribution obtained through marginalising definition 33 over variables associated to unobserved vertices.
### C.3 Equivalence of *p*-separation definitions

In a companion paper [FGV25], we provide a causal modelling framework for cyclic classical causal models. This follows the same ideas of this work in mapping a given cyclic functional model to a family of acyclic models with post-selections. However, the mapping in section 3 of [FGV25] is constructed on a slightly different family of acyclic graphs which simplifies the mapping in the classical case without relying on the quantum notation and formalism used in this paper<fnref n="20" />. The graph family of definition 6 plays a crucial role in defining *p*-separation (definition 21), while in [FGV25], *p*-separation defined with respect to this slightly modified family. Here, we show that the two definitions of *p*-separation, which only depend on the graph, are equivalent. Let us first comment on the two different graph families.

**One graph, two acyclic families.** Given a (causal) graph $G$, we constructed a family of acyclic causal graph as in definition 6. This is denoted as $\mathbb{G}_{\text{TP}}(G)$, and defined above. For completeness, let us repeat the definition of $\mathbb{G}_{\text{TP}}(G)$.

**Definition 6 (Family of acyclic causal graphs** $\mathbb{G}_{\text{TP}}(G)$).** Given a causal graph $G = (V, E)$, we define an associated family $\mathbb{G}_{\text{TP}}(G)$ of directed acyclic causal graphs, where each element $G_{\text{TP}} \in \mathbb{G}_{\text{TP}}(G)$ is obtained from the causal graph $G$ as follows.

1. Choose any subgraph $G' := (V', E')$ of $G = (V, E)$ with $V' = V$ and $E' \subseteq E$, such that $G'$ is acyclic.

2. Include in $G_{\text{TP}}$ all the vertices and edges of the subgraph $G'$ associated with the same vertex types (observed or unobserved) and edge types (classical or quantum) as the original causal graph $G$.

3. Denoting the set of so-called split edges $E_S(G_{\text{TP}}) := E \setminus E'$, for each edge $(v_i, v'_i) \in E_S(G_{\text{TP}})$, include in $G_{\text{TP}}$, two vertices $T_i$ and $R_i$ and three edges $(v_i, T_i)$, $(R_i, T_i)$ and $(R_i, v'_i)$.

4. The vertex $T_i$ is observed and $R_i$ unobserved. Outgoing edges from $R_i$, $(R_i, T_i)$ and $(R_i, v'_i)$, are quantum edges. The edge $(v_i, T_i)$ is of the same type of the edge $(v_i, v'_i)$ in the original causal graph $G$.

$G_{\text{TP}}$ constructed in this manner is thus a causal graph. It will be useful to refer to $T_i$ and $R_i$ as pre and post-selection vertices respectively and depict them with distinct vertex styles $T_i$ and $R_i$, as these will play a special role in our framework. This makes $G_{\text{TP}}$ identical to $G$ up to replacing each split edge $(v_i, v'_i) \in E_S(G_{\text{TP}}) \subseteq E$ with the following structure:

<figure><img src="image_21.png" /></figure>

We will refer to every $G_{\text{TP}} \in \mathbb{G}_{\text{TP}}(G)$ as a teleportation graph, as we will later associate teleportation protocols to such graphs. It will be useful to denote the set of all post-selection vertices and the set of all pre-selection vertices in $G_{\text{TP}}$ as $V_{\text{post}} := \{T_i\}_{(v_i, v'_i) \in E_S(G_{\text{TP}})}$ and $V_{\text{pre}} := \{R_i\}_{(v_i, v'_i) \in E_S(G_{\text{TP}})}$.

<footnote n="20">²⁰The simplification is possible as classical variables can be copied and broadcast to multiple parties, i.e., the children vertices of the variable.</footnote>
From the same underlying graph G, in definition 7 of [FGV25] we introduce a different family of acyclic graphs. We provide the definition of this family, which is denoted as $G_{c,TP}(G)$, here.

**Definition 38** (Family of acyclic classical teleportation graphs $G_{c,TP}(G)$). Given a directed graph $G = (V, E)$, we define an associated family $G_{c,TP}(G)$ of acyclic graphs, where each element $G_{c,TP} \in G_{c,TP}(G)$ is obtained from the graph $G$ as follows.

1. Choose any subset of vertices $V_S(G_{c,TP}) \subseteq V$, such that the subgraph $G' = (V', E')$ of $G$ with $V' = V$ and $E' = E \setminus (\bigcup_{v \in V_S(G_{c,TP})} \text{Out}(v))$ is acyclic. We refer to $V_S(G_{c,TP})$ as the split vertices of $G_{c,TP}$.

2. The vertices $V_{c,TP}$ of $G_{c,TP}$ consist of the vertices $V$ of the original graph $G$ together with new vertices $R_v, T_v$ for each split vertex $v \in V_S(G_{c,TP})$, i.e.,

$$V_{c,TP} = V \cup \{R_v\}_{v \in V_S(G_{c,TP})} \cup \{T_v\}_{v \in V_S(G_{c,TP})}. \quad (144)$$

3. The edges $E_{c,TP}$ of $G_{c,TP}$ consist of the edges $E'$ of the subgraph $G'$ together with the following new edges:

$$E_{c,TP} = E' \cup \{(v, T_v)\}_{v \in V_S(G_{c,TP})} \cup \{(R_v, T_v)\}_{v \in V_S(G_{c,TP})} \\ \cup \{(R_v, v')\}_{v \in V_S(G_{c,TP}), v' \in \text{Ch}(v)_G}, \quad (145)$$

where $\text{Ch}(v)_G$ refers to the children vertices of $v$ in the graph $G$.

For each $v \in V_S(G_{c,TP})$, we refer to the vertices $R_v$ and $T_v$ as pre- and post-selection vertices respectively and depict them with distinct vertex styles $T_v$ and $R_v$, as these will play a special role in our framework. This construction makes $G_{c,TP}$ identical to $G$ up to replacing all vertices $v \in V_S(G_{c,TP})$ with the following structure

<figure><img src="image_22.png" /></figure>

Each element of the family $G_{c,TP} \in G_{c,TP}(G)$ is called a classical teleportation graph. The set of all pre- and post-selection vertices in $G_{c,TP}$ are denoted as $V_{\text{pre}} = \{R_v\}_{v \in V_S(G_{c,TP})}$ and $V_{\text{post}} = \{T_v\}_{v \in V_S(G_{c,TP})}$.

Definition 6 is formulated in terms of causal graphs while definition 38 is not. However, we can equivalently consider a causal graph where all vertices are observed in definition 38 (consistently with the notation used in equation (146)) and compare the two definitions.

With this in mind, the family of acyclic graphs of definition 38 is constructed in a similar way to the family of acyclic teleportation graphs in definition 6. Specifically, both definitions involve considering an acyclic subgraph of $G$. The subgraphs that are used to construct definition 38 are a subset of those considered for definition 6. Indeed, in the latter, one can remove any subset of edges from $G$ to make graph acyclic, while in definition 38 one removes all outgoing edges from a subset of vertices $V_S(G_{c,TP})$. In addition, in the
classical case all outgoing edges are jointly substituted with one pair of pre- and post-selection vertices, while, in the teleportation graph family of definition 6, each split edge is associated to a pair of pre- and post-selection vertices.

To understand the difference between these two families, consider the cyclic graph, $G_{eg}$, and its subgraphs, $G'_{eg}$ and $G''_{eg}$.

$$ G_{eg} = \text{A} \xrightarrow{B} \text{C}, \quad G'_{eg} = \text{A} \xrightarrow{B} \text{C}, \quad G''_{eg} = \text{A} \xrightarrow{B} \text{C} \qquad (147) $$

The acyclic subgraph $G'_{eg}$ is a valid acyclic subgraph for both definitions 6 and 38, respectively with set of split edges $E_s(G'_{eg}) = \{(A, B), (A, C)\}$ and split vertices $V_s(G_{eg}) = \{A\}$. While, the acyclic subgraph $G''_{eg}$, which has set of split edges $E_s(G''_{eg}) = \{(A, B), (C, A)\}$, is not valid for definition 38.

The member of the families in definitions 6 and 38 constructed from $G'_{eg}$, are, respectively,

$$ G_{TP} = \text{A} \xrightarrow{B} \text{R}_1 \xrightarrow{T_1} \text{C}, \quad \text{and} \quad G_{c,TP} = \text{A} \xrightarrow{T} \text{R} \qquad (148) $$

which differ in the amount of pre- and post-selection vertices. Such difference reflects the fact that each vertex of a functional model broadcasts its value to all its children, thus allowing for simplification compared to the quantum definition.

**Equivalence of p-separation definitions.** Let us now show that p-separation can be equivalently defined using either of these graph families. This result is necessary to establish consistency of the present work with the results of [FGV25]. First we recall the two definitions of p-separation by repeating the definition given in section 5 and in [FGV25].

