REVIEW 4 major objections 6 minor 3 cited by
Cyclic quantum causal modelling with a graph separation theorem
T0 review · 4 major / 6 minor · reviewed 2026-08-08 · deepseek-v4-flash
Pith's one-line read This paper establishes p-separation as a sound and complete graph-theoretic criterion for all cyclic quantum and classical causal models in its framework.
desk verdict A genuine new framework for cyclic quantum causal models with a sound-and-complete graph-separation criterion relative to its post-selected teleportation probability rule; the central completeness proof is deferred, and the probability postulate deserves more scrutiny, but this deserves serious refereeing. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The carrying objects are the teleportation graphs $G_{tp}(G)$ and the graph relation p-separation. A teleportation graph is obtained from a cyclic causal graph by deleting enough edges to make it acyclic and replacing each deleted edge with a post-selected teleportation gadget: a shared entangled state, a measurement with one successful outcome, and the induced identity channel simulation. p-separation asks whether two vertex sets are d-separated in some teleportation graph after conditioning on all post-selection vertices. The probability rule is carried by the self-cycle composition $\mathrm{cycle}(C_x)$: the total post-selection success probability factorises as a product of teleportation probabilities times $\sum_x \mathrm{cycle}(C_x)$, and the observable probabilities are $\mathrm{cycle}(C_x)$ normalized by that sum; this formula is proven independent of the teleportation implementation and of the chosen acyclic subgraph.
What would settle it
A concrete observation would be to take any directed graph, choose a pair of vertex sets declared p-separated given a third, build a finite-dimensional causal model within the stated framework, compute its distribution through the self-cycle formula, and check whether the two variables are ever conditional-dependent despite p-separation; soundness would fail if so. Conversely, if every causal model yields conditional independence for a p-connected pair, completeness fails. Since the theorem declares both impossible, a computer search over small finite-dimensional models would settle it.
Extended reading notes
Core claim
The central claim is the p-separation theorem (Theorem 22). For any directed graph $G$, if $V_1$ and $V_2$ are p-separated given $V_3$, then every causal model in the framework on $G$ yields conditional independence $X_1 \perp\!\!\perp X_2 \mid X_3$ in the induced distribution; if $V_1$ is p-connected to $V_2$ given $V_3$, then some causal model yields conditional dependence. p-separation is defined by expanding the cyclic graph into an acyclic teleportation graph, obtained by replacing edges with post-selected teleportation protocols, and asking whether the original vertices are d-separated after conditioning on all post-selection vertices; the existential form of the definition is what lets it reduce to d-separation for acyclic graphs. The probability rule is defined as the conditional distribution of the acyclic teleportation model given that all post-selections succeed, and is shown independent of which edges are split and which teleportation implementation is used. This gives a single coherent semantics for cyclic quantum causal models, including classical functional models that are not uniquely solvable and cases where d-separation soundness fails.
Load-bearing premise
The probability rule postulates that a cyclic causal model's observable distribution is exactly the distribution of an associated acyclic teleportation model conditioned on successful post-selection, and a model whose post-selection success probability is zero is called inconsistent; under a different consistency condition for causal loops, p-separation would not be expected to remain sound.
Editorial extensions
If this is right
- Soundness and completeness of p-separation let one read conditional independences of the observed distribution directly from a cyclic graph, without solving the causal mechanisms, just as d-separation does for acyclic graphs.
- p-separation reduces to d-separation on acyclic graphs, so existing acyclic results are recovered as a special case.
- The framework supplies a causal-model semantics to post-selected closed timelike curves and to classical functional models that are not uniquely solvable, including cases where d-separation soundness fails.
- The probability rule and p-separation open the way to causal discovery and causal compatibility algorithms for cyclic quantum and classical structures, and to certifying non-classicality in cyclic networks.
Reading between the lines
- If p-separation is indeed complete, no strictly stronger graph-separation rule can be sound for this framework; completeness bounds what any graph criterion can infer from cyclic topology under the post-selection semantics.
- The normalized probabilities depend non-linearly on the causal mechanisms, so p-separation may help delineate cyclic effects that go beyond linear process-matrix descriptions and their indefinite-causal-order interpretations.
