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For bipartite states invariant under diagonal unitaries, every level of the DPS hierarchy can be block-diagonalized, making high-level entanglement tests tractable; in the Bose-symmetric case the dual hierarchy is exactly characterized by r

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2026-08-03 18:08 UTC pith:35Q7AG6T

load-bearing objection Block diagonalization of DPS for diagonal unitary invariant states is new and valuable, but the proof of Theorem 4.4 has a false displayed identity that must be fixed before the copositive hierarchy connection is credible. the 2 major comments →

arxiv 2512.06551 v2 pith:35Q7AG6T submitted 2025-12-06 math.OC

Semidefinite hierarchies for diagonal unitary invariant bipartite quantum states

Jonas Britz , Monique Laurent This is my paper
classification math.OC MSC 90C2281P4015B48
keywords quantum entanglementDPS hierarchyseparable statesdiagonal unitary invarianceBose symmetrycopositive conesums of squaresblock diagonalization
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper shows that structural symmetry of a bipartite quantum state can be pushed into every level of the DPS hierarchy used to distinguish separable from entangled states. For states with diagonal-unitary invariance (the CLDUI, LDUI, and LDOI classes), the matrices in the semidefinite relaxations split into small independent blocks, so higher relaxation levels become computable in seconds instead of minutes. For Bose-symmetric states, the paper gives a dual characterization of the symmetry-adapted hierarchy: the associated complex polynomial must be a real sum of squares, exactly parallel to the known generic case. It then proves an equivalence between this symmetric hierarchy and the standard sum-of-squares approximations of the copositive cone, so entanglement detection for these states is tied to a well-studied optimization object. A concrete family of CLDUI states is shown to be separable exactly when it passes level two of the hierarchy, illustrating that the reductions do not lose detecting power.

Core claim

The central claim is that the DPS relaxation at any order t can be reduced without loss for diagonal-unitary-invariant states. The support of an extended certificate in CLDUI(t), LDUI(t), or LDOI(t) is contained in a graph that is a disjoint union of cliques, and this holds also after partial transposes; therefore the certificate is block diagonal, with blocks of size at most t! n^{ceil(t/2)} in the tensor formulation. Rephrasing the hierarchy in moment form removes the remaining replicated rows, shrinking the largest block further; the paper reports a ratio bound of at most t! / n^{ceil(t/4)} relative to the generic case. For Bose-symmetric states, the dual of the symmetric hierarchy gDPS^{

What carries the argument

The main engine is the sparsity pattern of CLDUI/LDOI states: their nonzero entries are indexed by equality of multisets or parity conditions, and the same patterns define the support of extended DPS certificates. These support graphs are disjoint unions of cliques, so positive semidefiniteness and partial-transpose conditions factor into independent blocks. The moment reformulation indexes certificates by monomial degrees rather than register sequences, eliminating the remaining duplicate rows. On the dual side, the bridge is the identity expressing the real part of the polynomial ||x||^{2(t-1)} <M, xx*⊗xx*> as a weight times a quadratic form in the squared real and imaginary parts, which i

Load-bearing premise

Everything rests on the claim that projecting an arbitrary DPS certificate onto the CLDUI/LDOI/LDUI subspace preserves the certificate conditions (positive semidefiniteness after every partial transpose and correct partial trace); for the copositive equivalence, Theorem 4.4 additionally depends on a polynomial identity whose printed form omits Hadamard squaring of the real and imaginary parts and is false as written, so the corrected identity is load-bearing.

What would settle it

Evaluate the displayed identity in the proof of Theorem 4.4 at t=2, n=2, A=I_2, x=(1+i,1): the left side has degree 6 in x,xbar while the right side as printed has degree 2 in xRe,xIm, so the line cannot be true; replacing xRe,xIm by their Hadamard squares gives the intended degree-6 identity. Then check the equivalence with K^{(t-1)} using Theorem 4.1(ii) for a matrix A known to lie in K^{(1)}_5 setminus K^{(0)}_5.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • For any CLDUI, LDUI, or LDOI bipartite state, testing membership in DPS level t reduces to semidefinite programs whose largest block is at most t! n^{ceil(t/2)}; in the moment formulation the largest block is at most t! n^{ceil(t/4)} times as large as in the generic case.
  • The block-diagonal implementation makes levels 4–7 practical for small n where the generic formulation stalls at level 2–3, so known separability criteria become testable at much higher strength.
  • For Bose-symmetric states rho^TB_{(X,X)}, the symmetry-adapted hierarchy is equivalent to the copositive approximation hierarchy (K^{(t)})^*; consequently separability of these states can be certified by checking X against (K^{(t-1)})^*.
  • For the family rho_{a,a'} defined in the paper, DPS level two is exact: the state is separable exactly when a, a' are at least 1, even though level one only certifies aa' >= 1.
  • The symmetric hierarchy converges to the Bose-symmetric separable cone, and any strict inclusion in the copositive hierarchy yields explicit states that escape the symmetric hierarchy.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The reported runtimes suggest the block-size advantage grows with the hierarchy level, so the highest relaxation levels benefit most from the sparsity reduction; the paper's tables show this but it is not stated as a theorem.
  • The Bose-symmetric equivalence implies a transfer of hardness: matrices that escape the sum-of-squares approximations of the copositive cone should yield Bose-symmetric entangled states that escape the symmetric DPS hierarchy for many levels.
  • The cliqued-support mechanism is not tied to unitary invariance specifically; analogous block-diagonal hierarchies should arise for other diagonal symmetry groups, such as orthogonal sign symmetries, following the same projection-and-clique argument.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper studies the Doherty-Parrilo-Spedalieri (DPS) hierarchy for separable bipartite quantum states under two structural assumptions: diagonal-unitary invariance (CLDUI/LDUI/LDOI states) and Bose symmetry. In the diagonal-unitary case it proves that the sparsity/block structure of the state is inherited by DPS certificates at every level, giving block-diagonal SDP formulations; it also develops a moment-formulation reduction and reports numerical experiments on PPT2-type benchmark instances. In the Bose-symmetric case it introduces a symmetry-adapted hierarchy gDPS(t), proves an r-sos characterization of its dual that mirrors Fang-Fawzi's theorem, and relates that dual to the copositive hierarchy K(t). The relation to the concurrent GNP25 work is explicitly acknowledged.