**Definition 21 (p-separation).** Let G be a directed graph and $V_1, V_2$ and $V_3$ denote any three disjoint subsets of the vertices of G with $V_1$ and $V_2$ being non-empty. Then, we say that $V_1$ is p-separated from $V_2$ given $V_3$ in G, denoted $(V_1 \perp^p V_2|V_3)_G$, if and only if there exists $G_{TP} \in \mathbb{G}_{TP}(G)$ (definition 6) such that $(V_1 \perp^d V_2|V_3 \cup V_{\text{post}})_{G_{TP}}$, where $\perp^d$ denotes d-separation and $V_{\text{post}}$ denotes the set of all post-selection vertices in the teleportation graph $G_{TP} \in \mathbb{G}_{TP}(G)$. Otherwise, we say that $V_1$ is p-connected to $V_2$ given $V_3$ in G, and we denote it $(V_1 \not\perp^p V_2|V_3)_G$. To summarize,

$$ \begin{align} \textit{p-separation:} \quad (V_1 \perp^p V_2|V_3)_G &\iff \exists G_{TP} \in \mathbb{G}_{TP}(G) : (V_1 \perp^d V_2|V_3 \cup V_{\text{post}})_{G_{TP}}, \tag{75} \\ \textit{p-connection:} \quad (V_1 \not\perp^p V_2|V_3)_G &\iff \forall G_{TP} \in \mathbb{G}_{TP}(G) : (V_1 \not\perp^d V_2|V_3 \cup V_{\text{post}})_{G_{TP}}. \nonumber \end{align} $$
In section 4 of [FGV25], p-separation was defined with respect to the graph family definition 38.<fnref n="21" />

**Definition 39 (p-separation in [FGV25]).** Let $G$ be a directed graph and $V_1, V_2$ and $V_3$ denote any three disjoint subsets of the vertices of $G$ with $V_1$ and $V_2$ being non-empty. Then, we say that $V_1$ is p-separated from $V_2$ given $V_3$ in $G$, denoted $(V_1 \perp^{p,c} V_2|V_3)_G$, if and only if there exists $G_{c,TP} \in \mathbb{G}_{c,TP}(G)$ (definition 38) such that $(V_1 \perp^d V_2|V_3 \cup V_{post})_{G_{c,TP}}$, where $\perp^d$ denotes d-separation and $V_{post}$ denotes the set of all post-selection vertices in the teleportation graph $G_{c,TP} \in \mathbb{G}_{c,TP}(G)$. Otherwise, we say that $V_1$ is p-connected to $V_2$ given $V_3$ in $G$, and we denote it $(V_1 \not\perp^{p,c} V_2|V_3)_G$. To summarize,

$$ \begin{align} \textit{p-separation:} \quad (V_1 \perp^{p,c} V_2|V_3)_G &\iff \exists G_{c,TP} \in \mathbb{G}_{c,TP}(G) : (V_1 \perp^d V_2|V_3 \cup V_{post})_{G_{c,TP}}, \nonumber \\ \textit{p-connection:} \quad (V_1 \not\perp^{p,c} V_2|V_3)_G &\iff \forall G_{c,TP} \in \mathbb{G}_{c,TP}(G) : (V_1 \not\perp^d V_2|V_3 \cup V_{post})_{G_{c,TP}}. \tag{149} \end{align} $$

The following lemma shows equivalence of the two definitions of p-separation.

**Lemma 40 (Equivalence of p-separation in the two graph families).** Let us consider a directed graph $G$ and three disjoint subsets of vertices $V_1, V_2, V_3 \subseteq V$ with $V_1, V_2$ being not empty. Then, it holds

$$ (V_1 \perp^{p,c} V_2|V_3)_G \iff (V_1 \perp^p V_2|V_3)_G, \tag{150} $$

where $(V_1 \perp^{p,c} V_2|V_3)_G$ denotes p-separation with respect to the graph family $\mathbb{G}_{c,TP}(G)$ (definition 39) and $(V_1 \perp^p V_2|V_3)_G$ denotes p-separation with respect to the graph family $\mathbb{G}_{TP}(G)$ (definition 21).

*Proof.* $(\perp^{p,c} \implies \perp^p)$: By definition 39, $(V_1 \perp^{p,c} V_2|V_3)_G$ implies that $\exists G_{c,TP} \in \mathbb{G}_{c,TP}(G)$ (definition 38) such that $(V_1 \perp^d V_2|V_3 \cup V_{post})_{G_{c,TP}}$, where $V_{post}$ is the set of all post-selection vertices in $G_{c,TP}$. We now define a teleportation graph $G_{TP} \in \mathbb{G}_{TP}(G)$ (definition 6) by choosing $E_S(G_{TP}) := \text{Out}(V_S(G_{c,TP}))$. Then the graphs $G_{TP}$ and $G_{c,TP}$ are equivalent up to the following replacement (see equation 20 in [FGV25])

<figure><img src="image_23.png" /></figure>

The left hand side represents the structure of $G_{TP}$ and the right hand side, $G_{c,TP}$, and where the red vertices correspond to the post-selection vertices in both cases. Then it is easy to see that $(V_1 \perp^d V_2|V_3 \cup V_{post})_{G_{c,TP}}$ implies $(V_1 \perp^d V_2|V_3 \cup V'_{post})_{G_{c,TP}}$, where $V'_{post}$ is the set of all post-selection vertices in $G_{TP}$. By definition 21 of p-separation, $(V_1 \perp^d V_2|V_3 \cup V'_{post})_{G_{TP}}$ implies $(V_1 \perp^p V_2|V_3)_G$ as required.

<footnote n="21">²¹In what follows we modify the notation of p-separation with respect to $\mathbb{G}_{c,TP}(G)$ compared to [FGV25], which is denoted there as $(V_1 \perp^p V_2|V_3)_G$. Here, $(V_1 \perp^p V_2|V_3)_G$ denotes p-separation with respect to $\mathbb{G}_{TP}(G)$, while $(V_1 \perp^{p,c} V_2|V_3)_G$ is with respect to $\mathbb{G}_{c,TP}(G)$. This distinction is necessary to show equivalence of the two definitions. Once this is established the notation $(V_1 \perp^p V_2|V_3)_G$ will denote either definition.</footnote>
($\perp^p \implies \perp^{p,c}$): If $(V_1 \perp^p V_2 | V_3)_G$, then there exists $G_{\text{TP}} \in \mathbb{G}_{\text{TP}}(G)$ such that $(V_1 \perp^d V_2 | V_3 \cup V_{\text{post}})_{G_{\text{TP}}}$ where $V_{\text{post}}$ denotes the set of all post-selection vertices in $G_{\text{TP}}$ (definition 21). Let $E_S(G_{\text{TP}})$ be the set of split edges of $G_{\text{TP}}$ and define

$$V_S(G_{c,\text{TP}}) := \{v \in V \text{ s.t. } \exists e = (v, v') \in E_S(G_{\text{TP}})\}. \qquad (152)$$

Consider the graph $G'_{\text{TP}} \in \mathbb{G}_{\text{TP}}(G)$ such that $E_S(G'_{\text{TP}}) = \bigcup_{v \in V_S(G_{c,\text{TP}})} \text{Out}(v)$. We want to prove that $(V_1 \perp^d V_2 | V_3 \cup V'_{\text{post}})_{G'_{\text{TP}}}$, where $V'_{\text{post}}$ denotes the set of post-selection vertices in $G'_{\text{TP}}$.

Let us consider $v_1 \in V_1$ and $v_2 \in V_2$, if these are $d$-separated conditioned on $v_3$, $V_{\text{post}}$ in $G_{\text{TP}}$, then one of the following conditions is satisfied:

* There is no path (even undirected) between $v_1$ and $v_2$ in $G_{\text{TP}}$. $G'_{\text{TP}}$ can be obtained through substituting some edges of $G_{\text{TP}}$ with a pre- and post-selection edges and vertices. Such substitution does not create new paths, hence there is also no path between $v_1$ and $v_2$ in $G'_{\text{TP}}$.

* All paths between $v_1$ and $v_2$ are blocked by $V_3 \cup V_{\text{post}}$. Hence, for each path in $G_{\text{TP}}$, one of the following holds:

* There exists $W \in V_3 \cup V_{\text{post}}$ such that $A \leftarrow W \rightarrow B$ for some $A$ and $B$ in the path. Adding pre and post-selection edges and vertices does not change the path, hence, the difference between $G_{\text{TP}}$ and $G'_{\text{TP}}$ is that some edges of the path might be substituted with such vertices and edges. If this substitution does not involve the subgraph $A \leftarrow W \rightarrow B$, then the same $W$ blocks the path in $G'_{\text{TP}}$. If the edge $A \leftarrow W$ is substituted, we have $A \leftarrow Q \rightarrow P \leftarrow W \rightarrow B$. Since, the path between $v_1$ and $v_2$ contains $P \leftarrow W \rightarrow B$ and $W \in V_3 \cup V'_{\text{post}}$, then the path is blocked in $G'_{\text{TP}}$. The same considerations hold if the edge $W \rightarrow B$ is substituted.

* There exists $W \in V_3 \cup V_{\text{post}}$ such that $A \rightarrow W \rightarrow B$ for some $A$ and $B$ in the path. Adding pre and post-selection edges and vertices does not change the path, hence, the difference between $G_{\text{TP}}$ and $G'_{\text{TP}}$ is that some edges of the path might be substituted with a teleportation subgraph. If this substitution does not involve the subgraph $A \rightarrow W \rightarrow B$, then the same vertex $W$ blocks the path also in $G'_{\text{TP}}$. If the edge $A \rightarrow W$ is substituted, we have $A \rightarrow P \leftarrow Q \rightarrow W \rightarrow B$. Since the path between $v_1$ and $v_2$ contains $Q \rightarrow W \rightarrow B$ and $W \in V_3 \cup V'_{\text{post}}$, then the path is blocked in $G'_{\text{TP}}$. The same considerations hold if the edge $W \rightarrow B$ is substituted.

* There exists $U$ such that $A \rightarrow U \leftarrow B$ for some $A$ and $B$ in the path and $U$ nor any descendent of $U$ are in $V_3 \cup V_{\text{post}}$. Assume first that the substitution of edges with pre and post-selection vertices and edges does not involve the subgraph $A \rightarrow U \leftarrow B$, but that there is a descendent of $U$ in the conditioning set, i.e., $W \in V_3 \cup V'_{\text{post}}$ such that

$$A \rightarrow U \leftarrow B \\ \downarrow \\ \dots \\ \downarrow \\ W \qquad (153)$$

Since, $U$ had no descendents in the set $W \in V_3 \cup V_{\text{post}}$, $W$ is a post-selection vertex that was not in $G_{\text{TP}}$, i.e., $W \in V'_{\text{post}} \setminus V_{\text{post}}$. However, by construction
of $E_S(G'_{\text{TP}})$, this means that $v \in \text{In}(W)$ is also a split vertex, $v \in V_S(G_{c,\text{TP}})$. Therefore, there exists $e = (v, v') \in E_S(G_{\text{TP}})$. In $G'_{\text{TP}}$ we have

$$ \begin{array}{ccc} A \to U \leftarrow B & & A \to U \leftarrow B \\ \downarrow & & \downarrow \\ \vdots & & \vdots \\ \downarrow & & \downarrow \\ v \to W' & \text{and in } G_{\text{TP}} & v \to W' \\ \downarrow & & \downarrow \\ W & & \vdots \end{array} \qquad (154) $$

Hence, $W' \in V_3 \cup V_{\text{post}}$ and it is a descendent of $U$. This is not possible by hypothesis, meaning that even in $G'_{\text{TP}}$, $U$ has no descendants in the conditioning set. If the edge $A \to U$ is substituted, we have $A \to P \leftarrow Q \to U \leftarrow B$. The path between $v_1$ and $v_2$ contains $Q \to U \leftarrow B$, hence the path is blocked in $G'_{\text{TP}}$. The same considerations hold if the edge $U \leftarrow B$ is substituted.