- The finite-dimensional and per-edge tensor-factor assumptions are the likely pressure points: extending p-separation to infinite dimensions would need a measure-theoretic treatment of post-selection success, and lifting the tensor-factor restriction connects to the open causal-decomposition question for general channels.
- A testable classical extension is to compare p-separation predictions with sigma-separation on continuous-variable cyclic functional models, since the two criteria agree on finite classical models but can diverge when variables are continuous, pinpointing where the finite-dimensional boundary matters.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper develops a framework for cyclic quantum (and classical) causal models on finite-dimensional systems. It defines a probability rule by mapping a cyclic causal model to a family of acyclic teleportation graphs, using post-selected quantum teleportation to simulate removed edges, and conditioning on the success of all post-selections. The paper proves that the resulting probabilities are independent of the chosen teleportation graph and of the teleportation protocol. The central contribution is a new graph separation property, p-separation, defined via d-separation in the teleportation graphs, with Theorem 22 claiming that p-separation is sound and complete for all causal models admitted by the framework. The paper also relates the framework to other approaches, including BLO causal models, Costa-Shrapnel models, process matrices, post-selected closed timelike curves, and classical functional models.
Significance. If Theorem 22 holds, the paper provides the first graph-separation theorem applicable to a general class of cyclic quantum causal models, potentially enabling causal discovery and non-classicality certification in cyclic structures. The probability-rule independence results (Propositions 10 and 14, Corollaries 15 and 16) and the self-testing characterization of post-selected teleportation (Lemma 26) are proven in detail and are valuable in their own right. The paper is also careful in mapping classical functional models into the framework and in discussing connections to existing formalisms. However, the central theorem's proof is deferred to an appendix that is not available in the reviewed manuscript, and the soundness direction is largely by construction relative to the post-selection postulate; the completeness direction is the load-bearing element and cannot currently be certified.
major comments (4)
- [§5.3, Theorem 22] The proof of the p-separation theorem is deferred to Appendix E, which is not included in the manuscript provided for review. The completeness direction requires, for every p-connected triple (V1,V2,V3), an explicit causal model CmG on G with p✓>0 whose distribution violates the conditional independence. This is not automatic from the acyclic d-separation theorem, since a d-connecting path in every Gtp supplies dependence only in the acyclic teleportation model, and that dependence must survive conditioning on the success event {ti=✓} while p✓ remains strictly positive. Without inspecting Appendix E, the central claim of the paper cannot be verified. Please include the full proof in the revised version and state explicitly how the construction guarantees p✓>0 and dependence after conditioning.
- [§4.3 Definition 12 and §5.3 Definition 21] Both the probability rule and p-separation are defined in terms of the same family of teleportation graphs Gtp(G). Consequently, the soundness direction of Theorem 22 follows almost directly from the acyclic d-separation theorem after conditioning on the post-selection vertices; the substantive content is entirely in the completeness direction. The paper should state this explicitly and clarify that the theorem is a theorem about the post-selected teleportation consistency postulate, not about alternative consistency criteria such as Deutsch's D-CTCs. A short discussion of how p-separation would (or would not) be expected to behave under alternative consistency conditions would address the implicit circularity concern and delineate the scope of the claim.
- [§4.3 Definition 12] The probability rule declares a model inconsistent when p✓=0 and leaves probabilities undefined. This is a substantive modeling assumption that excludes a priori any cyclic model whose post-selection success probability vanishes. The completeness construction of Theorem 22 must therefore be shown to avoid p✓=0 for every p-connected triple. The visible text does not provide such an argument, and the paper does not discuss whether models with p✓=0 are physically meaningful or can be approximated by models with p✓>0. Please justify this exclusion and ensure that the completeness proof explicitly constructs models with p✓>0.
