REVIEW 1 major objections 5 minor 8 references
An upper bound for the purity of absolutely positive partial transpose states
T0 review · 1 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash
Pith's one-line read For every APPT state on $m\otimes n$ with $mn\ge8$, the purity is bounded by a piecewise formula in $mn\bmod 4$, and the bound is sharp for qubit-qudit systems.
desk verdict A clean convex-geometry proof of a new APPT purity bound that needs a one-line domain fix (m,n≥2) and a more careful abstract, but is otherwise solid and worth refereeing. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The working object is the convex polytope $S_k$ of ordered spectra $\lambda_1\ge\cdots\ge\lambda_k\ge0$ with $\sum_i\lambda_i=1$ and $\lambda_1\le\lambda_{k-2}+\lambda_{k-1}+\lambda_k$. It is obtained from the APPT necessary inequality $\lambda_1\le\lambda_{k-1}+2\sqrt{\lambda_{k-2}\lambda_k}$ by replacing $2\sqrt{ab}$ with $a+b$. Since the purity $F(\lambda)=\sum_i\lambda_i^2$ is strictly convex and $S_k$ is compact, the maximum is attained at a vertex; Lemma 2.1 lists all vertices as belonging to three types, and Theorem 2.2 evaluates $F$ at each type. This reduces a spectral optimization over continuous spectra to a finite comparison of rational values indexed by $k$ modulo 4.
What would settle it
Take $m=1$, $n\ge8$, and any pure state on $\mathbb{C}^1\otimes\mathbb{C}^n$: it is APPT with purity $1$, while the theorem's formula is $<1$, so the printed statement is false. For the intended $m,n\ge2$ version, search the $2\times4$ system numerically for APPT states with purity above $1/6$; the theorem predicts none, and the spectrum $(1/4,1/4,1/12,\ldots,1/12)$ attains $1/6$.
Extended reading notes
Core claim
The central claim, Theorem 1, is that for $mn\ge8$ every APPT state on $\mathbb{C}^m\otimes\mathbb{C}^n$ has purity at most $4/(3mn)$ when $mn\equiv0\pmod4$, and at most $4(3mn-2)/(3mn-1)^2$, $4(3mn+4)/(3mn+2)^2$, or $4(3mn+2)/(3mn+1)^2$ in the other three residue classes. The route is spectral: the proof keeps only the ordered eigenvalues $\lambda_1\ge\cdots\ge\lambda_k$, the trace condition $\sum_i\lambda_i=1$, and the inequality $\lambda_1\le\lambda_{k-2}+\lambda_{k-1}+\lambda_k$, which is a relaxation of a known necessary condition for APPT spectra. Maximizing $\sum_i\lambda_i^2$ over that polytope is the entire content of the argument, and because the objective is strictly convex the maximum is found at a vertex. The paper classifies the vertices into three families and compares the purity values, obtaining the piecewise formula. For qubit-qudit systems the bound is attained, and because absolute separability coincides with APPT there, it is the maximum purity of absolutely separable states as well.
Load-bearing premise
The whole proof rests on applying the known APPT spectral inequality $\lambda_1\le\lambda_{k-1}+2\sqrt{\lambda_{k-2}\lambda_k}$ (and its relaxation) to the state, which implicitly requires both local dimensions at least 2; the theorem as printed omits that hypothesis and is false for $m=1$.
Editorial extensions
If this is right
- For every absolutely separable state on $\mathbb{C}^m\otimes\mathbb{C}^n$ with $mn\ge8$, the same purity ceiling holds, improving the earlier general bound $2/(mn)$.
- For qubit-qudit systems with $m=2$ and $n\ge4$, the ceiling is attainable, so it gives the exact maximum purity of both APPT and absolutely separable states.
- The extremal spectra are essentially two-value spectra, so any state that reaches the bound is a mixture of two uniform components on different rank supports.
- For qutrit-qudit and larger systems the theorem gives an upper bound only; the exact maximum remains open, as the paper notes.
