slug: mallick-ganguly-majumdar-2026-pea-tensor-nonclosure
bibkey: mallickgangulymajumdar2026partialannihilating
doi: 10.1063/5.0274921
url: https://arxiv.org/abs/2411.19315v2
triage: theorem
motivation_gids:
  - D5/S3/Quantum/Entanglement/SchmidtNumber/PartiallyEntanglementAnnihilatingTensorRefutation.result

Non-local partial entanglement annihilation is not closed under aligned tensor products

Problem

B. Mallick, N. Ganguly and A. S. Majumdar, On the characterization of partially entanglement breaking and annihilating channels, arXiv:2411.19315v2; J. Math. Phys. 67, 082201 (2026), ask in Section VI, Conclusions, PDF page 16:

We have also shown that $r$-partially entanglement breaking channels are not closed under tensor products, motivating the investigation of whether a similar property holds for $r$-partially entanglement annihilating channels.

This is the verbatim sentence in main.tex, line 1167, from the v2 source archive (member MD5 02f5d47ee4475c5ba30a52b1e5931b9b). The definitions are in Section IV.A, PDF pages 9–10, main.tex lines 630–632; see also the literature note.

Write ${\mathrm{PEA}}_{d,r}(E)$ for the module’s IsPEA d r E: $E$ is a completely positive, trace-preserving complex-linear map on the matrices of ${\mathbb{C}}^d\otimes {\mathbb{C}}^d$, cannot be written as a tensor product of two individual CPTP maps, satisfies $r<d$, and sends every positive semidefinite trace-one input to a state of Schmidt number at most $r$.

Issue #21802 fixes the closure assertion: for every $d,r\in \mathbb{N}$ and every pair of complex-linear bipartite matrix maps $E,F$, ${\mathrm{PEA}}_{d,r}(E)$ and ${\mathrm{PEA}}_{d,r}(F)$ imply ${\mathrm{PEA}}_{d^2,r}(E{\otimes }_{\mathrm{aligned}}F)$. The aligned split groups $B_1{B}_1^{\prime }|B_2{B}_2^{\prime }$ when the two copies are $B_1{B}_2$ and $B_1^{\prime }{B}_2^{\prime }$. The threshold remains $r$.

This closure assertion is refuted. The source asks whether non-closure occurs; it does not assert closure. Thus refuting the registered assertion answers the source’s non-closure question affirmatively.

Motivation

The question tests whether a bound on the internal entanglement dimension of every single-use output remains valid for parallel uses across the aligned split. This matters when interpreting partial entanglement annihilation as a restriction on available high-dimensional entanglement. The definition bounds the output Schmidt number; it permits a channel to prepare an entangled state even from a separable input.

Gap

Issue #21802 records the bounded literature check before any Lean:

• The arXiv listing and v2 TeX retain the question.
• The reported web search combining “partially entanglement annihilating”, tensor products and Schmidt number returned the source and background on partially entanglement breaking channels, without a settlement.
• Citation and publisher screening are reported in the issue, without an enumerated corpus of citing works. The issue also records no existing repository settlement.

not-found-in-searched-scope.

The v2 TeX comments at lines 679–706 contain a non-closure proposal and an informal proof sketch at lines 691–699. They are absent from the rendered article. The sketch considers pure outputs of Schmidt number $r$ and their tensor product, but supplies no channels realizing those outputs and does not justify a product-output identity for an arbitrary joint input. Replacement channels supply a concrete realization for which the identity is valid on every input matrix.

Route

Set $d=3$, $r=2$, $\varphi =(|00⟩+|11⟩)/\sqrt{2}$, $\sigma =|\varphi ⟩⟨\varphi |$, and $R(X)=\mathrm{Tr}(X)\sigma$. These are the module’s fixed phi and replacement.