- [Appendices D and E] The proofs of Propositions 10 and 14 (Appendix D) and of Theorem 22 (Appendix E) are not included in the manuscript text provided for review. Proposition 14 is central to the probability rule, and Theorem 22 is the central result of the paper, so the absence of these proofs prevents verification of the main claims. Please ensure that the full appendices are part of the submitted manuscript, or indicate clearly where they can be found in the arXiv version, so that the proofs can be checked.
minor comments (6)
- [References] The references [LMGP+11a] and [LMGP+11b] are identical (same title and DOI); please correct the duplication or differentiate the two entries.
- [Definition 8(2)(a)] The definition identifies H(vi,Ti) and H(Ri,v'i) with the original edge Hilbert space H(vi,v'i), but does not specify the Hilbert space of the edge (Ri,Ti), which is part of the post-selected teleportation protocol. Please state that dim H(Ri,Ti) ≥ dim H(vi,v'i) (as implied by Lemma 26) and specify how the POVM element and state are assigned to the protocol.
- [§5.2] In the classical example, the sentence 'computing the probability through our rule, it can be checked that we would obtain Pr(x3,x4)G=0 whenever x3≠x4' should be qualified: this holds for the conditional probability rule of Definition 12 when the prior distributions assign nonzero probability to the consistent assignments; otherwise the model is inconsistent. Please clarify.
- [Abstract] The abstract states that the framework applies to 'all consistent quantum and classical cyclic causal models on finite-dimensional systems'. Since consistency is defined via p✓>0 in Definition 12, please make explicit in the abstract or introduction that 'consistent' is defined by the post-selected teleportation rule, so that readers do not infer a theory-independent notion of consistency.
- [Figure 1] Figure 1 is dense and the annotations are difficult to parse; please enlarge and/or provide a more detailed caption explaining each arrow and annotation, particularly the blue open arrow and the crossed red arrows.
- [§5.4] The statement that in the collider-with-descendant example 'applying our probability rule, the outcomes a and b ... will be conditionally independent even when conditioned on the colliders Vpost' is non-obvious and deserves a brief proof or reference, since a reader might expect conditioning on a descendant of a collider to create dependence.
Circularity Check
No significant circularity: p-separation soundness is definitionally aligned with the post-selection probability rule but rests on the external acyclic d-separation theorem; completeness is nontrivial and deferred, not circular.
full rationale
The paper's central claim, Theorem 22, has two halves. Soundness is obtained by combining Definition 21 (p-separation as existence of a teleportation graph Gtp with d-separation) with Definition 12 (cyclic probabilities as conditioned probabilities on the same Gtp family) and then invoking the external acyclic d-separation theorem (Theorem 20, from HLP14) plus the Gtp-independence result Proposition 10. This makes the soundness direction tightly aligned with the definitions, but it is not circular: p-separation is a purely graph-theoretic notion, and the independence statement follows from an independent theorem for acyclic causal models rather than from the claimed conclusion. Completeness is the substantive direction; it requires constructing, for every p-connection, a causal model with nonzero post-selection success and conditional dependence. That construction is deferred to Appendix E, whose text is not included in the material supplied, so it cannot be checked here; an unverified deferred proof is a correctness risk, not a circularity. The companion-paper citations to [FGV25] are used to connect the quantum formalism to classical functional models and to prove equivalence of two p-separation definitions; these are supported by proofs in Appendices C.3 and C.4 rather than used as unverified authority. There is no fitted parameter renamed as a prediction, no uniqueness theorem imported from the authors, and no empirical claim that would reduce to its own input. The main caveat is that the completeness proof is not available for inspection in the provided text, but that does not constitute circularity.
Assumptions & free parameters
assumptions (4)
- standard math Finite-dimensional quantum mechanics with CPTP maps, POVMs, and the Born rule.
- standard math d-separation soundness and completeness for acyclic causal models (Theorem 20, citing HLP14).
- domain assumption The probability of a cyclic causal model is given by conditioning on successful post-selection in a teleportation graph (Definition 12); models with zero success probability are inconsistent.
- domain assumption Each edge carries its own tensor factor, and all systems are finite-dimensional.
