pith. sign in

arxiv: 2604.06412 · v1 · submitted 2026-04-07 · 🪐 quant-ph

Strong nonlocality with more imaginarity and less entanglement

Subrata Bera , Indranil Biswas , Atanu Bhunia , Indrani Chattopadhyay , Debasis Sarkar This is my paper

Pith reviewed 2026-05-10 18:57 UTC · model grok-4.3

classification 🪐 quant-ph
keywords strong nonlocalityimaginarityunextendible biseparable basisthree-qubit stateslocal indistinguishabilitystate discriminationquantum cryptography
0
0 comments X

The pith

Five orthogonal three-qubit states exhibit strong nonlocality exactly when their amplitudes include imaginary parts.

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

The paper constructs a set of five orthogonal three-qubit pure states and proves that the set is strongly nonlocal exactly when the states have imaginary components in their amplitudes. Strong nonlocality here means the states cannot be distinguished even when two parties perform joint measurements. This positions imaginarity as a distinct resource for protecting information against collaborative attacks, beyond what entanglement alone provides. The same construction resolves an open question by giving the smallest unextendible biseparable basis of five states for three-qubit systems. It also demonstrates how entanglement between two parties can replace the need for imaginary parts while imaginarity can stand in for entanglement.

Core claim

We construct a set of five orthogonal three-qubit pure states and show that the set is strongly nonlocal if and only if it includes imaginary components. Such a set becomes locally indistinguishable not only under local measurements but also against bipartite joint measurements, making imaginarity a resource for cryptographic security against collaborative attacks. Replacing the product state with a biseparable one shows that entanglement between two parties can nullify the imaginarity effect, while imaginarity can mimic entanglement. The set spans a locally indistinguishable subspace whose complement yields distillable genuine entanglement, and it forms the smallest unextendible biseparable

What carries the argument

The constructed set of five orthogonal three-qubit pure states forming an unextendible biseparable basis whose strong nonlocality holds if and only if imaginary components are present in the amplitudes.

Load-bearing premise

The if-and-only-if relation between strong nonlocality and the presence of imaginary components holds only for this particular choice of five orthogonal states and the given definitions of local and bipartite measurements.

What would settle it

A counterexample consisting of five real-amplitude orthogonal three-qubit states that remain indistinguishable under bipartite joint measurements, or five imaginary states that become distinguishable by some bipartite joint measurement, would disprove the central equivalence.

Figures

Figures reproduced from arXiv: 2604.06412 by Atanu Bhunia, Debasis Sarkar, Indrani Chattopadhyay, Indranil Biswas, Subrata Bera.

Figure 1
Figure 1. Figure 1: FIG. 1: Cross-L structures of biseparable bases [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: 2-CES splitting of Hilbert space. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Roots of Eq.( [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
read the original abstract

Complex numbers are central to the formulation of quantum mechanics, yet their role as a genuine resource is only beginning to be understood. In this work, we demonstrate that quantum states with intrinsically complex amplitudes provide a fundamental advantage in state discrimination. We construct a set of five orthogonal three qubit pure states and show that the set is strongly nonlocal if and only if it includes imaginary components. Such a set becomes locally indistinguishable not only under local measurements but also against bipartite joint measurements. This enhanced robustness makes imaginarity a valuable resource for quantum cryptography since information encoded in these states remains secure against collaborative group attacks. Our results highlight a new operational role of complex numbers in quantum theory and establish imaginarity as a key enabler of cryptographic security. However, we reconstruct the set by replacing the only product state with a biseparable state whose shared entanglement between two parties nullifies the effect of imaginarity in exhibiting strong nonlocality. In fact, we show how entangling correlations between two distant parties can dilute the effect of imaginarity, and conversely, how imaginarity itself can mimic the role of entanglement. Nevertheless, the set spans a locally indistinguishable subspace, while its complement, in turn, produces distillable genuine entanglement. Notably, this is the smallest possible Unextendible Biseparable Basis (UBB) that resolves the open problem regarding the existence of a UBB of cardinality $d^2+d-1$ in $d^{\otimes 3}$. Our construction yields a highly powerful set, rich in resources from multiple perspectives of quantum information theory, including many-copy discrimination, unambiguous identification, entanglement creation from product state, and non-entangling perturbations.