This proves that if $(V_1 \perp^d V_2 | V_3 \cup V_{\text{post}})_{G'_{\text{TP}}}$, then $(V_1 \perp^d V_2 | V_3 \cup V'_{\text{post}})_{G'_{\text{TP}}}$. Using equation (151) to relate $G'_{\text{TP}}$ and $G_{c,\text{TP}}$, it holds that $(V_1 \perp^d V_2 | V_3 \cup V'_{\text{post}})_{G'_{\text{TP}}}$ implies $(V_1 \perp^d V_2 | V_3 \cup V_{\text{post}}^c)_{G_{c,\text{TP}}}$. $\square$

Thus, p-separation relations of a given graph $G$ can be equivalently deduced using definition 21 or definition 39.

### C.4 Equivalence of cyclic probabilities under the mapping in definition 36

In appendix C.2, we showed that applying definition 36 to map an acyclic functional model to an acyclic causal model preserves the probability distribution over observed events. In this section, we consider probabilities over cyclic functional models and show that the probability rule given in definition 12 of [FGV25] is equivalent to applying the probability rule of definition 12 to the image causal models through definition 36.

**Lemma 41.** Consider a functional model on a directed graph $G = (V, E)$, $f\text{Cm}_G$, and its image under the mapping of definition 36, $\text{Cm}(f\text{Cm})_G$. Let $x = \{x_v \in \mathcal{X}_v\}_{v \in V}$ be a joint observed event, then we have:

$$ \mathrm{Pr}_f(x)_G = \mathrm{Pr}(x)_G \qquad (155) $$

where $\mathrm{Pr}_f$ is the probability distribution of $f\text{Cm}_G$ evaluated with definition 12 of [FGV25] and $\mathrm{Pr}_G$ is the probability distribution given by definition 12 of $\text{Cm}(f\text{Cm})_G$.

*Proof.* In the proof the following two results are used.

**Lemma 42.** Consider a functional model on a directed graph $G = (V, E)$ with functional dependencies $f^v : \mathcal{X}_{\text{Pa}(v)} \times \mathcal{U}_v \mapsto \mathcal{X}_v$, associated to each vertex $v \in V$. The maps $\mathcal{M}_x^{f^v}$ obtained through definition 36, satisfy

$$ \mathcal{D}_{E_{\text{out}}} \circ \mathcal{M}_x^{f^v} \circ \mathcal{D}_{E_{\text{in}}} = \mathcal{M}_x^{f^v} \qquad (156) $$

for any subset of outgoing and ingoing edges to $v$, $E_{\text{out}} \subseteq \text{Out}(v)$ and $E_{\text{in}} \subseteq \text{In}(v)$. The decohering channel $\mathcal{D}_{E'} : \mathcal{L}(\mathcal{H}_{E'}) \mapsto \mathcal{L}(\mathcal{H}_{E'})$ is defined for any subset of edges $E' \subseteq E$ as $\mathcal{D}_{E'} = \bigotimes_{e \in E} \mathcal{D}_e$ acting as $\mathcal{D}_e(\rho) = \sum_{x \in \mathcal{X}_e} |x\rangle\langle x|\rho|x\rangle\langle x|$ for any $\rho \in \mathcal{L}(\mathcal{H}_e)$.

*Proof.* Let us prove that

$$ \mathcal{D}_e \circ \mathcal{M}_x^{f^v} = \mathcal{M}_x^{f^v} = \mathcal{M}_x^{f^v} \circ \mathcal{D}_{e'} \qquad (157) $$
for any $e \in \text{Out}(v)$ and $e' \in \text{In}(v)$. Once this is proven the general results follows from iterative application of this equality. The first equality follows immediately from the form of $\mathcal{M}_x^v$, since for all $\rho \in \mathcal{L}(\mathcal{H}_{\text{In}(v)})$, we have

$$ \mathcal{M}_x^v(\rho) = \text{Tr} \left[ E_x^{f^v}(\rho \otimes \sigma^v) \right] \bigotimes_{e \in \text{Out}(v)} |x\rangle\langle x|_{\mathcal{H}_e}, \qquad (158) $$

which is already coherence-free. The second follows from the form of $E_x^{f^v}$ and of the state $\sigma^v$ (see equation (135)), indeed

$$ \begin{aligned} \text{Tr} \left( E_x^{f^v} (\mathcal{D}_{e'}(\rho) \otimes \sigma^v) \right) &= \text{Tr} \left( E_x^{f^v} \mathcal{D}_{e' \cup U_v} (\rho \otimes \sigma^v) \right) \\ &= \text{Tr} \left( \sum_y \delta_{x, f^v(y)} |y\rangle\langle y|_{\mathcal{D}_{\tilde{e}}(\tilde{\rho})} \right) \\ &= \sum_y \sum_z \delta_{x, f^v(y)} \text{Tr} (|y\rangle\langle y|_z \langle z|_{\tilde{\rho}}|z\rangle\langle z|_z) \\ &= \sum_y \delta_{x, f^v(y)} \text{Tr} (|y\rangle\langle y|_z (\rho \otimes \sigma^v)) = \text{Tr} \left( E_x^{f^v} (\rho \otimes \sigma^v) \right). \end{aligned} \qquad (159) $$

where we denoted $\tilde{e} = e' \cup U_v$ and $\tilde{\rho} = \rho \otimes \sigma^v$ for brevity, and the sums run over $y = \{y' \in \mathcal{X}_{v'} : v' \in \text{Pa}(v) \cup \{u_v \in \mathcal{U}_v\}$ and $z = \{z_{v'} \in \mathcal{X}_{v'} : v' \in \text{Pa}(v) \cup \{u'_v \in \mathcal{U}_v\}\}$. $\square$

**Lemma 43.** Consider a Hilbert space $\mathcal{H}_{\text{In}(v)} = \bigotimes_{e \in \text{In}(v)} \mathcal{H}_e^{22}$. For any $E_x, \rho \in \mathcal{L}(\mathcal{H}_{\text{In}(v)})$ and $E_{\text{in}} \subseteq \text{In}(v)$, it holds

$$ \text{Tr} [E_x(\mathcal{D}_{E_{\text{in}}}(\rho))] = \text{Tr} [(\mathcal{D}_{E_{\text{in}}}(E_x))\rho]. \qquad (160) $$

*Proof.* The proof follows from the cyclicity of the trace

$$ \text{Tr} [E_x(\mathcal{D}_{E_{\text{in}}}(\rho))] = \sum_y \text{Tr} [E_x(|y\rangle\langle y|_{E_{\text{in}}} \otimes 1)\rho(|y\rangle\langle y|_{E_{\text{in}}} \otimes 1)] \qquad (161) $$

$$ = \sum_y \text{Tr} [(|y\rangle\langle y|_{E_{\text{in}}} \otimes 1) E_x(|y\rangle\langle y|_{E_{\text{in}}} \otimes 1)\rho] = \text{Tr} [(\mathcal{D}_{E_{\text{in}}}(E_x))\rho]. \qquad (162) $$

$\square$

Given these two results, we can proceed with the proof.

**Overview of the proof and relevant notation.** The goal of this proof is to connect the probabilities of a functional causal model on a possibly cyclic graph $G$, namely $f\text{Cm}_G$, and those of its image causal model through the embedding definition 36, namely $\text{Cm}(f\text{Cm}_G)$ (which, for brevity, we will simply denote as $\text{Cm}_G$ in what follows). To achieve this, we define auxiliary functional models (definition 32) and causal models (definition 2). Although the idea behind the proof is straightforward, given our formalism, it involves relating a number of different causal graphs and causal models. We therefore first sketch the main steps and notation of the proof, before detailing it.

**Step 0:** We start considering a functional causal model on $G$: $f\text{Cm}_G$ (definition 32).

The cyclic probability distribution for the observed event $x$ relative to this functional model is $\text{Pr}_f(x)_G$ (definition 12 in [FGV25]).

<footnote>22The lemma holds for any Hilbert space with such partition. We call the space $\mathcal{H}_{\text{In}(v)}$ since the lemma will be applied to subsystems in a causal model.</footnote>
**Step 1:** We define a classical teleportation causal model on a teleportation graph $G_{c,TP}$ induced by $f\mathbf{Cm}_G: f\mathbf{Cm}_{G_{c,TP}}$ (definition 9 of [FGV25]).

Let $V_{\text{post}}$ be the set of post-selection vertices in $G_{c,TP}$, which is acyclic by construction, and $V_s(G_{c,TP})$ the set of split vertices of $G_{c,TP}$. The probability distribution of the event $x$ and successful post-selections $\{t_v = 1\}_{T_v \in V_{\text{post}}}$ relative to this acyclic functional model is $\text{Pr}_{f,acyc}(x, \{t_v = 1\}_{T_v \in V_{\text{post}}} )_{G_{c,TP}}$ (definition 33).

By definition, the probability distribution of the cyclic model $f\mathbf{Cm}_G$ equals the conditional distribution of the acyclic model $f\mathbf{Cm}_{G_{c,TP}}$, i.e.,

$$ \text{Pr}_{f}(x)_{G} = \text{Pr}_{f,acyc}(x|\{t_v = 1\}_{T_v \in V_{\text{post}}} )_{G_{c,TP}} \quad (163) $$

for all observed events $x := \{x_v \in \mathcal{X}_v\}_{v \in V}$.

**Step 2:** We define an intermediate functional model on a graph $G'_{c,TP}$ which is constructed from the classical teleportation graph $G_{c,TP}$: $f\mathbf{Cm}_{G'_{c,TP}}$. The idea is to use the correspondence between a single broadcasted classical teleportation protocol and multiple copies of regular classical teleportation, in going from $G_{c,TP}$ to $G'_{c,TP}$ (equation (151)).

Let $V'_{\text{post}}$ be the set of post-selection vertices in $G'_{c,TP}$. The acyclic probability distribution of the event $x$ and successful post-selections $\{t_v^e = 1\}_{T_v^e \in V'_{\text{post}}}$ relative to this functional model is $\text{Pr}_{f,acyc}(x, \{t_v^e = 1\}_{T_v^e \in V_{\text{post}}} )_{G'_{c,TP}}$ (definition 33).

