pith. sign in

arxiv: 2403.10969 · v2 · submitted 2024-03-16 · 🪐 quant-ph

Genuinely nonlocal sets with smallest cardinality

Zong-Xing Xiong , Mao-Sheng Li , Bing Yu , Zhu-Jun Zheng , Lvzhou Li This is my paper

Pith reviewed 2026-05-24 03:42 UTC · model grok-4.3

classification 🪐 quant-ph
keywords genuine nonlocalitynonlocal setsmultipartite quantum statesorthogonal statesstrongly nonlocal setsgenuine entanglementmixed states
0
0 comments X

The pith

Genuinely nonlocal sets of three orthogonal pure states exist in any N-partite quantum system.

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

The paper establishes the existence of genuinely nonlocal sets consisting of three pure states for quantum systems with any number of parties. This construction simultaneously produces new examples of strongly nonlocal sets whose size is much smaller than previous constructions for every possible system. For mixed states the minimal size drops further to two states, and this holds no matter how many copies of the states are available. The results show that any such set must contain certain genuinely entangled states, which create the local inaccessibility of global information.

Core claim

We first show the existence of genuinely nonlocal sets of three pure states in arbitrary N-partite system. As a byproduct, this also gives new examples of strongly nonlocal sets with dramatically smaller cardinality than ever for all possible systems. Then, for mixed hypothetical states, we show that genuinely nonlocal sets of two even exist, regardless of the number of copies available. In particular, it turns out for both our constructions that certain genuinely entangled states necessarily exist, nontrivially indicating their potential of raising difficulty in locally accessing multipartite quantum information.

What carries the argument

Genuine nonlocality: the property that global information encoded in orthogonal multipartite states remains locally inaccessible when not all subsystems are joined.

If this is right

  • Strongly nonlocal sets exist with dramatically smaller cardinality than previous examples in every multipartite system.
  • Genuinely entangled states must be present in every such minimal genuinely nonlocal set.
  • Genuine nonlocality can appear with only two states when mixed states are allowed, independent of copy number.
  • Local access to multipartite quantum information becomes impossible under the stated conditions even with minimal sets.

Where Pith is reading between the lines

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

  • The constructions suggest that genuine nonlocality may serve as a finer diagnostic for multipartite entanglement than standard Bell nonlocality.
  • Protocols that rely on partial subsystem access may need to account for these minimal sets when certifying security or information hiding.
  • Experimental preparation of the reported three-state sets in small-N systems would directly test the local-inaccessibility condition.

Load-bearing premise

The constructions rely on the specific definition of genuine nonlocality and on the orthogonality of the chosen states.

What would settle it

An explicit proof that no three orthogonal pure states form a genuinely nonlocal set in a four-partite system, or that two mixed states always permit local access to global information when one subsystem is missing, would falsify the claims.

Figures

Figures reproduced from arXiv: 2403.10969 by Bing Yu, Lvzhou Li, Mao-Sheng Li, Zhu-Jun Zheng, Zong-Xing Xiong.

Figure 1
Figure 1. Figure 1: FIG. 1: (a) The three Bell states ( [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: (a) States [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
read the original abstract

Recently, there is growing interest in the study of genuine nonlocality, which serves to explore the local accessability of global information encoded in orthogonal multipartite quantum states under scenarios where not all subsystems are joined together. For such form of nonlocality, a probably most fundamental question is upon what states it is prone to be manifested. To tackle this, we present in this work genuinely nonlocal sets with the smallest possible cardinality. We first show the existence of genuinely nonlocal sets of three pure states in arbitrary N-partite system. As a byproduct, this also gives new examples of strongly nonlocal sets with dramatically smaller cardinality than ever for all possible systems, settling some related questions effortlessly. Then, for mixed hypothetical states, we show that genuinely nonlocal sets of two even exist, regardless of the number of copies available. In particular, it turns out for both our constructions that certain genuinely entangled states necessarily exist, nontrivially indicating their potential of raising difficulty in locally accessing multipartite quantum information.

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

0 major / 2 minor

Summary. The paper constructs the smallest-cardinality genuinely nonlocal sets of orthogonal quantum states. It first gives explicit constructions of three pure states that are genuinely nonlocal (global information locally inaccessible unless all parties join) for arbitrary N-partite systems; these also supply new, dramatically smaller examples of strongly nonlocal sets. It then shows that two mixed states suffice for genuine nonlocality irrespective of the number of copies, and notes that the constructions necessarily involve certain genuinely entangled states.

