Decoherence to quantum theory from a causally-indefinite post-quantum theory

These are equations, theorem steps, or proof moves where math specialists identified compressed or unverified derivations. They are shown once here as the canonical deduplicated list so the review stays auditable without repeating the same flags in every math specialist report.

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  Lemma 2 / Appendix C (equivalence Hypdec ≃ CPTP) - Key parts are asserted without a reproducible proof: (i) that F is well-defined on Split(QBox) morphisms (i.e., respects the idempotent constraints), and (ii) especially faithfulness—claiming that equality after inserting hyper-decoherence implies equality of original supermaps—requires a separation/tomography argument not provided.

  If wrong: Definition 11’s equivalence requirement is unmet; QBox would not be shown to hyper-decohere to CPTP, invalidating Theorem 1.

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  Lemma 2 / Appendix C (Equivalence Hypdec ≃ CPTP), faithfulness step - Faithfulness is argued as: if two supermaps have equal hyper-decohered versions then the original supermaps are equal. In Split(D), morphisms are already constrained to satisfy f=e'fe; equality after inserting idempotents does not generally imply equality of underlying maps without an additional characterization theorem.

  If wrong: The claimed equivalence to CPTP may fail; then the hyper-decohered subtheory might be a proper quotient/subcategory, and Theorem 1 ('QBox is post-quantum') is unsupported.

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  Lemma 2 and Appendix C, equivalence Hypdec ≃ CPTP - Lemma 2 and Appendix C sketch the functor F and assert full faithfulness; the faithfulness step relies on diagrammatic equalities rather than an explicit algebraic proof that hyper-decohered supermaps are completely determined by their image.

  If wrong: If F is not faithful or not full, the hyper-decohered subtheory need not be equivalent to CPTP, so the main hyper-decoherence claim fails.

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  Lemma 2 and Appendix C, proof of equivalence Hypdec ≃ CPTP - The equivalence functor F is defined diagrammatically and its properties are described as straightforward. The proof does not fully characterize morphisms satisfying f=e'fe or prove algebraically that equality of lower-order images implies equality of hyper-decohered supermaps.

  If wrong: If F is not full, faithful, monoidal, or well-defined on all hyper-decohered morphisms, the decohered subtheory is not proven equivalent to CPTP, invalidating the main theorem.

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  Lemma 3 (Ax3 purity copreservation) - Step from g(Id/d)=|φ><φ| to ‘for any orthonormal basis, g(|e_i><e_i|)=|φ><φ|, hence g is constant on every state’ is incomplete as written; dependence on off-diagonal terms/coherences is not explicitly ruled out using complete positivity/linearity on all operators.

  If wrong: Ax3 may fail; then hypdec would not qualify as a hyper-decoherence map under Definition 11, undermining Theorem 1.

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  Lemma 3, equations around g(I/d)=|φ><φ| - The proof that a preimage of a pure state is pure infers from g(I/d)=|φ><φ| that g is a constant discard-and-prepare channel, then uses Choi extremality. The argument is compressed for general multipartite non-signalling channels and does not explicitly prove extremality in the QBox deterministic state space.

  If wrong: If a pure hyper-decohered quantum state can arise from a mixed/extremally non-pure QBox state, Ax3 fails and QBox would not satisfy the paper's own definition of post-quantum hyper-decoherence.

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  Lemma 4, claimed maximally mixed QBox state - Lemma 4 states the maximally mixed state in QBox and says it is easily sent to itself, but does not prove that every deterministic QBox state appears in a convex decomposition of it or that it is invariant under all reversible QBox transformations.

  If wrong: If the displayed state is not maximally mixed under Definition 8, Ax4 is not proved and Theorem 1 does not follow.

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  Section V, definition of hypdec as a QBox process - hypdec is asserted to be a deterministic QBox process because it maps product CPTP maps to a CPTP map, but Definition 6/Definition 5 require preservation of arbitrary non-signalling channels and stability under ancillary systems; this stronger condition is not explicitly verified.

  If wrong: If hypdec fails the full Definition 5/6 determinism condition over arbitrary ancillas and non-signalling inputs, it would not be a legitimate QBox process and Theorem 1 (QBox is post-quantum) would be unsupported.

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  Section V, definition of hypdec before Lemma 1 - The text asserts that hypdec is a deterministic superchannel because hypdec(f1,...,fn) is CPTP when applied to CPTP maps. Definition 6/Definition 5 require preservation of non-signalling channels, with ancillary stability; the stronger condition is not explicitly verified.

  If wrong: If this sufficiency criterion is invalid, hypdec may not be a deterministic process of QBox, and the construction of the hyper-decoherence map fails at the first step.

