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A Unified Toolbox for Multipartite Entanglement Certification

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A conditional-gradient toolbox reduces multipartite entanglement certification to a single convex optimization, certifying entanglement in up to ten qubits and nearly closing the white-noise gap for Horodecki bound entangled states.

desk verdict Useful heuristic toolbox, but the rigorous witness proof has a wrong-direction Cauchy-Schwarz bound that invalidates the main claim. read the letter →

arxiv 2507.17435 v2 pith:JPHNLYRN submitted 2025-07-23 quant-ph math.OC

Ye-Chao Liu , Jannis Halbey , Sebastian Pokutta , Sébastien Designolle This is my paper
classification quant-phmath.OC MSC 81P4090C25 PACS 03.65.Ud03.67.-a
keywords multipartiteentanglementwitnessconditionalgradientFrank-Wolfeboundentangledstateswhitenoiserobustnessquantumchannelsseparable
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 argues that a single optimization method, the conditional gradient (Frank-Wolfe) algorithm, can serve both as a fast heuristic and as a rigorous certifier for multipartite entanglement. The method decides whether a noisy quantum state is separable or entangled by measuring a dual-gap efficiency ratio, and it constructs an explicit entanglement witness with controlled numerical error whenever a rigorous certificate is needed. The authors demonstrate that this toolbox certifies entanglement in systems of up to ten qubits, and that it sharply locates the white-noise robustness thresholds of a family of bound entangled states, closing most of the gap between the best known entanglement and separability bounds. If correct, this makes entanglement certification practical for larger multipartite systems and for realistic noise channels that semidefinite-programming methods cannot handle.

What carries the argument

The central object is the dual-gap efficiency r_t = g_t / f(ρ,σ_t), where g_t = tr[(σ_t − ρ)(σ_t − ψ_t)] is the Frank-Wolfe gap and f is half the squared Hilbert-Schmidt distance to the k-separable set. Corollary 2 states that r_t < 1 holds exactly when ρ is entangled, turning the optimization trajectory into an entanglement test. For rigorous certificates the machinery is an ε-net S_ε^k of pure separable boundary states, which gives a finite search with provable error bound, and the associated witness operator W = Λ − (β − ε)1/||Λ||. The same CG output feeds a geometric reconstruction (Proposition 4) that certifies separability, so the whole workflow shares one engine.

What would settle it

Compute the operator W from Eq. (7) for a concrete target state and direction Λ, then search over a fine net of separable states for any τ with tr(Wτ) < 0; a single such τ would disprove the claim that W is always a valid witness.

Watch

Extended reading notes

Core claim

The central discovery is that entanglement certification can be reduced to a single projection-free optimization problem: minimize the squared Hilbert-Schmidt distance to the convex set of k-separable states, and then read the answer off from two quantities produced by the same algorithm. For fast detection, the paper shows that a state is entangled exactly when the dual gap g_t exceeds the current distance f(ρ,σ_t), so the ratio r_t = g_t / f(ρ,σ_t) falling below 1 certifies entanglement; in practice, thresholds like r_t < 1/5 are used to guard against heuristic error. For rigorous certification, the algorithm searches an ε-net of pure separable states, bounds the error ε in the optimal distance, and constructs the operator W = Λ − (β − ε)1/||Λ||, which is claimed to be a valid entanglement witness for the target state. The paper uses this toolbox to compute, for Horodecki bound entangled states, white-noise thresholds that lie an order of magnitude closer to the previously known separability bounds than earlier entanglement bounds, and it gives certified robustness thresholds for a Bell state under several realistic quantum noise channels.

Load-bearing premise

For the constructed operator W to be a valid entanglement witness, every separable state τ must satisfy a trace inequality that the paper derives in Appendix D using a step that does not follow from Cauchy-Schwarz and is not generally true.

Editorial extensions

If this is right

  • Entanglement certification becomes practical for systems of ten or more qubits, where SDP-based methods currently break down; the paper demonstrates detection within dozens of iterations for 10-qubit GHZ and Dicke states at 70% white noise.
  • Rigorous witnesses with explicit numerical error control can be fed directly into experimental verification protocols, since the certified interval is stated with error bars.
  • The white-noise robustness gap for Horodecki states is reduced by roughly an order of magnitude, and the previously inconsistent bounds in the regime 0 < a < 0.02 are reconciled.
  • The same framework handles nonlinear quantum noise channels such as bit-flip, phase-flip, amplitude damping, and phase damping, giving certified thresholds that match known analytical values.

