bibkey: marquezgonzalez2026feasibility authors: Samuel A. Márquez González year: 2026 title: “Feasibility Ordering of Entanglement-Source Placement for Qubit Channels” doi: 10.48550/arXiv.2609.18803 url: https://arxiv.org/abs/2609.18803v1 claim: “Section VI, Eq. (47), asks whether every unital qudit channel is CP-filter equivalent to its channel transpose with CP inverses; Eq. (48) asks the weaker question whether the transpose factors as a positive map after the channel after a completely positive map.” strata_touched:

• D5/S3/Quantum/QuantumChannels/CPFilterTransposeRefutation
• D5/S3/Quantum/QuantumChannels/PositiveFilterTransposeRefutation license: citation-only triage: anchor

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• DOI: 10.48550/arXiv.2609.18803
• URL: https://arxiv.org/abs/2609.18803v1

Feasibility Ordering of Entanglement-Source Placement for Qubit Channels

Samuel A. Márquez González, arXiv:2609.18803v1. Page numbers refer to the printed PDF.

Page 2, Section II.A:

Consider completely positive trace-preserving (CPTP) maps $\Lambda :\mathcal{B}({\mathbb{C}}^2)\to \mathcal{B}({\mathbb{C}}^2)$.

Pages 2–3, Section II.D, Eqs. (15)–(16):

Fix a qubit basis. If a CP map has Kraus representation $\Lambda (X)={\sum }_i{K}_{i}X{K}_i^{†}$, its channel transpose is the CP map ${\Lambda }^T(X)={\sum }_i{K}_i^{T}X\overline{K_i}$.

This definition is independent of the chosen Kraus representation, although it depends on the underlying basis.

Page 5, Section VI, Eq. (47):

This separates the higher-dimensional problem into two levels. The strong question is whether every unital qudit channel ${\rm Y}$ is CP-filter equivalent to its transpose, i.e., whether there exist invertible CP maps $\mathcal{L}$ and $\mathcal{R}$, with CP inverses, such that ${{\rm Y}}^T=\mathcal{L}\circ {\rm Y}\circ \mathcal{R}$.

The dimension-three encoding uses complex linear maps on Matrix (Fin 3) (Fin 3) ℂ. Complete positivity means positivity under every finite matrix amplification. A CP inverse satisfies both composition identities on every matrix. The standard-basis coefficient transpose agrees with Eq. (16) for every finite Kraus representation. Trace duality and the nondegenerate matrix trace pairing identify the Heisenberg map across Kraus representations; transposing its value on $X^T$ identifies the channel transpose. Trace preservation and unitality mean preservation of matrix trace and of the identity matrix.

Section VI, Eq. (48) (label eq:quditquestion in the v1 TeX source), immediately after Eq. (47):

The weaker question, which is sufficient for the source-placement theorem, asks only whether there exist a positive map $\mathcal{F}$ and a completely positive map $\mathcal{E}$ such that ${{\rm Y}}^T=\mathcal{F}\circ {\rm Y}\circ \mathcal{E}$.

Qubit unital channels satisfy the strong condition. A positive answer to the weak condition for any class of unital qudit channels would, after the same Sinkhorn and regularization steps, extend the midpoint-feasibility theorem to the corresponding class of arbitrary qudit channels. Failure of the strong condition would not by itself preclude such an extension; failure of the weak condition would identify the precise obstruction to this proof strategy.

The strong question fails for a unital qutrit measure-and-prepare channel. Its transpose has diagonal range, while its range contains two noncommuting trace-one matrices. CP-invertible maps are invertible congruences; such a congruence cannot make the entire unital noncommuting range diagonal. The weak question also fails at dimension three: for the unital measure-and-prepare channel with states ${\rho }_j=(e_j{e}_j^{\mathsf{T}}+u_j{u}_j^{\mathsf{T}})/2$, $u_j$ the columns of $\frac{1}{7}\left[\begin{array}{ccc}3& -2& 6\\ 6& 3& -2\\ -2& 6& 3\end{array}\right]$, no positive $\mathcal{F}$ and completely positive $\mathcal{E}$ give ${{\rm Y}}^T=\mathcal{F}\circ {\rm Y}\circ \mathcal{E}$. The qubit theorems and the source’s feasibility conclusion are unaffected.