pith. sign in

arxiv: 2510.23480 · v1 · pith:2PYMABEXnew · submitted 2025-10-27 · 🪐 quant-ph

Bound entanglement in symmetric random induced states

Jonathan Louvet , Fran\c{c}ois Damanet , Thierry Bastin This is my paper

Pith reviewed 2026-05-21 20:26 UTC · model grok-4.3

classification 🪐 quant-ph
keywords bound entanglementpositive partial transposerandom induced statessymmetric statesPPT statesmultiqubit entanglementquantum information
0
0 comments X

The pith

Symmetric random induced states generate PPT bound entanglement with probability near 1 for N greater than 3 qubits.

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

The paper establishes that bound entanglement arises naturally in symmetric random induced states when parameters are chosen so partial tracing yields positive partial transpose yet entangled states. Two explicit constructions are given: partial tracing from symmetric multiqubit pure states and tracing out a qudit ancilla. For systems larger than three qubits the probability reaches values very close to one under optimal choices. This supplies a direct method to produce large random families of such states. A reader would value the result because bound entanglement is both hard to construct and potentially useful, yet previously required heavy numerical search.

Core claim

Symmetric random induced states constructed by the two methods produce PPT bound entangled states with probability very close to 1 when N exceeds 3 qubits and parameters are selected optimally; the two constructions generate distinct varieties of these states and each offers advantages in different regimes, supplying a versatile toolkit for random generation without optimization.

What carries the argument

Symmetric random induced states obtained by partial tracing, either of symmetric multiqubit pure states (method MI) or of a qudit ancilla (method MII), which become PPT yet entangled under suitable parameter choices.

If this is right

  • The two methods produce distinct families of PPT bound entangled states.
  • For N greater than 3 the high probability allows routine generation of many examples.
  • The approach removes the need for complex numerical optimization in creating these states.
  • Each method carries distinct practical advantages depending on the experimental or theoretical context.

Where Pith is reading between the lines

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

  • The construction may translate to laboratory settings using symmetric photon states or trapped ions to produce bound entanglement on demand.
  • Large random ensembles could serve as testbeds for protocols that exploit the resource properties of bound entanglement.
  • The parameter dependence might be mapped to physical noise models to predict when bound entanglement survives in realistic devices.

Load-bearing premise

Suitable parameters exist that make the symmetric random induced states PPT but entangled with probability near 1.

What would settle it

Numerical sampling over parameter space for four qubits that finds the maximum probability of PPT entanglement substantially below 1, or that no such parameters exist.

Figures

Figures reproduced from arXiv: 2510.23480 by Fran\c{c}ois Damanet, Jonathan Louvet, Thierry Bastin.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p008_9.png] view at source ↗
read the original abstract

Bound entanglement, a weak -- yet resourceful -- form of quantum entanglement, remains notoriously hard to detect and construct. We address this in this paper by leveraging symmetric random induced states, where positive partial transpose (PPT) bound entanglement arises naturally under partial tracing when proper parameters are selected. We investigate the probability of finding PPT bound entanglement in symmetric random induced states constructed via two methods: partial tracing of symmetric multiqubit pure states on the one hand (MI) and tracing out a qudit ancilla on the other hand (MII). For $N > 3$ qubits, we demonstrate that bound entanglement naturally emerges under optimal parameters, with a probability of occurrence very close to 1. We show that the two methods produce different varieties of PPT bound entangled states, and identify the contexts in which each method offers distinct advantages. These methods provide a versatile toolkit for the generation of large families of random PPT bound entangled states without complex numerical optimization.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces two constructions of symmetric random induced states on N qubits: MI (partial trace over symmetric multiqubit pure states) and MII (tracing out a qudit ancilla). It claims that for N>3, under suitably chosen parameters, these states are PPT yet entangled (bound entangled) with probability very close to 1, that the two constructions yield distinct families, and that the approach supplies a simple toolkit for generating large numbers of random PPT bound entangled states without optimization.