Cite this review
Pith. "Pith review of Cyclic quantum causal modelling with a graph separation theorem." pith.science (2026). https://pith.science/paper/Z55Z6PUA
@misc{pith2026250204168,
author = {Pith},
title = {Pith review of: Cyclic quantum causal modelling with a graph separation theorem},
year = {2026},
howpublished = {\url{https://pith.science/paper/Z55Z6PUA}},
note = {Machine review of arXiv:2502.04168}
}
read the original abstract
Causal modelling frameworks link observable correlations to causal explanations, which is a crucial aspect of science. These models represent causal relationships through directed graphs, with vertices and edges denoting systems and transformations within a theory. Most studies focus on acyclic causal graphs, where well-defined probability rules and powerful graph-theoretic properties like the d-separation theorem apply. However, understanding complex feedback processes and exotic fundamental scenarios with causal loops requires cyclic causal models, where such results do not generally hold. While progress has been made in classical cyclic causal models, challenges remain in uniquely fixing probability distributions and identifying graph-separation properties applicable in general cyclic models. In cyclic quantum scenarios, existing frameworks have focussed on a subset of possible cyclic causal scenarios, with graph-separation properties yet unexplored. This work proposes a framework applicable to all consistent quantum and classical cyclic causal models on finite-dimensional systems. We address these challenges by introducing a robust probability rule and a novel graph-separation property, p-separation, which we prove to be sound and complete for all such models. Our approach maps cyclic causal models to acyclic ones with post-selection, leveraging the post-selected quantum teleportation protocol. We characterize these protocols and their success probabilities along the way. We also establish connections between this formalism and other classical and quantum frameworks to inform a more unified perspective on causality. This provides a foundation for more general cyclic causal discovery algorithms and to systematically extend open problems and techniques from acyclic informational networks (e.g., certification of non-classicality) to cyclic causal structures and networks.
Figures
Forward citations
Cited by 3 Pith papers
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Cyclic functional causal models beyond unique solvability with a graph separation theorem
Cyclic functional causal models over finite variables get a unique probability rule and a sound and complete graph-separation property (p-separation) that reduces to d-separation in acyclic graphs.
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An equivalence between time-symmetry and cyclic causality in quantum theory
Every multi-time quantum state can be simulated by a post-selected closed timelike curve circuit with open slots, and vice versa.
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Events and their Localisation are Relative to a Lab
Events, their localisation, and even conclusions about indefinite causal order in the quantum switch are shown to depend on the choice of a Lab and its reference degrees of freedom.
Reference graph
Works this paper leans on
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[1]
A causal structure which corresponds to a directed graphG with verticesA1,...,A N
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[2]
For eachAi, a quantum channel (in CJ representation) σAi|Pa(Ai)∈L HAin i ⊗ ⨂ Ak∈Pa(Ai) H∗ Aout k where Pa(Ai) denotes the set of all parents ofAi in G, such that [σAi|Pa(Ai),σ Aj|Pa(Aj )] = 0 for alli,j and σA1,...,AN =∏N i=1σAi|Pa(Ai) is a valid process operator. Note that the original papers [BLO20, BLO21] useρAi|Pa(Ai) rather thanσAi|Pa(...
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[3]
A causal structure which corresponds to directed graphG with verticesA1,...,A N
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[4]
For eachAi, a tensor factorisation of its output space asHAout i =⨂ Ak∈Ch(Ai)HAout ik where Ch(Ai) denotes the set of all children ofAi inG andHAout ik are arbitrary finite dimensional Hilbert spaces
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[5]
For eachAi, a quantum channel σAi|Pa(Ai)∈L HAin i ⊗ ⨂ Ak∈Pa(Ai) H∗ Aout ki such thatσA1,...,AN =∏N i=1σAi|Pa(Ai) is a valid process operator. Since each channelσAi|Pa(Ai) acts on a distinct Hilbert space (distinct tensor factors of the parental Hilbert space), the commutation condition[σAi|Pa(Ai),σ Aj|Pa(Aj )] = 0 is satisfied for alli,j. H...
