{"id":"1284575c-0e9e-4c50-9454-ee57486c9ffa","arxiv_id":"2502.04168","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper introduces p-separation, a graph-separation property that is sound and complete for all finite-dimensional cyclic quantum and classical causal models, with probabilities defined via post-selected teleportation.","lead":"This paper introduces a framework that assigns probabilities to quantum systems with causal loops by unfolding cyclic causal graphs into acyclic ones with post-selected teleportation. It also defines a graph property, p-separation, and proves it is sound and complete for these models.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Completeness of Theorem 22 rests on the deferred Appendix E proof; it must construct consistent models with nonzero post-selection probability and conditional dependence for every p-connection, which is not yet verifiable.","rationale":"The paper makes a valuable and coherent contribution: it defines a probability rule for cyclic quantum causal models via post-selected teleportation, proves the rule is independent of the teleportation implementation and graph choice, and introduces p-separation with a claimed sound and complete theorem. The soundness half of Theorem 22 is secure: it follows directly from Definition 12 and the acyclic d-separation theorem applied to the teleportation graph, and conditioning on the success outcome preserves the independence. The completeness half is the load-bearing element. It is deferred to Appendix E, which was not visible in the reviewed text. The construction must overcome three non-trivial constraints: (i) the acyclic model on Gtp must be induced by a cyclic model, meaning the pre- and post-selection vertices implement a valid post-selected teleportation protocol rather than arbitrary mechanisms; (ii) the conditional dependence must occur for the success outcome ✓, not only for ✗; and (iii) the post-selection success probability must be strictly positive, since p✓=0 models are declared inconsistent and have no distribution. Any failure of these constraints for some p-connected triple would falsify the completeness claim. The reader's conditional verdict with moderate confidence is therefore appropriate. Our proposed check—reproducing the construction on concrete graphs and numerically searching small cyclic models—would settle whether the completeness proof holds. We do not see an internal inconsistency in the main text, and we credit the paper for making its assumptions explicit, but the central claim is only as strong as the Appendix E proof.","tokens_in":57541,"tokens_out":12618,"duration_ms":126032,"concrete_test":"Reproduce the completeness construction of Appendix E for the p-connected graph G of Eq. (74) with V3=∅: explicitly choose the classical functional model (x1=x2⊕x3, x2=x1⊕x4) embedded via Definition 36, compute the cyclic distribution using Proposition 14, and verify p✓>0 and Pr(x3,x4)≠Pr(x3)Pr(x4). Then run a numerical search over small random causal models (e.g., two-vertex cycles with random CPTP maps and random post-selected teleportation protocols satisfying Definition 4) to check that every p-connection admits at least one consistent model with dependence, and that the soundness condition holds for all p-separations.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The soundness direction of Theorem 22 is a direct consequence of Definition 12 and the acyclic d-separation theorem: if V1⊥pV2|V3 holds via some Gtp, then in that acyclic model X1⊥X2|X3,Tpost, and conditioning on Tpost=✓ preserves the independence. The completeness direction, however, requires that for every p-connected triple (V1,V2,V3), there exists a causal model CmG with p✓>0 in which X1 and X2 are dependent given X3. This is not automatic. A d-connecting path in every Gtp only ensures that the acyclic d-separation theorem can supply a model on Gtp with dependence; that model must then be realizable as a teleportation model of a cyclic model, meaning the mechanisms at R and T must implement a valid post-selected teleportation protocol (Definition 4), and the dependence must survive conditioning on the success outcome ✓ rather than only on ✗. Moreover, p✓ must be strictly positive; models with p✓=0 are declared inconsistent and have no probability distribution. If the Appendix E construction fails on any of these points for some p-connection, completeness collapses. The paper's own framework defines consistency through the same post-selected teleportation probability rule, so the theorem is relative to that postulate; under an alternative consistency condition (e.g., Deutsch's D-CTCs) p-separation is not expected to be sound. But the immediate unverified step is the completeness proof, which is deferred and not available for inspection.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":57765,"tokens_out":8857,"duration_ms":83683,"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":[{"comment":"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.","section":"§5.3, Theorem 22"},{"comment":"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.","section":"§4.3 Definition 12 and §5.3 Definition 21"},{"comment":"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.","section":"§4.3 Definition 12"},{"comment":"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.","section":"Appendices D and E"}],"minor_comments":[{"comment":"The references [LMGP+11a] and [LMGP+11b] are identical (same title and DOI); please correct the duplication or differentiate the two entries.","section":"References"},{"comment":"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.","section":"Definition 8(2)(a)"},{"comment":"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.","section":"§5.2"},{"comment":"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.","section":"Abstract"},{"comment":"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.","section":"Figure 1"},{"comment":"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.","section":"§5.4"}],"recommendation":"major_revision","confidential_remarks":"To the editor: The manuscript is ambitious and likely of interest to the quantum foundations and causal modelling communities. The main concern is the deferred Appendix E proof of Theorem 22; if it is valid and included in the arXiv version, the paper could become acceptable after revision. Please ensure the appendices are part of the review package, and note the duplicated reference. I would recommend asking the authors to provide the full proof and to clarify the scope of the consistency postulate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"I read this with real interest. It's the first general graph-separation theorem for cyclic quantum causal models that I know of, and the paper mostly delivers. The technical core is solid: the probability rule via post-selected teleportation is proven well-defined, independent of the choice of teleportation graph and protocol, and the self-testing results on post-selected teleportation are of independent value. The definitions are careful, the examples are illuminating, and p-separation correctly reduces to d-separation on DAGs while capturing the classic Neal feedback counterexample. That is a real contribution to the literature.