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REVIEW 3 major objections 4 minor 46 references

Nonequivalence between absolute separability and positive partial transposition in the symmetric subspace

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Symmetric absolutely-PPT states are not always symmetric absolutely separable: the paper constructs entangled five-qubit SAPPT states and analogous counterexamples for larger odd qubit numbers.

desk verdict Clean analytic SAPPT boundary plus a numerically backed counterexample that should pass review once the witness positivity is certified. read the letter →

arxiv 2411.16461 v2 pith:D5YU73SX submitted 2024-11-25 quant-ph

Jonathan Louvet , Eduardo Serrano-Ensástiga , Thierry Bastin , John Martin This is my paper · ORCID
classification quant-ph
keywords symmetricmultiqubitstatesabsoluteseparabilityabsolutelyPPTpositivepartialtransposeentanglementwitnessboundDickeGHZstate
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

In quantum entanglement theory, the sets of absolutely separable states and absolutely PPT states are conjecturally related, and whether they coincide for general systems is open. This paper studies the same question inside the permutation-symmetric subspace of multiqubit states, where both the states and the allowed unitary evolutions are restricted. It establishes that the symmetric analogues of the two sets do not coincide: for five qubits, and for all larger odd numbers of qubits tested, there exist symmetric states $\rho(p)$ that remain PPT under every symmetry-preserving unitary evolution yet are entangled. The counterexamples are explicit mixtures of the symmetric maximally mixed state and the GHZ state, and entanglement is certified by explicit witness operators; the result does not settle the general non-symmetric AS-versus-APPT question, but it removes one natural route to proving it by symmetry.

What carries the argument

The argument has two load-bearing parts. The first is the exact spectrum of the partial transpose of the symmetric maximally mixed state: using Dicke-state ladder operators $M_\pm, M_0$ (angular-momentum analogues), the paper proves $\lambda_{\min}(\rho_0^{T_A}) = [(N+1)\binom{N}{k}]^{-1}$ for the $k|N-k$ bipartition, giving the SAPPT threshold $p_{\min}$ when combined with the two-equal-Schmidt-coefficient spectrum of the GHZ state. The second is a family of explicit entanglement witnesses $W_N$, matrices in the Dicke basis whose expectation values on symmetric product states are nonnegative, so they certify entanglement of symmetric states; their negative expectation on $\rho(p)$ for $p \leq p_{W}^{\mathrm{ent}}$ proves the states are entangled despite being SAPPT.

What would settle it

Evaluate $\langle \theta,\phi | W_5 | \theta,\phi \rangle$ over a fine grid of the single-qubit angles $\theta \in [0,\pi]$, $\phi \in [0,2\pi)$ with rigorous error bounds; finding any negative value would invalidate $W_5$ as an entanglement witness and remove the explicit five-qubit counterexample.

Watch

Extended reading notes

Core claim

For odd $N \geq 5$, the paper identifies the state $\rho(p) = p\,\rho_0 + (1-p)\,|GHZ_N\rangle\langle GHZ_N|$, with $\rho_0$ the maximally mixed state on the $N+1$-dimensional symmetric subspace, as a symmetric absolutely PPT (SAPPT) state exactly for $p \geq p_{\min}$, where $p_{\min} = [1 + 2((N+1)\binom{N}{\lfloor N/2\rfloor})^{-1}]^{-1}$. It then proves that for $p \in [p_{\min}, p_{\mathrm{ent}}]$ the state is entangled, e.g. for five qubits $p_{\min}=30/31$ and $p_{\mathrm{ent}} \approx 0.96953$. Entanglement is shown by explicit witnesses $W_5, W_7, W_9$ in the Dicke basis that are nonnegative on every symmetric product state yet have negative expectation on $\rho(p)$. These are the first counterexamples showing that SAPPT does not imply symmetric absolute separability (SAS), while the analogous family for even $N$ up to 10 remains separable wherever it is SAPPT.

Load-bearing premise

The witness operators $W_N$ are only proven nonnegative on every symmetric product state by numerical minimization (for $W_5$ the minimum is approximately $0.00276$ at $(\theta,\varphi)=(\pi/2,0)$), not by an analytic or certified-interval argument; if any product state gives a negative value, the corresponding counterexample is not rigorously established.

Editorial extensions

If this is right

  • For odd $N \geq 5$, the interval $[p_{\min}, p_{\mathrm{ent}}]$ of entangled SAPPT states provides uniparametric families of bound entangled states, since being PPT for every bipartition makes any entanglement non-distillable under local operations and classical communication.
  • The SAPPT and SAS sets coincide for two and three qubits but differ from five qubits onward, so a symmetry-restricted PPT check cannot be used to certify symmetric absolute separability in larger systems.
  • The same construction is conjectured to hold for symmetric $N$-qudit systems with $D = \binom{N+d-1}{d-1}$, giving an explicit SAPPT threshold formula of the same form.
  • Because these counterexamples live only in the symmetric subspace, the paper's result leaves the general AS-versus-APPT equivalence open; it shows that permutation symmetry itself can break the link, so perfect indistinguishability of constituents can change entanglement behaviour.