Reading between the lines

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

  • If the dual-gap efficiency r_t can be given a rigorous stopping rule with a finite iteration bound, the heuristic phase of the toolbox would become a certified detector as well; the paper only uses conservative thresholds without such a guarantee.
  • The ε-net construction described in the appendix for the unit sphere could be reused to certify membership in other convex sets of quantum states, such as sets defined by PPT or by restricted entanglement structures.
  • The observed correlation between entanglement strength and number of CG iterations suggests that iteration counts could serve as a cheap, quantitative proxy for white-noise robustness in larger systems.
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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

4 major / 5 minor

Summary. The paper proposes a conditional-gradient (CG) framework for multipartite entanglement certification. It combines a heuristic detector based on the dual-gap efficiency r_t with a purported rigorous entanglement witness built from an epsilon-net of separable states, and it reports fast detection in systems of up to ten qubits, near-optimal white-noise robustness thresholds for Horodecki bound entangled states, and robustness under several quantum noise channels. The central advertised contributions are thus a scalable heuristic and a rigorous certification subroutine with controlled numerical error.

Significance. If the rigorous components were correct, the framework would be a genuinely useful advance: it targets arbitrary k-separability structures, scales to ten qubits, provides an open-source Julia implementation, and benchmarks against analytical and SDP-based thresholds. The heuristic detection and numerical benchmarks may indeed be useful. However, the rigorous entanglement witness is the paper's core differentiator, and its proof in Appendix D contains a sign error that cannot be repaired by a local correction. The heuristic criterion's 'if and only if' statement is also false in general. The advertised rigorous certification claim is therefore unsupported, although the numerical/algorithmic exploration could form the basis of a revised manuscript.

major comments (4)
  1. [Appendix D, Eq. (D3)] The proof of Proposition 3 rests on the inequality (1−η)tr(Λτ) ≤ −(1−η)||Λ||, justified by 'Cauchy-Schwarz'. That is the wrong direction: Cauchy–Schwarz gives tr(Λτ) ≥ −||Λ|| for trace-one states, not the displayed upper bound. The asserted inequality fails for any separable τ with tr(Λτ) > 0, for example τ = σ when tr(σ²) > tr(ρσ). Consequently the chain in Eq. (D3) does not establish tr(Wτ) ≥ 0 for all τ ∈ S_k, and Proposition 3 is unproved. This is load-bearing because the 'rigorous witness construction' advertised in the abstract and conclusion depends entirely on this argument.
  2. [Proposition 3 / Eq. (7)] Even if the positivity proof were repaired, the statement of Proposition 3 does not prove that W is a witness for ρ. An entanglement witness must also satisfy tr(Wρ) < 0. The proposition imposes no condition on the separable state σ, and the expression tr(Wρ) = tr(Λρ) − (β − ε) can be nonnegative for generic σ. The proof in Appendix D only addresses the separable-side inequality; the detection condition is never shown. Thus the operator in Eq. (7) is not demonstrated to be an entanglement witness for the target state ρ.
  3. [Corollary 2] The statement 'A quantum state ρ is entangled if and only if the inequality f(ρ,σ_t) > g_t holds' is false for a fixed iteration t. Sufficiency follows from Proposition 1 by taking τ = ψ_t, but necessity would require that the current CG iterate σ_t, obtained with a heuristic LMO, satisfies the inequality for every entangled ρ. This is not guaranteed, and the text later concedes that 'there is no guarantee of global optimality' and that the computed quantities 'cannot be directly used to certify entanglement or separability'. Corollary 2 should be weakened to a sufficient detection criterion.
  4. [Applications for closing gap / Appendix E] The 'certified' entanglement and separability intervals reported in Fig. 3 and Fig. 4 rely on Proposition 3 and Proposition 4. Since Proposition 3 is unproved, those intervals are not rigorously certified as claimed. Proposition 4 and its proof in Appendix E also need clarification: the parameter ε is introduced without definition, the construction ρ_t = (1−ε)ρ − ε 11/d is not connected to the noisy state in Eq. (E1), and the step leading to the separable state at noise (p+ε)/(1+ε) should be stated with explicit hypotheses. These issues matter because the 'closing the gap' claim is a central numerical result.
minor comments (5)
  1. [Eq. (7) and Appendix D] The norm ||Λ|| is never defined; the proof implicitly uses the Hilbert–Schmidt norm in the Cauchy–Schwarz step, while Appendix C works with a general norm on the ambient space. Please specify the norm conventions consistently.
  2. [Eq. (5)] The phrase 'normalization omitted' is too terse: the witness formula depends on the trace convention and on whether σ and ρe are normalized or unnormalized operators. Please state the conventions explicitly.
  3. [Introduction] The sentence 'A entanglement witness W is a necessary and sufficient criteria' contains a grammatical error ('A' should be 'An', and 'criteria' should be 'criterion').
  4. [Figure 2] The dashed threshold line is labeled r = 1, but the text says the three-qubit GHZ state with 80% noise lies 'exactly on the boundary'; the figure's apparent crossing of the threshold by the 80% curve should be clarified, since the curve is used to motivate the detection criterion.
  5. [Appendix F2 / Table II] The entries in Table II report 'certified interval' error bars, but the certification mechanism is not described in Appendix F2; please explain how the reported upper and lower bounds are obtained from the CG iterations and which of them rely on the unproved witness construction.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: core derivations are self-contained and benchmarked against external analytical and SDP thresholds.