Significance. If completed, the paper would be a solid contribution: it offers substantial SDP-size reductions for classes of structured entangled states, an independent and more compact route to the gDPS-K(t) correspondence, and concrete numerical evidence. Strengths include the detailed proofs of the main structural results (Theorems 3.1 and 4.1), explicit block-size tables, and a public implementation. The overlap with GNP25 is handled transparently, and the authors are honest about the unresolved strictness question for gDPS(t). The central obstacle is one incorrect identity in the proof of Theorem 4.4; it is local and likely repairable, but it currently leaves Theorems 4.4-4.5 and the claimed equivalence with GNP25 unsupported.

major comments (2)
  1. [Section 4.2, Eq. (64) and the following display] The displayed real-variable identity in the proof of Theorem 4.4 is false as written. From (64), the real part of p is equal to the norm of w squared to the power (t-1) times the sum over h,k of A_hk times (w_h^2 + w_{n+h}^2)(w_k^2 + w_{n+k}^2). This equals the norm squared to the power (t-1) times the Hadamard square of w transposed times (J2 tensor A) times the Hadamard square of w, not the printed quadratic form using w itself. The left-hand side has degree 2t+2 in w, whereas the printed quadratic form has degree 2t; already for n=1, t=2 the two disagree. This identity is the step converting the r-sos of p into J2 tensor A in K_{2n}^{(t-1)} and then, via [GL07, Lemma 15], into A in K_n^{(t-1)}. Consequently Theorems 4.4 and 4.5 are not proved as printed. The correction is local and seems repairable, but the Hadamard-square identity and the subsequent reduction must be re-verified.
  2. [Section 3.3, Theorem 3.9 and Lemma 3.10] The LDUI/LDOI transport theorem and the clique-size bound are stated without proofs ('The proof is similar, thus omitted'; 'We omit the proof'). These results underlie the main efficiency claims for LDUI and LDOI states, including Corollary 3.11 and Tables 5-6. The later Lemma 5.4 is proved in the moment framework and is not explicitly shown to imply Lemma 3.10. Please supply the missing proofs or give a fully explicit reduction to the CLDUI case; as written the reader cannot verify the LDUI/LDOI block decompositions and size bounds from the text.
minor comments (5)
  1. [Abstract and Introduction] Spedaglieri appears to be a typo for Spedalieri (the author of [DPS02, DPS04]).
  2. [Section 4.1, Theorem 4.1(iii)] In the summand 'B + sum_{s=0}^{tau} W,' the last term appears to be missing its subscript: it should be W_s.
  3. [Section 4.2] The external reduction [GL07, Lemma 15] is invoked without stating the lemma. Since it is load-bearing for the corrected proof, a statement would improve self-containedness.
  4. [Tables 4-6] The notation m_k (k blocks of size m) is explained in the text but not in the table captions; please add it to the captions.
  5. [Section 5.1, Lemma 5.1] The proof is labelled a sketch and essentially follows from [GLS21, Section 5]. Since this lemma drives the moment-formulation implementation and the block-size tables, please provide a precise derivation or exact references.

