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REVIEW 3 major objections 5 minor 36 references

Every PPT channel has finite entanglement-breaking index

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Repeatedly applying any PPT quantum channel eventually produces an entanglement-breaking channel.

desk verdict Theorem 1.1 is likely correct and closes the full-rank gap; Section 5's proof has a genuine hole, and the reader's counterexample to it misfires because identity is not PPT. read the letter →

arxiv 2608.13551 v1 pith:6KA4SHIY submitted 2026-08-13 quant-ph math-phmath.MPmath.OA

Sang-Jun Park This is my paper · ORCID
classification quant-phmath-phmath.MPmath.OA MSC 81P4515B4847B65 PACS 03.67.-a
keywords entanglement-breakingindexPPTmapseventuallyentanglementbreakingPerron–FrobeniustheorySchmidtnumbersuperpositivePPT-squaredconjecturequantumchannels
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proves that every completely positive and completely copositive linear map on a finite-dimensional matrix algebra—the class of PPT maps—has finite entanglement-breaking index. Equivalently, for every PPT map $\Phi : M_d \to M_d$ there is an integer $N$ such that $\Phi^n$ is entanglement breaking for every $n \ge N$, meaning the Choi matrix of the iterate is separable. The result removes the unital, trace-preserving, and full-rank assumptions that earlier results needed. The proof splits a PPT map with a singular Perron eigenmatrix into two lower-dimensional PPT blocks and uses induction, while the full-support case is imported from a known theorem. A second set of results gives uniform bounds: PPT maps in $\mathrm{DSP}_2$ become entanglement breaking at the third iterate, and those in $\mathrm{DSP}_3$ at the fifth, independently of dimension.

What carries the argument

The central object is the Perron–Frobenius support splitting of a PPT map. Given a positive eigenmatrix $\rho$ of $\Phi$, the paper writes $\Phi = A + B$ with $A$ supported on the range of $\rho$ and $B$ on its orthogonal complement, so that $B \circ A = 0$; powers of $A$ and $B$ then factor through the restricted corner maps $\Phi_\rho$ and $\Phi_{\rho^\perp}$. The key identity is $n_{\mathrm{EB}}(\Phi) \le n_{\mathrm{EB}}(\Phi_\rho) + n_{\mathrm{EB}}(\Phi_{\rho^\perp}) + 1$, which drives the induction on dimension. A complementary estimate bounds the Schmidt number of the Choi matrix of $\Phi$ by $\max(\operatorname{rank} \rho, d - \operatorname{rank} \rho)$, forcing the map into $\mathrm{SP}_{d-1}$ when $\rho$ is singular.

What would settle it

A single PPT map on some $M_d$ whose Choi matrix is PPT but whose iterates $\Phi^n$ all remain entangled for every $n$ would refute Theorem 1.1; a concrete search would be to compute the Choi matrices of high powers for a parametrized family of PPT maps with singular Perron eigenmatrices and test separability numerically.

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Extended reading notes

Core claim

The central claim is Theorem 1.1: for every linear map $\Phi : M_d \to M_d$ that is completely positive and completely copositive, there exists an integer $N$ such that $\Phi^n \in \mathrm{EB}$ for all $n \ge N$. The proof proceeds by induction on dimension, using Perron–Frobenius theory to find a positive eigenmatrix $\rho$. When $\rho$ is singular, the support splitting of Section 3 factors the iterates through two strictly smaller PPT maps, reducing the problem; the full-support case is covered by a previously known theorem. The paper also proves that PPT maps in the classes $\mathrm{DSP}_2$ and $\mathrm{DSP}_3$ satisfy the dimension-independent bounds $n_{\mathrm{EB}}(\Phi) \le 3$ and $n_{\mathrm{EB}}(\Phi) \le 5$, respectively.

Load-bearing premise

The argument leans on a quoted theorem saying that a PPT map whose two Perron eigenmatrices are strictly positive has finite entanglement-breaking index; if that theorem has hidden assumptions, or fails for non-trace-preserving maps, the induction collapses.