We prove that the conditional probabilities in $f\mathbf{Cm}_{G_{c,TP}}$ (from **Step 1**) and $f\mathbf{Cm}_{G'_{c,TP}}$ are equal, i.e.,

$$ \text{Pr}_{f,acyc}(x|\{t_v^e = 1\}_{T_v^e \in V'_{\text{post}}} )_{G'_{c,TP}} = \text{Pr}_{f,acyc}(x|\{t_v = 1\}_{T_v \in V_{\text{post}}} )_{G_{c,TP}}. \quad (164) $$

**Step 3:** We map the acyclic functional model $f\mathbf{Cm}_{G'_{c,TP}}$ to a causal model (definition 36): $\mathbf{Cm}_{G'_{c,TP}}$.

The acyclic probability distribution of the event $x$ and successful post-selections $\{t_v^e = 1\}_{T_v^e \in V'_{\text{post}}}$ relative to this functional model is $\text{Pr}_{acyc}(x|\{t_v^e = 1\}_{T_v^e \in V'_{\text{post}}} )_{G'_{c,TP}}$.

As a consequence of lemma 37, the acyclic probability rule is preserved through the mapping, i.e.,

$$ \text{Pr}_{f,acyc}(x, \{t_v^e = 1\}_{T_v^e \in V'_{\text{post}}} )_{G'_{c,TP}} = \text{Pr}_{acyc}(x, \{t_v^e = 1\}_{T_v^e \in V'_{\text{post}}} )_{G'_{c,TP}}. \quad (165) $$

Before proceeding, let us sketch the results of the first three steps. The following diagram summarises the (functional) causal models that we introduced so far and their conditional probabilities. Each step establishes equivalence between the (conditional) distributions of these causal models.

$$ \begin{array}{ccc} \text{Step 1} & \rightarrow & \text{Step 2} \\ \text{fCm}_G & \rightarrow & \text{fCm}_{G_{c,TP}} \\ \text{fCm}_{G'_{c,TP}} & \rightarrow & \text{Step 3} \\ \text{fCm}_{G'_{c,TP}} & \rightarrow & \text{Cm}_{G'_{c,TP}} \end{array} \quad (166) $$

$$ \text{Pr}_{f}(x) := \text{Pr}_{f,acyc}(x|t) = \text{Pr}_{f,acyc}(x|\{t^e\}_e) = \text{Pr}_{acyc}(x|\{t^e\}_e) $$

**Step 4:** We map the initial functional model $f\mathbf{Cm}_G$ (from **Step 0**) to a causal model (definition 2): $\mathbf{Cm}_G$ (as specified by the mapping in definition 36).

The cyclic probability distribution for the observed event $x$ relative to this causal model is $\text{Pr}(x)_G$ (definition 12).
**Step 5:** We define a teleportation causal model on a specific teleportation graph $G_{TP}$ induced by $\mathbf{Cm}_G$: $\mathbf{Cm}_{G_{TP}}$ (definition 6 and definition 8). Specifically, we choose $G_{TP}$ with split edges correspond to the outgoing edges of split vertices of $G_{c,TP}$ (see **Step 1**), i.e., $E_S(G_{TP}) = \cup_{v \in V_S(G_{c,TP})} \text{Out}(v)$.

Let us denote the set of post-selection vertices of $G_{TP}$ as $\tilde{V}_{post}$. The acyclic probability distribution of the event $x$ and successful post-selections $\{t_v^e = \check{\check{\check{\check{v}}}}\}_{T_v^e \in \tilde{V}_{post}}$ relative to this causal model is $\text{Pr}_{acyc}(x, \{t_v^e = \check{\check{\check{\check{v}}}}\}_{T_v^e \in \tilde{V}_{post}})_{G_{TP}}$ (definition 3).

By definition, the cyclic probability of $\mathbf{Cm}_G$ equals the acyclic conditional distribution of $\mathbf{Cm}_{G_{TP}}$, i.e.,

$$ \text{Pr}(x)_G = \text{Pr}_{acyc} \left( x | \{t_v^e = \check{\check{\check{\check{v}}}}\}_{T_v^e \in \tilde{V}_{post}} \right)_{G_{TP}}. \qquad (167) $$

Let us sketch what the previous two steps achieve. The diagram summarises the causal models introduced in **Step 4-5** together with the original functional causal model introduced in **Step 0**, and their (conditional) probabilities. **Step 5** established the equivalence between the two probabilities of the newly introduced causal models. The equivalence between $\text{Pr}_f(x)_G$ and $\text{Pr}(x)_G$ is what we aim to prove.

$$ \begin{array}{ccc} & \text{Step 4} & \text{Step 5} \\ \text{fCm}_G & \longrightarrow & \text{Cm}_G \\ & & \text{Cm}_{G_{TP}} \\ \text{Pr}_f(x) & & \text{Pr}(x) := \text{Pr}_{acyc}(x|t) \end{array} \qquad (168) $$

**Step 6:** We prove that the conditional probability distribution of the causal models $\mathbf{Cm}_{G'_{c,TP}}$ (from **Step 3**) and $\mathbf{Cm}_{G_{TP}}$ (from **Step 5**) are equal, i.e.,

$$ \text{Pr}_{acyc} \left( x | \{t_v^e = \check{\check{\check{\check{v}}}}\}_{T_v^e \in \tilde{V}_{post}} \right)_{G_{TP}} = \text{Pr}_{acyc} \left( x | \{t_v^e = 1\}_{T_v^e \in V'_{post}} \right)_{G'_{c,TP}}. \qquad (169) $$

Combining together all the equalities established concludes the proof. The following diagram combines those of equations (166) and (168) through **Step 6**.

$$ \begin{array}{ccc} & \text{Step 1} & \text{Step 2} \\ \text{fCm}_G & \longrightarrow & \text{fCm}_{G_{c,TP}} \\ & & \text{Step 3} \\ & & \left. \begin{array}{c} \text{Step 6} \\ \text{Cm}_G \end{array} \right| \\ & & \text{Step 5} \\ & & \text{Cm}_{G_{TP}} \\ \text{Pr}(x) & & \text{Pr}_{acyc}(x|t) \end{array} = \text{Pr}_{acyc}(x|\{t^e\}_e) \qquad (170) $$

**Details of the proof.** In what follows, we prove in detail the results of **Steps 1-6**.

**Step 1:** We define a classical teleportation causal model on a teleportation graph $G_{c,TP} \in \mathbb{G}_{c,TP}(G)$ with split vertices $V_S(G_{c,TP}) \subseteq V$ (see definition 38), as prescribed in definition 9 of [FGV25]. For completeness, let us briefly remind what this amounts
to. The classical teleportation graph is obtained through performing the following substitution for all $v \in V_S(G_{c,TP})$:

<figure><img src="image_24.png" /></figure>

Then, we define a classical teleportation functional model on $G_{c,TP}$ through definition 9 of [FGV25]. In particular, this construction prescribes that the same finite sets, functions and probabilities of the original functional model are associated to vertices and edges that are preserved from $G$ to $G_{c,TP}$. To pre- and post-selection vertices we associate uniform prior classical teleportation protocols$^{23}$, i.e., for all $v \in V_S(G_{c,TP})$ we associate to the corresponding pre-selection vertex $R_v$ the finite set $\mathcal{X}_{R_v} = \mathcal{X}_v$, a uniform distribution to its error variable $p^{R_v}(u) = |\mathcal{X}_v|^{-1}$ for all $u \in \mathcal{X}_v$ and a function $f^{R_v}(u) = u$. Notice that this amounts to having the variable associated to pre-selection vertices $R_v$ distributed as $p^{R_v}$ (for more details see the remark in section 2 of [FGV25]), thus in what follows we will refer to $R_v$ and its distribution. We associate to the corresponding post-selection vertex $T_v$ the finite set $\mathcal{X}_{T_v} = \{0,1\}$ and a (deterministic) delta function $f^{T_v}(x, y) = \delta_{x,y}$ for all $x, y \in \mathcal{X}_v$.

By definition 12 of [FGV25], given an observed event $x := \{x_v \in \mathcal{X}_v\}_{v \in V}$, the probability of $f\mathfrak{C}m_G$ is defined as

$$ \mathrm{Pr}_f(x)_G = \mathrm{Pr}_{f,acyc} (x | \{t_T = 1\}_{T \in V_{\mathrm{post}}} )_{G_{c,TP}} \quad (172) $$

where $V_{\mathrm{post}}$ denotes the set of post-selection vertices of $G_{c,TP}$.

**Step 2:** We define an intermediate functional causal model over a graph $G'_{c,TP}$, $f\mathfrak{C}m_{G'_{c,TP}}$. The graph $G'_{c,TP}$ is constructed from $G_{c,TP}$ through performing the following substitution of edges and vertices for $v \in V_S(G_{c,TP})$

<figure><img src="image_25.png" /></figure>

This substitution is motivated by equation 20 of [FGV25] which proves that $n$ copies of a classical teleportation protocol (on the right) are equivalent to one classical teleportation protocol which is broadcast to $n$ parties (on the left). For each split vertex $v \in V_S(G_{c,TP})$ and each outgoing edge $e \in \mathrm{Out}(v)$, let us denote with $R_v^e$ and

<footnote>$^{23}$Any other choice of classical post-selected teleportation protocol would yield the same probability as this canonical choice, as shown in [FGV25] paper.</footnote>
$T_v^e$ the pre- and post-selection vertices in $G'_{c,TP}$ associated to the outgoing edge $e$ of $v$ (see equation (173)). Let us denote with $V'_{post}$ the set of post-selection vertices of $G'_{c,TP}$.

We define a functional causal model over $G'_{c,TP}$, $f\mathbf{Cm}_{G'_{c,TP}}$, as follows:

1. To each vertex present in both $G_{c,TP}$ and $G'_{c,TP}$ assign the same finite set, functional dependency (for endogenous vertices) or probability (for exogenous vertices), of the causal model $f\mathbf{Cm}_{G_{c,TP}}$<fnref n="24" />.

2. For all $v \in V_S(G_{c,TP})$ and $e \in \text{Out}(v)$, associate to $R_v^e$ the finite set $\mathcal{X}_{R_v^e} = \mathcal{X}_v$ and a uniform distribution to its error variable $p_{R_v^e}(u) = |\mathcal{X}_v|^{-1}$ for all $u \in \mathcal{X}_v$ and a function $f_{R_v^e}(u) = u$. Notice that this amounts to having the variable associated to pre-selection vertices $R_v^e$ uniformly distributed (for more details see the remark in section 2 of [FGV25]).

