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Partial entanglement annihilation fails under aligned tensor products

Abstract

Two non-local channels with Schmidt-number-two outputs can have an aligned tensor product with a Schmidt-rank-four output.

Definition 1.1 (Excluding products of individual channels).

Formalization. D5/S3/Quantum/Entanglement/SchmidtNumber/PartiallyEntanglementAnnihilatingTensorRefutation.NonLocal (✓ std3).

Citation. Bivas Mallick; Nirman Ganguly; A. S. Majumdar (2026). On the characterization of partially entanglement breaking and annihilating channels. DOI: 10.1063/5.0274921. URL: https://arxiv.org/abs/2411.19315v2.

Commentary.

A linear map on the bipartite d by d system is non-local when it cannot be expressed as the tensor product of two completely positive, trace-preserving maps on the individual d-dimensional systems. This is the distinction in Section IV, main.tex line 630. The predicate only excludes such a factorization; complete positivity and trace preservation of the bipartite map are separate conditions.

Definition 1.2 (Non-local partial entanglement annihilation).

Formalization. D5/S3/Quantum/Entanglement/SchmidtNumber/PartiallyEntanglementAnnihilatingTensorRefutation.IsPEA (✓ std3).

Citation. Bivas Mallick; Nirman Ganguly; A. S. Majumdar (2026). On the characterization of partially entanglement breaking and annihilating channels. DOI: 10.1063/5.0274921. URL: https://arxiv.org/abs/2411.19315v2.

Commentary.

A non-local r-partially entanglement annihilating channel is completely positive and trace preserving, is non-local, has r strictly below d, and sends every positive semidefinite trace-one input to an operator of Schmidt number at most r. This is Section IV, main.tex line 632. SchmidtNumberLE expresses the existence of a finite convex decomposition into normalized pure states whose coefficient matrices all have rank at most r, corresponding to the Schmidt-number definition in Section II, lines 261–265.

Definition 1.3 (The tensor product across the aligned split).

Formalization. D5/S3/Quantum/Entanglement/SchmidtNumber/PartiallyEntanglementAnnihilatingTensorRefutation.alignedTensor (✓ std3).

Source. Repository-derived.

Acknowledgement. Bivas Mallick; Nirman Ganguly; A. S. Majumdar (2026). On the characterization of partially entanglement breaking and annihilating channels. DOI: 10.1063/5.0274921. URL: https://arxiv.org/abs/2411.19315v2.

Commentary.

The equivalence a first exchanges the middle two factors of the fourfold product and then identifies each pair of Fin d indices with Fin (d * d). The linear equivalence t reindexes both matrix indices by a. Conjugating the tensor product of e and f by t makes the two first subsystems one party and the two second subsystems the other party. Thus both parties have dimension d * d.

Definition 1.4 (Closure at the same Schmidt-number threshold).

Formalization. D5/S3/Quantum/Entanglement/SchmidtNumber/PartiallyEntanglementAnnihilatingTensorRefutation.claim (✓ std3).

Source. Repository-derived.

Acknowledgement. Bivas Mallick; Nirman Ganguly; A. S. Majumdar (2026). On the characterization of partially entanglement breaking and annihilating channels. DOI: 10.1063/5.0274921. URL: https://arxiv.org/abs/2411.19315v2.

Commentary.

For all natural dimensions d and thresholds r, the assertion says that any two non-local r-partially entanglement annihilating channels remain in that class after taking their aligned tensor product. The threshold stays r while the local dimension becomes d * d. This is a precise closure assertion for the tensor-product question in the Conclusion, main.tex line 1167; the source asks the question without asserting this closure.

Theorem 1.5 (Two qutrit channels disprove closure).

Proof. Machine-checked in Lean as D5/S3/Quantum/Entanglement/SchmidtNumber/PartiallyEntanglementAnnihilatingTensorRefutation.result (✓ std3). ∎

Resolves. Problems/mallick-ganguly-majumdar-2026-pea-tensor-nonclosure (refuted) by D5/S3/Quantum/Entanglement/SchmidtNumber/PartiallyEntanglementAnnihilatingTensorRefutation.result.

Source. Repository-derived.

Acknowledgement. Bivas Mallick; Nirman Ganguly; A. S. Majumdar (2026). On the characterization of partially entanglement breaking and annihilating channels. DOI: 10.1063/5.0274921. URL: https://arxiv.org/abs/2411.19315v2.

Commentary.

Take d = 3 and r = 2. Let φ be the normalized vector (|00⟩ + |11⟩)/√2 and let R send X to Tr(X)|φ⟩⟨φ|. The coefficient matrix of φ is diagonal with entries 1/√2, 1/√2 and 0. Its two nonzero diagonal entries give rank two.

For each coordinate vector |j⟩ of the nine-dimensional input, use the Kraus operator |φ⟩⟨j|. Summing their actions gives exactly R, so R is completely positive. Its output trace equals the input trace because |φ⟩⟨φ| has trace one; hence R is trace preserving.

Suppose R were a tensor product of two individual channels. Evaluating it on |00⟩⟨00| would express |φ⟩⟨φ| as A ⊗ B, where A and B are the individual outputs on |0⟩⟨0|. The diagonal entries at (0,0) and (1,1) are both (1/√2)², which is nonzero, whereas the entry at (0,1) is zero. Thus A₀₀B₁₁ = 0. If A₀₀ = 0 then A₀₀B₀₀ is zero, and if B₁₁ = 0 then A₁₁B₁₁ is zero. Either case contradicts one of the two nonzero diagonal entries, so R is non-local.

On every trace-one input, R outputs |φ⟩⟨φ|. The one-term convex decomposition with weight one therefore gives Schmidt number at most two. Together with complete positivity, trace preservation, non-locality and 2 < 3, this makes R a non-local 2-partially entanglement annihilating channel.

Let ψ be φ ⊗ φ regrouped across the aligned split. In the Fin 9 coordinates its coefficient matrix is diagonal, with entry 1/2 at indices 0, 1, 3 and 4 and zero elsewhere. Its rank is therefore four.

For any normalized bipartite vector u, a convex decomposition of |u⟩⟨u| with coefficient ranks at most r forces the coefficient rank of u to be at most r. Indeed, nonnegative weights summing to one are individually at most one and include a positive weight. The pure-state decomposition property then identifies that component with z times u for a complex scalar z of norm one. Its coefficient matrix is z times the coefficient matrix of u; since z is nonzero, the ranks agree, giving the required rank bound.

Expanding the tensor product of the two trace-and-prepare maps on matrix units shows that it sends X to Tr(X) times (|φ⟩⟨φ| ⊗ |φ⟩⟨φ|). Reindexing preserves trace. On the aligned input |ψ⟩⟨ψ| this trace is one, and the reindexed output is |ψ⟩⟨ψ| itself, as follows by multiplying the vector coordinates.

If the aligned tensor product were non-local 2-partially entanglement annihilating, its Schmidt-number bound on the positive semidefinite trace-one input |ψ⟩⟨ψ| would give Schmidt number at most two for that same pure output. The pure-state rank bound would then imply 4 ≤ 2, a contradiction. Applying the closure assertion to d = 3, r = 2 and e = f = R would give precisely this impossible conclusion.

References