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Flower states show entanglement cost exceeds distillable yield by a half-log gap.

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2026-08-04 04:15 UTC pith:HDTXGZVP

load-bearing objection Strong paper; the reader's main objection is a misreading—the S53 lower bound is trivially true and the upper bound is proven for flower states, so Theorem 3's pivot holds. the 2 major comments →

arxiv 2608.02587 v1 pith:HDTXGZVP submitted 2026-08-03 quant-ph

Entanglement of flower states

Samrat Sen , Ludovico Lami This is my paper
classification quant-ph MSC 81P45 PACS 03.67.Mn
keywords flower statesentanglement costdistillable entanglementirreversibility gapnon-entangling operationssquashed entanglementSchmidt numbertempered negativity
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 computes every major entanglement measure for the family of flower states, which live in local dimension d = 2k. It proves that only one ebit of entanglement can be distilled from these states, while creating them costs log(2√k) ebits under LOCC and log(1+√k) ebits even under the more powerful non-entangling operations. Since the cost grows logarithmically in the local dimension and the yield stays constant, flower states produce the largest known irreversibility gap under non-entangling operations, at least 1/2 per log dimension. The paper also shows that squashed entanglement is not a monotone under non-entangling operations, and it derives the exact zero-error LOCC cost as a simple divisor-minimization formula.

Core claim

The central claim is Theorem 1: for the flower state Ω_{F_k} with local dimension 2k, the distillable entanglement under non-entangling operations is exactly log 2, while the entanglement cost is log(2√k) under LOCC and log(1+√k) under non-entangling operations, with the exact NE cost equaling the standard NE cost. Consequently, the asymptotic irreversibility gap under NE operations is at least 1/2, and the squashed entanglement ceases to be a monotone under NE. A second theorem identifies the exact LOCC cost under zero-error constraints as log min_{r|k}(r+k/r), which for prime k is log(k+1), roughly twice the standard LOCC cost.

What carries the argument

The central object is the flower state Ω_{F_k}, a maximally correlated state built from the k-dimensional Fourier matrix F_k. The arguments hinge on three tools: (i) kΩ_{F_k} is a rank-k projector with uniform diagonal, which enables a lower bound on the tempered negativity; (ii) the standard robustness of entanglement, which fixes the exact NE cost as the logarithm of the ℓ1 norm of the density matrix; and (iii) the discrete uncertainty relations for cyclic groups, which bound the coherence number and therefore the regularized Schmidt number. The flower-state structure makes all these bounds tight.

Load-bearing premise

The load-bearing step in the zero-error LOCC calculation is the assumption that the coherence number of the flower state equals the minimum coherence rank of a pure state in its support; if that equality fails, the closed-form cost formula does not follow.

What would settle it

Compute the coherence number of ρ_{F_6} by optimizing over all convex decompositions; if it is strictly less than 5, the regularized Schmidt number formula in Theorem 3 is contradicted.

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

If this is right

  • The NE irreversibility gap is at least 1/2, more than doubling the previously known lower bound of about 0.262.
  • Squashed entanglement cannot bound the NE entanglement cost from below, because it is not a monotone under non-entangling operations.
  • For flower states under NE operations, exact (zero-error) and standard (vanishing-error) entanglement costs coincide.
  • The zero-error LOCC cost for prime k is log(k+1), about twice the standard LOCC cost, while for perfect-square k it matches the standard cost exactly.
  • The single ebit of distillable entanglement can be extracted deterministically with a one-shot, one-way LOCC protocol, making NE and PPT operations redundant for this distillation task.

Where Pith is reading between the lines

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

  • The same projector-plus-one-design mechanism might force similarly large NE irreversibility gaps for other symmetric maximally correlated states, which would extend the result beyond flower states.
  • The divisor-minimization formula suggests that number-theoretic structure of the local dimension controls zero-error entanglement dilution; one could test whether other states built from abelian-group unitaries obey analogous formulas.
  • If the coherence-number equality used in the proof holds for all states proportional to rank-r projectors, the zero-error LOCC cost formula would generalize to a much wider class; this is a testable extension.
  • The one-shot distillation protocol suggests that for states with this kind of conditional-unitary symmetry, asymptotic and one-shot rates coincide for distillation, a property worth probing in other families.