3. For all $v \in V_S(G_{c,TP})$ and $e \in \text{Out}(v)$, associate to $T_v^e$ the finite set $\mathcal{X}_{T_v^e} = \{0,1\}$ and a (deterministic) delta function $f_{T_v^e}(x, y) = \delta_{x,y}$ for all $x, y \in \mathcal{X}_v$.

Hence, each outgoing edge from a vertex $v \in V_S(G_{c,TP})$ in $G$ has been replaced with a uniform prior classical teleportation protocol (definition 6 in [FGV25]).

Since this model is acyclic, we can use the acyclic probability rule (definition 33) to evaluate probabilities. The equivalence between broadcasting a classical teleportation protocol to $n$ parties and $n$ copies of the same classical teleportation protocol (established in equation 20 of [FGV25]) implies that

$$ \mathrm{Pr}_{f,acyc}(x|\{t_v^e = 1\}_{T_v^e \in V'_{post}})_{G'_{c,TP}} = \mathrm{Pr}_{f,acyc}(x|\{t_v = 1\}_{T_v \in V_{post}})_{G_{c,TP}}, \quad (174) $$

i.e., probabilities in $f\mathbf{Cm}_{G_{c,TP}}$ and $f\mathbf{Cm}_{G'_{c,TP}}$, conditioned on all post-selections being successful, are equal. Notice that the variables associated to pre-selection vertices are considered unobserved.

**Step 3:** We map the acyclic functional causal model $f\mathbf{Cm}_{G'_{c,TP}}$ to a causal model using definition 36, $\mathbf{Cm}_{G'_{c,TP}}$. As prescribed by the mapping, $G'_{c,TP}$ is decorated to form a causal graph whose edges are all classical. In addition, we consider the pre-selection vertices $V'_{pre}$ to be unobserved<fnref n="25" /> and the remaining ones, $V \cup V'_{post}$ to be observed. The associations prescribed by the mapping in definition 36 together with the specific form of uniform prior teleportation protocols, give the following assignments:

1. To each vertex that is present in both $G$ and $G'_{c,TP}$, i.e., $v \in V$, we associate the set of CP maps acting on $\rho \in \mathcal{L}(\mathcal{H}_{\text{In}(v)})$

$$ \mathcal{M}_x^v(\rho) = \mathrm{Tr} \left[ E_x^{f^v}(\rho \otimes \sigma^v) \right] \otimes_{e \in \text{Out}(v)} |x\rangle\langle x|_{\mathcal{H}_e} \quad (175) $$

where $\{E_x^{f^v}\}_{x \in \mathcal{X}_v}$ is the POVM obtained through applying definition 35 to the function $f^v$ associated to $v$ in the original functional model $f\mathbf{Cm}_G$ and $\sigma^v$ is the state associated to the error variable (see equation (135)).

<footnote n="24">²⁴By construction, these are also the same association of the original causal model $f\mathbf{Cm}_G$ on $G$.</footnote>

<footnote n="25">²⁵In definition 34, all vertices are considered observed. In what follows, pre-selection vertices are always marginalised, hence unobserved. In the remark of appendix C.2, we showed that in this case we can consider such vertices unobserved and modify the associated maps and states.</footnote>
2. For all $v \in V_S(G_{c,TP})$ and $e \in \text{Out}(v)$, associate to $T_v^e$ and outcome $t_v^e = 1$ the map acting on $\rho \in \mathcal{L}(\mathcal{H}_{\text{In}(T_v^e)}) \cong \mathcal{L}(\mathcal{H}_{(v,T_v^e)} \otimes \mathcal{H}_{(R_v^e,T_v^e)})$

$$ \mathcal{M}_1^{T_v^e}(\rho) = \text{Tr} \left[ E_{t=1}^{T_v^e} \rho \right] = \text{Tr} \left[ \sum_{x \in \mathcal{X}_v} \left( |x\rangle\langle x|_{\mathcal{H}_{(v,T_v^e)}} \otimes |x\rangle\langle x|_{\mathcal{H}_{(R_v^e,T_v^e)}} \right) \rho \right], \quad (176) $$

where $E_{t=1}^{T_v^e}$ is the POVM element obtained through applying definition 35 to a delta function and outcome 1, i.e., $f_v^{T_v^e}(x, y) = \delta_{x,y} = 1$.

3. For all $v \in V_S(G_{c,TP})$ and $e = (v, v') \in \text{Out}(v)$, associate to $R_v^e$, which is considered unobserved, the state

$$ \sigma_{v}^{R_v^e} = \sum_{x \in \mathcal{X}_v} \frac{1}{|\mathcal{X}_v|} \bigotimes_{e \in \text{Out}(R_v^e)} |x\rangle\langle x|_{\mathcal{H}_e} = \sum_{x \in \mathcal{X}_v} \frac{1}{|\mathcal{X}_v|} |x\rangle\langle x|_{\mathcal{H}_{(R_v^e, T_v^e)}} \otimes |x\rangle\langle x|_{\mathcal{H}_{(R_v^e, v')}}. \quad (177) $$

Lemma 37 establishes that probabilities are preserved through mapping an acyclic functional model to an acyclic causal model through definition 36. Since $G'_{c,TP}$ is acyclic, this implies that

$$ \text{Pr}_{\text{f,acyc}}(x, \{t_v^e = 1\}_{T_v^e \in V_{\text{post}}'})_{G'_{c,TP}} = \text{Pr}_{\text{acyc}}(x, \{t_v^e = 1\}_{T_v^e \in V_{\text{post}}'})_{G'_{c,TP}}. \quad (178) $$

**Step 4:** We map the original functional model $f\mathfrak{Cm}_G$ to a causal model (definition 2), $\mathfrak{Cm}_G$, using definition 36. As prescribed by the mapping, the graph $G$ is decorated to form a causal graph such that all edges are classical and all vertices are observed. The causal model is then defined through the associations of definition 36. In particular, to each vertex $v \in V$ we associate the set of CP maps acting on $\rho \in \mathcal{L}(\mathcal{H}_{\text{In}(v)})$

$$ \tilde{\mathcal{M}}_x^v(\rho) = \text{Tr} \left[ E_x^{f^v}(\rho \otimes \sigma^v) \right] \bigotimes_{e \in \text{Out}(v)} |x\rangle\langle x|_{\mathcal{H}_e} \quad (179) $$

where $\{E_x^{f^v}\}_{x \in \mathcal{X}_v}$ is the POVM obtained through applying definition 35 to the function $f^v$ associated to $v$ in the functional model $f\mathfrak{Cm}_G$.

**Step 5:** We define a teleportation causal model induced by $\mathfrak{Cm}_G$. Firstly, let us consider the graph $G_{\text{TP}}$ which is constructed using definition 6 and has split edges

$$ E_S(G_{\text{TP}}) = \bigcup_{v \in V_S(G_{c,TP})} \text{Out}(v), \quad (180) $$

i.e., the split edges of $G_{\text{TP}}$ correspond to the outgoing edges of split vertices in $G_{c,TP}$ (see **Step 2**). As given by the definition, this amounts to adding pre and post-selection vertices for each split edge. For $v \in V_S(G_{c,TP})$ and $e \in \text{Out}(v)$ let us denote the pre and post-selection vertices respectively as $R_v^e$ and $T_v^e$. Thus, the graph $G_{\text{TP}}$ is obtained from $G$ through performing the following substitution for all $v \in V_S(G_{c,TP})$

<figure><img src="image_26.png" /></figure>
Let us denote with $\tilde{V}_{\text{post}}$ the set of post-selection vertices of $G_{\text{TP}}$.

Then, we define a teleportation causal model on $G_{\text{TP}}$ through definition 8. In particular, this construction prescribes that the same maps and Hilbert spaces are associated to vertices and edges that are preserved from $G$ to $G_{\text{TP}}$. To pre- and post-selection vertices we associate (without loss of generality, corollary 16) Bell teleportation protocols (definition 5), i.e., for all $v \in V_S(G_{c,\text{TP}})$ and $e = (v, v') \in \text{Out}(v)$ we associate to the corresponding pre-selection vertex $R_v^e$ the Bell state

$$ \tilde{\sigma}_{v}^{R_v^e} = |\Phi^+\rangle\langle\Phi^+|_{\text{Out}(R_v^e)} = \frac{1}{|\mathcal{X}_v|} \sum_{x,y \in \mathcal{X}_v} |x\rangle\langle y|_{(R_v^e, T_v^e)} \otimes |x\rangle\langle y|_{(R_v^e, v')} \quad (182) $$

and to the corresponding post-selection vertex $T_v^e$ and outcome $t = \check{\text{✓}}$ the CP map acting on all $\rho \in \mathcal{L}(\mathcal{H}_{\text{In}(T_v^e)}) \cong \mathcal{L}(\mathcal{H}_{(v, T_v^e)} \otimes \mathcal{H}_{(R_v^e, T_v^e)})$ as

$$ \begin{aligned} \tilde{\mathcal{M}}_{\check{v}}^{T_v^e}(\rho) &= \text{Tr}(|\Phi^+\rangle\langle\Phi^+|_{\text{In}(T_v^e)}\rho) \\ &= \frac{1}{|\mathcal{X}_v|} \text{Tr}\left[\sum_{x \in \mathcal{X}_v} (|x\rangle\langle y|_{\mathcal{H}_{(v, T_v^e)}} \otimes |x\rangle\langle y|_{\mathcal{H}_{(R_v^e, T_v^e)}}) \rho\right]. \end{aligned} \quad (183) $$

By definition 12, given an observed event $x := \{x_v \in \mathcal{X}_v\}_{v \in V}$, the probability of $\text{Cm}_G$ (see Step 4) is defined as

$$ \text{Pr}(x)_G = \text{Pr}_{\text{acyc}} \left( x \middle| \{t_v^e = \check{\text{✓}}\}_{T_v^e \in \tilde{V}_{\text{post}}} \right)_{G_{\text{TP}}}, \quad (184) $$

where $\tilde{V}_{\text{post}}$ denotes the set of post-selection vertices of $G_{\text{TP}}$.