Reading between the lines

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

  • The nonnegativity of the witnesses could likely be turned from a numerical check into a rigorous certificate by searching for a sum-of-squares decomposition of $W_N(\theta,\varphi)$; the matrices are small enough that such a certification appears feasible.
  • One testable extension is to check whether entangled SAPPT states exist for every odd $N$, and whether the gap $p_{\mathrm{ent}} - p_{\min}$ shrinks with $N$; the tabulated values suggest the window narrows but could be quantified.
  • In bosonic experiments that prepare symmetric multiqubit states, for example multiphoton or spin-condensate systems, the family $\rho(p)$ is a natural candidate for observing bound entanglement that is robust under all symmetric unitary operations.
  • A parallel question could be posed for $k$-copy symmetric extensions: whether states that admit symmetric extensions for all $k$ but are not SAS exist, refining the boundary between separable and bound-entangled symmetric states.
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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 whether, within the symmetric subspace of N qubits, every symmetric absolutely PPT (SAPPT) state is necessarily symmetric absolutely separable (SAS). It introduces a uniparametric family ρ(p) = p ρ0 + (1−p)|ψ0⟩⟨ψ0|, where ρ0 is the maximally mixed symmetric state, and proves a necessary and sufficient SAPPT condition for the corresponding spectrum (Theorem 1), based on an analytic eigendecomposition of the partial transpose of ρ0. For odd N ≥ 5 with |ψ0⟩ = |GHZ_N⟩, the paper claims that for a range of p above the SAPPT threshold the states are entangled, using explicit entanglement witnesses W5, W7, W9 and numerical symmetric-extension checks. The central conclusion is that SAPPT does not imply SAS in the symmetric subspace.

Significance. If the counterexamples are rigorously established, the result settles an open question in the symmetric setting and has implications for entanglement in bosonic systems and for spin-j absolutely classical states. The analytic part—Theorem 1 and the spectral analysis of ρ0^TA (Appendices B and C)—is clean, self-contained, and appears mathematically sound; it also provides a new SAPPT criterion that improves on earlier bounds. The construction of explicit witnesses is valuable, but the verification of those witnesses is currently numerical rather than certified, which leaves the main claim in need of a rigorous foundation.

major comments (3)
  1. [Section III, Eq. (14)] The claim that W5 is a valid entanglement witness rests on the assertion that W5(θ,φ) ≥ 0 for all symmetric product states, supported only by a numerical minimization (reported minimum ≈ 0.00276 at (θ,φ) = (π/2,0)) and a logarithmic plot. Since W5 contains a large negative off-diagonal coefficient c = −9.31947 and a negative diagonal entry b = −0.134595, positivity is not structurally evident, and a numerical scan over a discretized parameter space cannot exclude a small negative region elsewhere. No analytic proof, interval-arithmetic bound, or code is provided. This is a load-bearing gap: without a rigorous certificate of global positivity, the existence of an entangled SAPPT state for N = 5 is not mathematically established. The same issue affects the witnesses W7 and W9 in Appendix D.
  2. [Section III, QETLAB paragraph] The statement that ρ(p) is entangled because it lacks a 2-copy PPT symmetric extension is based on a numerical SDP check implemented in QETLAB. As reported, no dual certificate, numerical tolerances, or code are given, so this evidence cannot serve as a rigorous proof of entanglement either. The manuscript should provide a certified numerical proof (e.g., via interval arithmetic, a rational SOS decomposition, or a verifiable dual solution) or explicitly qualify the conclusion as a numerical finding rather than a proof.
  3. [Conclusions] The conclusions state that the paper 'proved the existence of entangled SAPPT states.' Given that the witness verification and the symmetric-extension checks are numerical, this overstates the rigor of the argument. The conclusion should be conditioned on a certified verification of W5 positivity, or the proof should be upgraded to an analytic or interval-arithmetic certificate.
minor comments (4)
  1. [Section III, Figure 1] The top panel of Figure 1 is a logarithmic plot of W5(θ,φ); the axis ranges, color scale, and the method used for the minimization are not described, which makes it difficult to assess the uniqueness claim for the minimum at (θ,φ) = (π/2,0).
  2. [Appendix C, Eq. (C1)] The operator K0 is defined with eigenvalues (N/2 − α), and the eigenvalue of M0 is later stated as k + β − α − N/2. The arithmetic is correct, but the presentation would be clearer if the eigenvalue of M0 were written as (k/2 − α) − ((N−k)/2 − β), making the cancellation explicit.
  3. [Section I] The acronyms APPT and SAPPT are used extensively; spelling out 'absolutely positive partial transposed' and 'symmetric absolutely positive partial transposed' at first use would improve readability.
  4. [General] The paper does not provide the code used for the numerical minimization, the QETLAB checks, or the truncated-moment-method calculations. Supplying this code (or at least detailed parameters and tolerances) would aid reproducibility and allow the numerical claims to be independently audited.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the SAPPT threshold comes from an analytic spectral bound, and the entanglement witness is a standard state-tailored certificate whose validity is checked separately.