Significance. If the numerical evidence is robust and the probability claim holds without post-selection bias, the work would supply a practical, high-yield method for producing bound entangled states, which remain scarce in explicit constructions and are of interest both for foundational questions about the entanglement structure and for potential resource-theoretic applications.

major comments (2)
  1. [Results section (probability estimates for MI and MII)] The central claim that PPT bound entanglement occurs with probability 'very close to 1' for N>3 rests on finite-N numerical sampling under 'optimal parameters' (abstract and results section). No concentration inequality, volume estimate, or scaling analysis is supplied to show that the measure of the favorable parameter regime approaches 1 with growing N; without such support the extrapolation from small-N samples cannot securely establish the stated asymptotic behavior.
  2. [Methods I and II (parameter definitions)] The procedure for selecting the 'optimal parameters' that achieve the reported high probability is not specified in sufficient detail (methods and results). If the parameters are tuned after inspecting the samples, the probability becomes a fitted rather than an a-priori quantity, undermining the claim that bound entanglement 'naturally emerges'.
minor comments (2)
  1. [Preliminaries] Notation for the symmetric subspace and the induced measure should be introduced once with a clear equation reference rather than repeated descriptively.
  2. [Figures 2 and 3] Figure captions for the probability plots should state the number of samples, the precise definition of 'optimal', and any error bars or confidence intervals used.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. We address each major comment below, indicating whether revisions will be made.

read point-by-point responses

  1. Referee: [Results section (probability estimates for MI and MII)] The central claim that PPT bound entanglement occurs with probability 'very close to 1' for N>3 rests on finite-N numerical sampling under 'optimal parameters' (abstract and results section). No concentration inequality, volume estimate, or scaling analysis is supplied to show that the measure of the favorable parameter regime approaches 1 with growing N; without such support the extrapolation from small-N samples cannot securely establish the stated asymptotic behavior.

    Authors: We agree that the manuscript relies on numerical sampling for finite N to support the claim that the probability is very close to 1. No concentration inequality, volume estimate, or scaling analysis is provided to establish the asymptotic behavior rigorously as N grows. In the revised manuscript we will add explicit language in the Results and Discussion sections clarifying that the reported probabilities are empirical observations from sampling (for N up to 10 in our experiments) and that a mathematical proof of the limit remains beyond the present scope. We will also moderate the phrasing in the abstract to avoid implying a proven asymptotic result. revision: partial

  2. Referee: [Methods I and II (parameter definitions)] The procedure for selecting the 'optimal parameters' that achieve the reported high probability is not specified in sufficient detail (methods and results). If the parameters are tuned after inspecting the samples, the probability becomes a fitted rather than an a-priori quantity, undermining the claim that bound entanglement 'naturally emerges'.

    Authors: The optimal parameters are identified by scanning a discrete grid of values for the relevant dimensions and ranks and retaining those that maximize the observed fraction of PPT entangled states in preliminary runs. This process is mentioned briefly in the Methods but lacks the necessary detail. We will expand the Methods section to describe the exact grid, the sample size per parameter combination, and the selection criterion. We will also clarify that parameters are fixed after the scan and that all reported probabilities are obtained from fresh, independent sampling runs, thereby avoiding post-selection bias in the final statistics. revision: yes

Circularity Check

0 steps flagged

No significant circularity; numerical probability estimates are independent of inputs

full rationale

The paper constructs symmetric random induced states via two explicit methods (partial trace of multiqubit symmetric pure states and tracing out a qudit ancilla), then reports the empirically sampled fraction that are PPT yet entangled for N>3 under selected parameter regimes. This is a direct Monte-Carlo measurement on the generated ensemble rather than a derivation that reduces the reported probability to a fitted quantity or self-citation by construction. No equations equate the output probability to an input fit, no uniqueness theorem is invoked from prior self-work, and the 'optimal parameters' phrasing describes the regime under study rather than smuggling an ansatz or renaming a known result. The central claim therefore remains an independent numerical observation.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are identifiable from the provided text.

pith-pipeline@v0.9.0 · 5689 in / 1018 out tokens · 33257 ms · 2026-05-21T20:26:04.889326+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

40 extracted references · 40 canonical work pages · 2 internal anchors

  1. [1]

    , N, where the sum runs over all possible permutation operatorsP σ of theNqubits

    1|{z} α (1) forα= 0, . . . , N, where the sum runs over all possible permutation operatorsP σ of theNqubits. Two ma- jor simplifications emerge when investigating the entan- glement of symmetric states. First, a symmetric state is either fully separable or genuinely multipartite entan- gled. Secondly, for any bipartitionA|Bof theN-qubits, the explicit tra...

    work page

  2. [2]

    Figure 6 depicts how the mean distance for MII decreases dramatically faster to 0 than for MI, forN= 4,5,6

    = 1/(N+ 1). Figure 6 depicts how the mean distance for MII decreases dramatically faster to 0 than for MI, forN= 4,5,6. The yellow background on the panels indicates where the probability to get PPT BE states is the highest compared to the probability to get NPT or SEP RIS. These observations indicates that MII FIG. 7.V ariety of bound entangled RIS.Proba...

    work page 2000

  3. [3]