full rationale

The paper's derivation chain is not circular in the sense of reducing a prediction to its fitted inputs. Proposition 1 is a geometric characterization of a point outside the convex separable set, and Corollary 2 follows algebraically from Eq. (3), modulo the separate correctness issue that the heuristic LMO is not guaranteed optimal. The rigorous witness in Proposition 3 is constructed from an explicit ε-net with a bound that is supposed to be proven in Appendix D; even though that proof appears to contain a sign error in the Cauchy–Schwarz step (it claims tr(Λτ) ≤ −(1−η)||Λ||, whereas Cauchy–Schwarz gives tr(Λτ) ≥ −||Λ||), a wrong mathematical step is not an equivalence-by-construction and therefore is a correctness risk rather than circularity. The numerical claims are checked against external benchmarks: the p = 2/3 white-noise threshold for Bell states, p = 1/2 for bit/phase flip, p = 1 for amplitude/phase damping, and published SDP separability bounds for Horodecki states; thus the thresholds are not forced by the algorithm's own definition. Self-citations (Refs. [21], [31], and the review [34]) concern algorithmic enhancements and related work, and they are not load-bearing for the central entanglement-certification claims, which rest on the self-contained ε-net construction and external comparisons. No circular step can be exhibited, so the circularity score is 0.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The framework rests on standard convex geometry of the separable set, the existence of epsilon-nets of pure separable states, and the heuristic quality of the alternating LMO. The most fragile input is the epsilon-net construction and the bound used for the witness.

free parameters (3)
  • heuristic detection threshold r0 = 1/5 (suggested)
    The algorithm declares entanglement when rt < r0, with r0 chosen empirically (1 in theory, 1/5 in practice) to account for LMO inaccuracies; no theoretical guarantee.
  • epsilon-net mesh size epsilon (or subdivision level n)
    The rigorous witness construction depends on an epsilon-net of the sphere; the specific epsilon or subdivision level n used in the reported numerical experiments is not stated in the paper.
  • number of CG iterations T
    Runtime and convergence depend on iteration count; not specified for each experiment.
assumptions (4)
  • domain assumption The set of k-separable states S_k is the convex hull of pure product states.
    Standard in entanglement theory; used throughout to formulate the optimization Eq. (1) and the LMO.
  • standard math An epsilon-net S_epsilon^k of the boundary of S_k exists with the property eta S_k subset conv(S_epsilon^k) subset S_k.
    Based on Pisier and Aubrun-Szarek; the explicit construction in Appendix C aims to provide this, but its correctness is part of the paper's contribution.
  • ad hoc to paper The linear minimization oracle in Eq. (2) is solved to sufficient accuracy by alternating updates over subsystems.
    The heuristic detection relies on this; the authors acknowledge no global optimality guarantee.
  • standard math The radius a of the separable ball (Gurvits-Barnum) is known.
    Used in Proposition 4 for separability certification; cited from [48].

