Supporting hyperplanes for Schmidt numbers and Schmidt number witnesses

Kyung Hoon Han; Seung-Hyeok Kye

REVIEW 1 major objections 5 minor 23 references

Supporting hyperplanes for Schmidt numbers and Schmidt number witnesses

T0 review · 1 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper proves a duality that pins down the exact Schmidt-number-1 boundary along pure-state one-parameter families and locates the corresponding supporting hyperplanes to the witness set.

desk verdict A clean convex-duality result with explicit witness positions for pure-state families; the main gap is one missing eigenvalue computation for the PPT threshold, which is easy to fill. read the letter →

arxiv 2506.03733 v3 pith:WE3MF6MP submitted 2025-06-04 quant-ph

Kyung Hoon Han , Seung-Hyeok Kye This is my paper

classification quant-ph MSC 15A3081P1546L0546L07

keywords supportinghyperplanesSchmidtnumberk-blockpositivematriceswitnessesWernerstatesisotropicseparabledecompositionpositivepartialtranspose

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 establishes that, for one-parameter families of bipartite states running through the maximally mixed state, finding supporting hyperplanes to the convex set of Schmidt-number witnesses is exactly the same problem as determining the boundary of the set of states of Schmidt number at most k. For the family generated by a pure state with Schmidt coefficients p0 ≥ p1 ≥ ⋯, it finds the exact boundary: Schmidt number 1 prevails up to λ = 1/(1+$n^{2}$ p0 p1), and the supporting hyperplane to the witness set sits at the negative value −($n^{2}$ p0 p1 + 1)/($n^{2}$ − 1). It also supplies an explicit decomposition of the separable Werner state as a sum of product states. A reader interested in entanglement quantification or in the convex geometry of state spaces would care because these are exact, not heuristic, boundaries.

What carries the argument

The one-parameter family X_λ = (1−λ)I/(mn) + λρ, together with the affine function f_λ(X) = ⟨X|X_λ⟩ whose level sets are precisely the hyperplanes perpendicular to the family. Theorem 2.4 is the load-bearing mechanism: it says the boundary interval [γ−[C], γ+[C]] of a compact convex set C and the supporting-hyperplane positions β̃±[C^∘] of its dual are tied by the zero-pair condition ⟨X_ν|X_μ⟩ = 0. In the pure-state computation, the key constructive tool is averaging product states |η_α⟩ = (Σ √p_i α_i |i⟩) ⊗ (its conjugate) over α_i ∈ {±1, ±i}, which yields the separable decomposition for the endpoint state X_μ.

What would settle it

For any concrete n and Schmidt coefficients, compute the smallest eigenvalue of X_λ^Γ as a function of λ. The claim predicts that this eigenvalue has its last zero precisely at λ = 1/(1 + $n^{2}$ p0 p1); finding a λ greater than this value with X_λ^Γ positive semidefinite, or finding the eigenvalue still negative exactly at this value, would refute Theorem 3.1.

Watch

Extended reading notes

Core claim

The central discovery is Theorem 2.4, a duality principle: for a compact convex set C in the affine space of trace-one Hermitian matrices with the maximally mixed state as an interior point, a hyperplane perpendicular to the one-parameter family {X_λ} supports the dual set C^∘ exactly when its partner hyperplane, through the orthogonal point X_μ with ⟨X_ν|X_μ⟩ = 0, passes through the boundary point of C. Applying this with C = S_1, the set of states of Schmidt number at most 1, and C^∘ = BP_1, the trace-one 1-blockpositive witnesses, and with ρ = |ξ⟩⟨ξ| a pure state, the paper proves Theorem 3.1: the supporting hyperplane to BP_1 lies at β̃^-_1 = −($n^{2}$ p0 p1 + 1)/($n^{2}$ − 1), while the Schmidt-number-1 interval ends at σ^+_1 = 1/(1 + $n^{2}$ p0 p1). The state at that endpoint is separable, and the touching witness is the Choi matrix of an explicit completely copositive map. In the isotropic and Werner special cases this recovers the known boundaries and gives a new product-state decomposition of the separable Werner state.

Load-bearing premise

The proof that σ^+_1 equals 1/(1 + $n^{2}$ p0 p1) relies on the unproved assertion that this value is exactly the largest λ for which the partial transpose X_λ^Γ is positive semidefinite, so if that PPT threshold were smaller or larger, the equality and the derived supporting-hyperplane value would need adjustment.