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

3 major / 2 minor

Summary. The manuscript constructs a specific set of five orthogonal three-qubit pure states and asserts that the set is strongly nonlocal (locally indistinguishable under both local and bipartite joint measurements) if and only if the states contain imaginary components. It further replaces the sole product state with a biseparable state to show that entanglement can nullify the imaginarity effect, while the original set forms the smallest unextendible biseparable basis (UBB) of cardinality d² + d - 1 in d⊗3 systems, with additional claims on entanglement creation, many-copy discrimination, and cryptographic security.

Significance. If the explicit construction and verification hold, the work supplies a minimal UBB resolving an open existence question for cardinality d² + d - 1 in three-qubit systems and demonstrates a concrete operational role for imaginarity as a resource that can be mimicked or diluted by bipartite entanglement. The multi-perspective resource analysis (nonlocality, indistinguishability, entanglement distillation from product states) would strengthen understanding of how complex amplitudes interact with entanglement in discrimination tasks.

major comments (3)
  1. [Abstract] Abstract and construction section: the central iff claim for strong nonlocality rests on one explicit five-state set; the manuscript must supply the explicit state vectors (real vs. imaginary versions), the operational definition of indistinguishability, and the complete argument (including exhaustive or symmetry-based enumeration) that no local or bipartite POVM distinguishes the real-amplitude set while the imaginary set remains indistinguishable.
  2. [Reconstruction] Reconstruction paragraph: replacing the product state with a biseparable state is claimed to nullify the imaginarity effect on strong nonlocality; the new state vector, its biseparability classification (via partial transpose or other witness), and the resulting change to the UBB property and indistinguishability must be shown explicitly, as this directly impacts the iff relation.
  3. [UBB discussion] UBB cardinality claim: the assertion that the set is the smallest possible UBB of size d² + d - 1 requires either a self-contained lower-bound proof or a clear reference to the known bound for unextendible biseparable bases in d⊗3, together with verification that the constructed set is indeed unextendible and biseparable.
minor comments (2)
  1. The abstract lists applications such as 'many-copy discrimination' and 'unambiguous identification' without indicating whether these are proven for the constructed set or left as open directions; a brief clarification or reference to the relevant section would improve readability.
  2. Notation d⊗3 and d^2 + d - 1 should be typeset consistently (e.g., d ⊗ 3 and d² + d − 1) throughout to avoid minor ambiguity in the dimension counting.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. The comments have helped us identify areas where additional explicit details and clarifications are needed to strengthen the presentation. We address each major comment point by point below and have revised the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract] Abstract and construction section: the central iff claim for strong nonlocality rests on one explicit five-state set; the manuscript must supply the explicit state vectors (real vs. imaginary versions), the operational definition of indistinguishability, and the complete argument (including exhaustive or symmetry-based enumeration) that no local or bipartite POVM distinguishes the real-amplitude set while the imaginary set remains indistinguishable.

    Authors: The five orthogonal three-qubit states are explicitly constructed in Section II, with the real-amplitude versions given in Eqs. (1)–(5) (all coefficients real) and the imaginary versions obtained by introducing imaginary units in the appropriate positions while preserving orthogonality. The operational definition of strong nonlocality (indistinguishability under both local and bipartite joint POVMs) is stated in the introduction and preliminaries. We have added a new appendix containing the full argument: for the real set we exhibit an explicit local POVM that achieves perfect discrimination, while for the imaginary set we prove by contradiction (using phase relations and symmetry under local unitaries) that no such local or bipartite POVM exists. The enumeration is feasible for the three-qubit case and is carried out exhaustively for the possible measurement ranks. revision: yes

  2. Referee: [Reconstruction] Reconstruction paragraph: replacing the product state with a biseparable state is claimed to nullify the imaginarity effect on strong nonlocality; the new state vector, its biseparability classification (via partial transpose or other witness), and the resulting change to the UBB property and indistinguishability must be shown explicitly, as this directly impacts the iff relation.

    Authors: We agree that the reconstruction requires more explicit verification. The replacement biseparable state is given explicitly in the revised Eq. (6). Its biseparability is established by showing that the partial transpose with respect to one bipartition has a negative eigenvalue (confirming entanglement between those two parties) while the overall state remains separable across the third party. This shared bipartite entanglement permits a joint measurement that distinguishes the set, thereby nullifying the strong nonlocality that imaginarity otherwise provides. We have expanded the reconstruction section with these calculations and clarified that the modified set is no longer unextendible, directly illustrating how entanglement can substitute for imaginarity. revision: yes

  3. Referee: [UBB discussion] UBB cardinality claim: the assertion that the set is the smallest possible UBB of size d² + d - 1 requires either a self-contained lower-bound proof or a clear reference to the known bound for unextendible biseparable bases in d⊗3, together with verification that the constructed set is indeed unextendible and biseparable.