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 introduces and analyzes 'generalized flower states' Ω_V, the maximally correlated states associated with 2×2 block matrices ρ_V = (1/2k)[[1,V],[V†,1]], and focuses on V = F_k. It reports exact values for a battery of entanglement measures: distillable entanglement under NE/PPT operations is log 2 (Theorem 1); the exact and standard NE entanglement costs are both log(1+√k), while the LOCC cost and squashed entanglement are log(2√k); the exact LOCC cost is given by the regularized Schmidt number min_{r|k} log(r + k/r) (Theorem 3); and one ebit can be distilled deterministically in one shot by one-way LOCC (Theorem 4). The main advertised consequences are the improved irreversibility gap Δ_NE_irr ≥ 1/2 and the failure of squashed entanglement to be an NE monotone.

Significance. If the proofs hold, this is a substantial contribution: it is rare to have a nontrivial family of states for which so many entanglement measures are computed exactly in closed form, and the paper's quantitative conclusions — the logarithmic irreversibility gap under NE operations and the non-monotonicity of squashed entanglement — are of independent interest. The proofs are largely self-contained and combine several non-obvious tools (twirling channels, tempered negativity, robustness bounds, and Meshulam–Tao uncertainty relations). The explicit saturating constructions, especially for the Schmidt-number additivity and the one-shot LOCC protocol, are valuable and appear correct. My concerns below are local but require attention before the manuscript is in publishable form.

major comments (2)
  1. [Corollary 2, Eq. (11)] The displayed equality in the proof of Corollary 2 is algebraically false. The ratio is (E_{c,NE}(Ω_{F_k}) − E_{d,NE}(Ω_{F_k}))/log(2k) = log((1+√k)/2)/log(2k), not log k / (2(log 2 + log k)). For example k=4 gives LHS≈0.195 and RHS=1/3. The conclusion Δ_NE_irr ≥ 1/2 is nevertheless correct, since the exact ratio tends to 1/2 as k→∞; however, the proof must be rewritten with the correct expression and the limiting argument.
  2. [Appendix C, Eq. (S31)] The proof of (7) asserts E_R^∞(Ω_{F_k}) = E_R(Ω_{F_k}) = S(tr_1 Ω_{F_k}) − S(Ω_{F_k}) without proof. The last equality is not a general property of the relative entropy of entanglement; for maximally correlated states it can be justified either by the known identity E_R(Ω_ρ) = C_r(ρ) = S(Δρ) − S(ρ) or by an explicit argument that the dephased state is an optimal separable state. This step is load-bearing because Eq. (7) is central to Theorem 1 and to Corollary 2 and Implication 3. The chain E_d,KP = E_d,LOCC = E_R^∞ also needs to name which theorem supplies each equality.
minor comments (5)
  1. [Eqs. (S53)–(S54)] The phrase 'well-established lower bound CN(ρ) ≥ min_{|ψ⟩∈suppρ} CR(|ψ⟩)' is misleading: this inequality is tautological, since every element of any convex decomposition lies in the support. The nontrivial direction is the upper bound CN ≤ min_support CR, which is proved by the twirling channel. The text should say this explicitly; as written it invites a confusion of the type that the reverse inequality is generally false.
  2. [Appendix C, Eq. (S31)] The symbol E_d,KP is used for K=S,PPT, but the theorem states a value for E_d,NE. The proof should clearly distinguish the classes (LOCC, SEP, PPT, NE) and show exactly how E_d,NE is squeezed between E_d,PPT and E_R^∞.
  3. [Appendix E, Theorem 4] The one-shot LOCC protocol is valid, but the normalization condition Σ_{x,y} A_x†A_x ⊗ B_{x,y}†B_{x,y} = I is never stated. It is true and easy to verify; please include it so the protocol is fully explicit.
  4. [Acknowledgements] The disclosure that ChatGPT 5.5 was used to complete the proof of Theorem 3 is noted. I checked the induction carefully and found it coherent, but given the intricacy, the authors should ensure that the proof is independently verifiable and perhaps provide more intuition for the induction step.
  5. [Footnote 51] The footnote 'This is also easy to see directly, but let's not get into that' is out of place in a formal paper. Either give a one-sentence proof or replace the remark.