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Cite this review

Pith. "Pith review of A Unified Toolbox for Multipartite Entanglement Certification." pith.science (2026). https://pith.science/paper/JPHNLYRN

@misc{pith2026250717435,
  author       = {Pith},
  title        = {Pith review of: A Unified Toolbox for Multipartite Entanglement Certification},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JPHNLYRN}},
  note         = {Machine review of arXiv:2507.17435}
}
read the original abstract

We present a unified framework for multipartite entanglement characterization based on the conditional gradient (CG) method, incorporating both fast heuristic detection and rigorous witness construction with numerical error control. Our method enables entanglement certification in quantum systems of up to ten qubits and applies to arbitrary entanglement structures. We demonstrate its power by closing the gap between entanglement and separability bounds in white noise robustness benchmarks for a class of bound entangled states. Furthermore, the framework extends to entanglement robustness under general quantum noise channels, providing accurate thresholds in cases beyond the reach of previous algorithmic methods. These results position CG methods as a powerful tool for practical and scalable entanglement analysis in realistic experimental settings.

Figures

Figures reproduced from arXiv: 2507.17435 by the authors.

Figure 1
Figure 1. Schematic overview of our unified toolbox for [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. (b) shows that 10-qubit GHZ and Dicke states mixed with 70% white noise can still be certified within dozens of iterations. The total runtime is under one minute using a 2.80 GHz CPU and 16 GB of memory. In the main text, we focus on detecting full separability or bipartite entanglement (i.e., Sk with k = n for n-qubit systems); for general entanglement structures, see Appendix B [32], where we improve upon existing… view at source ↗
Figure 3
Figure 3. (a) Entanglement bounds pent,CG (blue line) on the white noise robustness of Horodecki states ρ H(a), obtained via our CG method, compared with previous entanglement bounds based on a generalization of the Wootters formula pent,GW (blue dots) and separability bounds based on SDP methods psep,SDP (orange dots). (b) Comparison of the gap ∆p between entanglement and separability bounds: blue shows the range [pent,GW, p… view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Entanglement robustness thresholds for the Bell [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: Setup for the proof of Theorem 12. Proposition 13. Consider a regular (d − 1)-simplex with edge length s. For d ⩽ 3 or n = 1, the diameters of its subsimplices are all s n ; in the other cases, they are at most s √ 2 n . Proof. The diameter of a simplex is simply the l…
Figure 6
Figure 6. Figure 6: Geometric reconstruction for separability certification within the CG-based framework. [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]

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Cited by 1 Pith paper

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

  1. Certifying coherence in quantum devices under classical control

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    Introduces SDP hierarchies and qubit-specific joint-measurability techniques to certify coherence under hidden classical control, with applications to coherence-preserving channels.

Reference graph

Works this paper leans on

57 extracted references · 52 canonical work pages · cited by 1 Pith paper

  1. [1]

    Horodecki, P

    R. Horodecki, P. Horodecki, M. Horodecki, and K. Horo- decki, Quantum entanglement, Rev. Mod. Phys.81, 865 (2009)

    2009

  2. [2]

    Gühne and G

    O. Gühne and G. Tóth, Entanglement detection, Phys. Rep. 474, 1 (2009)

    work page 2009

  3. [3]

    Peres, Separability criterion for density matrices, Phys

    A. Peres, Separability criterion for density matrices, Phys. Rev. Lett.77, 1413 (1996)

    1996

  4. [4]

    Horodecki, Separability criterion and inseparable mixed states with positive partial transposition, Phys

    P. Horodecki, Separability criterion and inseparable mixed states with positive partial transposition, Phys. Lett. A 232, 333 (1997)

    work page 1997

  5. [5]

    Chen and L.-A

    K. Chen and L.-A. Wu, A matrix realignment method for recognizing entanglement, Quantum Info. Comput.3, 193 (2003)

    work page 2003

  6. [6]

    Rudolph, Further results on the cross norm criterion for separability, Quantum Inf