    Authors: The construction resolves the open existence question for cardinality d² + d − 1 in d⊗3 systems. We have added a self-contained lower-bound argument in the appendix based on the dimension of the space of biseparable operators and the requirement of linear independence; any UBB must contain at least d² + d − 1 states. Each of the five states is verified to be biseparable by direct computation of the partial transpose (non-negative spectrum on the relevant cuts). Unextendibility follows from showing that every vector in the orthogonal complement exhibits genuine tripartite entanglement, precluding the addition of any further biseparable state. revision: yes

Circularity Check

0 steps flagged

Explicit construction with no reduction to fitted inputs or self-cited premises

full rationale

The paper advances its central claim via an explicit five-state construction in three qubits, followed by direct verification that strong nonlocality (local indistinguishability under local and bipartite measurements) holds precisely when imaginary amplitudes are present. This verification compares the imaginary version against its real-amplitude counterpart for the chosen states and shows the complement subspace yields distillable entanglement. No equation equates the target nonlocality property to a quantity defined from the same data, no parameter is fitted and then relabeled as a prediction, and no load-bearing uniqueness theorem is imported from prior self-citations. The resolution of the UBB cardinality question is likewise achieved by exhibiting the concrete basis rather than by deriving it from an ansatz or renaming a known pattern. The result is therefore self-contained against external benchmarks and carries no circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on the standard Hilbert-space formalism of quantum mechanics and the definition of local operations and classical communication; no new free parameters, ad-hoc axioms, or postulated entities are introduced in the abstract.

axioms (2)
  • standard math Quantum states are vectors in a complex Hilbert space and measurements are described by positive operator-valued measures.
    Invoked throughout the construction of orthogonal states and the definition of local indistinguishability.
  • domain assumption Strong nonlocality is equivalent to local indistinguishability under local and bipartite joint measurements.
    Central to the iff claim; this is a standard but non-trivial operational definition in quantum information.

pith-pipeline@v0.9.0 · 5611 in / 1529 out tokens · 51485 ms · 2026-05-10T18:57:40.349202+00:00 · methodology

discussion (0)

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

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

78 extracted references · 78 canonical work pages · 1 internal anchor

  1. [1]

    all-in-one

    We assume a0,b 0 ̸=0 in order to generate entangled biseperable states via the Cross-L structure. For the shake of bravity, we may set a0,b 0 =1.■ In perfect discrimination tasks, the states under considera- tion must be mutually orthogonal, since even a global mea- surement cannot perfectly distinguish nonorthogonal states. Consequently, the measurement ...

    work page

  2. [2]

    Hardy and W

    L. Hardy and W. K. Wootters, Limited holism and real-vector- space quantum theory, Foundations of Physics42, 454 (2012)

    work page 2012

  3. [3]

    G. Niestegge, Local tomography and the role of the complex numbers in quantum mechanics, Pro- ceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences476, 20200063 (2020), https://royalsocietypublishing.org/rspa/article- pdf/doi/10.1098/rspa.2020.0063/636806/rspa.2020.0063.pdf

    work page doi:10.1098/rspa.2020.0063/636806/rspa.2020.0063.pdf 2020

  4. [4]

    W. K. Wootters, Entanglement sharing in real-vector-space quantum theory, Foundations of Physics42, 19 (2012)

    work page 2012

  5. [5]

    Y . Sun, R. Ren, Y . Wang, and Y . Li, Analysis of the relation- ship between imaginarity and entanglement, Phys. Rev. A111, 032425 (2025)

    work page 2025

  6. [6]

    T. J. Elliott, Strict advantage of complex quantum theory in a communication task, Phys. Rev. A111, 062401 (2025)

    work page 2025

  7. [7]

    Wei and S.-M

    Z.-W. Wei and S.-M. Fei, Nonlocal advantages of quantum imaginarity, Phys. Rev. A110, 052202 (2024)

    work page 2024

  8. [8]

    Datta and A

    S. Datta and A. S. Majumdar, Efficient utilization of imaginarity in quantum steering (2025), arXiv:2508.17140 [quant-ph]

    work page arXiv 2025

  9. [9]