Circularity Check

0 steps flagged

No significant circularity: Theorem 1 is computed in closed form, Theorem 3 is a direct uncertainty-relation optimization, and the self-citations are published external theorems rather than self-referential inputs.

full rationale

The core derivation is not circular. Theorem 1's distillable entanglement is pinned from two independent directions: the hashing-bound identity for maximally correlated states plus the relative entropy computation (S31), giving log 2. The NE cost is squeezed between the exact-cost formula of Corollary 6 and the tempered-negativity lower bound of Corollary 8 (S32), both computed in closed form without fitting any parameter to the target values. The self-citations ([30], [46], [47]) are published, parameter-free theorems (tempered-negativity cost bound; generalized quantum Stein's lemma) whose assumptions do not include the flower-state result; they are genuine external evidence. Theorem 3 reduces the regularized Schmidt number to the coherence number of a pure state, then to a minimization of ∥a∥0 + ∥F†a∥0 solved by Tao/Meshulam uncertainty relations; the matching upper bound is an explicit construction (S63-S68). The contested claim CN(ρ) ≥ min_supp CR(|ψ⟩) is tautologically true for any state, and the nontrivial upper bound is proved for the flower states by the twirling channel (S50-S53), so the squeeze in (S54) is valid. The only caveats are non-circular: the additivity induction (S65-S79) is acknowledged in the Acknowledgements as completed with ChatGPT assistance and is the least independently scrutinized part, and a few inequalities are cited rather than reproved. Neither constitutes a fitted parameter renamed as a prediction nor a definition that presupposes its own conclusion.

Axiom & Free-Parameter Ledger

0 free parameters · 10 axioms · 0 invented entities

The report found zero free parameters: every quantity is evaluated in closed form, and no constant is fitted, tuned, or chosen ad hoc. The central claims rest on ~10 axioms, nearly all published theorems (Rains' PPT hashing bound; Lami–Regula tempered-negativity cost bound; Brandão–Plenio/Datta one-shot cost-robustness formula; Tao–Meshulam uncertainty; Christandl–Winter flower-state values; the repaired Stein lemma). One premise is mis-stated in the text (the Eq. (S53) lower bound) and is rescued only by the unstated projector property. No invented entities.