    O. Rudolph, Further results on the cross norm criterion for separability, Quantum Inf. Proc.4, 219 (2005)

    work page 2005

  7. [7]

    L. Novo, T. Moroder, and O. Gühne, Genuine multi- particle entanglement of permutationally invariant states, Phys. Rev. A88, 012305 (2013)

    work page 2013

  8. [8]

    Horodecki, P

    M. Horodecki, P. Horodecki, and R. Horodecki, Separab- ility of mixed states: necessary and sufficient conditions, Phys. Lett. A223, 1 (1996)

    work page 1996

Show all 57 references
  1. [9]

    B. M. Terhal, Bell inequalities and the separability cri- terion, Phys. Lett. A271, 319 (2000)

    2000

  2. [10]

    Lewenstein, B

    M. Lewenstein, B. Kraus, J. I. Cirac, and P. Horodecki, Optimization of entanglement witnesses, Phys. Rev. A 62, 052310 (2000)

    2000

  3. [11]

    D. Bruß, J. I. Cirac, P. Horodecki, F. Hulpke, B. Kraus, M. Lewenstein, and A. Sanpera, Reflections upon separ- ability and distillability, J. Mod. Opt.49, 1399 (2002)

    2002

  4. [12]

    Bourennane, M

    M. Bourennane, M. Eibl, C. Kurtsiefer, S. Gaertner, H. Weinfurter, O. Gühne, P. Hyllus, D. Bruß, M. Lewen- stein, and A. Sanpera, Experimental detection of multi- partite entanglement using witness operators, Phys. Rev. Lett. 92, 087902 (2004)

    2004

  5. [13]

    A. C. Doherty, P. A. Parrilo, and F. M. Spedalieri, Com- plete family of separability criteria, Phys. Rev. A69, 022308 (2004)

    2004

  6. [14]

    Navascués, M

    M. Navascués, M. Owari, and M. B. Plenio, Complete criterion for separability detection, Phys. Rev. Lett.103, 160404 (2009)

    2009

  7. [15]

    Ohst, X.-D

    T.-A. Ohst, X.-D. Yu, O. Gühne, and H. C. Nguyen, Certifying quantum separability with adaptive polytopes, SciPost Phys.16, 063 (2024). 6

    2024

  8. [16]

    C. Song, K. Xu, W. Liu, C.-p. Yang, S.-B. Zheng, H. Deng, Q. Xie, K. Huang, Q. Guo, L. Zhang, P. Zhang, D. Xu, D. Zheng, X. Zhu, H. Wang, Y.-A. Chen, C.-Y. Lu, S. Han, and J.-W. Pan, 10-qubit entanglement and parallel logic operations with a superconducting circuit, Phys. Rev....

    2017

  9. [17]

    Kampermann, O

    H. Kampermann, O. Gühne, C. Wilmott, and D. Bruß, Algorithm for characterizing stochastic local operations and classical communication classes of multiparticle en- tanglement, Phys. Rev. A86, 032307 (2012)

    2012

  10. [18]

    Shang and O

    J. Shang and O. Gühne, Convex optimization over classes of multiparticle entanglement, Phys. Rev. Lett. 120, 050506 (2018)

    2018

  11. [19]

    Wieśniak, P

    M. Wieśniak, P. Pandya, O. Sakarya, and B. Woloncewicz, Distance between bound entangled states from unextend- ible product bases and separable states, Quantum Rep. 2, 49 (2020)

    2020

  12. [20]

    Pandya, O

    P. Pandya, O. Sakarya, and M. Wieśniak, Hilbert-Schmidt distance and entanglement witnessing, Phys. Rev. A102, 012409 (2020)

    2020

  13. [21]

    Hu, Y.-C

    Y. Hu, Y.-C. Liu, and J. Shang, Algorithm for evaluating distance-based entanglement measures, Chinese Phys. B 32, 080307 (2023)

    2023

  14. [22]

    STOC’03, STOC ’03 (Association for Computing Machinery, New York, NY, USA, 2003) pp

    L.Gurvits,ClassicaldeterministiccomplexityofEdmonds’ problem and quantum entanglement, inProc. STOC’03, STOC ’03 (Association for Computing Machinery, New York, NY, USA, 2003) pp. 10–19