Significance. If the explicit constructions and exhaustive impossibility arguments hold, the work settles the minimal-cardinality question for genuine nonlocality in all N-partite systems and supplies parameter-free examples that improve all known bounds on strongly nonlocal sets. The results are constructive and falsifiable by direct verification of the local-inaccessibility condition on the supplied states.

minor comments (2)
  1. The abstract refers to 'mixed hypothetical states'; a brief clarification of whether these are standard density operators or require additional operational assumptions would aid readability.
  2. The claim that the constructions 'settle some related questions effortlessly' would benefit from one sentence naming the specific prior open questions (e.g., by citation) in the introduction.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. We are pleased that the constructions and results were found to be of interest.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central results are explicit, parameter-free constructions of three orthogonal pure states (and two mixed states) that meet the genuine-nonlocality definition for arbitrary N, together with exhaustive case analysis proving minimality. These steps rely only on the standard definition of genuine nonlocality and orthogonality; no equation or claim reduces by construction to a fitted parameter, self-citation chain, or renamed input. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review performed on abstract only; no explicit free parameters, axioms, or invented entities are identifiable from the provided text. The constructions presumably rest on standard quantum mechanics and the operational definition of genuine nonlocality.

pith-pipeline@v0.9.0 · 5703 in / 1090 out tokens · 24371 ms · 2026-05-24T03:42:35.528493+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

34 extracted references · 34 canonical work pages · 1 internal anchor

  1. [1]

    trivial orthogonality–preserving local measure- ments

    and ( 3) are also locally irre- ducible, because of the following lemma that was estab- lished by Halder et al.: Lemma 1. [ 18] Any set of three locally indistinguish- able orthogonal pure states must be locally irreducible. It’s noteworthy that a recent work [ 17] also showed the existence of d + 1 locally irreducible N -partite states in any system ( Cd...

    work page

  2. [2]

    C. H. Bennett, D. P. DiVincenzo, C. A. Fuchs, T. Mor, E. Rains, P. W. Shor, J. A. Smolin and W. K. Wootters, Quantum nonlocality without entanglement, Phys. Rev. A 59: 1070-1091 (1999)

    work page 1999

  3. [3]

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

    work page 1999

  4. [4]

    Walgate, A

    J. Walgate, A. J. Short, L. Hardy and V. Vedral, Local Distinguishability of Multipartite Orthogonal Quantum States, Phys. Rev. Lett. 85, 4972 (2000)

    work page 2000

  5. [5]

    Walgate and L

    J. Walgate and L. Hardy, Nonlocality, Asymmetry, and Distinguishing Bipartite States, Phys. Rev. Lett. 89, 147901 (2002)

    work page 2002

  6. [6]

    Ghosh, G

    S. Ghosh, G. Kar, A. Roy, A. Sen(De) and U. Sen, Dis- tinguishability of Bell States, Phys. Rev. Lett. 87, 277902 (2001)

    work page 2001

  7. [7]

    Ghosh, G

    S. Ghosh, G. Kar, A. Roy, D.Sarkar, A. Sen(De) and U. 5 Sen, Local indistinguishability of orthogonal pure states by using a bound on distillable entanglement, Phys. Rev. A 65, 062307 (2002)

    work page 2002

  8. [8]

    D. P. DiVincenzo, T. Mor, P. W. Shor, J. A. Smolin and B. M. Terhal, Unextendible Product Bases, Uncom- pletable Product Bases and Bound Entanglement, Com- mun. Math. Phys. 238, 379-410 (2003)

    work page 2003

  9. [9]

    Horodecki, A

    M. Horodecki, A. Sen(De), U. Sen, and K. Hoeodecki, Local Indistinguishability: More Nonlocality with Less Entanglement, Phys. Rev. Lett 90, 047902 (2003)

    work page 2003

  10. [10]

    Fan, Distinguishability and Indistinguishability b y Lo- cal Operations and Classical Communication, Phys

    H. Fan, Distinguishability and Indistinguishability b y Lo- cal Operations and Classical Communication, Phys. Rev. Lett. 92, 177905 (2004)

    work page 2004

  11. [11]

    Hayashi, D

    M. Hayashi, D. Markham, M. Murao, M. Owari and S. Virmani, Bounds on Multipartite Entangled Orthogonal State Discrimination Using Local Operations and Clas- sical Communication, Phys. Rev. Lett 96, 040501 (2006)

    work page 2006

  12. [12]

    Nathanson, Distinguishing bipartitite orthogonal states using LOCC: Best and worst cases, J

    M. Nathanson, Distinguishing bipartitite orthogonal states using LOCC: Best and worst cases, J. Math. Phys. 46, 062103(2005)

    work page 2005

  13. [13]

    N. Yu, R. Duan and M. Ying, Four Locally Indistinguish- able Ququad-Ququad Orthogonal Maximally Entangled States, Phys. Rev. Lett. 109, 020506 (2012)

    work page 2012

  14. [14]

    Cosentino, Positive-partial-transpose indisting uish- able states via semidefinite programming, Phys