Editorial extensions

If this is right

  • Every PPT channel is eventually entanglement breaking: no PPT map can preserve entanglement indefinitely under iteration.
  • For a completely positive map, eventual PPT-ness is equivalent to eventual entanglement-breaking, with the index bound $n_{\mathrm{EB}}(\Phi) \le n_{\mathrm{PPT}}(\Phi) \, n_{\mathrm{EB}}(\Phi^{n_{\mathrm{PPT}}})$.
  • Compositions of PPT maps through a smaller intermediate algebra are eventually entanglement breaking, with $n_{\mathrm{EB}}(\Phi_2 \circ \Phi_1) \le 1 + n_{\mathrm{EB}}(\Phi_1 \circ \Phi_2) < \infty$.
  • Any PPT map in $\mathrm{DSP}_2$ satisfies $\Phi^3 \in \mathrm{EB}$, and any PPT map in $\mathrm{DSP}_3$ satisfies $\Phi^5 \in \mathrm{EB}$, uniformly in dimension.
  • The inclusion $\mathrm{PPT} \circ \mathrm{DSP}_2 \circ \mathrm{PPT} \subseteq \mathrm{EB}$ gives a large, explicitly parameterized family of triples of maps whose composition is immediately entanglement breaking.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The recursion underlying Theorem 1.1 suggests an algorithmic way to compute an explicit upper bound on $N(\Phi)$: repeatedly split along singular Perron eigenmatrices until reaching full-support blocks; the paper states the bound but does not optimize or implement this recursion.
  • If every PPT map on $M_d$ were eventually shown to lie in $\mathrm{DSP}_2$ or $\mathrm{DSP}_3$ (the paper's Question 5.6), Theorem 1.2 would supply a dimension-independent uniform bound, settling the open uniformity question affirmatively for all $d$.
  • The full-support case is imported as a black box; replacing it with a constructive proof would make the entanglement-breaking time of concrete channels computable in practice.
  • A natural numerical test is to generate random PPT channels on $M_4$ or $M_5$, check membership in $\mathrm{DSP}_2$ by semidefinite programming, and verify $\Phi^3 \in \mathrm{EB}$; failures would indicate where a uniform bound could break.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper claims two main results. First, Theorem 1.1 states that every completely positive and completely copositive (PPT) linear map on M_d has finite entanglement-breaking index, removing full-rank, unital, and trace-preserving assumptions. The proof in Section 4 proceeds by induction on dimension: singular right or left Perron eigenmatrices are handled by a support-splitting decomposition developed in Section 3, and the full-support case is delegated to a quoted theorem of Hanson–Rouzé–Stilck França (Theorem 2.3). Second, Section 5 introduces cones DSP_k = SP_k + ⊤∘SP_k and claims, via Theorem 5.5, that the triples (PPT, DSP2, PPT) and the fivefold tuple (PPT, DSP3, PPT, DSP3, PPT) are EB-composable, yielding uniform EB-index bounds 3 and 5 for PPT maps in DSP2 and DSP3 respectively.

Significance. If correct, Theorem 1.1 would be a substantial improvement over earlier eventual-entanglement-breaking results: it would settle the qualitative finite-index problem for all PPT maps without any ancillary assumption. The Perron-support splitting argument in Section 3 is elegant and the induction in Section 4 is coherent and plausible. However, the secondary results in Section 5 are not merely unproven; Theorem 5.5(1) is false as stated, and the proof of Theorem 5.5 contains a concrete composability error involving the transpose map. The advertised consequences in Theorem 1.2 are therefore unsupported. The main theorem deserves serious consideration, but the manuscript in its current form cannot be accepted.

major comments (3)
  1. [§5, Theorem 5.5(1)] The statement is false. Take d=2, let Φ1 = Φ2 = id, and let Λ = ⊤. Since id ∈ SP2, the transpose map satisfies ⊤ = ⊤∘id ∈ ⊤∘SP2 ⊆ DSP2. Also id is PPT. Yet Φ2 ∘ Λ ∘ Φ1 = ⊤, which is not completely positive for d=2 and hence is not entanglement breaking. Thus the triple (PPT, DSP2, PPT) is not EB-composable. This directly invalidates the claimed derivation of Theorem 1.2(1). Note that this counterexample does not refute Theorem 1.2(1) itself, because ⊤ is not PPT; but the proof mechanism claimed for that theorem is false.
  2. [§5, proof of Theorem 5.5(1), after Eq. (11)] The proof's grouping of summands is invalid. For the transpose summands, the outer factor Φ2 ∘ ⊤ ∘ Ad_{L_{l,β}^*} need not be PPT: composing a PPT map with the transpose map does not preserve complete positivity. A concrete instance is d=3, L_{l,β}^* = [I_2; 0] (embedding M_2 into the top-left block of M_3), and Φ2(X) = PXP with P = diag(1,1,0). Then Φ2 ∘ ⊤ ∘ Ad_{L^*}(X) equals the transpose of X embedded into M_3; its Choi matrix is the embedded swap on C^2 ⊗ C^3, which has negative eigenvalues and is therefore not CP. Hence the assertion that all summands are compositions of PPT maps factoring through M_2 is false, and Theorem 5.1 cannot be applied to that summand.
  3. [§5, proof of Theorem 5.5(2)] The same composability error appears in the proof of part (2). The four displayed 'middle linear maps' are asserted to be PPT maps on M_3, but the two types involving ⊤, for example Ad_{L_{r,β'}^{(2)}} ∘ Φ2 ∘ ⊤ ∘ Ad_{L_{l,β}^{(1)*}}, are not generally PPT. Taking Φ2 = id makes an identity-composed middle factor contain the transpose map, which is not CP and hence not 2-superpositive. Consequently the application of part (1) to these summands is unjustified, and Theorem 1.2(2) is not established by the given argument.
minor comments (5)
  1. [§1.2 and §5] The reference to 'Theorem??' before Eq. (2) is a placeholder; it should be replaced by the actual theorem number (presumably Theorem 5.5(1)).
  2. [Throughout] There are several typographical errors, including 'Furhtermore' (Corollary 4.1), 'EB-comosable' (Theorem 5.5(2)), 'trasnpose' (Section 2), 'eventaully' (Definition of EB index), and 'entangelment-breaking' (proof of Theorem 5.5(2)). These should be corrected.
  3. [§5, proof of Theorem 5.5] The definition of the rectangular matrices K_l, K_r, L_l, L_r is ambiguous: the text says 'd×k rectangular matrices' and also writes 'M_{k,d}'. Please fix the orientation so that the factorizations K = K_l^* K_r and L = L_l^* L_r have the stated ranks.
  4. [§4, Theorem 1.1, Case 4] The proof of the full-support case relies entirely on the quoted Theorem 2.3 from [HRSF20]. Please verify that the statement reproduced here exactly matches the original theorem, and add a sentence confirming that no unital or trace-preserving hypothesis is needed; the induction in Theorem 1.1 depends on this black-box result.
  5. [§5, Example 5.4(1)] The extremality argument for id_d ∉ DSP_k for k < d is terse. Consider spelling out why the two summands Ψ1 and ⊤∘Ψ2 must both be scalar multiples of id_d before concluding that ⊤ would be a scalar multiple of id_d.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the main proof reduces to external theorems and its own induction, with only incidental self-citations.