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  Section V: claim 'hypdec is deterministic ... maps CPTP maps f_i to CPTP' - Determinism in Definition 5 is stronger than 'maps CPTP inputs to CPTP output': it quantifies over ancillas L and all allowed discarding effects in Σ_{K⊗L}. The text does not show hypdec satisfies this stronger condition in QBox’s multi-environment structure.

  If wrong: hypdec may not be a morphism in the deterministic subtheory QBox (as required), undermining Lemma 1 onward and invalidating the main construction.

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  Appendix A (No superluminal signalling in QBox) - Uses a cited result that any multipartite non-signalling channel can be written as an affine linear combination of localized channels. The argument then linearly evaluates effects across this affine combination. It is not shown that this affine expansion is legitimate within the state/effect semantics of QBox (affine coefficients may be negative, leaving the state cone).

  If wrong: The claim that QBox is non-signalling (in the stated multi-environment sense) would not be established; this would weaken consistency of the operational interpretation and could affect which maps qualify as deterministic in QBox.

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  Appendix A, use of [37] affine decomposition - Appendix A invokes an affine decomposition of multipartite non-signalling channels into localized channels and manipulates the resulting affine combination, but does not spell out the legitimacy of this extension for the process-theoretic semantics.

  If wrong: If the affine decomposition cannot be used in the required categorical/state-space setting, the Appendix A proof of no-superluminal-signalling for general QBox bipartite states is incomplete.

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  Appendix B, non-unique purifications example - Appendix B shows two bipartite states with the same reduction and argues no reversible comb relates them, but does not explicitly prove that both states are purifications, i.e. pure in the Definition 7 convex-extremal deterministic state space.

  If wrong: If the displayed bipartite states are not pure/extremal deterministic QBox states, Appendix B does not establish non-unique purifications in the paper’s stated sense, weakening the claimed explanation of how Lee–Selby is evaded.

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  Lemma 1 (no-backwards-signalling of hypdec) - The proof is diagrammatic and asserts that for any multi-environment element W (process matrix), discarding after hypdec yields the same input discard. This does not explicitly track the quantification in Definition 4 over all Σ_K effects nor show the required equality for arbitrary allowed discards on the output.

  If wrong: Ax2 would not be established for hypdec; then Definition 11 is not satisfied and Theorem 1 fails.

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  Lemma 1 (no-backwards-signalling) + Definition 4 quantifiers - No-backwards-signalling is asserted diagrammatically ('for any process matrix W') without explicitly matching Definition 4's requirement: existence of an input discard in Σ_H such that for every output discard in Σ_K the stated equality holds. The proof sketch does not show the existential choice on the input side nor its dependence/independence from W.

  If wrong: Ax2 might not actually be satisfied by hypdec as defined; then the proposed hyper-decoherence would fail the stated axioms and Theorem 1 would not follow.

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  Lemma 1, no-backwards-signalling diagram - No-backwards-signalling is shown by a single diagrammatic equality for arbitrary process matrix W, without an explicit algebraic derivation from the process-matrix normalization constraints.

  If wrong: If the equality does not hold for all process-matrix effects with the required typing, Ax2 no-backwards-signalling is not established for the hyper-decoherence map.

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  Lemma 2 / Appendix C (faithfulness) - Full faithfulness and essential surjectivity of F: Hypdec -> CPTP are argued diagrammatically; the algebraic characterization of hyper-decohered QBox morphisms needed to make the equivalence reproducible is not spelled out.

  If wrong: If the equivalence fails, the hyper-decohered subtheory would not be CPTP and Theorem 1's central identification with quantum theory would be invalid.

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  Lemma 3 - The inference that g must be a constant discard-and-prepare channel is argued only via diagonal basis projectors g(I/d)=|phi><phi|; dependence on off-diagonal coherences is not explicitly excluded, and the extension from CPTP extremality to extremality in the multipartite non-signalling deterministic state space is compressed.

  If wrong: If the extension does not hold, Ax3 (purity copreservation) would not be established and Theorem 1 would lose one of its four required axioms.

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  Lemma 3 (purity copreservation), step: g(I/d)=|φ⟩⟨φ| ⇒ g is constant prepare |φ⟩⟨φ| - The proof uses decomposition of I/d into diagonal basis projectors and extremality of |φ⟩⟨φ| to infer g(|e_i⟩⟨e_i|)=|φ⟩⟨φ|, then asserts this 'applies to every state' without explicitly treating off-diagonal operators/coherences or using a full operator-basis linearity argument under CP/TP constraints.

  If wrong: Ax3 may fail: a non-pure QBox state could hyper-decohere to a pure quantum state, contradicting the claimed copreservation of purity and weakening Theorem 1.

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  Lemma 3, inference from g(I/d)=|φ><φ| to extremality of g - Lemma 3 compresses the argument from a pure hyper-decohered output state to g being a constant pure-state preparation channel, and then from Choi extremality in CPTP to purity in the QBox deterministic state space.