**Step 6:** Finally, we show that the causal models $\text{Cm}_{G'_{c,\text{TP}}}$ (introduced in Step 3) and $\text{Cm}_{G_{\text{TP}}}$ (introduced in Step 5) are equivalent, in the sense that for all observed events $x := \{x_v \in \mathcal{X}_v\}_{v \in V}$ it holds

$$ \text{Pr}_{\text{acyc}} \left( x \middle| \{t_v^e = \check{\text{✓}}\}_{T_v^e \in \tilde{V}_{\text{post}}} \right)_{G_{\text{TP}}} = \text{Pr}_{\text{acyc}} \left( x \middle| \{t_v^e = 1\}_{T_v^e \in V'_{\text{post}}} \right)_{G'_{c,\text{TP}}}. \quad (185) $$

Firstly, we notice that the construction of $G'_{c,\text{TP}}$ and $G_{\text{TP}}$ give the same causal graphs, indeed, the subsequent substitutions in equation (171) and equation (173) are equivalent to equation (181). Let us show that the maps associated to each vertex are equivalent in the two causal models, $\text{Cm}_{G'_{c,\text{TP}}}$ and $\text{Cm}_{G_{\text{TP}}}$. We have:

1. For all vertices which are preserved from $G$ to $G'_{c,\text{TP}}$ (and $G_{\text{TP}}$), $v \in V$ the maps associated in $\text{Cm}_{G_{\text{TP}}}$ and $\text{Cm}_{G'_{c,\text{TP}}}$ are equal (see equations (175) and (179)), i.e., for all $\rho \in \mathcal{L}(\mathcal{H}_{\text{In}(v)})$ it holds

$$ \tilde{\mathcal{M}}_x^v(\rho) = \mathcal{M}_x^v(\rho) \quad (186) $$

where we recall that $\tilde{\mathcal{M}}_x^v$ is the map associated to $v$ in $\text{Cm}_{G_{\text{TP}}}$ and $\mathcal{M}_x^v$ is the map associated to $v$ in $\text{Cm}_{G'_{c,\text{TP}}}$.

2. For all post-selection vertices, i.e., for all $v \in V_S(G_{c,\text{TP}})$ and $e \in \text{Out}(v)$, we have (by the decoherence condition for observed vertices in definition 2)

$$ \tilde{\mathcal{M}}_{\check{v}}^{T_v^e} \circ \tilde{\mathcal{M}}_x^v = \tilde{\mathcal{M}}_{\check{v}}^{T_v^e} \circ \mathcal{D}_{(v, T_v^e)} \circ \tilde{\mathcal{M}}_x^v \quad (187) $$
where all maps $\tilde{\mathcal{M}}$ are in $\mathbf{Cm}_{G_{\text{TP}}}$ and $\mathcal{D}_{(v,T_v^e)}$ is a decohering channel acting as $\mathcal{D}_e(\rho) = \sum_{x \in \mathcal{X}_e} |x\rangle\langle x|\rho|x\rangle\langle x|$ for any $\rho \in \mathcal{L}(\mathcal{H}_e)$. In order to evaluate probabilities, the maps of $\mathbf{Cm}_{G_{\text{TP}}}$ are composed as described in definition 3. In particular, the map $\tilde{\mathcal{M}}_{\check{v}}^{T_v^e}$ is always composed with $\tilde{\mathcal{M}}_v^e$ as in equation (187), hence, it can be replaced with $\tilde{\mathcal{M}}_{\check{v}}^{T_v^e} \circ \mathcal{D}_{(v,T_v^e)}$ without affecting probabilities. Thus, we have for all $\rho \in \mathcal{L}(\mathcal{H}_{\text{In}(T_v^e)})$

$$ \tilde{\mathcal{M}}_{\check{v}}^{T_v^e} \circ \mathcal{D}_{(v,T_v^e)}(\rho) = \text{Tr} \left( |\Phi^+\rangle\langle\Phi^+|_{\text{In}(T_v^e)} \mathcal{D}_{(v,T_v^e)}(\rho) \right) \qquad (188) \\ \text{lem. } 43 \text{Tr} \left[ \mathcal{D}_{(v,T_v^e)} \left( |\Phi^+\rangle\langle\Phi^+|_{\text{In}(T_v^e)} \right) \rho \right] $$

and

$$ \mathcal{D}_{(v,T_v^e)}(|\Phi^+\rangle\langle\Phi^+|_{\text{In}(T_v^e)}) = \frac{1}{|\mathcal{X}_v|} \sum_{x \in \mathcal{X}_v} |x\rangle\langle x|_{(v,T_v^e)} \otimes |x\rangle\langle x|_{(R_v^e,T_v^e)}. \quad (189) $$

Thus we can compare this map, associated to the post-selection vertex $T_v^e$ in $\mathbf{Cm}_{G_{\text{TP}}}$, to the map $\mathcal{M}_1^{T_v^e}$ associated to the same post-selection vertex in $\mathbf{Cm}_{G'_{c,\text{TP}}}$ (see equation (176)). It holds

$$ \tilde{\mathcal{M}}_{\check{v}}^{T_v^e} \circ \mathcal{D}_{(v,T_v^e)} = \frac{1}{|\mathcal{X}_v|} \mathcal{M}_1^{T_v^e}. \quad (190) $$

3. For all pre-selection vertices, i.e., for all $v \in V_S(G_{c,\text{TP}})$ and $e = (v, v') \in \text{Out}(v)$ we have

$$ \tilde{\mathcal{M}}_{x'}^{v'} \circ \tilde{\sigma}^{R_v^e} \stackrel{\text{lem. } 42}{\equiv} \tilde{\mathcal{M}}_{x'}^{v'} \circ \mathcal{D}_{(R_v^e, v')} \circ \tilde{\sigma}^{R_v^e} \quad (191) $$

where $\tilde{\sigma}$ denotes associations in $\mathbf{Cm}_{G_{\text{TP}}}$ and $\mathcal{D}_{(R_v^e, v')}$ is a decohering channel. In order to evaluate probabilities, the maps of $\mathbf{Cm}_{G_{\text{TP}}}$ are composed as described in definition 3. In particular, the state $\tilde{\sigma}^{R_v^e}$ is always composed with the map $\tilde{\mathcal{M}}_{x'}^{v'}$ as in eq. (191), hence, the state $\tilde{\sigma}^{R_v^e}$ can be replaced with $\mathcal{D}_{(R_v^e, v')} \circ \tilde{\sigma}^{R_v^e}$ without affecting probabilities. Thus, we can compare the decohered state $\mathcal{D}_{(R_v^e, v')} \circ \tilde{\sigma}^{R_v^e}$, associated to the pre-selection vertex $R_v^e$ in $\mathbf{Cm}_{G_{\text{TP}}}$, to the state $\sigma^{R_v^e}$ associated to the same pre-selection vertex in $\mathbf{Cm}_{G'_{c,\text{TP}}}$ (see eq. (177)). It holds:

$$ \mathcal{D}_{(R_v^e, v')} \circ \tilde{\sigma}^{R_v^e} = \mathcal{D}_{(R_v^e, v')} \left( |\Phi^+\rangle\langle\Phi^+|_{\text{Out}(R_v^e)} \right) \stackrel{\text{eq. (189)}}{\equiv} \sigma^{R_v^e}. \quad (192) $$

Using the probability rule in definition 3, we can easily see that the conditional probabilities in $\mathbf{Cm}_{G_{\text{TP}}}$ and $\mathbf{Cm}_{G'_{c,\text{TP}}}$ satisfy

$$ \text{Pr}_{\text{acyc}}(x|\{t_v^e = \check{v}\}_{T_v^e \in \tilde{V}_{\text{post}}} \}_{G_{\text{TP}}} = \text{Pr}_{\text{acyc}}(x|\{t_v^e = 1\}_{T_v^e \in V'_{\text{post}}} \}_{G'_{c,\text{TP}}}, \quad (193) $$

notice that the scaling factor between the two maps in equation (190) drops out once we condition on successful post-selections.

Finally, we can combine the results of **Steps 1-6** to complete the proof. Indeed, we have

$$ \begin{align*} \text{Pr}_f(x)_G &\stackrel{\text{S1}}{=} \text{Pr}_{f, \text{acyc}}(x|\{t_v = 1\}_{T_v \in V_{\text{post}}})_{G_{c, \text{TP}}} \\ &\stackrel{\text{S2}}{=} \text{Pr}_{f, \text{acyc}}(x|\{t_v^e = 1\}_{T_v^e \in V'_{\text{post}}})_{G'_{c, \text{TP}}} \\ &\stackrel{\text{S3}}{=} \text{Pr}_{\text{acyc}}(x, \{t_v^e = 1\}_{T_v^e \in V'_{\text{post}}} \}_{G'_{c, \text{TP}}} \\ &\stackrel{\text{S6}}{=} \text{Pr}_{\text{acyc}}(x|\{t_v^e = \check{v}\}_{T_v^e \in \tilde{V}_{\text{post}}})_{G_{\text{TP}}} \\ &\stackrel{\text{S5}}{=} \text{Pr}(x)_G \end{align*} \quad (194) $$
where $Pr_{fG}$ is the probability in $f\mathbf{Cm}_G$ (Step 0), $Pr_{f,acycG_{c,TP}}$ in $f\mathbf{Cm}_{G_{c,TP}}$ (Step 1), $Pr_{f,acycG'_{c,TP}}$ in $f\mathbf{Cm}_{G'_{c,TP}}$ (Step 2), $Pr_{acycG'_{c,TP}}$ in $\mathbf{Cm}_{G'_{c,TP}}$ (Step 3), $Pr_G$ in $\mathbf{Cm}_G$ (Step 4) and $Pr_{acycG_{TP}}$ in $\mathbf{Cm}_{G_{TP}}$ (Step 5). □

# D Proofs of section 4

**Lemma 7 (Acyclicity of teleportation graphs).** Each directed graph $G_{TP}$ obtained from a directed graph $G$ as described in definition 6 is acyclic.