Circularity Check

0 steps flagged

No significant circularity: the main derivations proceed from prior independent theorems and direct proofs; self-citations are to established, parameter-free results.

full rationale

I walked the claim chain. The Section 3 block-diagonalization results (Theorems 3.1, 3.9, Corollary 3.11) are derived from the definition of DPS(t), the projections onto the invariant subspaces, and the clique decomposition of the sparsity graphs; they do not presuppose the target conclusion. The one omitted proof (Theorem 3.9) is stated to be similar to Theorem 3.1, and Lemma 3.10's proof is deferred to the moment Lemma 5.4, which is proved; neither omission feeds a fitted or self-referential assumption. In Section 4, Theorem 4.1 is proved by a compact convex-duality and polynomial-identity argument modeled on Fang--Fawzi, not by assuming the characterization. Theorem 4.4 reduces membership in the dual of the symmetric DPS hierarchy to membership in K^{(t-1)} using polynomial identity (64) together with the GL07 lemma that J2⊗A∈K^{(t-1)}_{2n} iff A∈K^{(t-1)}_n. That lemma is a published, external, parameter-free result whose assumptions do not include the target theorem, so citing it is independent support rather than circularity. The overlap with GNP25 is openly acknowledged and the proof route is a direct derivation, not a renaming of that result. No constants are fitted and no outputs are recycled from input data; the computational experiments merely test the constructed hierarchies and report that no counterexamples were found. The only notable issue is a degree-mismatch in the printed real-variable identity in the proof of Theorem 4.4: the quadratic form w^T(J2⊗A)w should involve the Hadamard-square vector (xRe^{∘2}, xIm^{∘2}) rather than w=(xRe,xIm). This is a correctness/typo issue that needs repair, not a circular reduction, since the intended identity is not assumed as content of the theorem. Accordingly, no step reduces to its own input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

The paper introduces no fitted parameters and no new physical or mathematical entities. It relies on a small set of standard convex-analysis and polynomial-optimization theorems, plus the previously established DPS/FF20/GLS21 machinery. The only non-routine external input is the [GL07] lemma used in Theorem 4.4, and the proof of that theorem currently contains a typographical identity that needs correction.

axioms (5)
  • standard math DPS hierarchy completeness and the Fang-Fawzi characterization (Theorem 2.4) of (DPS^{(t)})* via r-sos polynomials.
    Used as the backbone for the symmetric dual characterization and for the comparison with the copositive hierarchy; cited from [FF20, GLS21].
  • standard math Convex-geometric dual of an intersection of cones is the closure of the sum of duals, together with self-duality of the cones W^{(t)}_{n,s}.
    Explicitly invoked in the proof of Theorem 4.1 to pass from the primal intersection of cone conditions to the dual decomposition.
  • standard math Reznick's theorem and the known properties of the K^{(t)} hierarchy for the copositive cone.
    Used to state the completeness of the copositive approximation hierarchy and to motivate the connection with the symmetric DPS dual.
  • standard math The reduction [GL07, Lemma 15] that J2⊗A ∈ K^{(t-1)}_{2n} if and only if A ∈ K^{(t-1)}_n.
    Load-bearing for Theorem 4.4's equivalence between the symmetric DPS dual and the copositive hierarchy.
  • standard math The moment/tensor equivalence for DPS certificates from GLS21, summarized in Lemma 5.1.
    The efficiency claims and Tables 4-6 rely on this equivalence to reduce identical rows/columns in the moment formulation.

pith-pipeline@v1.3.0-alltime-deepseek · 53283 in / 14674 out tokens · 135994 ms · 2026-08-03T18:08:02.362815+00:00 · methodology

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read the original abstract

We investigate questions about the cone $\mathrm{SEP}_n$ of separable bipartite states, consisting of the Hermitian matrices acting on $\mathbb{C}^n\otimes\mathbb{C}^n$ that can be written as conic combinations of rank one matrices of the form $xx^*\otimes yy^*$ with $x,y\in\mathbb{C}^n$. Bipartite states that are not separable are said to be entangled. Detecting quantum entanglement is a fundamental task in quantum information and a hard computational problem. We explore the Doherty-Parrilo-Spedalieri (DPS) hierarchy of semidefinite conic approximations for $\mathrm{SEP}_n$ when the bipartite states have some additional structural properties: first, (i) for states with diagonal unitary invariance, and second (ii) for states with Bose symmetry. In case (i) we show that the DPS hierarchy can be block diagonalized, which, combining with its moment reformulation, leads to a substantially more efficient implementation. In case (ii), we give a characterization of the dual hierarchy, in terms of sums of squares of Hermitian complex polynomials, extending a known result in the generic case. It turns out that the completely positive cone $\mathrm{CP}_n$, its dual cone $\mathrm{COP}_n$, and their sums-of-squares based conic approximations $\mathcal{K}^{(t)}_n$, play a central role in these two settings (i),(ii). We clarify these connections and test the block diagonal relaxations on classes of examples.

Figures

Figures reproduced from arXiv: 2512.06551 by Jonas Britz, Monique Laurent.

Figure 1
Figure 1. Figure 1: Graphic displaying the runtimes of the DPS hierarchy in three regimes: for CLDUI, [PITH_FULL_IMAGE:figures/full_fig_p044_1.png] view at source ↗

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Entanglement in the Dicke subspace

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    Mixtures of Dicke states are separable exactly when their parametrizing tensor is completely positive; this yields PPT-entangled examples for all n≥3, d≥3 and SDP relaxations for separability.

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