\n\nThe principal caveat is the one the stress-test flags: the proof of Theorem 22 lives in Appendix E, which was not visible. The soundness direction is straightforward given the definitions. Completeness is the load-bearing claim—it requires constructing, for every p-connection, a teleportation model with nonzero post-selection success and genuine conditional dependence. That is not automatic, and I cannot verify it from the main text. If the construction works, the theorem is a genuine advance; if it fails, the framework still has value but the headline claim is weaker.\n\nThe second caveat is the modeling status of the probability rule. Definition 12 postulates that the cyclic distribution is the post-selected distribution of an acyclic teleportation model. If you adopt a different consistency condition for causal loops, such as Deutsch's D-CTCs, p-separation would not be expected to hold. The paper is upfront about this, but it means the theorem is relative to the framework's own consistency criterion, not a fully theory-independent statement. The circularity concern about defining p-separation through the same teleportation graph family is real but, in my view, not fatal: the soundness direction is partly built in, but the completeness direction carries the content, and that is where the scrutiny should go.\n\nThis paper is for quantum foundations people working on causal modelling and, through the companion classical paper, for the classical causality community. It deserves a serious referee. I would send it to peer review with a request to include the appendix proof or a detailed sketch, and to discuss how the post-selected teleportation rule relates to other consistency conditions for causal loops. I'd be willing to write the report.","headline":"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.","tokens_in":58358,"tokens_out":2424,"would_cite":true,"duration_ms":24977,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper establishes p-separation as a sound and complete graph-theoretic criterion for all cyclic quantum and classical causal models in its framework.","keywords":["cyclic causal models","quantum causal models","p-separation","graph separation theorem","post-selected teleportation","causal loops","closed timelike curves","conditional independence"],"falsifier":"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.","tokens_in":57271,"feed_emoji":"🌀","tokens_out":5902,"duration_ms":60220,"temperature":0.7,"pith_summary":"The paper aims to give cyclic directed causal graphs the same graph-theoretic handle that d-separation gives to acyclic ones. It proposes a probability rule for any finite-dimensional quantum, and by embedding also classical, causal model on a cyclic graph, by simulating each missing edge with a post-selected teleportation protocol and conditioning on the protocol's success. It then defines a new graph-separation relation, p-separation, and proves it sound and complete: p-separated vertex sets are conditionally independent in every causal model on the graph, while p-connected vertex sets are dependent in at least one causal model. If right, this opens causal discovery and non-classicality arguments to cycles and feedback rather than requiring acyclicity. The load-bearing cost is an explicit modeling assumption: a cyclic model is consistent only when the defining post-selection succeeds with nonzero probability.","feed_headline":"p-separation: one graph rule for causal loops","feed_subtitle":"It reads conditional independences from cyclic graphs, holding for every finite-dimensional causal model in the framework.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the original quantum teleportation protocol used as the canonical post-selected teleportation implementation.","marker":"[BBC+93]"},{"why":"Supplies the acyclic quantum causal-model probability rule and d-separation theorem that the paper extends to cyclic graphs.","marker":"[HLP14]"},{"why":"Supplies a known uniquely solvable classical cyclic model where d-separation soundness fails, motivating a new separation criterion.","marker":"[Nea00]"},{"why":"Supplies the loop-composition or self-cycle composition formalism used to write the cyclic probability rule independent of teleportation choices.","marker":"[PMM+17]"},{"why":"Supplies the post-selected closed-timelike-curve model that motivates the probability rule and is given a causal-model semantics here.","marker":"[LMGP+11a]"},{"why":"The companion classical functional-model paper whose classical post-selection formulation and p-separation definition are shown equivalent to the quantum ones.","marker":"[FGV25]"}],"fun_headline_variants":["One graph rule for causal loops: p-separation","p-separation: sound and complete for cyclic causal models","Cyclic quantum causality gets a separation rule","A complete separation rule for cyclic quantum causality"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["One graph rule for causal loops: p-separation","p-separation: sound and complete for cyclic causal models","Cyclic quantum causality gets a separation rule","A complete separation rule for cyclic quantum causality"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001094,"raw_usage":{"total_tokens":4630,"prompt_tokens":1069,"completion_tokens":3561,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":685,"completion_tokens_details":{"reasoning_tokens":3501}},"tokens_in":685,"tokens_out":3561,"duration_ms":25331,"temperature":1.0,"reasoning_tokens":3501,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T23:16:52.753896+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}