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The geometry of absolute separability and other convex matrix properties from spectrum

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Pith's one-line read The spectra of absolute-PPT bipartite states form a spectrahedron for every $m\le n$, with all faces exposed and a rank formula for face dimensions.

desk verdict Solid new geometry for APPT spectra; the main LMI theorem rests on a cited lemma and the exponential volume claim outruns the proof. read the letter →

arxiv 2608.03390 v1 pith:6KSKWTVF submitted 2026-08-04 quant-ph math-phmath.MP

Jennifer Ahiable , Naga Bhavya Teja Kothakonda , Andreas Winter This is my paper · ORCID
classification quant-phmath-phmath.MP MSC 52A2014P1090C2281P40
keywords absoluteseparabilityPPTspectrahedronconvexgeometryofspectrasemialgebraicsetexposedfacesquantumentanglementvonNeumannentropy
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

Quantum states that keep the positive-partial-transpose property under every global unitary rotation are called absolute PPT, and their reachable eigenvalue distributions form a convex body $\mathrm{APPT}_{m,n}$ inside the probability simplex. The paper's central achievement is a permutation-symmetric reformulation of the known absolute-PPT criterion that makes $\mathrm{APPT}_{m,n}$ a spectrahedron: a spectrum belongs to the set exactly when every matrix $L_\pi(\lambda)$ in a permutation-indexed family is positive semidefinite. Because spectrahedra have only exposed faces, the paper obtains a complete face and extreme-point description of $\mathrm{APPT}_{m,n}$, with face dimension $(mn-1)-\mathrm{rank}(C)$ and maximal proper faces of dimension $mn-m-1$. It also proves that $\mathrm{ASEP}_{m,n}$ is semialgebraic for all $m\le n$ and that $\mathrm{ASEP}_{2,n}$ is itself a spectrahedron, which matters because spectral geometry may help decide whether absolute separability coincides with absolute PPT. The paper adds an inscribed polytope $\mathcal{P}_{m,n}$ giving rigorous purity, entropy, and volume bounds, and it explicitly notes that the converse of its extreme-point projection theorem is proven only for $m=2,3$.

What carries the argument

The carrying object is the permutation-indexed block matrix $L(\lambda)=\bigoplus_{\pi\in\widetilde{S}} L_\pi(\lambda)$, where $\widetilde{S}$ is a reduced set of permutations of the $mn$ eigenvalues and each $L_\pi(\lambda)$ is the $m\times m$ symmetric matrix defined by the index maps $p(i,j)$ and $q(i,j)$. These maps split the $m^2$ active spectral coordinates into diagonal and off-diagonal entries, and permutations acting only on the remaining $mn-m^2$ coordinates produce redundant blocks. This object converts the quantifier over all global unitaries in the definition of absolute PPT into a finite family of linear matrix inequalities, yields the topological boundary as a union of irreduc

What would settle it

A concrete check would be to take a small case such as $m=n=3$, generate spectra that satisfy the known absolute-PPT inequalities of [16], and test whether $L_\pi(\lambda)\ge 0$ for every permutation; a single violation would refute Theorem 4.1. Independently, the face-dimension formula would fail if any proper face's constraint matrix $C$ had rank outside $\{0\}\cup\{m,\ldots,mn-1\}$, and Conjecture 6.7 could be tested by a global SDP or branch-and-bound optimization for $m=5$ searching for an APPT spectrum with purity higher than the polytope's closed-form maximum.

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Extended reading notes

Core claim

On its own terms, the paper establishes that for every $m\le n$ the set $\mathrm{APPT}_{m,n}$ of absolute-PPT spectra is a spectrahedron: $$\mathrm{APPT}_{m,n}=\Delta_{mn-1}\cap $L^{{-1}}$\left(\bigoplus_{\pi\in\widetilde{S}} S_m^+\right),$$ where $L(\lambda)$ is the block diagonal matrix formed by the $m\times m$ matrices $L_\pi(\lambda)$. Each $L_\pi(\lambda)$ has diagonal entries $2\lambda_{\pi(p(i,i))}$ and off-diagonal entries $\lambda_{\pi(p(i,j))}-\lambda_{\pi(q(i,j))}$, with explicit index maps $p,q$ selecting $m^2$ of the $mn$ spectral coordinates. The paper then uses the kernel subspaces of these matrices to characterize faces: a face is determined by demanding $U_\pi\subseteq\ker L_\p