Reading between the lines
- Because the argument uses only the relaxed constraint, the same vertex classification should bound other convex spectral functions of APPT states, such as $\sum_i\lambda_i^p$ for $p>1$, yielding dimension-mod-4 ceilings for Rényi-type purities.
- The relaxation is lossy for systems beyond qubit-qudit; the gap between the theorem's value and the true maximum could be explored by testing the classified vertices against the full APPT inequalities rather than the relaxed one.
- For $k=8$ one of the two maximizing spectra in $S_8$ violates the original APPT inequality, so the attainable maximum in the $2\times4$ case is pinned to the other spectrum; checking this explicitly for $3\times3$ and $3\times4$ systems would show how tight the general bound is.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper gives an upper bound for the purity Tr(ρ²) of bipartite absolutely positive partial transpose (APPT) states in terms of the total dimension k=mn. The proof translates the APPT spectral condition into the relaxed eigenvalue inequality λ₁ ≤ λ_{k−2}+λ_{k−1}+λ_k, defines the polytope S_k of ordered spectra satisfying this inequality, enumerates its vertices, and maximizes the strictly convex purity functional over S_k. The resulting closed-form bound for k≥8 depends on k modulo 4 and matches the known qubit-qudit maximal purity results of Song and Chen for the covered range. The argument is transparent and self-contained apart from Hildebrand's necessary condition and standard convex analysis.
Significance. If the missing domain hypothesis is repaired, the result is a useful, clean improvement over the general purity bound 2/(mn) for absolutely separable states: for example, the new bound is 1/6 for k=8 and 1/9 for k=12, compared with 1/4 and 1/6 from the earlier bound. It also confirms the qutrit-qudit conjecture of Dũng and Khoi as an upper bound. The proof has genuine strengths: it reduces a quantum-information problem to a small, explicit vertex-enumeration problem, the polytope computation is transparent and checkable, and the type-3 maximizing spectra satisfy Hildebrand's stronger inequality with equality, so the bound is tight for nontrivial bipartite systems in the covered range.
major comments (1)
- [Theorem 1, Section 1] Theorem 1 is stated for mn≥8 without requiring m,n≥2. This is false as stated: for m=1 (or n=1), every state is APPT because partial transposition on a one-dimensional subsystem is trivial, and a pure one-qudit state has purity 1, which exceeds the claimed bound (for k=8 the claimed bound is 1/6). The proof in Section 2 uses Hildebrand's inequality λ₁ ≤ λ_{mn−1} + 2√(λ_{mn−2}λ_{mn}), which is a necessary condition for APPT spectra only when both local dimensions are at least 2. The statement should explicitly assume dim H_A, dim H_B ≥ 2 (equivalently m,n ≥ 2); with this additional hypothesis the upper-bound argument goes through.
minor comments (5)
- [Abstract and Section 1] The sentence 'For qubit-qudit systems, this upper bound becomes the maximum purity of APPT (and absolutely separable) states' should be qualified by the theorem's condition mn≥8; otherwise the two-qubit case (mn=4, true maximum 3/8) and the qubit-qutrit case (mn=6) are not covered by the theorem as stated.
- [Theorem 2.2, k=8] For k=8, the maximum over S_8 is also attained at the type-1 vertex (1/6,...,1/6,0,0), which does not satisfy Hildebrand's stronger inequality. This does not affect the upper-bound argument because the type-3 vertex achieves the same value, but a short remark clarifying that only the type-3 maximizer is APPT-admissible would prevent confusion.
- [Lemma 2.1] The proof uses the fact that every segment between two vertices of T_k is an edge; this is true because T_k is a simplex, but the claim is not stated or justified. A one-sentence explanation would make the vertex enumeration fully self-contained.
- [Proof of Theorem 2.2] The comparisons such as M₃ > M₂ for k≥12, k≥9, etc., are asserted without derivation. The inequalities are plausible and can be verified algebraically, but adding a short verification or an appendix would strengthen the paper.
- [References] Reference [1] contains a typographical error in the author name: 'D˜ ung' should be 'Dũng'.