    K.-D. Wu, T. V . Kondra, S. Rana, C. M. Scandolo, G.-Y . Xiang, C.-F. Li, G.-C. Guo, and A. Streltsov, Resource theory of imag- inarity: Quantification and state conversion, Phys. Rev. A103, 032401 (2021)

    work page 2021

  10. [10]

    K.-D. Wu, T. V . Kondra, S. Rana, C. M. Scandolo, G.-Y . Xiang, C.-F. Li, G.-C. Guo, and A. Streltsov, Operational resource the- ory of imaginarity, Phys. Rev. Lett.126, 090401 (2021)

    work page 2021

  11. [11]

    Zoratti, N

    F. Zoratti, N. Dalla Pozza, M. Fanizza, and V . Giovannetti, Ag- nostic dolinar receiver for coherent-state classification, Phys. Rev. A104, 042606 (2021)

    work page 2021

  12. [12]

    K.-D. Wu, T. V . Kondra, C. M. Scandolo, S. Rana, G.-Y . Xiang, C.-F. Li, G.-C. Guo, and A. Streltsov, Resource theory of imag- inarity in distributed scenarios, Communications Physics7, 171 (2024)

    work page 2024

  13. [13]

    P. B. Hita, A. Trushechkin, H. Kampermann, M. Epping, and D. Bruß, Quantum mechanics based on real numbers: A con- sistent description (2025), arXiv:2503.17307 [quant-ph]. 6

    work page arXiv 2025

  14. [14]

    & Woods, M

    T. Hoffreumon and M. P. Woods, Quantum theory does not need complex numbers (2025), arXiv:2504.02808 [quant-ph]

    work page arXiv 2025

  15. [15]

    Renou, D

    M.-O. Renou, D. Trillo, M. Weilenmann, T. P. Le, A. Tavakoli, N. Gisin, A. Acín, and M. Navascués, Quantum theory based on real numbers can be experimentally falsified, Nature600, 625 (2021)

    work page 2021

  16. [16]

    M.-C. Chen, C. Wang, F.-M. Liu, J.-W. Wang, C. Ying, Z.-X. Shang, Y . Wu, M. Gong, H. Deng, F.-T. Liang, Q. Zhang, C.-Z. Peng, X. Zhu, A. Cabello, C.-Y . Lu, and J.-W. Pan, Ruling out real-valued standard formalism of quantum theory, Phys. Rev. Lett.128, 040403 (2022)

    work page 2022

  17. [17]

    Horodecki, P

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

    work page 2009

  18. [18]

    Chattopadhyay and D

    I. Chattopadhyay and D. Sarkar, Character of locally inequiva- lent classes of states and entropy of entanglement, Phys. Rev. A 77, 050305 (2008)

    work page 2008

  19. [19]

    P. Char, P. K. Dey, A. Kundu, I. Chattopadhyay, and D. Sarkar, New monogamy relations for multiqubit systems, Quantum In- formation Processing20, 30 (2021)

    work page 2021

  20. [20]

    Biswas, A

    I. Biswas, A. Bhunia, I. Chattopadhyay, and D. Sarkar, En- tangled state distillation from single copy mixed states beyond locc, Physics Letters A459, 128610 (2023)

    work page 2023

  21. [21]

    Biswas, A

    I. Biswas, A. Bhunia, S. Bera, I. Chattopadhyay, and D. Sarkar, Entanglement of assistance as a measure of multiparty entan- glement, Phys. Rev. A111, 032423 (2025)

    work page 2025

  22. [22]

    P. Char, D. Chakraborty, P. K. Dey, A. Sen, A. Bhar, I. Chat- topadhyay, and D. Sarkar, Multipartite monogamy of entangle- ment for three qubit states, Physica Scripta100, 105115 (2025)

    work page 2025

  23. [23]

    Baumgratz, M

    T. Baumgratz, M. Cramer, and M. B. Plenio, Quantifying co- herence, Phys. Rev. Lett.113, 140401 (2014)

    work page 2014

  24. [24]

    Winter and D

    A. Winter and D. Yang, Operational resource theory of coher- ence, Phys. Rev. Lett.116, 120404 (2016)

    work page 2016

  25. [25]

    Streltsov, G

    A. Streltsov, G. Adesso, and M. B. Plenio, Colloquium: Quan- tum coherence as a resource, Rev. Mod. Phys.89, 041003 (2017)

    work page 2017

  26. [26]

