Exact classification of elementary channel divisibility

Frederik vom Ende, A Sufficient Criterion for Divisibility of Quantum Channels, arXiv:2407.17103v3, Section V, asks:

An open question at this point is whether our sufficient condition is also necessary, and whether our necessary condition is also sufficient.

The source is arXiv v3, with publication identifier DOI 10.1063/5.0231812. Definition 1 and Lemmas 2–3 specify the elementary channel and inverse; Proposition 1 and Theorem 1 specify the necessary and sufficient conditions. This dossier anchors the combined classification of the two reversals, including their fixed-pair and existential-pair readings.

Let the complex dimension be any finite n ≥ 2, let Φ be an actual CPTP map, and let (Kⱼ) be any finite Kraus representation of Φ. For an ordered pair of orthogonal unit vectors (x,y), E(x,y) means that Φ composed on the right with the inverse of the elementary channel Ψₜ(x,y) is completely positive for some real 0 < t < 1. N(x,y) means ⟨y,KⱼGx⟩ = 0 for every index j and every complex matrix G in the Hilbert–Schmidt orthogonal complement of the whole complex Kraus span. S(x,y) means N(x,y) together with the additional diagonal tests ⟨x,KⱼGx⟩ = 0 over the same indices and complement.

The classification established by the designated result has four cells:

| Reading | Answer | | --- | --- | | Fixed ordered pair: N(x,y) iff E(x,y) | Proved for every map and representation in the stated scope. | | Existential ordered pair: some N pair iff some E pair | Proved for the same map and representation. | | Fixed ordered pair: E(x,y) implies S(x,y) | Refuted in every dimension n ≥ 2. | | Existential ordered pair: some E pair implies some S pair | Refuted in every dimension n ≥ 2; the counterexample has no S pair in any finite Kraus representation. |

The Scribe claim's Proved kind refers to this combined classification, whose statement includes the counterexample. It does not label either E ⇒ S assertion as proved.

The necessary test and the stronger sufficient test offer different Kraus-space descriptions of elementary right division. Identifying which one is exact determines whether the diagonal tests are essential to existence of a divisor. The formal anchor is D5/S3/Quantum/QuantumChannels/ElementaryChannelDivisibility.result.

The source supplies E ⇒ N and S ⇒ E. The converse N ⇒ E must cover singular Choi matrices, and failure of E ⇒ S at one chosen pair does not settle the existential reading. The classification therefore also requires a channel with an E pair and no S pair at all, uniformly in dimension and independently of its finite Kraus representation.

Use the source's input-first Choi convention and the complex Hilbert–Schmidt complement. With A = C(Φ) and Z = (|y⟩⟨x|)ᵀ ⊗ I, the necessary tests give the support inclusion ker A ⊆ ker(ZA Z*). Finite-dimensional positive-semidefinite domination then provides t > 0 with A − t ZA Z* positive semidefinite. The exact inverse-filter congruence is the Choi matrix of the residual. Both inverse identities and trace preservation give an actual residual CPTP right factorization.

For distinct basis vectors eᵣ,eₛ, take Φ = Ψ₁₅⁄₁₆(eᵣ,eₛ) and Q = Ψ₃⁄₄(eᵣ,eₛ). The identity Q composed with Q = Φ gives an E pair. The diagonal Hilbert–Schmidt tests exclude every S pair. The intrinsic kernel description transfers this obstruction to every finite Kraus representation of Φ. Thus one dimension-uniform construction refutes both readings of the stronger converse.

A mismatch with the source's composition order, elementary channel, complex Kraus span, complement or ordered-pair quantifiers would invalidate the claimed interpretation. For the affirmative cells a counterexample would be an actual CPTP map and Kraus representation with N but no E at the same pair. For the negative cells an S pair in any finite representation of the displayed Φ would invalidate the universal no-S-pair claim. No conclusion about the separate rank-reduction assertion of Theorem 1 is included.

The kernel-checked theorem D5/S3/Quantum/QuantumChannels/ElementaryChannelDivisibility.result contains finite Kraus existence for every actual CPTP map, both N iff E readings, the elementary inverse and residual-factorization facts, and the counterexample with no S pair in every finite representation. Complete positivity ranges over every finite ancillary amplification; trace preservation ranges over every matrix. The Blueprint states the same quantifiers and the Library note vomende2024divisibility binds the source locators and quoted question.

The proof uses only the accepted standard axioms Classical.choice, Quot.sound and propext. These mathematical statements establish no physical gate implementation, local-operations protocol, resource cost, preparation yield, NativeQ bridge, or AURIC/QFT realization.

The combined classification is a theorem under the declared formal interpretation. Admission uses the external open-problem-resolution exception, with proof_shape: bind-only and no escape-witness claim. The preregistration is issue 17562. This classification answers the source question in the stated scope; it makes no broader literature-priority or originality claim.

The journal full text has not been checked against arXiv v3; the DOI identifies the publication and does not assert journal/arXiv equivalence. The literature check is bounded to the supplied source and prior independent adjudication; no fresh exhaustive literature or priority search is claimed. Source-to-formal-statement fidelity is a reviewed interpretation, rather than a theorem about natural-language equivalence.

The escape-registration audit remains unfinished under the existing linked issue exception, issue 20941. No Reg source or declared_validated registration is supplied. The issue is an unfinished-audit disclosure and supplies no registration evidence.