{"id":"001618fd-5121-43fb-b3ed-9caa4efea65e","arxiv_id":"2411.16461","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A family of symmetric five-qubit states that remain PPT under all symmetric unitary evolutions are shown to be entangled, refuting the SAPPT equals SAS equivalence for odd N≥5.","lead":"For systems of five or more identical quantum particles, some states stay positive under every allowed rotation and under the standard entanglement test, yet are still entangled. This disproves a natural conjecture that these two properties coincide for symmetric states, a focused step toward a broader open problem in quantum information.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"N=5 counterexample hinges on numerically verified witness positivity; without an analytic certificate the main conclusion is not rigorously established.","rationale":"The reader's weakest assumption correctly identifies the numerical verification of W5 positivity as the principal gap. I re-read the construction: Theorem 1 and the derivation of λmin(ρ0^TA) in Appendix C are analytic and internally consistent; the GHZ-based necessity also checks out. The only point at which 'entangled SAPPT' could fail is the validity of the entanglement witnesses. The paper gives no executable code and no certified bound, and the witnesses for N=7,9 have the same status. This does not rise to rejection because the claim is concrete and independently checkable, and the numerical evidence is substantial; it does justify a conditional verdict pending a certified positivity proof or shipped verification script. My recommendation is therefore to leave the reader's CONDITIONAL verdict unchanged.","tokens_in":22882,"tokens_out":5131,"duration_ms":50954,"concrete_test":"Verify W5 positivity rigorously: express F(y,φ)=⟨θ,φ|W5|θ,φ⟩ with y=sin^2(θ/2) and note the φ-dependent off-diagonal term is minimized at cos(5φ)=1, leaving a one-variable function F(y). Use interval arithmetic or Bernstein polynomial bounds on y∈[0,1] (e.g. with 10^-8 tolerances) to certify F(y)≥0.00276, and also solve the derivative polynomial exactly or via Sturm sequences to locate the global minimum. If such a certificate is produced, the N=5 counterexample becomes rigorous; if a negative interval is found, the witness and the counterexample fail.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is the claim that W5 in Eq. (14) is a valid entanglement witness, i.e. W5(θ,φ)≥0 for all symmetric product states. The paper supports this only by a numerical minimization (minimum ≈ 0.00276 at (θ,φ)=(π/2,0)) and a logarithmic plot; no code, interval bound, or analytic proof is given. Since W5 contains a large negative off-diagonal coefficient c=-9.31947 and a negative diagonal entry b=-0.134595, positivity is not structurally obvious and a small negative region away from the reported minimum would invalidate the witness. The QETLAB 2-copy symmetric-extension check is likewise a numerical SDP and not a mathematical proof. Thus the central counterexample for N=5 (and by extension the N=7,9 witnesses in Appendix D) rests on numerical evidence rather than a certified argument. Theorem 1 and the spectral analysis of ρ0^TA appear analytically sound, and the claimed quantitative entanglement range is plausible, but the rigorous status of 'entangled SAPPT states exist' is not settled by the manuscript as written.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":74,"tokens_out":8698,"duration_ms":141478,"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":[{"comment":"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.","section":"Section III, Eq. (14)"},{"comment":"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.","section":"Section III, QETLAB paragraph"},{"comment":"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.","section":"Conclusions"}],"minor_comments":[{"comment":"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).","section":"Section III, Figure 1"},{"comment":"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.","section":"Appendix C, Eq. (C1)"},{"comment":"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.","section":"Section I"},{"comment":"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.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"To the editor: The paper addresses a meaningful open question and the analytic portion (Theorem 1 and the spectral analysis) appears solid. The decisive issue is that the claimed counterexample for N=5 relies on numerical verification of witness positivity with a small margin (≈0.00276), and the same applies to the N=7 and N=9 witnesses. Because the witness has large negative coefficients, the risk of a missed negative region is real, so this is not a mere presentational issue. I recommend major revision, asking the authors to provide a certified proof of W_N(θ,φ) ≥ 0 (e.g., via interval arithmetic or an explicit sum-of-squares decomposition), or to clearly reframe the paper as presenting strong numerical evidence rather than a proof. If a certified argument cannot be supplied, the claim that SAPPT ⊄ SAS should be weakened accordingly."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe short version: this paper gives the first explicit entangled SAPPT states for odd N≥5, and if the witness checks hold up, it settles the SAPPT vs SAS question in the symmetric subspace by counterexample. That is a real result, and it is not in the prior literature.\n\nThe analytic core is the strongest part. Theorem 1 identifies a uniparametric spectrum that is SAPPT exactly for p≥p_min, with a clean spectral decomposition of rho_0^TA (the CSCO argument in Appendix C is careful and I see no gap there). The necessity proof via the GHZ state saturating the bound is also neat. That part deserves credit.