    Horodecki, P

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

    work page 2009

  4. [4]

    Gühne and G

    O. Gühne and G. Tóth, Entanglement detection, Physics Reports474, 1 (2009)

    work page 2009

  5. [5]

    Peres, Separability criterion for density matrices, Phys

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

    work page 1996

  6. [6]

    Horodecki, P

    M. Horodecki, P. Horodecki, and R. Horodecki, Horodecki, m., horodecki, p. & horodecki, r. separabil- ity of mixed states: Necessary and sufficient conditions. phys. lett. a 223, 1-8, Physics Letters A223, 1 (1996)

    work page 1996

  7. [7]

    Horodecki, P

    M. Horodecki, P. Horodecki, and R. Horodecki, Mixed- state entanglement and distillation: Is there a “bound” entanglement in nature?, Phys. Rev. Lett.80, 5239 (1998)

    work page 1998

  8. [8]

    Ishizaka, Bound entanglement provides convertibility of pure entangled states, Phys

    S. Ishizaka, Bound entanglement provides convertibility of pure entangled states, Phys. Rev. Lett.93, 190501 (2004)

    work page 2004

  9. [9]

    Horodecki, L

    K. Horodecki, L. Pankowski, M. Horodecki, and P. Horodecki, Low-dimensional bound entanglement with one-way distillable cryptographic key, IEEE Transactions on Information Theory54, 2621 (2008)

    work page 2008

  10. [10]

    Tóth and T

    G. Tóth and T. Vértesi, Quantum states with a posi- tive partial transpose are useful for metrology, Phys. Rev. Lett.120, 020506 (2018)

    work page 2018

  11. [11]

    K. F. Pál, G. Tóth, E. Bene, and T. Vértesi, Bound en- tangled singlet-like states for quantum metrology, Phys. Rev. Res.3, 023101 (2021)

    work page 2021

  12. [12]

    Horodecki, Separability criterion and inseparable mixed states with positive partial transposition, Physics Letters A232, 333 (1997)

    P. Horodecki, Separability criterion and inseparable mixed states with positive partial transposition, Physics Letters A232, 333 (1997)

    work page 1997

  13. [13]

    Horodecki, M

    P. Horodecki, M. Horodecki, and R. Horodecki, Bound entanglement can be activated, Phys. Rev. Lett.82, 1056 (1999)

    work page 1999

  14. [14]

    C. H. Bennett, D. P. DiVincenzo, T. Mor, P. W. Shor, J. A. Smolin, and B. M. Terhal, Unextendible product basesandboundentanglement,Phys.Rev.Lett.82,5385 (1999)

    work page 1999

  15. [15]

    Halder and R

    S. Halder and R. Sengupta, Construction of noisy bound entangled states and the range criterion, Physics Letters A383, 2004 (2019)

    work page 2004

  16. [16]

    Skowronek, Three-by-three bound entanglement with general unextendible product bases, Journal of Mathe- matical Physics52, 10.1063/1.3663836 (2011)

    L. Skowronek, Three-by-three bound entanglement with general unextendible product bases, Journal of Mathe- matical Physics52, 10.1063/1.3663836 (2011)

    work page doi:10.1063/1.3663836 2011

  17. [17]

    D. P. DiVincenzo, T. Mor, P. W. Shor, J. A. Smolin, and B. M. Terhal, Unextendible product bases, un- completable product bases and bound entanglement, Communications in Mathematical Physics238, 379–410 (2003)

    work page 2003

  18. [18]

    Bej and S

    P. Bej and S. Halder, Unextendible product bases, bound entangled states, and the range criterion, Physics Letters A386, 126992 (2021)

    work page 2021

  19. [19]

    Bandyopadhyay, S

    S. Bandyopadhyay, S. Ghosh, and V. Roychowdhury, Non-full-rankboundentangledstatessatisfyingtherange criterion, Phys. Rev. A71, 012316 (2005)

    work page 2005

  20. [20]

    P.CabanandB.C.Hiesmayr,Boundentanglementisnot lorentz invariant, Scientific Reports13, 10.1038/s41598- 023-38217-3 (2023)

    work page doi:10.1038/s41598- 2023

  21. [21]

    Augusiak, J

    R. Augusiak, J. Grabowski, M. Kuś, and M. Lewenstein, Searching for extremal ppt entangled states, Optics Com- munications283, 805–813 (2010)

    work page 2010

  22. [22]

    Halder, M

    S. Halder, M. Banik, and S. Ghosh, Family of bound en- tangled states on the boundary of the peres set, Physical 10 Review A99, 10.1103/physreva.99.062329 (2019)

    work page doi:10.1103/physreva.99.062329 2019

  23. [23]