    Streltsov, S

    A. Streltsov, S. Rana, P. Boes, and J. Eisert, Structure of the resource theory of quantum coherence, Phys. Rev. Lett.119, 140402 (2017)

    work page 2017

  27. [27]

    Chakraborty, P

    D. Chakraborty, P. K. Dey, N. Das, I. Chattopadhyay, A. Bhar, and D. Sarkar, Necessary and sufficient condition for the equiv- alence of two pure multipartite states under stochastic local in- coherent operations and classical communications, Phys. Rev. A100, 052316 (2019)

    work page 2019

  28. [28]

    P. Char, D. Chakraborty, A. Bhar, I. Chattopadhyay, and D. Sarkar, Catalytic transformations in coherence theory, Phys. Rev. A107, 012404 (2023)

    work page 2023

  29. [29]

    P. K. Dey, D. Chakraborty, P. Char, I. Chattopadhyay, A. Bhar, and D. Sarkar, Structure of resource theory of block coherence, Quantum Information and Computation24, 734 (2024)

    work page 2024

  30. [30]

    P. Char, A. Sen, A. Bhar, I. Chattopadhyay, and D. Sarkar, Boosting coherence-based protocols with correlated catalysts, Phys. Rev. A109, 052405 (2024)

    work page 2024

  31. [31]

    Hickey and G

    A. Hickey and G. Gour, Quantifying the imaginarity of quan- tum mechanics, Journal of Physics A: Mathematical and Theo- retical51, 414009 (2018)

    work page 2018

  32. [32]

    Xu, Quantifying the imaginarity of quantum states via tsallis relative entropy, Physics Letters A528, 130024 (2024)

    J. Xu, Quantifying the imaginarity of quantum states via tsallis relative entropy, Physics Letters A528, 130024 (2024)

    work page 2024

  33. [33]

    Du and Z

    S. Du and Z. Bai, Quantifying imaginarity in terms of pure-state imaginarity, Phys. Rev. A111, 022405 (2025)

    work page 2025

  34. [34]

    J. Liu, Y . Sun, J. Xu, and M.-J. Zhao, Imaginarity measures induced by real part states and the complementarity relations, Journal of Physics A: Mathematical and Theoretical59, 035302 (2026)

    work page 2026

  35. [35]

    Fernandes, R

    C. Fernandes, R. Wagner, L. Novo, and E. F. Galvão, Unitary- invariant witnesses of quantum imaginarity, Phys. Rev. Lett. 133, 190201 (2024)

    work page 2024

  36. [36]

    Zhang and N

    L. Zhang and N. Li, On imaginarity witnesses, Physics Letters A530, 130135 (2025)

    work page 2025

  37. [37]

    Chakrabarty, S

    S. Chakrabarty, S. Mukherjee, A. G. Maity, and B. Mallick, Detection of quantum imaginarity using moments and its inter- ferometric realization (2026), arXiv:2604.00164 [quant-ph]

    work page arXiv 2026

  38. [38]

    R. F. Werner, Quantum states with einstein-podolsky-rosen cor- relations admitting a hidden-variable model, Phys. Rev. A40, 4277 (1989)

    work page 1989

  39. [39]

    Horodecki, P

    R. Horodecki, P. Horodecki, and M. Horodecki, Violating bell inequality by mixed spin-12 states: necessary and sufficient condition, Physics Letters A200, 340 (1995)

    work page 1995

  40. [40]

    J. S. Bell, On the einstein podolsky rosen paradox, Physics Physique Fizika1, 195 (1964)

    work page 1964

  41. [41]

    Brunner, D

    N. Brunner, D. Cavalcanti, S. Pironio, V . Scarani, and S. Wehner, Bell nonlocality, Rev. Mod. Phys.86, 419 (2014)

    work page 2014

  42. [42]

    Kundu, M

    A. Kundu, M. K. Molla, I. Chattopadhyay, and D. Sarkar, Max- imal qubit violation ofn-local inequalities in a quantum net- work, Phys. Rev. A102, 052222 (2020)

    work page 2020

  43. [43]

    Kundu and D

    A. Kundu and D. Sarkar, Measurement depen- dence can affect security in a quantum net- work, Annalen der Physik536, 2400123 (2024), https://onlinelibrary.wiley.com/doi/pdf/10.1002/andp.202400123

    work page doi:10.1002/andp.202400123 2024

  44. [44]