\n\nThe soft spot is the entanglement witness. W5 in Eq. (14) is claimed to be a valid witness because W5(θ,φ)≥0 on all symmetric product states, but the support is a numerical minimization (minimum ≈0.00276) and a log plot. No code, no interval bound, no analytic certificate. Same for W7 and W9 in Appendix D. The margin is small enough that a missed negative region could wipe out the tight N=5 example. That said, the QETLAB 2-copy symmetric-extension check is independent numerical evidence for entanglement, and p_ent from the truncated moment method is a separate numerical handle. So the conclusion is plausible, but the manuscript as written does not rigorously establish the counterexample.\n\nAlso worth noting: the paper is honest about scope. It does not claim to resolve the general AS vs APPT question, and it explicitly says even N yields no counterexample. The generalization to qudits is stated as a conjecture, which is fine.\n\nIf I were refereeing, I would ask for one of three things: an analytic positivity proof for W5, a certified interval bound, or at least shipped code reproducing the minimization and the QETLAB check. That is a tractable revision, not a change of direction.\n\nBottom line: this is a genuinely useful paper for people working on absolute separability, symmetric states, or bosonic entanglement. Give it a proper review; the main claim is probably right, but the witness step needs to be made rigorous before publication.","headline":"Clean analytic SAPPT boundary plus a numerically backed counterexample that should pass review once the witness positivity is certified.","tokens_in":23630,"tokens_out":2010,"would_cite":true,"duration_ms":18473,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["symmetric multiqubit states","absolute separability","absolutely PPT","positive partial transpose","entanglement witness","bound entanglement","Dicke states","GHZ state"],"falsifier":"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.","tokens_in":22670,"feed_emoji":"🔗","tokens_out":10303,"duration_ms":88768,"temperature":0.7,"pith_summary":"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.","feed_headline":"Odd-qubit symmetric PPT states can be entangled","feed_subtitle":"A five-qubit family stays PPT under every symmetric rotation yet is entangled, breaking the expected equivalence.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Gives the linear-matrix-inequality characterization of APPT states that motivates the spectrum-based approach used here.","marker":"[16]"},{"why":"Is the earlier study of whether absolute separability is determined by the partial transpose, whose symmetric analogue this paper disproves.","marker":"[17]"},{"why":"Proves the AS/APPT equivalence for qubit-qudit systems, the nonsymmetric positive case that the counterexamples leave intact.","marker":"[18]"},{"why":"Defines the SAPPT setting and supplies an earlier SAPPT criterion that Theorem 1 improves.","marker":"[20]"},{"why":"Supplies the truncated-moment semidefinite method that determines the full entangled range $p_{\\mathrm{ent}}$.","marker":"[37]"},{"why":"Provides the symmetric-extension criterion whose failure detects entanglement and yields the witness operators.","marker":"[38]"},{"why":"The numerical software used for the two-copy symmetric-extension check and for extracting the explicit witnesses.","marker":"[39]"},{"why":"Show separable symmetric states are convex combinations of symmetric product states, so nonnegativity on product states makes $W_N$ a valid witness.","marker":"[40, 41]"}],"fun_headline_variants":["SAPPT states entangled for odd qubits","Five-qubit SAPPT state is entangled","Symmetric PPT not always separable for odd qubits","Odd-qubit SAPPT states break absolute separability","Entangled SAPPT states found for odd qubit counts"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["SAPPT states entangled for odd qubits","Five-qubit SAPPT state is entangled","Symmetric PPT not always separable for odd qubits","Odd-qubit SAPPT states break absolute separability","Entangled SAPPT states found for odd qubit counts"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000368,"raw_usage":{"total_tokens":1941,"prompt_tokens":880,"completion_tokens":1061,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":496,"completion_tokens_details":{"reasoning_tokens":987}},"tokens_in":496,"tokens_out":1061,"duration_ms":9932,"temperature":1.0,"reasoning_tokens":987,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T13:05:28.096478+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Tura i Brugués, Characterizing Entanglement and Quantum Correlations Constrained by Symmetry (Springer International Publishing, 2017)","cited_arxiv_id":null,"evidence_quote":"Gives the linear-matrix-inequality characterization of APPT states that motivates the spectrum-based approach used here."},{"cited_title":"Horodecki, P","cited_arxiv_id":null,"evidence_quote":"Is the earlier study of whether absolute separability is determined by the partial transpose, whose symmetric analogue this paper disproves."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proves the AS/APPT equivalence for qubit-qudit systems, the nonsymmetric positive case that the counterexamples leave intact."},{"cited_title":"Arunachalam, N","cited_arxiv_id":null,"evidence_quote":"Defines the SAPPT setting and supplies an earlier SAPPT criterion that Theorem 1 improves."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the truncated-moment semidefinite method that determines the full entangled range $p_{\\mathrm{ent}}$."},{"cited_title":"Johnston and E","cited_arxiv_id":null,"evidence_quote":"The numerical software used for the two-copy symmetric-extension check and for extracting the explicit witnesses."}],"review_version":1}