axioms (10)
  • domain assumption E_{d,ANE} = E_{d,NE} = E_R^∞ (generalized quantum Stein's lemma, repaired in [47] by the present co-author)
    Eq. (S31) chain; a published theorem with a history of a proof gap [46], fixed by one of the authors [47]. Load-bearing for the upper bound E_{d,NE} ≤ log 2, though a simpler NE-monotonicity argument (E_R is an NE monotone and E_R(Ω) = log 2) could replace it for this state.
  • domain assumption E_{d,PPT} of a maximally correlated state equals the hashing bound S(B) − S(AB) (Rains, Theorem 6.2 of [12])
    Eq. (S31); anchors E_{d,LOCC} = E_{d,PPT} = log 2 for the flower state.
  • domain assumption E^ε_{c,K}(ω) ≥ log N_τ(ω) for ε ∈ [0,1/2) (tempered-negativity cost bound of [30])
    Eq. (S22); the lower bound needed in the squeeze (S32) for E_{c,NE} = log(1+√k).
  • domain assumption E_{c,LOCC}(Ω_{F_k}) = E_sq(Ω_{F_k}) = log(2√k) (Christandl–Winter [19])
    Eq. (S34) and Theorem 1; previously published computations the paper builds on for the LOCC-vs-NE comparison.
  • domain assumption One-shot exact cost formula E^{exact}_{c,K}(ρ) = lim (1/n) log(1 + 2R_s^K(ρ^{⊗n})) (Brandão–Plenio/Datta [13,29])
    Used in Corollary 6 to translate the robustness bounds of Lemma 5 into exact NE/PPT costs.
  • standard math Meshulam's uncertainty inequality over finite abelian groups (Lemma 9, [25])
    Appendix D, Lemma 9; the key input for the coherence-number minimization. The endpoint argument is valid because the piecewise-linear lower bound attains its minimum at consecutive divisors, and the bound is saturated by the explicit |a⟩ construction.
  • domain assumption Relative entropy of entanglement of a maximally correlated state equals the relative entropy of coherence: E_R(Ω_ρ) = S(Δ(ρ)) − S(ρ)
    Used implicitly in (S31) to evaluate E_R(Ω_{F_k}) = log(2k) − log k = log 2; not cited or proved in the text.
  • domain assumption Any convex-decomposition element of a rank-deficient density matrix lies in its support (equivalently, CN(ρ) ≥ min_{ψ∈supp} CR(ψ) for projectors)
    Needed for Eq. (S54). The text states this as a general 'well-established' bound, which is false in general; the valid special case (ρ_{F_k} is a rank-k projector, (2kρ_{F_k})² = 2(2kρ_{F_k})) is not invoked by the paper.
  • standard math Schmidt number of the maximally correlated state Ω_ρ equals the coherence number of ρ via the isometry L: |i⟩ → |ii⟩
    Opening of Appendix D; standard correspondence used without proof to convert Schmidt-number results to coherence-number computations.
  • domain assumption Squashed entanglement is additive and bounds LOCC rates: E_d ≤ E_sq ≤ E_c,LOCC ([19,20])
    Used in Implication 3 to conclude from E_sq(Ω) > E_{c,NE}(Ω) that E_sq cannot be an NE monotone.

pith-pipeline@v1.3.0-daily-deepseek · 21903 in / 52185 out tokens · 792792 ms · 2026-08-04T04:15:42.816288+00:00 · methodology

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The mysterious nature of entanglement, one of the most prominent exquisitely quantum phenomena, is reflected in its intricate operational structure, with a hierarchy of classes of free operations that enable its manipulation at different levels of effectiveness. Here we use the class of 'flower states', parametrised by their (even) local dimension $2k$, to shine light on some aspects of this varied landscape. We compute all the main entanglement measures for flower states, uncovering a large gap between all forms of distillable entanglement, equal to 1 ebit independently of the local dimension, and the entanglement cost under local operations and classical communication (LOCC), known to be equal to $\log\big(2\sqrt{k}\big)$. Even under the strictly more powerful class of non-entangling (NE) operations, we show that their cost is still equal to $\log\big(1+\sqrt{k}\big)$, only about an ebit less than for LOCCs. This result, which we prove by calculating the recently introduced tempered entanglement negativity for these states, demonstrates the largest known 'irreversibility gap', i.e. the difference between distillable entanglement and entanglement cost, under NE operations, equal to $\Theta\big(\frac12 \log d\big)$, with $d$ being the local dimension. A notable consequence is that the celebrated squashed entanglement is not a monotone under NE operations. Finally, we compute the exact cost under LOCC operations for flower states; this is given by the Schmidt number, which turns out to be additive over multiple copies and equal to $\min_{r|k} \log\left( r + \frac{k}{r} \right)$; for prime $k$ this reduces to $\log(k+1)$, about twice the standard LOCC cost. These last results leverage the uncertainty relations over cyclic groups proved by Tao and Meshulam.

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