Load-bearing premise

The load-bearing external premise is the known absolute-PPT criterion, specifically the lemma cited as [16, Lemma 3], which Theorem 4.1 invokes by reference: if that lemma, or its applicability to all $m\le n$, is incomplete, then the spectrahedron representation of $\mathrm{APPT}_{m,n}$, the exposed-face corollary, and the rank-based dimension formula all inherit the failure.

Editorial extensions

If this is right

  • If the spectrahedral description is correct, membership of a spectrum in $\mathrm{APPT}_{m,n}$ is a semidefinite feasibility problem, so optimizing linear functionals over absolute-PPT spectra becomes a conic-optimization question.
  • Every face of $\mathrm{APPT}_{m,n}$ is exposed; the dimension of a face is $(mn-1)-\mathrm{rank}(C)$, extreme points are exactly the spectra with $\mathrm{rank}(C)=mn-1$, and maximal proper faces have dimension $mn-m-1$.
  • For qubit-qudit systems the maximal-face equations become explicit linear relations, recovering and organizing known extreme-point structure for $\mathrm{ASEP}_{2,n}$.
  • Since $\mathrm{ASEP}_{m,n}$ is semialgebraic while $\mathrm{APPT}_{m,n}$ is a spectrahedron, exhibiting any non-exposed extreme point of $\mathrm{ASEP}_{m,n}$ would prove absolute separability is strictly smaller than absolute PPT.
  • The inscribed polytope $\mathcal{P}_{m,n}$ supplies closed-form lower bounds on maximum purity, upper bounds on minimum von Neumann entropy, and an exact relative volume; numerical estimates indicate the relative volume of $\mathrm{APPT}_{m,n}$ decays exponentially in $n$ at nearly the polytope's rate.

Reading between the lines

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

  • Beyond the paper, the spectrahedral description suggests that the maximum-purity conjecture could be attacked with symmetry-reduced semidefinite programming; the numerics reported stop at $m=4$, so a certified SDP computation for $m=5$ would be a natural next check.
  • Inference: the rank formula for face dimensions invites a purely combinatorial enumeration of faces by assigning a kernel subspace to each active permutation, which could produce the full face lattice of $\mathrm{APPT}_{m,n}$ for small $m,n$ without sampling spectra.
  • Inference: if the polytope $\mathcal{P}_{m,n}$ is eventually shown to lie inside $\mathrm{ASEP}_{m,n}$—a containment the paper leaves open—then its exact purity, entropy, and volume bounds immediately become bounds on absolute separability, and the comparison of the two sets reduces to their boundaries outside the polytope.
  • Inference: the exposedness of every APPT face gives a concrete separation strategy: search for non-exposed extreme points of $\mathrm{ASEP}_{m,n}$. The paper's explicit face equations provide a finite algebraic system on which such a search could be based.
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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 / 4 minor

Summary. The paper studies the convex geometry of the sets of spectra of bipartite absolutely separable (ASEP) and absolute-PPT (APPT) states. The central technical claim is a permutation-symmetric reformulation of Hildebrand's absolute PPT criterion: for m≤n, λ∈APPT_{m,n} iff L_π(λ)≥0 for all π∈S_{mn}, where L_π is an m×m matrix whose entries are linear forms in the permuted spectrum. This yields the advertised spectrahedral representation APPT_{m,n}=Δ_{mn-1}∩{⊕_π L_π(λ)≥0}. From it the paper derives a determinantal description of the boundary, a kernel-based description of all faces, a rank formula for face dimensions, and the conclusion that every face is exposed. In the quantitative part, an inscribed polytope P_{m,n} is defined via Gershgorin-type sufficient conditions; its maximal purity, minimal von Neumann entropy and relative volume are computed. The paper also gives a Monte-Carlo study suggesting that the relative volume of APPT_{m,n} decays exponentially in n with a rate close to that of the polytope, and states a conjecture that the maximal purity of APPT_{m,n} equals that of P_{m,n} for all m,n except m=n=2.

