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A qutrit obstruction to CP-filter equivalence with the channel transpose

Abstract

A unital measure-and-prepare channel on three-dimensional complex matrices has diagonal transpose range and noncommuting original range. Complete positivity of a map and its inverse forces an invertible congruence, which cannot identify these ranges.

Definition 1.1 (Invertibility with a completely positive inverse).

Formalization. D5/S3/Quantum/QuantumChannels/CPFilterTransposeRefutation.CPInvertible (✓ std3).

Citation. Samuel A. Márquez González (2026). Feasibility Ordering of Entanglement-Source Placement for Qubit Channels. DOI: 10.48550/arXiv.2609.18803. URL: https://arxiv.org/abs/2609.18803v1.

Commentary.

MatrixMap(Fin 3, Fin 3, complex) is the type of complex linear maps on 3 by 3 matrices. IsCompletelyPositive is the existing all-amplification predicate: for every natural n, the tensor product with the identity on Fin n maps every positive semidefinite matrix to a positive semidefinite matrix. Both inverse identities hold on every matrix. The maps need not preserve trace. PDF p. 5, Section VI, Eq. (47): “This separates the higher-dimensional problem into two levels. The strong question is whether every unital qudit channel Υ is CP-filter equivalent to its transpose, i.e., whether there exist invertible CP maps ℒ and ℛ, with CP inverses, such that Υᵀ = ℒ ∘ Υ ∘ ℛ.”

Definition 1.2 (Unital qutrit channels).

Formalization. D5/S3/Quantum/QuantumChannels/CPFilterTransposeRefutation.UnitalChannel (✓ std3).

Citation. Samuel A. Márquez González (2026). Feasibility Ordering of Entanglement-Source Placement for Qubit Channels. DOI: 10.48550/arXiv.2609.18803. URL: https://arxiv.org/abs/2609.18803v1.

Commentary.

PDF p. 2, Section II.A: “Consider completely positive trace-preserving (CPTP) maps”. PDF p. 3, Lemma 2: the scaled channel “is trace preserving and unital.” IsCPTP is the existing conjunction of all-amplification complete positivity and preservation of the matrix trace on every matrix; unitality is F(1)=1. The displayed conjunction expands that reused predicate. The carrier here is Fin 3, the qutrit instance of the dimensional question.

Definition 1.3 (The strong question at dimension three).

Formalization. D5/S3/Quantum/QuantumChannels/CPFilterTransposeRefutation.claim (✓ std3).

Citation. Samuel A. Márquez González (2026). Feasibility Ordering of Entanglement-Source Placement for Qubit Channels. DOI: 10.48550/arXiv.2609.18803. URL: https://arxiv.org/abs/2609.18803v1.

Commentary.

PDF p. 5, Section VI, Eq. (47): “This separates the higher-dimensional problem into two levels. The strong question is whether every unital qudit channel Υ is CP-filter equivalent to its transpose, i.e., whether there exist invertible CP maps ℒ and ℛ, with CP inverses, such that Υᵀ = ℒ ∘ Υ ∘ ℛ.” F encodes Υ, and L and R encode ℒ and ℛ. The composition symbols denote composition of complex linear maps. PDF pp. 2–3, Section II.D: “Fix a qubit basis. If a CP map has Kraus representation Λ(X)=∑ᵢ Kᵢ X Kᵢ†, its channel transpose is the CP map Λᵀ(X)=∑ᵢ Kᵢᵀ X K̅ᵢ.” PDF p. 3: “This definition is independent of the chosen Kraus representation, although it depends on the underlying basis”. ofKraus denotes the Lean function MatrixMap.of_kraus. MatrixMap.of_kraus K K is literally X↦∑ᵢ Kᵢ X Kᵢ†. Applying that same operator to the transposed family gives the displayed full sum ∑ᵢ Kᵢᵀ X (Kᵢᵀ)†. Since (Kᵢᵀ)† = K̅ᵢ, this is the source’s Eq. (16) form Kᵀ X K̅, in the fixed standard basis. Definition-fidelity table: CPInvertible is CP with a two-sided CP inverse; UnitalChannel is IsCPTP together with F 1 = 1; claim uses MatrixMap.of_kraus K K for the Kraus representation and MatrixMap.of_kraus on the transposed family for the channel transpose. Trace duality determines the Heisenberg map Y↦∑ᵢ Kᵢ† Y Kᵢ from F: tr(Y F(X))=tr((∑ᵢ Kᵢ† Y Kᵢ) X). Nondegeneracy of the trace pairing identifies this map for any two finite Kraus representations. Transposing its value on Xᵀ gives ∑ᵢ Kᵢᵀ X K̅ᵢ, so the channel transpose is independent of the Kraus family. This is the dimension-three instance of the universal dimensional question; a failure here refutes that question.

Theorem 1.4 (Refutation of the strong question).

Proof. Machine-checked in Lean as D5/S3/Quantum/QuantumChannels/CPFilterTransposeRefutation.result (✓ std3). ∎

Resolves. Problems/marquez-gonzalez-2026-cp-filter-transpose-refutation (refuted) by D5/S3/Quantum/QuantumChannels/CPFilterTransposeRefutation.result.

Source. Repository-derived.

Acknowledgement. Samuel A. Márquez González (2026). Feasibility Ordering of Entanglement-Source Placement for Qubit Channels. DOI: 10.48550/arXiv.2609.18803. URL: https://arxiv.org/abs/2609.18803v1.

Commentary.

Take the three trace-one positive semidefinite matrices ρ₀=diag(1/2,1/3,1/6), ρ₁=[[1/6,1/12,0],[1/12,1/3,0],[0,0,1/2]], and ρ₂=[[1/3,−1/12,0],[−1/12,1/3,0],[0,0,1/3]]. They sum to the identity. The channel F(X)=∑ⱼ Xⱼⱼρⱼ is completely positive, trace preserving, and unital. Its channel transpose has diagonal range. Applying complete positivity to the unnormalized maximally entangled projector gives a positive semidefinite Choi matrix, whose spectral decomposition supplies the finite Kraus operators. The identity Kraus scalarity theorem then forces a CP map with CP inverse to be an invertible congruence X↦AXA†. Surjectivity of R would make A F(Y) A† diagonal for every Y. Commutation of these diagonal matrices, including the image of the identity, and cancellation of A and A† force ρ₀ρ₁=ρ₁ρ₀. Their commutator has entry (0,1) equal to 1/72, a contradiction. The weaker positive/CP question in Eq. (48) is refuted separately by PositiveFilterTransposeRefutation; the source’s qubit conclusions are unaffected.

References