full rationale

The derivation of the SAPPT condition is self-contained. Theorem 1's pmin follows from the bound lambda_min(A+B) >= lambda_min(A)+lambda_min(B), the analytic eigendecomposition of rho0^TA in Appendix C, the exact Schmidt spectrum of rho_psi0^TA, and an explicit GHZ-state saturation; no parameter is fitted to the entanglement conclusion. The counterexample is then supported by two independent checks: absence of a 2-copy PPT symmetric extension (QETLAB) and the explicit witness W5. Constructing a witness tailored to the state is standard practice, and W5's validity is a separate positivity condition over all symmetric product states, not merely a restatement of Tr(W5 rho) < 0. The claimed range p in [pmin, pW5_ent] is just the linear dependence of the witness functional on p. Self-citations (Refs. [25], [26], [30]) appear only as background on SAS parametrization and existing witnesses and are not load-bearing; no uniqueness theorem from the authors' prior work is invoked. The main caveat is one of rigor rather than circularity: W5 positivity is verified only numerically (minimum approximate to 0.00276) and QETLAB is a numerical SDP, so the existence claim rests on numerical evidence. That is a completeness concern, not a case in which a prediction is equivalent to its input by construction.

Assumptions & free parameters 4 free parameters · 7 assumptions · 0 invented entities

No new physical entities are introduced; all ingredients are known objects (Dicke states, GHZ states, entanglement witnesses). The central claim depends on fitted witness coefficients and on a numerical positivity check that is not backed by an analytic proof or shipped code.

free parameters (4)
  • W5 witness coefficients (a,b,c) = a=0.0366656, b=-0.134595, c=-9.31947
    Numerical values chosen so that Tr(ρ(p_min)W5)<0 while W5(θ,φ)≥0 on symmetric product states; no closed form or code provided.
  • W7 witness coefficients (a,b,c,d) = a=0.00197514, b=0.0643064, c=-0.189017, d=-31.2405
    Appendix D witness for 7 qubits, constructed analogously to W5.
  • W9 witness coefficients (a,b,c,d,e) = a=0.00235791, b=-0.013747, c=0.0621661, d=-0.1636915, e=-114.305
    Appendix D witness for 9 qubits, constructed analogously to W5.
  • p_ent thresholds from truncated moment method = N=5: 0.96953, N=7: 0.99329, N=9: 0.99849 (precision 1e-5)
    Numerically estimated boundaries between entangled and separable SAPPT states for the GHZ-based family; not derived analytically.
assumptions (7)
  • standard math Minimum eigenvalue of a sum of Hermitian matrices is at least the sum of their minimum eigenvalues.
    Invoked in Eqs. (4)-(5) to lower bound λ_min(ρ^TA).
  • standard math For a pure bipartite state, the minimal eigenvalue of its partial transpose equals -sqrt(Γ_1Γ_2), where Γ_i are the two largest Schmidt coefficients.
    Used after Eq. (9) to compute λ_min(ρ_ψ^TA).
  • standard math Binomial identities, including Vandermonde convolution, hold as stated.
    Used in Appendix C to prove the eigenspectrum of ρ_0^TA.
  • domain assumption A symmetric N-qubit state is permutation invariant and its bipartite reductions have support on H^∨k ⊗ H^∨(N−k).
    Foundation of the symmetric subspace treatment in Section II.
  • domain assumption Any separable symmetric state can be written as a convex combination of symmetric product states |θ,φ⟩⟨θ,φ|^⊗N.
    Used to justify checking W5 on product states only, citing Refs. [40,41].
  • domain assumption The nonexistence of a 2-copy PPT symmetric extension implies entanglement.
    Used in Section III to detect entanglement via QETLAB, citing Ref. [38].
  • ad hoc to paper The numerical check showing min W5(θ,φ)≈0.00276>0 is treated as proof of positivity for all product states.
    The paper states the minimum value but provides no analytic certificate or code, making this the load-bearing numerical assumption.

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Pith. "Pith review of Nonequivalence between absolute separability and positive partial transposition in the symmetric subspace." pith.science (2026). https://pith.science/paper/D5YU73SX

@misc{pith2026241116461,
  author       = {Pith},
  title        = {Pith review of: Nonequivalence between absolute separability and positive partial transposition in the symmetric subspace},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/D5YU73SX}},
  note         = {Machine review of arXiv:2411.16461}
}
read the original abstract

The equivalence between absolutely separable states and absolutely positive partial transposed (PPT) states in general remains an open problem in quantum entanglement theory. In this work, we study an analogous question for symmetric multiqubit states. We show that symmetric absolutely PPT (SAPPT) states (symmetric states that remain PPT after any symmetry-preserving unitary evolution) are not always symmetric absolutely separable by providing explicit counterexamples. More precisely, we construct a family of entangled five-qubit SAPPT states. Similar counterexamples for larger odd numbers of qubits are identified.

Figures

Figures reproduced from arXiv: 2411.16461 by the authors.

Figure 1
Figure 1. FIG. 1. Top panel: Logarithm of the expectation value of [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Schematic representation of the common eigenvectors [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

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