full rationale

The paper's central derivation is not circular. Theorem 1.1 is proved by induction on dimension, with four cases. The full-support terminal case explicitly invokes Theorem 2.3, quoted from HRSF20's Theorem 3.14, an external result with no author overlap with the present paper; the singular-support cases are reduced to lower-dimensional corner maps via the paper's own Perron–Frobenius splitting (Theorem 3.5, Proposition 3.6, Corollary 3.7) and Theorem 3.8's Schmidt-number estimate. No fitted parameter is renamed as a prediction, and no definition is built from the conclusion it is supposed to establish. The only self-citations ([Par26], [NP26], [PY24]) appear in the introduction or as supporting references for Example 5.4(2), and are not load-bearing for Theorem 1.1 or Theorem 1.2. The skeptical objection to the proof of Theorem 5.5(1) is a mathematical error in a step (the outer factor Φ2∘⊤∘Ad_{L*}^* need not be PPT), but an error is distinct from circularity; no circular reduction is exhibited. Accordingly, the honest finding is no significant circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 1 invented entities

No fitted constants or data appear; the paper is a pure proof. The axioms are standard matrix and algebra facts plus three prior theorems used as black boxes. The only newly introduced mathematical object, the DSP_k cone, is defined in Section 5 and is the site of the false claims.

assumptions (6)
  • standard math Finite-dimensional Perron-Frobenius theorem for positive maps: spectral radius is an eigenvalue with nonzero positive semidefinite right and left eigenvectors (Theorem 2.2).
    Invoked in Section 4 to obtain the Perron eigenmatrix whose support is split.
  • domain assumption Full-support PPT maps have finite EB index ([HRSF20, Theorem 3.14], restated as Theorem 2.3).
    Used as the base of the induction in Case 4 of Section 4; the proof does not rederive it.
  • domain assumption PPT maps on M_3 are 2-superpositive ([CYT19], [YLT16]).
    Used in the proof of Theorem 5.5(2) to replace middle M3 maps with SP2 maps.
  • domain assumption Every PPT composition through a qubit intermediate is entanglement breaking ([Hen26], Theorem 5.1).
    Load-bearing for the Section 5 triple-composition claims; cited as a Zenodo research note.
  • standard math Schur complement lemma for 2x2 positive block matrices with Moore-Penrose inverse (Lemma 3.2).
    Used in Proposition 3.3 to derive support properties from the PPT Choi matrix.
  • standard math Cayley-Hamilton theorem for the spectral-radius-zero case.
    Used in Section 4, Case 1 to conclude Φ^{d^2}=0.
invented entities (1)
  • DSP_k = SP_k + ⊤∘SP_k
    purpose: Defines a family of maps claimed to have absolute EB-index bounds in Theorem 1.2 and Theorem 5.5.
    A new mathematical cone, not a physical entity; it has no independent empirical handle, and the theorems built on it are false as stated.

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Pith. "Pith review of Every PPT channel has finite entanglement-breaking index." pith.science (2026). https://pith.science/paper/6KA4SHIY

@misc{pith2026260813551,
  author       = {Pith},
  title        = {Pith review of: Every PPT channel has finite entanglement-breaking index},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6KA4SHIY}},
  note         = {Machine review of arXiv:2608.13551}
}
read the original abstract

We prove that every PPT linear map has finite entanglement-breaking index, thereby establishing the eventual entanglement-breaking property of PPT channels in full generality. Furthermore, by utilizing completely positive maps with low entanglement dimensionality, we show that a large family of PPT maps, which strictly containing the class of 2-superpositive maps, has entanglement-breaking index bounded above by 3, uniformly in the dimension. In particular, these results provide strong evidence that the PPT-cubed conjecture may hold in full generality.

Discussion (0). Continue with ORCID to comment.

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