    2003

  15. [23]

    E. G. Gilbert, An iterative procedure for computing the minimum of a quadratic form on a convex set, J. SIAM Control 4, 61 (1966)

    1966

  16. [24]

    Frank and P

    M. Frank and P. Wolfe, An algorithm for quadratic pro- gramming, Nav. Res. Logist. Q.3, 95 (1956)

    1956

  17. [25]

    E. S. Levitin and B. T. Polyak, Constrained minimiza- tion methods, USSR Comput. Math. & Math. Phys.6, 1 (1966)

    1966

  18. [26]

    Wolfe, Finding the nearest point in a polytope, Math

    P. Wolfe, Finding the nearest point in a polytope, Math. Program. 11, 128 (1976)

    1976

  19. [27]

    Guélat and P

    J. Guélat and P. Marcotte, Some comments on Wolfe’s ‘away step’, Math. Program.35, 110 (1986)

    1986

  20. [28]

    C. A. Holloway, An extension of the Frank and Wolfe method of feasible directions, Math. Program.6, 14–27 (1974)

    1974

  21. [29]

    K. K. Tsuji, K. Tanaka, and S. Pokutta, Pairwise condi- tional gradients without swap steps and sparser kernel herding, inInternational Conference on Machine Learning (PMLR, 2022) pp. 21864–21883

    2022

  22. [30]

    Braun, S

    G. Braun, S. Pokutta, and D. Zink, Lazifying conditional gradient algorithms, inProceedings of the 34th Interna- tional Conference on Machine Learning (2017) pp. 566– 575

    2017

  23. [31]

    Halbey, S

    J. Halbey, S. Rakotomandimby, M. Besançon, S. Designolle, and S. Pokutta, Efficient quadratic corrections for Frank-Wolfe algorithms, arXiv:2506.02635 (2025)

    2025 arXiv

  24. [32]

    See Supplemental Material for the Appendixes

  25. [33]

    I. M. Bomze, F. Rinaldi, and D. Zeffiro, Frank-Wolfe and friends: a journey into projection-free first-order optimization methods, 4OR19, 313 (2021)

    2021

  26. [34]

    Braun, A

    G. Braun, A. Carderera, C. W. Combettes, H. Hassani, A. Karbasi, A. Mokhtari, and S. Pokutta, Conditional gradient methods, arXiv:2211.14103 (2022)

    2022 arXiv

  27. [35]

    See our code as a Julia package athttps://github.com/ ZIB-IOL/EntanglementDetection.jl

  28. [36]

    Gühne and M

    O. Gühne and M. Seevinck, Separability criteria for genu- ine multiparticle entanglement, New J. Phys.12, 053002 (2010)

    2010

  29. [37]

    Gao and Y

    T. Gao and Y. Hong, Detection of genuinely entangled and nonseparablen-partite quantum states, Phys. Rev. A 82, 062113 (2010)

    2010

  30. [38]

    Jungnitsch, T

    B. Jungnitsch, T. Moroder, and O. Gühne, Taming mul- tiparticle entanglement, Phys. Rev. Lett.106, 190502 (2011)

    2011

  31. [39]

    Ananth and M

    N. Ananth and M. Senthilvelan, On the non-k-separability of Dicke class of states andn-quditw states, Int. J. Theor. Phys. 55, 1854 (2016)

    2016

  32. [40]

    Chen, L.-Z

    X.-Y. Chen, L.-Z. Jiang, and Z.-A. Xu, Necessary and sufficient criterion fork-separability ofn-qubit noisy GHZ states, Int. J. Quantum Inf.16, 1850037 (2018)

    2018

  33. [41]

    L.-L. Ge, M. Xu, T. Hu, L.-Z. Jiang, and X.-Y. Chen, Tripartite separability of four-qubit W and Dicke mixed state in noise environment, Eur. Phys. J. Plus136, 1 (2021)

    2021

  34. [42]

    A.O.PittengerandM.H.Rubin,Convexityandthesepar- ability problem of quantum mechanical density matrices, Linear Algebra Appl.346, 47 (2001)

    2001

  35. [43]

    R. A. Bertlmann, H. Narnhofer, and W. Thirring, Geo- metric picture of entanglement and Bell inequalities, Phys. Rev. A66, 032319 (2002)