    Cheng and F

    W. Cheng and F. Xu, A general scheme for construction of bound entangled states by convex linear combination, 2010 IEEE International Conference on Information The- ory and Information Security , 434 (2010)

    work page 2010

  24. [24]

    A Class of PPT Entangled States Arbitrary Far From Separable States

    A. Rutkowski and M. Studziński, A class of ppt entan- gled states arbitrary far from separable states (2019), arXiv:1906.10023 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  25. [25]

    Piani and C

    M. Piani and C. E. Mora, Class of positive-partial- transpose bound entangled states associated with almost any set of pure entangled states, Phys. Rev. A75, 012305 (2007)

    work page 2007

  26. [26]

    H. Zhao, S. Guo, N. Jing, and S. Fei, Construction of bound entangled states based on permutation operators, Quantum Information Processing15, 1529–1538 (2015)

    work page 2015

  27. [27]

    Vasmer and D

    P. Badzia¸g, K. Horodecki, M. Horodecki, J. Jenkin- son, and S. J. Szarek, Bound entangled states with ex- tremal properties, Physical Review A90, 10.1103/phys- reva.90.012301 (2014)

    work page doi:10.1103/phys- 2014

  28. [28]

    J. Tura, R. Augusiak, P. Hyllus, M. Kuś, J. Samsonow- icz, and M. Lewenstein, Four-qubit entangled symmetric states with positive partial transpositions, Physical Re- view A85, 10.1103/physreva.85.060302 (2012)

    work page doi:10.1103/physreva.85.060302 2012

  29. [29]

    Popp and B

    C. Popp and B. C. Hiesmayr, Almost complete solu- tion for the np-hard separability problem of bell diagonal qutrits, Scientific Reports12, 10.1038/s41598-022-16225- z (2022)

    work page doi:10.1038/s41598-022-16225- 2022

  30. [30]

    W. Li, R. Han, J. Shang, H. K. Ng, and B.-G. Englert, Sequentially constrained monte carlo sampler for quan- tum states (2021), arXiv:2109.14215 [quant-ph]

    work page arXiv 2021

  31. [31]

    Sindici and M

    E. Sindici and M. Piani, Simple class of bound entan- gled states based on the properties of the antisymmetric subspace, Phys. Rev. A97, 032319 (2018)

    work page 2018

  32. [32]

    Clarisse, Construction of bound entangled edge states with special ranks, Physics Letters A359, 603 (2006)

    L. Clarisse, Construction of bound entangled edge states with special ranks, Physics Letters A359, 603 (2006)

    work page 2006

  33. [33]

    J.Moerland, N.Wyderka, H.Kampermann,andD.Bruß, Bound entangled bell diagonal states of unequal local di- mensions, and their witnesses, Phys. Rev. A109, 012217 (2024)

    work page 2024

  34. [34]

    Aubrun, S

    G. Aubrun, S. J. Szarek, and D. Ye, Entanglement thresholds for random induced states, Communications on Pure and Applied Mathematics67, 129–171 (2013)

    work page 2013

  35. [35]

    Knill, Open quantum problems,https://oqp.iqoqi

    E. Knill, Open quantum problems,https://oqp.iqoqi. oeaw.ac.at/undistillability-implies-ppt

    work page

  36. [36]

    Horodecki, P

    M. Horodecki, P. Horodecki, and R. Horodecki, Mixed- state entanglement and distillation: Is there a “bound” entanglement in nature?, Physical Review Letters80, 5239–5242 (1998)

    work page 1998

  37. [37]

    Partial transposition of random states and non-centered semicircular distributions

    G. Aubrun, Partial transposition of random states and non-centered semicircular distributions (2012), arXiv:1011.0275 [math.PR]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  38. [38]

    Życzkowski, K

    K. Życzkowski, K. A. Penson, I. Nechita, and B. Collins, Generating random density matrices, Journal of Mathe- matical Physics52, 10.1063/1.3595693 (2011)

    work page doi:10.1063/1.3595693 2011

  39. [39]

    Bohnet-Waldraff, D

    F. Bohnet-Waldraff, D. Braun, and O. Giraud, Entangle- ment and the truncated moment problem, Phys. Rev. A 96, 032312 (2017)

    work page 2017

  40. [40]

    Louvet, E

    J. Louvet, E. Serrano-Ensástiga, T. Bastin, and J. Mar- tin, Nonequivalence between absolute separability and positive partial transposition in the symmetric subspace, Phys. Rev. A111, 042418 (2025)

    work page 2025