Circularity Check
No circularity: the proof is self-contained, relying on an external spectral necessary condition and standard convex optimization.
full rationale
The derivation chain is self-contained relative to external inputs. The only link from APPT states to the optimization problem is Hildebrand's necessary spectral inequality λ1 ≤ λ_{mn−1} + 2√(λ_{mn−2}λ_{mn}), which is an external published result, not a restatement of the target purity bound. The relaxation to λ1 ≤ λ_{k−2} + λ_{k−1} + λ_k follows directly from the AM-GM inequality 2√(ab) ≤ a+b, so it is not a new assumption. The remainder of the proof is a vertex enumeration of the convex polytope S_k and a finite comparison of candidate maxima, with no fitted parameter renamed as a prediction and no load-bearing self-citation. The theorem's omission of the hypothesis m,n ≥ 2 is a genuine correctness defect, as the m=1 counterexample shows, but that is a failure of applicability of an external condition, not circularity. Likewise, the agreement with the known qubit-qudit maximum from Song and Chen is cross-validation of an independently derived upper bound, not construction from the known result.
Assumptions & free parameters
assumptions (4)
- domain assumption Hildebrand's necessary condition: lambda_1 <= lambda_{mn-1} + 2 sqrt(lambda_{mn-2} lambda_{mn}) for every APPT state
- standard math AM-GM inequality 2 sqrt(ab) <= a+b for nonnegative a and b
- standard math The purity functional F(lambda) = sum lambda_i^2 is strictly convex, so its maximum on the polytope S_k is attained at a vertex
- standard math The vertices of the ordered simplex T_k are nu_i = (1/i,...,1/i,0,...) and every pair of vertices spans an edge
Cite this review
Pith. "Pith review of An upper bound for the purity of absolutely positive partial transpose states." pith.science (2026). https://pith.science/paper/TYWA6QBD
@misc{pith2026260809832,
author = {Pith},
title = {Pith review of: An upper bound for the purity of absolutely positive partial transpose states},
year = {2026},
howpublished = {\url{https://pith.science/paper/TYWA6QBD}},
note = {Machine review of arXiv:2608.09832}
}
read the original abstract
A quantum state is called absolutely separable (resp. absolutely positive partial transpose (APPT)) if it remains separable (resp. positive partial transpose (PPT)) under any global unitary transformation. It is known that the set of all absolutely separable states is a subset of the set of all APPT states. Moreover, these two sets are identical for qubit-qudit systems. In this note, we give an upper bound for the purity of APPT bipartite states (and therefore for the purity of absolutely separable bipartite states). For qubit-qudit systems, this upper bound becomes the maximum purity of APPT (and absolutely separable) states.
Reference graph
Works this paper leans on
-
[1]
H. D˜ ung and V. Khoi,On the maximum purity of absolutely separable bipartite states, preprint 2025, arXiv:2510.19508
arXiv 2025
-
[2]
L. Gurvits and H. Barnum,Largest separable balls around the maximally mixed bipartite quantum state, Phys. Rev. A66(2002), 062311
work page 2002
-
[3]
Hildebrand,Positive partial transpose from spectra, Phys
R. Hildebrand,Positive partial transpose from spectra, Phys. Rev. A76(2007), 052325
work page 2007
-
[4]
Johnston,Separability from spectrum for qubit-qudit states, Phys
N. Johnston,Separability from spectrum for qubit-qudit states, Phys. Rev. A88(2013), 062330
work page 2013
- [5]
- [6]
-
[7]
Peres,Separability criterion for density matrices, Phys
A. Peres,Separability criterion for density matrices, Phys. Rev. Lett.77(1996), 1413
work page 1996
-
[8]
Z. Song and L. Chen,Extreme points of sets of absolutely separable and positive partial transpose states, Phys. Rev. A112(2025), 022409. Department of Mathematical Sciences, The University of Texas at Dallas, Richard- son, TX 75080, USA Email address:att140830@utdallas.edu
work page 2025
Reviewed August 11, 2026 · model on record in the stance chip above.
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