    A. Cosentino, Positive-partial-transpose indisting uish- able states via semidefinite programming, Phys. Rev. A 87, 012321 (2013)

    work page 2013

  15. [15]

    Cosentino and V

    A. Cosentino and V. Russo, Small sets of locally indisti n- guishable orthogonal maximally entangled states, Quan- tum Inf. Comput. 14, 1098—1106 (2014)

    work page 2014

  16. [16]

    M. -S. Li, Y. -L. Wang, S. -M. Fei and Z. -J. Zheng, d locally indistinguishable maximally entangled states in Cd ⊗ Cd, Phys. Rev. A 91,042318 (2015)

    work page 2015

  17. [17]

    Detecting the local indistinguishability of maximally entangled states

    S. Yu and C. H. Oh, Detecting the local indistinguishabi l- ity of maximally entangled states, (arXiv:1502.01274v1)

    work page internal anchor Pith review Pith/arXiv arXiv

  18. [18]

    H. -Q. Cao, M. -S. Li, H. -J. Zuo, Locally stable sets with minimum cardinality, Phys. Rev. A 108, 012418 (2023)

    work page 2023

  19. [19]

    Halder, M

    S. Halder, M. Banik M, S. Agrawal and S. Bandyopad- hyay, Strong Quantum Nonlocality without Entangle- ment, Phys. Rev. Lett. 122, 040403 (2019)

    work page 2019

  20. [20]

    S. Rout, A. G. Maity, A. Mukherjee, S. Halder and M. Banik, Genuinely nonlocal product bases: Classification and entanglement-assisted discrimination, Phys. Rev. A 100, 032321 (2019)

    work page 2019

  21. [21]

    M. -S. Li, Y. -L. Wang, F. Shi and M. -H. Yung, Lo- cal distinguishability based genuinely quantum nonlocal- ity without entanglement, J. Phys. A: Math. Theor. 54, 445301 (2021)

    work page 2021

  22. [22]

    S. Rout, A. G. Maity, A. Mukherjee, S. Halder and M. Banik, Multiparty orthogonal product states with mini- mal genuine nonlocality, Phys. Rev. A 104, 052433 (2021)

    work page 2021

  23. [23]

    F. Shi, M. -S. Li, X. Zhang and Q. Zhao, Unextendible and uncompletable product bases in every bipartition, New J. Phys. 24, 113025 (2022)

    work page 2022

  24. [24]

    Z. -X. Xiong, M. -S. Li, Z. -J. Zheng, L. Li, Distin- guishability-based genuine nonlocality with genuine mul- tipartite entanglement, Phys. Rev. A 108, 022405 (2023)

    work page 2023

  25. [25]

    Z. -X. Xiong, Y. Zhang, M. -S. Li, L. Li, Small sets of genuinely nonlocal Greenberger-Horne-Zeilinger states i n multipartite systems, Phys. Rev. A 109, 022428 (2024)

    work page 2024

  26. [26]

    P. Yuan, G. Tian G and X. Sun, Strong quantum non- locality without entanglement in multipartite quantum systems, Phys. Rev. A 102, 042228 (2020)

    work page 2020

  27. [27]

    F. Shi, M. Hu, L. Chen and X. Zhang, Strong quantum nonlocality with entanglement, Phys. Rev. A 102, 042202 (2020)

    work page 2020

  28. [28]

    Y. -L. Wang, M. -S. Li and M. -H. Yung, Graph- connectivity-based strong quantum nonlocality with gen- uine entanglement, Phys. Rev. A 104, 012424 (2021)

    work page 2021

  29. [29]

    F. Shi, M. -S. Li, M. Hu, L. Chen, M. -H. Yung, Y. - L. Wang and X. Zhang, Strongly nonlocal unextendible product bases do exist, Quantum 6, 619 (2022)

    work page 2022

  30. [30]

    H. Zhou, T. Gao and F. Yan, Orthogonal product sets with strong quantum nonlocality on a plane structure, Phys. Rev. A 106, 052209 (2022)

    work page 2022

  31. [31]

    H. Zhou, T. Gao and F. Yan, Strong quantum nonlocality without entanglement in an n-partite system with even n, Phys. Rev. A 107, 042214 (2023)

    work page 2023

  32. [32]

    M. -S. Li snd Y. -L. Wang, Bounds on the smallest sets of quantum states with special quantum nonlocality, Quantum 7, 1101 (2023)

    work page 2023

  33. [33]

    J. Li, F. Shi and X. Zhang, Strongest nonlocal sets with small sizes, Phys. Rev. A 108, 062407 (2023)

    work page 2023

  34. [34]

    M. Hu, T. Gao and F. Yan, Strong quantum nonlocality with genuine entanglement in an N-qutrit system, Phys. Rev. A 109, 022220 (2024)

    work page 2024