Significance. If the central equivalence is correct, the paper makes a substantial contribution: it gives a finite spectrahedral description of APPT_{m,n} for all m≤n, with all faces exposed, an explicit kernel-based characterization of faces and extreme points, and a dimension formula governed by the rank of a permutation-block matrix. The use of Gershgorin arguments to obtain a concrete inscribed polytope and the exact computation of its purity, entropy and volume are valuable and in part already go beyond previous work. The numerical fits for the volume decay are interesting and provide a concrete testable conjecture. However, the paper's headline volume statement in the abstract is stronger than what is rigorously proved, and the main spectrahedral representation relies on an external lemma whose proof is not reproduced. Both points need to be addressed before the paper can be accepted as is.

major comments (3)
  1. [Theorem 4.1, Eqs. (7)-(8)] The step from the continuum of unitary constraints Tr(U diag(λ) U† (|ψ⟩⟨ψ|)^Γ)≥0 to the finite family L_π(λ)≥0 for all π∈S_mn is the load-bearing point of the whole paper, but the proof only says 'see the proof of [16, Lemma 3]' and does not reproduce it. Since this equivalence is what makes APPT_{m,n} a spectrahedron and is inherited by Theorem 4.3, Proposition 5.4, Corollaries 5.7, 5.10, and 5.12, the reader cannot verify that the LMI family is exactly the absolute PPT condition, especially without any ordering assumption on λ. Please either give a self-contained proof of the equivalence, or state precisely which statement of [16] is being used and why it applies to the unordered, permutation-symmetric setting.
  2. [Abstract and Section 6.3] The abstract states that the relative spectral volume of APPT_{m,n} decays exponentially in n 'by a constant multiplicative factor of the relative volume of the inscribed polytope'. The rigorous content of Section 6.3 is Theorem 6.16, which only gives the lower bound vol_rel(APPT_{m,n})≥max{vol_rel(BALL), vol_rel(P_{m,n})}. The exponential decay statement is supported by the numerical log-linear fits in Eq. (71) and Table 1, not by a proof. No upper bound of vol_rel(APPT) relative to vol_rel(P) is established. Please either prove the claimed asymptotic relation or weaken the abstract and Section 7 to state the lower bound and the numerical evidence.
  3. [Proposition 6.14] The derivation of the exact relative volume of P_{m,n} applies Lasserre's formula (62) to coefficients h(u_i) that are not pairwise distinct. The text says that an 'identical weights condition' from [40, Sec. 2.1] applies, but the limiting expression is not stated. Since the formula (63) is used as a rigorous lower bound and feeds into the quantitative claims, the limiting argument should be written out explicitly, or the statement should be restricted to the cases where the formula can be obtained by a limiting argument from (62).
minor comments (4)
  1. [Introduction, p. 2] There is a typo 'and and identify' in the first paragraph of the introduction.
  2. [Figure 2 caption] The caption reads 'and its the convex hull'; this should be 'and its convex hull'.
  3. [Eq. (5)] The index functions p(i,j) and q(i,j) are central but somewhat opaque; a small example or one-line enumeration for m=2 would help the reader.
  4. [Section 6.3, Eq. (71)] The fit log10 vol ≈ C_m + γ_m n is presented without error bars or number of Monte Carlo samples; adding these would make the numerical comparison more reliable.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the spectrahedral description imports an external, independently established criterion (Hildebrand) and the quantitative bounds are proven from Gershgorin, not from fitted values.