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[6]
A random variableXv taking valuesxv from a non-empty finite setXv. We will use a notation where ifV′⊆V is a non-empty subset of vertices, XV′ = ∏ v∈V′ Xv (130) where∏here denotes the Cartesian product
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An error random variableUv taking values uv from a finite setUv, distributed as pv :Uv↦→[0, 1]
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A functionf v :XPa(v)×U v↦→Xv. In literature, the probability is considered to be well-defined for all functional models on acyclic graphs [Pea09] and for a restricted set ofcyclic functional models [FM17, BFPM21]. In definition 12 of [FGV25], we provided a probability rule over the vertices of any — except for a handful of pathological cases — cyclic fun...
Show all 29 references
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[9]
To each edgee∈ E = (v,v′) associate the finite dimensional Hilbert spaceHe = H(Xv), whereH(Xv) is defined as in equation(1)
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[10]
To each error variableUv of v∈ V associate the finite dimensional Hilbert space H(Uv), whereH(Uv) is defined as in equation(1), and the state σv = ∑ uv∈Uv pv(uv)|uv⟩⟨uv|H(Uv); (135)
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[11]
To each vertexv∈V associate the finite setXv as outcome set
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[12]
To each vertexv∈V associate the set of CP maps { Mv x :L ( HIn(v) ) ↦→L ( HOut(v) )} x∈Xv (136) defined as follows: for allρ∈L (HIn(v)), Mv x(ρ) = Tr [ Ef v x (ρ⊗σv) ] ⨂ e∈Out(v) |x⟩⟨x|He, (137) where{Ef v x }x∈Xv is the POVM obtained through applying definition 35 to the func...
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[13]
Choose any subgraphG′ := (V′,E′) of G = (V,E ) with V′ = V and E′⊆ E, such that G′ is acyclic
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[14]
Include inGtp all the vertices and edges of the subgraphG′ associated with the same vertex types (observed or unobserved) and edge types (classical or quantum) as the original causal graphG
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[15]
Denoting the set of so-called split edgesEs(Gtp) := E\E′, for each edge(vi,v′ i)∈ Es(Gtp), include inGtp, two verticesTi andRi and three edges(vi,T i), (Ri,T i) and (Ri,v′ i)
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[16]
Outgoing edges fromRi, (Ri,T i) and (Ri,v′ i), are quantum edges
The vertexTi is observed andRi unobserved. Outgoing edges fromRi, (Ri,T i) and (Ri,v′ i), are quantum edges. The edge(vi,T i) is of the same type of the edge(vi,v′ i) in the original causal graphG. Gtp constructed in this manner is thus a causal graph. It will be useful to ref...
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We refer toVs(Gc,tp) as the split vertices ofGc,tp
Choose any subset of verticesVs(Gc,tp)⊆V, such that the subgraphG′ = (V′,E′) of G with V′ =V and E′ =E\ (⋃ v∈Vs(Gc,tp) Out(v) ) is acyclic. We refer toVs(Gc,tp) as the split vertices ofGc,tp
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The verticesVc,tp of Gc,tp consist of the verticesV of the original graphG together with new verticesRv,T v for each split vertexv∈Vs(Gc,tp), i.e., Vc,tp =V∪{Rv}v∈Vs(Gc,tp)∪{Tv}v∈Vs(Gc,tp). (144)
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The edgesEc,tp of Gc,tp consist of the edgesE′ of the subgraphG′ together with the following new edges: Ec,tp =E′∪{ (v,T v)}v∈Vs(Gc,tp)∪{ (Rv,T v)}v∈Vs(Gc,tp) ∪{ (Rv,v′)}v∈Vs(Gc,tp),v′∈Ch(v)G, (145) where Ch(v)G refers to the children vertices ofv in the graphG. For eachv ∈ Vs...
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To each vertex present in bothGc,tp andG′ c,tp assign the same finite set, func- tional dependency (for endogenous vertices) or probability (for exogenous ver- tices), of the causal modelfCmGc,tp 24
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[21]
Notice that this amounts to having the variable associated to pre-selection verticesRe v uniformly distributed (for more details see the remark in section 2 of [FGV25])
For allv∈ Vs(Gc,tp) and e∈ Out(v), associate toRe v the finite setXRev =Xv and a uniform distribution to its error variablepRe v(u) =|Xv|−1 for allu∈X v and a functionf Re v(u) = u. Notice that this amounts to having the variable associated to pre-selection verticesRe v unifor...