1. A channel with a rank-two output. phi_rank gives coefficient-matrix rank two for this normalized $\varphi$. replacement_cptp proves CPTP for this fixed $R$, using the nine Kraus operators $|\varphi ⟩⟨j|$ indexed by the input basis.
2. Non-locality. replacement_nonlocal proves NonLocal replacement. A proposed product factorization evaluated on $|00⟩⟨00|$ would give $\sigma =A\otimes B$. Its diagonal entries require $A_{00}{B}_{00}=A_{11}{B}_{11}=(1/\sqrt{2}{)}^2\ne 0$ and $A_{00}{B}_{11}=0$, which are incompatible.
3. Membership. For every bipartite qutrit matrix $\rho$ with $\mathrm{Tr}\rho =1$, replacement_schmidt gives SchmidtNumberLE (replacement ρ) 2 by the one-term decomposition with weight one and vector $\varphi$. replacement_pea combines this with CPTP, non-locality and $2<3$ to establish IsPEA 3 2 replacement, including the required bound for every positive semidefinite trace-one input.
4. The aligned output. For every matrix $X$ on the two-copy input, replacement_kron gives $(R\otimes R)(X)=\mathrm{Tr}(X)(\sigma \otimes \sigma )$. Let $\psi$ be the normalized aligned tensor of the fixed $\varphi$ with itself, the module’s psi. tensor_output gives $(R{\otimes }_{\mathrm{aligned}}R)(|\psi ⟩⟨\psi |)=|\psi ⟩⟨\psi |$. psi_rank gives coefficient-matrix rank four: in the $\mathrm{Fin}9$ coordinates the matrix is diagonal with entry $1/2$ at indices $0,1,3,4$ and zero elsewhere.
5. Excluding a different decomposition. For all $d,r\in \mathbb{N}$ and every normalized vector $u\in {\mathbb{C}}^d\otimes {\mathbb{C}}^d$, pure_rank_le assumes SchmidtNumberLE (proj u) r and concludes that the coefficient matrix of $u$ has rank at most $r$. Applying it to the positive semidefinite trace-one input and output $|\psi ⟩⟨\psi |$, with tensor_output and psi_rank, makes an assumed IsPEA 9 2 imply $4\le 2$. This is tensor_not_pea.
6. Refutation. result applies the universal claim at $d=3$, $r=2$, $E=F=R$, using replacement_pea for both hypotheses, and contradicts tensor_not_pea.

Falsifier

The correspondence requires the source’s output Schmidt-number condition, its non-locality condition and the registered bipartition. The module uses:

• SchmidtNumberLE, the existence of a finite convex decomposition into normalized pure states with nonnegative weights summing to one and coefficient-matrix ranks at most $r$; a large rank in one decomposition alone would not exclude another decomposition;
• NonLocal, excluding a tensor product of individual CPTP maps, with CPTP of the bipartite map separately required by IsPEA;
• alignedTensor, exchanging the middle factors and reindexing each party by $\mathrm{Fin}(d^2)$;
• the unchanged threshold $r$, and the bound on all positive semidefinite trace-one inputs.

The failure is witnessed by the output Schmidt-number condition itself, independently of whether the tensor channel satisfies the other clauses of IsPEA.

Evidence

The canonical source is D5/S3/Quantum/Entanglement/SchmidtNumber/PartiallyEntanglementAnnihilatingTensorRefutation.lean (253 lines).

• Proof. It proves result : ¬ claim.
• Reuse. It uses MatrixMap and the finite Kraus API of FiniteKrausChannel, IsCPTP of ChannelAssemblageLOSRRefutation, and coeff, proj and SchmidtNumberLE of TavakoliMorelliIncompleteEAM. The pure-state decomposition step uses positive_phase of UniqueBasisFormationRefutation.
• Axioms. It uses only the standard axioms propext, Classical.choice and Quot.sound; no sorry, native_decide, or new axiom.
• Escape registrations. result is the designated refutation result of the utility header and is exempt. The module has no other public theorem.

Frozen state: Golden/Frozen/state/D5/S3/Quantum/Entanglement/SchmidtNumber/PartiallyEntanglementAnnihilatingTensorRefutation.lean.json, statement sha256:629c340faa7d88f500f198ec22a86cdcdc93034999c3052a16f5a37a112e3123.

Declaration statements:

• result: sha256:fde14a3bf65ec27d118ecda36c6cc23f12590b3a61291dc7bd91a6c51f2518aa;
• claim: sha256:6d93a41a70d4ae87aaee89f186840a903fba0c44409c38e1bc977aa662e9b237.

Triage

Tier 1 question of a 2026 journal article, preregistered in issue #21802 before any Lean. theorem; resolution refuted.

Every declaration is bind-only. The bind-only form is admitted under the open-problem-resolution basis of CLAUDE.md §3.2; no escape witness is claimed. The refutation is a certified instance with basis refutes. There is no digestion atom.