full rationale

The paper's central claim that APPT_{m,n} is a spectrahedron is grounded in Theorem 4.1, which takes Hildebrand's absolute-PPT criterion as external input. The proof explicitly delegates the key reduction from continuous unitary orbits to finite permutations: 'see the proof of [16, Lemma 3]'. This is a genuine external citation, not a self-citation, and the cited result does not depend on the present paper's conclusions. All subsequent facial and dimensional statements (Theorems 5.2-5.4, Corollaries 5.7-5.10) are derived from standard spectrahedra kernel theory applied to the LMI representation, and no equation reduces to a quantity that was fitted earlier in the paper. The polytope P_{m,n} bounds on purity, entropy, and volume are proven directly via Gershgorin's theorem and simplex geometry, with numerical results explicitly presented as numerics rather than as analytical predictions. The only concern is the load-bearing external lemma [16, Lemma 3], which is not reproduced in the paper; this is a correctness or verification risk, not a circularity, because it is an independent result and does not presuppose the target claims. No uniqueness theorem from the authors is invoked, and no self-citation chain carries the argument. Therefore the derivation is self-contained relative to its stated external premises, and no circular step is present.

Assumptions & free parameters 1 free parameters · 8 assumptions · 0 invented entities

The paper introduces no new physical entities. The key axioms are standard convex-geometric and real-algebraic results plus the external absolute PPT criterion of Hildebrand. The only numerically fitted quantity is the log-linear volume slope used for an interpretive claim rather than a proof. The polytope P_{m,n} is a new mathematical object, but its containment in APPT_{m,n} is proven, so it is not an unsupported invented entity.

free parameters (1)
  • γ_m, log-linear decay slope of APPT_{m,n} relative volume = m=2: -0.7625; m=3: -1.7454; m=4: -2.9033
    Fitted by linear regression to Monte Carlo log-volume estimates in Eq. (71) and Table 1, and used to support the abstract's exponential-decay claim.
assumptions (8)
  • standard math Krein-Milman theorem and conic version
    Used in Sections 2.1 and 3 to represent compact convex sets and pointed cones via extreme points and extreme rays.
  • standard math Hardy-Littlewood-Pólya majorization and Schur convexity
    Used in Theorems 3.1-3.5 to express a dephased unitary orbit as a convex combination of permuted diagonals.
  • standard math Tarski-Seidenberg quantifier elimination
    Used in Theorem 3.6 to show AM and spec(AM) are semialgebraic.
  • standard math Lasserre's simplex section volume formula
    Used in Proposition 6.14 to compute the exact relative volume of the inscribed polytope P_{m,n}, cited as [40].
  • standard math Determinant preserver theorem for symmetric matrices
    Used in Lemma 4.5 to characterize determinant-preserving linear maps on symmetric matrices as congruence by signed permutation matrices, citing [43], [44].
  • domain assumption Hildebrand's absolute PPT criterion, including the lemma used in Theorem 4.1
    The paper asserts λ∈APPT_{m,n} iff all L_π(λ)≥0 based on [16, Lemma 3] and does not reproduce the proof. This is the load-bearing external input for the spectrahedral representation.
  • domain assumption Johnston's equality ASEP_{2,n}=APPT_{2,n}
    Used to conclude ASEP_{2,n} is a spectrahedron and to derive the qubit-qudit maximal face equations, citing [17].
  • domain assumption Semialgebraicity of the set of separable states
    Used in Theorem 4.2 to derive that ASEP_{m,n} is semialgebraic, citing [22].

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Pith. "Pith review of The geometry of absolute separability and other convex matrix properties from spectrum." pith.science (2026). https://pith.science/paper/6KSKWTVF

@misc{pith2026260803390,
  author       = {Pith},
  title        = {Pith review of: The geometry of absolute separability and other convex matrix properties from spectrum},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6KSKWTVF}},
  note         = {Machine review of arXiv:2608.03390}
}
abstract