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Hence, each outgoing edge from a vertexv∈Vs(Gc,tp) in G has been replaced with a uniform prior classical teleportation protocol (definition 6 in [FGV25])
For allv∈Vs(Gc,tp) ande∈ Out(v), associate toT e v the finite setXT ev ={0, 1} and a (deterministic) delta functionf T e v (x,y ) =δx,y for allx,y∈X v. Hence, each outgoing edge from a vertexv∈Vs(Gc,tp) in G has been replaced with a uniform prior classical teleportation protoc...
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24By construction, these are also the same association of the original causal modelfCmG on G
To each vertex that is present in bothG and G′ c,tp, i.e.,v∈ V, we associate the set of CP maps acting onρ∈L (HIn(v)) Mv x(ρ) = Tr [ Ef v x (ρ⊗σv) ] ⨂ e∈Out(v) |x⟩⟨x|He (175) where{Ef v x }x∈Xv is the POVM obtained through applying definition 35 to the functionf v associated t...
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For allv∈Vs(Gc,tp) and e∈ Out(v), associate toT e v and outcomete v = 1 the map acting onρ∈L ( HIn(T ev ) ) ∼=L ( H(v,T ev )⊗H (Rev,T ev ) ) MT e v 1 (ρ) = Tr [ Ef T ev t=1ρ ] = Tr ∑ x∈Xv ( |x⟩⟨x|H(v,T ev )⊗|x⟩⟨x|H(Rev ,T ev ) ) ρ , (176) where Ef T ev t=1 is the POVM el...
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(177) Lemma 37 establishes that probabilities are preserved through mapping an acyclic functional model to an acyclic causal model through definition 36
For all v ∈ Vs(Gc,tp) and e = (v,v′) ∈ Out(v), associate to Re v, which is considered unobserved, the state σRe v = ∑ x∈Xv 1 |Xv| ⨂ e∈Out(Rev) |x⟩⟨x|He = ∑ x∈Xv 1 |Xv||x⟩⟨x|H(Rev ,T ev )⊗|x⟩⟨x|H(Rev ,v′). (177) Lemma 37 establishes that probabilities are preserved through mapp...
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For all vertices which are preserved fromG toG′ c,tp (andGtp),v∈V the maps associated in CmGtp and CmG′c,tp are equal (see equations (175) and (179)), i.e., for allρ∈L (HIn(v)) it holds ˜Mv x(ρ) =Mv x(ρ) (186) where we recall that ˜Mv x is the map associated tov in CmGtp andMv...
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In order to evaluate probabili- ties, the maps ofCmGtp are composed as described in definition 3
For all post-selection vertices, i.e., for allv∈Vs(Gc,tp) ande∈ Out(v), we have (by the decoherence condition for observed vertices in definition 2) ˜MT e v ✓ ◦ ˜Mv x = ˜MT e v ✓ ◦D (v,T ev )◦ ˜Mv x (187) 76 where all maps ˜M are in CmGtp andD(v,T ev ) is a decohering channel ...
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42= ˜Mv′ x′◦D (Rev,v′)◦ ˜σRe v (191) where˜denotes associations in CmGtp andD(Rev,v′) is a decohering channel
For all pre-selection vertices, i.e., for allv∈Vs(Gc,tp) and e = (v,v′)∈ Out(v) we have ˜Mv′ x′◦ ˜σRe v lem. 42= ˜Mv′ x′◦D (Rev,v′)◦ ˜σRe v (191) where˜denotes associations in CmGtp andD(Rev,v′) is a decohering channel. In order to evaluate probabilities, the maps ofCmGtp are ...
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D Proofs of section 4 Lemma 7(Acyclicity of teleportation graphs)
and PracycGtp in CmGtp (Step 5). D Proofs of section 4 Lemma 7(Acyclicity of teleportation graphs). Each directed graphGtp obtained from a directed graphG as described in definition 6 is acyclic. Proof. In the first step of definition 6, the graphG′ is acyclic. The second step...
Reviewed August 8, 2026 · model on record in the stance chip above.
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