Editorial extensions

If this is right

For every pure state in C^n ⊗ C^n with ordered Schmidt coefficients, the Schmidt-number-1 interval along the line through the maximally mixed state ends exactly at 1/(1 + n^2 p0 p1), and the supporting hyperplane to the witness set is at −(n^2 p0 p1 + 1)/(n^2 − 1).
The same duality applies to every k: once the boundary interval for k-blockpositivity is known, the Schmidt-number-k boundary and the supporting hyperplane position are determined by a single orthogonality relation.
Along this one-parameter family, the PPT condition and separability coincide for λ ≥ 0: the state is separable exactly when its partial transpose is positive semidefinite.
The averaging construction with α ∈ {±1, ±i} gives a direct, explicit product-state decomposition of the separable Werner state, complementing earlier existence arguments.

Reading between the lines

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

The same zero-pair duality should extend to other one-parameter families, including those generated by non-pure states, as long as the boundary interval for block-positivity can be computed.
The method of averaging over phase choices could serve as a template for constructing separable decompositions of boundary states for other symmetric families, not only the isotropic and Werner lines.
The unproved PPT threshold used in the paper suggests a testable operational criterion: along this family, the largest λ with PPT is conjecturally the exact Schmidt-number boundary, which could be checked numerically in higher dimensions.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The paper is a short, well-written note. The main result, Theorem 2.4, is a clean equivalence: for a compact convex set C with the maximally mixed state as an interior point, the supporting hyperplane to the dual set C° perpendicular to a one-parameter family sits at the point orthogonal to an endpoint of the interval for C. Applied to Schmidt number sets and their witness sets, this turns the problem of finding σ±1 and the supporting hyperplane values into the same interval computation. That is a genuine conceptual simplification, though not a deep surprise; it is essentially bipolar duality in the specific affine geometry of this family.

The paper does the concrete work well. For a pure state with Schmidt coefficients p0 ≥ p1 ≥ …, it derives closed forms for σ+1 = 1/(1+n²p0p1), the supporting hyperplane position β̃−1 = −(n²p0p1+1)/(n²−1), and also the lower interval endpoint σ−1 = −1/(n²−1). The product-state decomposition of X_μ (the separable state at the upper endpoint) is explicit and verifiable—I checked the averaging phase factors and the diagonal remainder is nonnegative. The decomposition of the separable Werner state at the lower endpoint is also explicit. These are the paper's real contributions.

The soft spots are real but minor. The biggest is the assertion, without proof, that μ = 1/(1+n²p0p1) is the largest λ for which X_λ is PPT. This is load-bearing: it is what excludes separability for λ > μ and fixes σ+1 = μ and, via Theorem 2.4, β̃−1. The computation is indeed simple—the partial transpose decomposes into 2×2 blocks with eigenvalues (1−λ)/n² ± λ p_i p_j—so the gap is closing a line of algebra, not a conceptual hole. Still, the authors should put the computation in the paper.

There are two smaller dings: the proof of Proposition 2.2(iii) has an "it is easy to see" step before invoking a book citation, and the value β−1 = −1/(n²p0²−1) is taken from the authors' companion preprint [5] rather than proved here. Neither undermines the argument; they are the kind of thing a referee would ask the authors to make self-contained.

The paper is worth a serious referee. The intended reader is a specialist in quantum entanglement who works with convex sets and positive maps; it is not a general-audience paper. I would recommend sending it to review with a request that the authors add the PPT eigenvalue computation and make the dependency on [5] explicit.

Referee Report

1 major / 5 minor

Summary. The paper studies the compact convex sets S_k of states with Schmidt number at most k and BP_k of trace-one k-blockpositive matrices, which are dual to each other. For a one-parameter family X_λ = (1−λ)ρ_* + λρ through the maximally mixed state, the authors prove a general duality theorem (Theorem 2.4) stating that determining the interval on which X_λ lies in S_k is equivalent to locating the supporting hyperplanes to BP_k perpendicular to the family. They then specialize to the case where ρ is a pure state with Schmidt coefficients p_0 ≥ p_1 ≥ ... and supply explicit formulas for σ_1^±, β_1^±, and the tilted supporting hyperplane numbers \tildeβ_1^± and \tildeσ_1^±. The proofs use convex duality, explicit product-state decompositions of the state X_μ at the separability threshold, and the positive partial transpose criterion.