We investigate the geometric structure of the set of spectra of bipartite absolute separable states ($\mathrm{ASEP}_{m,n}$) and absolute positive partial transpose states ($\mathrm{APPT}_{m,n}$), i.e., bipartite quantum states that remain separable or PPT respectively, under all global unitary transformations. First, we establish general geometric properties of absolute convex sets of matrices, their spectra and extreme points. Regarding absolute separability, we present a permutation-symmetric reformulation of the absolute PPT criterion and use it to demonstrate that $\mathrm{APPT}_{m,n}$ is a spectrahedron for all $m\leq n$: in particular, all its faces are exposed. In contrast, while $\mathrm{ASEP}_{2,n}$ is also a spectrahedron, we prove that in general $\mathrm{ASEP}_{m,n}$ is a semialgebraic set for all $m\leq n$. Furthermore, we provide a complete characterization of the faces and extreme points of $\mathrm{APPT}_{m,n}$ and demonstrate that the dimension of a face is determined by the rank of a certain matrix, with maximal proper faces having dimension $(mn-m-1)$. In the quantitative setting, we provide a rigorous lower bound on the maximal attainable purity of $\mathrm{APPT}_{m,n}$ via an inscribed polytope $\mathcal{P}_{m,n}$ and conjecture that the maximal purity of $\mathrm{APPT}_{m,n}$ (along with its spectra) coincides with the polytope for arbitrary dimensions except when $m=n=2$. Additionally, we also provide a rigorous upper bound on the minimal von Neumann entropy of $\mathrm{APPT}_{m,n}$ and demonstrate numerically that the minimum entropy eventually coincides with the polytope $\mathcal{P}_{m,n}$ as the local system dimension $n$ increases. Finally, we show that the relative spectral volume of $\mathrm{APPT}_{m,n}$ decays exponentially in $n$ by a constant multiplicative factor of the relative volume of the inscribed polytope $\mathcal{P}_{m,n}$.

Figures

Figures reproduced from arXiv: 2608.03390 by the authors.

Figure 1
Figure 1. The absolute separable spectrahedron ASEP [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. On the left (Fig. 2a), the inscribed object (orange) depicts the set of normalized spectra of two-qubit decomposable block-positive matrices [18, Theorem 3] and the enclosing shell (purple), is the convex hull of this object. This convex hull is exactly the normalized polar witness set ASEP◦ 2,2 (Fig. 2b) of the spectrahedron in [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. A cross-section of ASEP2,2 at λπ(1) = 0.25 whose boundary is generated by κ2,2 = 6 irreducible hypersurfaces, each of dimension 2 (brightly colored). The interior (yellow) includes the center u4 = 1 4 (1, 1) corresponding to the spectra of the maximally mixed state. Each colored curve is a hypersurface ZVmn (detLπi ) for i = 1, . . . , 6 in the slice. The feasible regions of the topological boundary are indicated by… view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: Some extreme points and maximal faces of ASEP [PITH_FULL_IMAGE:figures/full_fig_p019_4.png]
Figure 5
Figure 5. Figure 5: Schematic illustration of how faces of ASEP [PITH_FULL_IMAGE:figures/full_fig_p020_5.png]
Figure 6
Figure 6. Figure 6: Plots of estimated maximum purity of APPT [PITH_FULL_IMAGE:figures/full_fig_p030_6.png]
Figure 7
Figure 7. Figure 7: The polytope P2,2 inscribed in ASEP2,2 with their respective purity maximizers and entropy minimizers. minλ∈Pm,n S(λ)− minλ∈APPTm,n S(λ) so that up to numerical error, ∆Smin ≥ 0 since Pm,n ⊆ APPTm,n. If ∆Smin = 0, then the entropies of the two sets coincide and if ∆Smi…
Figure 8
Figure 8. Figure 8: The minimal von Neumann entropy of APPTm,n and Pm,n where m = 2, 3, 4. For each m, the minimum of APPTm,n eventually coincides with the polytope after the discrete transition value n ∗ S [PITH_FULL_IMAGE:figures/full_fig_p035_8.png]
Figure 9
Figure 9. Figure 9: The minimal von Neumann entropy gap of APPT [PITH_FULL_IMAGE:figures/full_fig_p035_9.png]
Figure 10
Figure 10. Figure 10: The estimated relative volume of APPTm,n for m = 2, 3, 4 (red) in comparison with the exact polytope relative volume (purple) and exact separable ball volume (dotted blue). Our two-qubit relative spectral volume estimate of approximately 0.32618 agrees with the spectr…
Figure 11
Figure 11. Figure 11: Log-linear regression of the relative volume for APPT [PITH_FULL_IMAGE:figures/full_fig_p039_11.png]

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Pith tools

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