*Proof.* In the first step of definition 6, the graph $G'$ is acyclic. The second step introduces pre- and post-selection vertices, $R_i$ and $T_i$, and associated edges according to equation (47). To show that this step preserves acyclicity, notice that a directed graph is acyclic if and only if it can be drawn on a page with all directed edges oriented from bottom to top of the page$^{26}$. Since $G'$ is acyclic, we can represent it on a page through such a diagram. We now add the pre- and post-selection vertices to this diagram by drawing all the $R_i$ at the bottom of the page, below all other vertices, and all the $T_i$ at the top of the page, above all the other vertices. Since all pre-selection vertices $R_i$ only have outgoing edges and all post-selection vertices $T_i$ only have incoming arrows, both of which will be oriented from bottom to top in our diagram, it follows that $G_{TP}$ is acyclic. □

**Proposition 10 (Equivalent probabilities from different teleportation graphs).** Let $\mathbf{Cm}_G$ be a causal model on a causal graph $G$, and let $V_o \subseteq V$ be the set of all observed vertices of $G$. Consider any $G_{TP,1}, G_{TP,2} \in \mathbb{G}_{TP}(G)$, and for $i \in \{1, 2\}$, let $\mathbf{Cm}_{G_{TP,i}}$ be a causal model on the teleportation graph $G_{TP,i}$ that is associated to $\mathbf{Cm}_G$ (definition 8). Then, we have

$$ \mathrm{Pr}_{\mathrm{acyc}}(x | \{t_i = \checkmark\}_{T_i \in V_{\mathrm{post}}^{1}})_{G_{\mathrm{TP},1}} = \mathrm{Pr}_{\mathrm{acyc}}(x | \{t_i = \checkmark\}_{T_i \in V_{\mathrm{post}}^{2}})_{G_{\mathrm{TP},2}}, \quad (52) $$

where we denoted the set of all post-selection vertices of $G_{TP,1}, G_{TP,2}$ as $V_{\mathrm{post}}^1$ and $V_{\mathrm{post}}^2$ respectively, and the joint observed event of $V_o$ in short as $x := \{x_v \in \mathcal{X}_v\}_{v \in V_o}$.

*Proof.* *Proof in a special case.* Consider the special case where $G_{TP,1}$ and $G_{TP,2}$ were constructed by choosing

$$ E_S(G_{TP,2}) = E_S(G_{TP,1}) \cup \{e_0\}, \quad (195) $$

i.e., $G_{TP,2}$ splits the same edges as $G_{TP,1}$ plus the edge $e_0$. Let $V_{\mathrm{post}}^1$ denote the set of post-selection vertices of $G_{TP,1}$ and $t := \{t_i \in \{\checkmark, \mathcal{X}\}_{T_i \in V_{\mathrm{post}}^1}\}$. Then, taking $t_0$ to be the outcome of the post-selection vertex associated to the edge $e_0$, we have $t \cup t_0 := \{t_i \in \{\checkmark, \mathcal{X}\}_{T_i \in V_{\mathrm{post}}^2}\}$.

Consider the probability $\mathrm{Pr}(x, t)_{G_{TP,1}}$ associated to the causal model $\mathbf{Cm}_{G_{TP,1}}$ on the causal graph underlying the teleportation graph $G_{TP,1}$. As this is an acyclic causal model by construction (c.f., lemma 7), this probability is immediately given by applying the acyclic probability rule of definition 3 to this causal model. Here, $x = \{x_v \in \mathcal{X}_v\}_{v \in V_o}$ denotes an observed event on $G_{TP,1}$ (which is also an event on $G_{TP,2}$, since these graphs have the same observed vertices). We can write

$$ \mathrm{Pr}(x, t)_{G_{TP,1}} = \mathrm{Tr}[F^{(x,t)}\rho^{(x,t)}], \quad (196) $$

<footnote>$^{26}$This representation is equivalent to a Hasse diagram for partially ordered sets, recognising the fact that every directed acyclic graph induces a partial order and vice versa.</footnote>
where we defined the subnormalized density matrix and POVM element $\rho^{(x,t)}$, $F^{(x,t)} \in \mathcal{L}(\mathcal{H}_{e_0})$ obtained from composing all the channels of the causal model $\mathbf{Cm}_{G_{TP,1}}$, but leaving the composition along the $e_0$ edge as the last step.<fnref n="27" /> We then consider the post-selected teleportation protocol that the causal model $\mathbf{Cm}_{G_{TP,2}}$ implements in place of the edge $e_0 =: (v_0, v'_0)$, with associated pre- and post-selection vertices $R_0$ and $T_0$. The associated teleportation probability is denoted $p_{TP} \in (0,1]$. For convenience, we label as A, B and C the Hilbert spaces involve in this teleportation protocol:

$$ \mathcal{H}_A = \mathcal{H}_{(v_0, T_0)}, \quad \mathcal{H}_B = \mathcal{H}_{(R_0, T_0)}, \quad \mathcal{H}_C = \mathcal{H}_{(R_0, v'_0)}. \qquad (197) $$

This allows us to denote the post-selected teleportation implementation as $(E_{AB}, \varphi_{BC})$. Visually, the situation is as follows:

<figure><img src="image_27.png" /></figure>

Thanks to the Hilbert space identities of equation (49), we have $\mathcal{H}_{e_0} = \mathcal{H}_A = \mathcal{H}_C$. We can thus rewrite equation (196) as

$$ \begin{aligned} \mathrm{Pr}(x, t)_{G_{\mathrm{TP},1}} &= \mathrm{Tr}_A[F_A^{(x,t)} \rho_A^{(x,t)}] \\ &= \frac{1}{p_{\mathrm{TP}}} \mathrm{Tr}_{AB} [(E_{AB} \otimes F_C^{(x,t)})(\rho_A^{(x,t)} \otimes \varphi_{BC})] \\ &= \frac{1}{p_{\mathrm{TP}}} \mathrm{Pr}(x, t, t_0 = \checkmark)_{G_{\mathrm{TP},2}}, \end{aligned} \qquad (198) $$

where $\mathrm{Pr}(x, t, t_0)_{G_{TP,2}}$ denotes the probabilities associated to the causal model $\mathbf{Cm}_{G_{TP,2}}$ on the causal graph underlying the teleportation graph $G_{TP,2}$. In the last equation, we used that $\mathbf{Cm}_{G_{TP,1}}$ and $\mathbf{Cm}_{G_{TP,2}}$ can be chosen to have the same associated $\rho^{(x,t)}$ and $F^{(x,t)}$ (to see this, one may for simplicity restrict all the post-selected teleportation protocols to be implemented as in definition 5 — if we establish proposition 10 in this case, the general result follows from corollary 16). We can now relate the success probability $p_{\checkmark}^{(i)}$ of the causal model $\mathbf{Cm}_{G_{TP,i}}$ on the teleportation graph $G_{TP,i}$ for $i \in \{1,2\}$. By definition 9, we have

$$ p_{\checkmark}^{(1)} = \sum_{x} \mathrm{Pr}(x, t = \checkmark)_{G_{\mathrm{TP},1}} \qquad (199) $$

$$ = \frac{1}{p_{\mathrm{TP}}} \sum_{x} \mathrm{Pr}(x, t = \checkmark, t_0 = \checkmark)_{G_{\mathrm{TP},2}} \qquad (200) $$

$$ = \frac{1}{p_{\mathrm{TP}}} p_{\checkmark}^{(2)}, \qquad (201) $$

where the sum $\sum_x$ runs over all $x = \{x_v \in \mathcal{X}_v\}_{v \in V_o}$, and where we denote with $t = \checkmark$ the event $\{t_i = \checkmark\}_{T_i \in V_{\mathrm{post}}^1}$. We see that the two success probabilities differ by a multiplicative constant (the success probability $p_{TP}$ of the post-selected teleportation protocol

<footnote n="27">²⁷This decomposition is not unique for teleportation graphs with disconnected components, in which case we can pick any such decomposition.</footnote>
defined by $\mathbf{Cm}_{G_{\text{TP},2}}$ for the split edge $e_0$), and hence, $p_{\check{\gamma}}^{(1)}$ is zero if and only if $p_{\check{\gamma}}^{(2)}$ is zero. Hence, the probabilities $\text{Pr}_{G_{\text{TP},1}}$ are defined if and only if the probabilities $\text{Pr}_{G_{\text{TP},2}}$ are defined. Furthermore, if they are defined, we can relate them as follows (again appealing to definition 9):

$$ \text{Pr}(x)_{G_{\text{TP},1}} = \frac{\text{Pr}(x, t = \check{\gamma})_{G_{\text{TP},1}}}{p_{\check{\gamma}}^{(1)}} = \frac{p_{\text{TP}} \text{Pr}(x, t = \check{\gamma}, t_0 = \check{\gamma})_{G_{\text{TP},2}}}{p_{\text{TP}} p_{\check{\gamma}}^{(2)}} = \text{Pr}(x)_{G_{\text{TP},2}}, \quad (202) $$

which completes the proof in this case.

**Proof in the general case.** Consider two general elements of $\mathbb{G}_{\text{TP}}(G)$, $G_{\text{TP},1}$, $G_{\text{TP},2} \in \mathbb{G}_{\text{TP}}(G)$. We can prove the general statement by first noticing that the repeated application of the above argument can be used to prove that the probabilities of both $G_{\text{TP},1}$ and $G_{\text{TP},2}$ are equivalent to those of $G_{\text{TP},0}$, where the latter corresponds to the graph where all the edges have been split (i.e., $E_S(G_{\text{TP},0}) = E$, where $G = (V, E)$). The result follows by transitivity. $\square$

**Proposition 14** (General probability rule in terms of self-cycle composition). Consider a causal model $\mathbf{Cm}_G$ on any causal graph $G$, associated with the sets $V_o$ and $V_u$ of observed and unobserved vertices. Let $G_{\text{TP}} \in \mathbb{G}_{\text{TP}}(G)$ be a teleportation graph and $\mathbf{Cm}_{G_{\text{TP}}}$ a corresponding teleportation causal model on $G_{\text{TP}}$ constructed from $\mathbf{Cm}_G$. By construction, pre and post-selection vertices in $G_{\text{TP}}$ come in pairs $R_i$ and $T_i$, and are associated in particular with edges $e_i := (v_i, T_i)$ and $e'_i := (R_i, v'_i)$. Suppose that the cardinality of the split edges set $E_S(G_{\text{TP}})$ is $k$, then we have the edges $\{e_1, \dots, e_k\}$ and $\{e'_1, \dots, e'_k\}$ defined as above. We let a joint observed event associated with $V_o$ be denoted as $x := \{x_v \in \mathcal{X}_v\}_{v \in V_o}$, and define the collection of CP maps

$$ \{C_x : \mathcal{L}(\bigotimes_{i=1}^k \mathcal{H}_{e'_i}) \mapsto \mathcal{L}(\bigotimes_{i=1}^k \mathcal{H}_{e_i})\}_x \quad (55) $$

through

$$ C_x = \bigcirc_{v \in V_o} \mathcal{M}_{x_v}^v \bigcirc_{u \in V_u} \mathcal{E}^u, \quad (56) $$

where $\mathcal{M}_{x_v}^v$ and $\mathcal{E}^u$ refer to the maps of the causal model $\mathbf{Cm}_G$ (see definition 2), and where the composition rule is dictated by the acyclic subgraph $G'$ of $G$ which is used in definition 6 to construct $G_{\text{TP}}$ $^{28}$. It holds that the success probability $p_{\check{\gamma}}$ and the probabilities $\text{Pr}(x)_G$ of definition 12, defined if and only if $p_{\check{\gamma}} \neq 0$, satisfy

$$ p_{\check{\gamma}} = \left( \prod_{i=1}^{k} p_{\text{TP}}^{(i)} \right) \sum_{x} \text{cycle}(C_x), \quad \text{Pr}(x)_G = \frac{\text{cycle}(C_x)}{\sum_{x} \text{cycle}(C_x)}, \quad (57) $$

where the cycle operation was introduced in definition 13 and $p_{\text{TP}}^{(i)}$ is the teleportation probability associated to the post-selected teleportation protocol implemented by the vertices $R_i$ and $T_i$ (definition 8).