Significance. The paper gives a clean general principle that reduces the search for supporting hyperplanes perpendicular to a fixed line to the computation of Schmidt-number intervals on that line. The main application to pure-state families yields closed-form values that interpolate between the isotropic and product-state extremes, and the product-state decompositions are explicit and hand-checkable. The central derivation is algebraic and does not rely on numerical fits; if the identified missing justification is supplied, the results are a useful addition to the literature on Schmidt-number witnesses.

major comments (1)

[Section 3, proof of Theorem 3.1] The statement "We note that µ is the maximum of λ’s such that X_λ is of PPT" is load-bearing: it is the only step that excludes separability for λ > µ, and it fixes σ_1^+ = µ and, through Theorem 2.4, the value \tildeβ_1^−. The eigenvalue computation for X_λ^Γ is not shown. Please supply it explicitly: after the partial transpose, each off-diagonal pair {|ij⟩,|ji⟩} with i≠j gives eigenvalues (1−λ)/n² ± λ p_i p_j, and each diagonal entry is (1−λ)/n² + λ p_i²; positivity of all eigenvalues is exactly equivalent to −1/(n²p_0²−1) ≤ λ ≤ 1/(1+n²p_0p_1). Without this computation, the equality σ_1^+ = µ is not established in the text.

minor comments (5)

[Section 3, proof of Theorem 3.1] The decomposition (n²p_0²−1)X_{β_1^−} = Σ_{i>j} ρ_ij^Γ + D with D a diagonal matrix with nonnegative entries is not correct for general Schmidt coefficients. Explicitly D_ii = p_0² − p_i Σ_j p_j, which can be negative, for example when (p_0,p_1,p_2) = (0.9,0.3,0.3) after normalization. Since this claim is used only as motivation for the candidate witnesses and does not affect the final values, please correct it or replace it with a statement that does not assert nonnegativity.
[Proposition 2.2(iii)] The step "it is easy to see the relation (4)" is compressed. Please expand the argument: for Y ∈ (\tilde C)^°, the positivity of Tr(Y) = mn⟨Y|ρ_*⟩ from [12, Proposition 2.3.1] gives Y = Tr(Y) · (Y/Tr(Y)) with Y/Tr(Y) ∈ C^°, which yields the identification (\tilde C)^° = \widetilde{C^°}.
[End of Section 3] The bullet list asserting that X_λ^Γ is a state exactly for −1/(n²p_0²−1) ≤ λ ≤ 1/(n²p_0p_1+1) and is separable exactly on −1/(n²−1) ≤ λ ≤ 1/(n²p_0p_1+1) is stated without proof. The first bullet is the missing PPT threshold from the major comment; the second should be justified by citing the positive partial transpose criterion together with the earlier separability arguments.
[Theorem 2.4 proof] In the proof of Theorem 2.4 the notation "H^0_ν \ H^−_ν" is used where the set-theoretic difference is meant; writing H^0_ν ∪ H^+_ν would avoid any ambiguity, since for µ > 0 the condition ⟨W−X_ν|X_µ⟩ ≥ 0 places W in H^0_ν ∪ H^+_ν.
[Throughout] The text contains several OCR-type artifacts (e.g., "á", "q", "suppress" in the reference list) that should be cleaned in the final version.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction: the principal new endpoint σ+1 is established by an explicit product-state decomposition and convex duality; the reliance on the authors' earlier preprint [5] for β−1 and the unproved PPT-threshold assertion are gaps, not circularity.

full rationale

The derivation chain is not circular. The new quantities σ+1 and β~−1 are obtained by explicit convex-geometric arguments: ν is chosen as the minimum of the candidates −(n^2 p_i p_j + 1)/(n^2 − 1), μ is defined by the orthogonality condition 〈X_ν|X_μ〉 = 0, and X_μ is then shown separable by an explicit average of product states. This is a constructive derivation, not an imposition of the answer. The assertion that μ is the largest λ for which X_λ is PPT is load-bearing for the equality σ+1 = μ, but it is an omitted eigenvalue computation rather than a circular definition: the PPT threshold is a property of X_λ^Γ independent of the claimed Schmidt-number endpoint. Similarly, the use of the authors' own preprint [5] for β−1 = −1/(n^2 p0^2 − 1) is self-citation, and it is load-bearing for part of Theorem 3.1, but it is a parameter-free prior formula with stated assumptions and does not restate the new conclusion. No fitted parameter is renamed as a prediction, and no uniqueness theorem from the authors is used to forbid alternatives. The main central claim therefore has independent, checkable content; the shortcomings are missing justifications, not circularity. Score 2 reflects the notable but non-circular self-citation and the unproved PPT-threshold step.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claims rest on three kinds of inputs: standard convex geometry, standard quantum information duality, and two formulas from the authors' companion preprint [5]. There are no fitted free parameters. The paper's contribution is the derivation of new endpoint and supporting-hyperplane values from these ingredients.