What the settlement shows

Proved by result: the universal closure assertion is false: it is not the case that, for every natural $d,r$ and every pair $E,F$ satisfying IsPEA d r, their aligned tensor product satisfies IsPEA (d * d) r.

Computed, not formalized. Exact rational Gaussian elimination for $r=1,\dots ,6$, $d=r+1$, used $D_{d,r}=\mathrm{diag}(1,\dots ,1,0,\dots ,0)$ with $r$ ones. For these six tested pairs, the ranks of $D_{d,r}$ were $1,2,3,4,5,6$, and those of $(D_{d,r}\otimes D_{d,r})/r$ were $1,4,9,16,25,36$. Each latter coefficient matrix had squared entry norm one. These computations concern this finite family only.

Argued, not formalized.

• Mechanism and missing structure. For the fixed $R$ and every two-copy matrix $X$, replacement_kron gives $(R\otimes R)(X)=\mathrm{Tr}(X)(\sigma \otimes \sigma )$. Thus even product density inputs produce the problematic output. A bound on each output’s Schmidt rank supplies no bound by the same number on the product of the ranks. The definition permits preparing a pure entangled output, so restricting input correlations cannot remove this obstruction. A surviving same-threshold statement needs an additional tensor budget or a restriction on the allowed prepared states.
• A family beyond the formal instance. For any integers $d>r\ge 2$, take ${\varphi }_{d,r}=r^{-1/2}{\sum }_{j=0}^{r-1}|jj⟩$ and the replacement onto its pure density. Kraus operators $|{\varphi }_{d,r}⟩⟨j|$ have adjoint products summing to the identity, so the map is CPTP. Its constant output has a one-term decomposition of rank $r$. Non-locality follows from the same diagonal obstruction: a product output would require $A_{00}{B}_{00}=A_{11}{B}_{11}=1/r$ but $A_{00}{B}_{11}=0$. The aligned two-copy vector is $r^{-1}{\sum }_{i,j<r}|(i,j)⟩\otimes |(i,j)⟩$, with $r^2$ nonzero orthogonal Schmidt terms. A decomposition of a pure density cannot lower that rank: for any vector perpendicular to its support, the expectation is a sum of nonnegative weighted squared overlaps and equals zero, forcing every positive-weight component into the same ray. Its Schmidt number is therefore $r^2>r$. This gives a mathematical family of counterexamples, while the compiled settlement uses only $d=3,r=2$.
• A surviving restricted statement. For the canonical pure replacement channels of ranks $s,t$ with $2\le s,t\le r<d$, the aligned output has $st$ orthogonal Schmidt terms, each of amplitude $1/\sqrt{st}$. It is the constant output on every density input, and its pure-state decompositions have that same Schmidt rank by the preceding support argument. On input matrix units, each replacement vanishes off the diagonal and prepares its density on the diagonal; linearity makes the joint output the input trace times the tensor of the prepared densities. Tensoring the Kraus operators gives CPTP, and the entangled pure output excludes a product of individual channels. Consequently these replacements satisfy same-threshold closure precisely when $st\le r$; allowing threshold $st$ always gives the output bound for this family. This is not a bound on arbitrary joint outputs of general PEA channels.
• Effect on the paper. The rendered article leaves tensor closure as a question and uses no affirmative closure hypothesis in its other statements. The settlement answers the tensor-product part of the Conclusions for the registered non-local class and aligned split. It does not establish the separate topological, detection, series-concatenation or class-inclusion assertions. The other questions in main.tex line 1167, concerning representations, capacities and multipartite extensions, remain unsettled by this result.

ASSUMED-UNVERIFIED

The bounded literature search and citation screening in #21802 do not establish exhaustive worldwide novelty or priority. The individual citing works and a separately hosted publisher full text have not been independently checked here; the source reading is the arXiv v2 TeX and PDF. Crossref metadata confirms the title, authors, DOI and journal reference.

The coefficient-matrix computations are finite checks. The family extension, restricted tensor-budget criterion and interpretations in “Argued, not formalized” are not additional compiled theorems. General non-local PEA tensor closure at $r=1$, and an $r^2$ output bound for general PEA channels on arbitrary correlated joint inputs, are not decided by this settlement.