  If wrong: If the inferred channel is not necessarily constant discard-and-prepare, or if extremality in CPTP does not transfer to the relevant QBox state space, Ax3 is not fully established.

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  Lemma 4 - The maximally mixed state in QBox is simply asserted to be the displayed discard/prepare diagram and 'easily seen' to be preserved; the required decomposition and invariance under all reversible QBox transformations are not shown.

  If wrong: If the displayed state is not maximally mixed in QBox under Definition 8, or if hypdec does not preserve the appropriate maximally mixed state, Ax4 is not established.

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  Lemma 4 (maximally mixed preserved) - The maximally mixed state in QBox is asserted to have a particular diagrammatic form and to satisfy Definition 8 (universality across convex decompositions + invariance under all invertibles) without proof.

  If wrong: Ax4 may not hold; then the proposed hypdec would not satisfy the hyper-decoherence axioms and Theorem 1 would not follow.

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  Lemma 4 (maximally mixed state in QBox and its preservation) - The maximally mixed state of QBox is asserted diagrammatically and claimed to map to the quantum maximally mixed state under the equivalence. Given Definition 8’s strong requirement (every deterministic state appears in some decomposition and invariance under invertibles), this identification is not proved.

  If wrong: Ax4 may not hold; without Ax4 the hyper-decoherence axioms are not satisfied and Theorem 1 would be unsupported.

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  Appendix A (No superluminal signalling in QBox) using [37] - Uses an affine linear combination decomposition of non-signalling channels into local channels to conclude equality under different discard effects. It is not shown that these affine combinations are legitimate within the deterministic state space notion used (affine combinations can exit the physical cone unless carefully interpreted as equalities in an ambient ordered vector space).

  If wrong: Non-signalling of QBox (as formulated here) might not be established by the given argument, potentially affecting claims about the tensor product/multi-environment structure but not directly Ax1–Ax4 unless used implicitly.

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  Appendix A (no-signalling proof) - Uses the affine decomposition of multipartite non-signalling channels [37]; it is not shown that the required affine/linear extension remains within the physical state cone in the categorical state semantics used.

  If wrong: If the affine combination leaves the physical state cone, the no-superluminal-signalling property of QBox (a supporting claim) would be unproven, though this is peripheral to Theorem 1.

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  Appendix A, use of [37] in no-superluminal-signalling proof - Appendix A invokes the cited affine decomposition of multipartite non-signalling channels and then applies arbitrary QBox effects to the localized decomposition. The cited result is legitimate, but the matching of hypotheses to the present multi-environment effects is compressed.

  If wrong: If the affine decomposition does not apply with the required arity, quantum specialization, or effect structure, the proof that QBox is non-superluminal-signalling is incomplete.

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  Appendix A: extension of no-signalling proof from atomic to general bipartite states via affine decomposition from [37] - The proof uses an affine decomposition of arbitrary-arity multi-partite non-signalling channels into localised quantum channels from [37]. It is not explicitly verified that the affine combination respects the categorical semantics and physical state space of QBox, though the cited result is intended exactly for this purpose in generalised probabilistic theories.

  If wrong: If the decomposition fails or extends outside the state space, the no-signalling property might not be established for general bipartite states in QBox. This does not affect the hyper-decoherence construction (the main result), only the auxiliary claim that QBox is non-signalling.

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  Appendix B (Non-uniqueness of purifications in QBox) - Constructs two purifications using unitaries U0,U1 and argues no reversible comb can connect them. The proof sketches equalities by plugging classical values 0/1; it assumes the reversible comb acts in a way that forces equality of the induced channels F(U0)=F(U1). The categorical typing/allowed interventions that justify these substitutions are not fully spelled out.

  If wrong: The narrative explanation for how Lee–Selby is evaded (via non-unique purifications) would be weakened, though Theorem 1 could still stand if Ax1–Ax4 and Lemma 2 were fully proven.

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  Appendix B, non-unique purification example - Appendix B gives a diagrammatic proof of non-unique purifications, but the typing and purity/extremality of the displayed bipartite states and the characterization of reversible environment transformations are not fully spelled out.

  If wrong: If the displayed states are not valid pure QBox states or the reversible-comb argument is incomplete, the discussion of non-unique purifications would be unsupported, though the main theorem primarily needs failure of the Lee–Selby assumptions rather than this exact example.

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  Lemma 3: inference from g(|e_i⟩⟨e_i|) = |φ⟩⟨φ| for basis projectors to g being constant prepare-|φ⟩⟨φ| - The argument uses only an orthonormal basis and appeals to extremality. An explicit extension from basis states to all input states would require closure under convex combinations and the fact that CPTP maps are affine. This is standard and expected to hold, but the text compresses this step.

  If wrong: If the extension genuinely fails, Lemma 3's conclusion that the pre-image of a pure state is pure might be unsupported for certain channels, weakening the verification of Ax3.