assumptions (4)

domain assumption The duality BP_k^o = S_k and S_k^o = BP_k between trace-one k-blockpositive matrices and Schmidt number at most k states holds under the trace pairing.
Used throughout to translate between witness hyperplanes and state intervals; it is standard in the quantum information literature (refs. [6,18] and duality in Section 2).
standard math rho_* (maximally mixed state) is an interior point of S_1, so S_1 and BP_1 are compact and satisfy the conditions of Theorem 2.4.
Cited to [3] (largest separable balls) in the paragraph before Proposition 2.3; needed for the dual convex set to be compact and for the interval representation.
domain assumption beta-_1 = -1/(n^2 p0^2 - 1) and the PPT threshold mu = 1/(1+n^2 p0 p1) are taken from the authors' companion preprint [5].
The first formula is explicitly quoted from [5] in the proof of Theorem 3.1; the second is stated as a note without proof. Both are load-bearing inputs to Theorem 3.1.
standard math Standard closed-convex-cone duality (C = tilde(C) intersect H and (tilde(C))^o^o = tilde(C)) is invoked in Proposition 2.2(iii).
Used to prove C = C^o^o; cited to [12, Proposition 2.3.1].

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Cite this review

Pith. "Pith review of Supporting hyperplanes for Schmidt numbers and Schmidt number witnesses." pith.science (2026). https://pith.science/paper/WE3MF6MP

@misc{pith2026250603733,
  author       = {Pith},
  title        = {Pith review of: Supporting hyperplanes for Schmidt numbers and Schmidt number witnesses},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WE3MF6MP}},
  note         = {Machine review of arXiv:2506.03733}
}

abstract

We consider the compact convex set of all bi-partite states of Schmidt number less than or equal to $k$, together with that of $k$-blockpositive matrices of trace one, which play the roles of Schmidt number witnesses. In this note, we look for hyperplanes which support those convex sets and are perpendicular to a one parameter family through the maximally mixed state. We show that this is equivalent to determining the intervals for the dual objects on the one parameter family. We illustrate our results for the one parameter families including Werner states and isotropic states. Through the discussion, we give a simple decomposition of the separable Werner state into the sum of product states.

Figures

Figures reproduced from arXiv: 2506.03733 by the authors.

**Figure 1.** Thick lines represent supporting hyperplanes to the convex set. In this note, we look for supporting hyperplanes to BPk which are perpendicular to the one parameter family (1) Xλ “ p1 ´ λq̺˚ ` λ̺ P Mm b Mn, ´8 ă λ ă `8 of Hermitian matrices, where ̺˚ “ 1 mn Imn denotes the maximally mixed state, and ̺ P Mm b Mn is a bi-partite state different from ̺˚. This will be, in fact, equivalent to know to what extent Xλ is a … view at source ↗

**Figure 2.** The horizontal line represents the one parameter family tXλu, and the vertical lines represent supporting hyperplane to the convex sets BPk and Sk. Proof. By the definition, we have xXγ˜`rC˝s |Xγ´rCsy “ 0. Since the affine map λ ÞÑ xXγ˜`rC˝s |Xλy is injective, ν “ γ˜ `rC ˝ s implies µ “ γ ´rCs. The reverse direction is just the definition of ˜γ `rC ˝ s. For the statement (ii), we note that Xµ P C if and only if W P… view at source ↗

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Reference graph

Works this paper leans on

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [5]

K. H. Han and S.-H. Kye, Global locations of Schmidt number witnesses, preprint. arXiv 2505.10288

work page Pith review arXiv

[2] [1]

Azuma and M

H. Azuma and M. Ban, Another convex combination of product states for the separa ble Werner state, Phys. Rev. A 73 (2006), 032315

work page 2006

[3] [2]

M. A. Graydon and D. M. Appleby, Quantum conical designs, J. Phys. A: Math. Theor. 49 (2016), 085301

work page 2016

[4] [3]

Gurvits and H

L. Gurvits and H. Barnum, Largest separable balls around the maximally mixed biparti te quantum state, Phys. Rev. A 66 (2002), 062311

work page 2002

[5] [4]

Ha and S.-H

K.-C. Ha and S.-H. Kye, Optimality for indecomposable entanglement witnesses, Phys. Rev. A 86 (2012), 034301

work page 2012

[6] [6]

Jamio/suppress lkowski,Linear transformations which preserve trace and positive s emideﬁnite operators, Rep

A. Jamio/suppress lkowski,Linear transformations which preserve trace and positive s emideﬁnite operators, Rep. Math. Phys. 3 (1972), 275–278

work page 1972

[7] [7]

Kye, Facial structures for positive linear maps between matrix a lgebras, Canad

S.-H. Kye, Facial structures for positive linear maps between matrix a lgebras, Canad. Math. Bull. 39 (1996), 74–82

work page 1996

[8] [8]

Kye, Boundaries of the cone of positive linear maps and subcones i n matrix algebras, J

S.-H. Kye, Boundaries of the cone of positive linear maps and subcones i n matrix algebras, J. Korean Math. Soc. 33 (1996), 669–677

work page 1996