**Proof.** We start by considering a teleportation graph $G_{\text{TP}} \in \mathbb{G}_{\text{TP}}(G)$ associated with our given graph $G$, and the causal model $\mathbf{Cm}_{G_{\text{TP}}}$ on the teleportation graph $G_{\text{TP}}$, associated with a set $V_{\text{post}} := \{T_i\}_{i=1}^k$ of post-selection vertices.

<footnote>$^{28}$This is similar to how the composition works in definition 3, but here specified by the chosen graph $G'$.</footnote>
The associated probability distribution $\Pr(x, \{t_i = \checkmark\}_{T_i \in V_{\text{post}}})_{G_{\text{TP}}}$ is then given by definition 3 since $\text{Cm}_{G_{\text{TP}}}$ is an acyclic causal model by construction (as $G_{\text{TP}}$ is a directed acyclic graph, lemma 7).

$$ \Pr(x, \{t_i = \checkmark\}_{T_i \in V_{\text{post}}} )_{G_{\text{TP}}} = \text{Tr} \left[ \bigotimes_{i=1}^{k} E^{(i)} C_x \left( \bigotimes_{i=1}^{k} \varphi^{(i)} \right) \right], \quad (203) $$

where the state $\otimes_i \varphi^{(i)}$ acts on $\otimes_i \mathcal{H}_{(R_i, T_i)} \otimes \mathcal{H}_{e'_i}$, and consists of the tensor product of the pre-selection states of each post-selected teleportation protocol. The POVM element $\otimes E^{(i)}$ acts on $\otimes_i \mathcal{H}_{(R_i, T_i)} \otimes \mathcal{H}_{e_i}$ and consists of the tensor product of the post-selection POVM elements of each post-selected teleportation protocol. Recall that the map $C_x$ takes inputs in $\mathcal{L}(\otimes_i \mathcal{H}_{e'_i})$ and outputs in $\mathcal{L}(\otimes_i \mathcal{H}_{e_i})$. Let us define the following Hilbert spaces:

$$ \mathcal{H}_A = \bigotimes_{i=1}^{k} \mathcal{H}_{e_i}, \quad \mathcal{H}_B = \bigotimes_{i=1}^{k} \mathcal{H}_{(R_i, T_i)}, \quad \mathcal{H}_C = \bigotimes_{i=1}^{k} \mathcal{H}_{e'_i}. \quad (204) $$

By construction (definition 8), we have that $\mathcal{H}_{e_i} = \mathcal{H}_{e'_i}$, so that we also have $\mathcal{H}_A = \mathcal{H}_C$. We denote $E_{AB} = \bigotimes_{i=1}^{k} E^{(i)}$ and $\varphi_{BC} = \bigotimes_{i=1}^{k} \varphi^{(i)}$, and we denote the map $C_x$ as $(C_x)_{A|C}$. The probabilities of equation (203) can be rewritten as

$$ \Pr(x, \{t_i = \checkmark\}_{T_i \in V_{\text{post}}} )_{G_{\text{TP}}} = \text{Tr}_{AB}[E_{AB}(C_x)_{A|C}(\varphi_{BC})]. \quad (205) $$

It is easy to prove that $(E_{AB}, \varphi_{BC})$ implements a post-selected teleportation protocol. Indeed, this follows from the linearity of the post-selected teleportation condition equation (45), which allows to successfully teleport part of a multipartite state (see also lemma 25), together with the fact that $(E_{AB}, \varphi_{BC})$ corresponds to $k$ parallel independent post-selected teleportation protocols. Furthermore, it is easy to see that the teleportation success probability of $(E_{AB}, \varphi_{BC})$ is the product of the individual teleportation probabilities. Using lemma 28, we see that the probabilities of equation (205) can be rewritten as

$$ \Pr(x, \{t_i = \checkmark\}_{T_i \in V_{\text{post}}} )_{G_{\text{TP}}} = \left( \prod_{i=1}^{k} p_{\text{TP}}^{(i)} \right) \text{cycle}(C_x). \quad (206) $$

Finally, by recalling definition 9 of the post-selection success probability $p_\checkmark$ and definition 12 of the probability $\Pr(x)_G$ of the observed outcomes of the original causal model $\text{Cm}_G$ in terms of the probability $\Pr(x, \{t_i = \checkmark\}_{T_i \in V_{\text{post}}} )_{G_{\text{TP}}}$ of the teleportation causal model $\text{Cm}_{G_{\text{TP}}}$ (on the graph $G_{\text{TP}}$), we see that equation (206) immediately implies the required equation (57), $\square$

# E Proofs of section 5

**Theorem 22 (p-separation theorem).** Consider a directed graph $G$ and let $V_1, V_2$ and $V_3$ be any three disjoint sets of the vertices of $G$ with $V_1$ and $V_2$ being non-empty. Then, the following holds:

(Soundness) For any causal model $\text{Cm}_G$ on $G$ where the sets $V_i$ are observed, we have that p-separation between the vertex sets $V_i$ implies conditional independence for the corresponding sets of random variables $X_i := \{X_v\}_{v \in V_i}$ where $i \in \{1, 2, 3\}$, i.e.,

$$ (V_1 \perp^p V_2 | V_3)_G \implies (X_1 \perp X_2 | X_3)_{\text{Pr}_G}. \quad (76) $$
(Completeness) If the $p$-connection $(V_1 \not\perp^p V_2|V_3)_G$ holds in $G$, then there exists a causal model $\mathbf{Cm}_G$ such that the sets $V_i$ are observed and $(X_1 \not\perp X_2|X_3)_{\mathrm{Pr}_G}$.

The above conditional (in)dependence statements are relative to the marginal $\mathrm{Pr}(x_1, x_2, x_3)_G$ on $X_1 \cup X_2 \cup X_3$, where $x_i = \{x_v \in \mathcal{X}_v\}_{v \in V_i}$ of the observed distribution $\mathrm{Pr}(x)_G$, where $x = \{x_v\}_{v \in V}$, in the causal model $\mathbf{Cm}_G$.

*Proof. (Soundness)* From definition 21, we have $(V_1 \perp^p V_2|V_3)_G \Leftrightarrow \exists G_{\mathrm{TP}} \in \mathbb{G}_{\mathrm{TP}}(G)$, such that $(V_1 \perp^d V_2|V_3 \cup V_{\mathrm{post}})_{G_{\mathrm{TP}}}$, where $V_{\mathrm{post}}$ is the set of all post-selection vertices in the chosen teleportation graph $G_{\mathrm{TP}}$. From theorem 20, it then follows that the conditional independence $(X_1 \perp X_2|X_3 \cup V_{\mathrm{post}})_{\mathrm{Pr}_{\mathrm{acyc}G_{\mathrm{TP}}}}$ holds since $G_{\mathrm{TP}}$ is a directed acyclic graph (lemma 7). Finally, by definition 12, the probability distribution associated with a causal model on the possibly cyclic causal graph $G$ is given by the corresponding probability computed in any representative teleportation graph $G_{\mathrm{TP}} \in \mathbb{G}_{\mathrm{TP}}(G)$ with an additional conditioning on the associated post-selection vertices $V_{\mathrm{post}}$. Recall from proposition 10 that this distribution is the same, independently of which representative $G_{\mathrm{TP}} \in \mathbb{G}_{\mathrm{TP}}(G)$ is chosen. Therefore, using definition 12, we know that $(X_1 \perp X_2|X_3 \cup V_{\mathrm{post}})_{\mathrm{Pr}_{\mathrm{acyc}G_{\mathrm{TP}}}}$ is equivalent to $(X_1 \perp X_2|X_3)_{\mathrm{Pr}_G}$ by definition 21, which completes the proof.

(Completeness) Suppose we have a $p$-connection $(V_1 \not\perp^p V_2|V_3)_G$ in the given directed graph $G$ according to definition 21. Then by lemma 40 this implies $p$-connection $(V_1 \not\perp^{p,c} V_2|V_3)_G$ according to the alternative definition, definition 39, of this notion introduced in [FGV25]. By the completeness result for $\perp^{p,c}$ proven in [FGV25] (Theorem 20)<fnref n="29" />, it follows that there exists a functional causal model (definition 32) $f\mathbf{Cm}_G$ on $G$ where the conditional dependence $(X_1 \not\perp X_2|X_3)_{\mathrm{Pr}_G}$ holds. By lemma 37, $f\mathbf{Cm}_G$ induces a causal model $\mathbf{Cm}(f\mathbf{Cm})_G$ on $G$ with equivalent probabilities, and hence the same conditional dependence holds in this causal model of $G$, $(X_1 \not\perp X_2|X_3)_{\mathrm{Pr}_G}$ as required. This completes the proof. □

<footnote n="29">²⁹We note that the notion of definition 39 is referred to as $\perp^p$ in [FGV25], but we use $\perp^{p,c}$ here to distinguish it from the definition 21 introduced here.</footnote>