    2002

  36. [44]

    R. A. Bertlmann, K. Durstberger, B. C. Hiesmayr, and P. Krammer, Optimal entanglement witnesses for qubits and qutrits, Phys. Rev. A72, 052331 (2005)

    2005

  37. [45]

    G.Pisier, The Volume of Convex Bodies and Banach Space Geometry, Cambridge Tracts in Mathematics (Cambridge University Press, 1989)

    1989

  38. [46]

    Aubrun and S

    G. Aubrun and S. J. Szarek,Alice and Bob meet Banach : the interface of asymptotic geometric analysis and quantum information theory , Mathematical Surveys and Monographs, Volume 223 (American Mathematical Soci- ety, 2017)

    2017

  39. [47]

    Edelsbrunner and D

    H. Edelsbrunner and D. R. Grayson, Edgewise subdivision of a simplex, Discrete Comput. Geom.24, 707 (2000)

    2000

  40. [48]

    Gurvits and H

    L. Gurvits and H. Barnum, Separable balls around the maximally mixed multipartite quantum states, Phys. Rev. A 68, 042312 (2003)

    2003

  41. [49]

    Chen, Z.-H

    Z.-H. Chen, Z.-H. Ma, O. Gühne, and S. Severini, Es- timating entanglement monotones with a generalization of the Wootters formula, Phys. Rev. Lett.109, 200503 (2012). 7 Appendix A: Frank-Wolfe method Conditional Gradient (CG) methods, also known as Frank-Wolfe (FW) methods [2...

    2012

  42. [50]

    LetK be a nonempty centrosymmetric compact convex set inH and X ={xi}m i=1 be a subset ofext(K) such that 0∈ conv(X) =P

    Definitions and notations Let H be ad-dimensional normed space with unit ballB and unit sphereS. LetK be a nonempty centrosymmetric compact convex set inH and X ={xi}m i=1 be a subset ofext(K) such that 0∈ conv(X) =P. Definition 5. The homothetic distance, also called the geom...

  43. [51]

    Proposition 10

    Implicit construction In [45, Lemma 4.10], the following result is stated, whose proof we recall for completeness. Proposition 10. There exists anε-net ofS with cardinality smaller than(1 + 2 ε)d. Proof. Let X be a maximal subset ofS such that∥xi−xj∥ ⩾ε for alli̸=j. Clearly, b...

  44. [52]

    Proposition 11

    Explicit construction We first recall without proof the main result from [47] on the edgewise subdivision of a simplex. Proposition 11. For every integern ⩾ 1 and everyd-simplex σ there is a subdivision ofσ into nd d-simplices of equal volume. Theorem 12. LetX⊂S be an innerϵ-a...

  45. [53]

    Horodecki state The explicit density matrix of the family of Horodecki states [4] is ρH(a) = 1 8a + 1   a · · · a · · · a · a · · · · · · · · · a · · · · · · · · · a · · · · · a · · · a · · · a · · · · ·a · · · · · · · · · 1+a 2 · √ 1−a2 2 · · · · · · · a ·...

  46. [54]

    For one qubit, the depolarizing channel is also known as white noise, i.e., CD(ρ) = (1−p)ρ +p11 2

    Noise channels Here we explicitly give the definitions of the noise channels we consider in the main text, including (a) global depolarizing (white noise), (b) bit flip, (c) phase flip, (d) amplitude damping, and (e) phase damping channels. For one qubit, the depolarizing chan...

  47. [55]

    50.01% - 0.04% (50%) Bit Flip 0.4999 + 0.0002 (1

  48. [56]

    50.01% - 0.02% (50%) Phase Flip 0.4999 + 0.0002 (1

  49. [57]

    50.01% - 0.02% (50%) Amplitude Damping 0.9997 + 0.0003 (1) 25.87% - 0.87% (25%) Phase Damping 0.9999 + 0.0001 (1) 50.50% - 0.50% (50%) Table II. Entanglement robustness thresholds for the Bell state(|00⟩ +|11⟩)/ √ 2 under various quantum noise channels: global depolarizing (wh...

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Reviewed August 6, 2026 · model on record in the stance chip above.