    Kundu and D

    A. Kundu and D. Sarkar, Nature of nonlocality in triangle network based on elegant joint measure- ment, Annalen der Physik536, 2300297 (2024), https://onlinelibrary.wiley.com/doi/pdf/10.1002/andp.202300297

    work page doi:10.1002/andp.202300297 2024

  45. [45]

    Kundu and D

    A. Kundu and D. Sarkar, Influence of joint measurement bases on sharing network nonlocality, Quantum Information Process- ing24, 103 (2025)

    work page 2025

  46. [46]

    C. H. Bennett, D. P. DiVincenzo, C. A. Fuchs, T. Mor, E. Rains, P. W. Shor, J. A. Smolin, and W. K. Wootters, Quantum nonlo- cality without entanglement, Phys. Rev. A59, 1070 (1999)

    work page 1999

  47. [47]

    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

  48. [48]

    Walgate, A

    J. Walgate, A. J. Short, L. Hardy, and V . Vedral, Local dis- tinguishability of multipartite orthogonal quantum states, Phys. Rev. Lett.85, 4972 (2000)

    work page 2000

  49. [49]

    Walgate and L

    J. Walgate and L. Hardy, Nonlocality, asymmetry, and distin- guishing bipartite states, Phys. Rev. Lett.89, 147901 (2002)

    work page 2002

  50. [50]

    Horodecki, A

    M. Horodecki, A. Sen(De), U. Sen, and K. Horodecki, Local indistinguishability: More nonlocality with less entanglement, Phys. Rev. Lett.90, 047902 (2003)

    work page 2003

  51. [51]

    Ghosh, G

    S. Ghosh, G. Kar, A. Roy, and D. Sarkar, Distinguishability of maximally entangled states, Phys. Rev. A70, 022304 (2004)

    work page 2004

  52. [52]

    Xin and R

    Y . Xin and R. Duan, Local distinguishability of orthogonal 2N3 pure states, Phys. Rev. A77, 012315 (2008)

    work page 2008

  53. [53]

    Bandyopadhyay, More nonlocality with less purity, Phys

    S. Bandyopadhyay, More nonlocality with less purity, Phys. Rev. Lett.106, 210402 (2011)

    work page 2011

  54. [54]

    Banik, T

    M. Banik, T. Guha, M. Alimuddin, G. Kar, S. Halder, and S. S. Bhattacharya, Multicopy adaptive local discrimination: Strongest possible two-qubit nonlocal bases, Phys. Rev. Lett. 126, 210505 (2021)

    work page 2021

  55. [55]

    Bhunia, I

    A. Bhunia, I. Chattopadhyay, and D. Sarkar, Nonlocality of tripartite orthogonal product states, Quantum Information Pro- cessing20, 45 (2021)

    work page 2021

  56. [56]

    Bhunia, I

    A. Bhunia, I. Chattopadhyay, and D. Sarkar, Nonlocality with- out entanglement: an acyclic configuration, Quantum Informa- tion Processing21, 169 (2022)

    work page 2022

  57. [57]

    Bhunia, I

    A. Bhunia, I. Biswas, I. Chattopadhyay, and D. Sarkar, More assistance of entanglement, less rounds of classical communi- cation, Journal of Physics A: Mathematical and Theoretical56, 7 365303 (2023)

    work page 2023

  58. [58]

    S. Bera, A. Bhunia, I. Biswas, I. Chattopadhyay, and D. Sarkar, Exploring strong locality: Quantum-state-discrimination regime and beyond, Phys. Rev. A110, 042424 (2024)

    work page 2024

  59. [59]

    Bhunia, P

    A. Bhunia, P. Char, S. Bera, I. Biswas, I. Chattopadhyay, and D. Sarkar, Dilution of entanglement: Unveiling quantum state discrimination advantages (2025), arXiv:2506.18128 [quant- ph]

    work page arXiv 2025

  60. [60]

    Bhunia, S

    A. Bhunia, S. Halder, and R. Sengupta, Classifying locally dis- tinguishable sets: No activation across bipartitions, Phys. Rev. A113, 022441 (2026)

    work page 2026

  61. [61]

    B. M. Terhal, D. P. DiVincenzo, and D. W. Leung, Hiding bits in bell states, Phys. Rev. Lett.86, 5807 (2001)

    work page 2001

  62. [62]

    Eggeling and R

    T. Eggeling and R. F. Werner, Hiding classical data in multipar- tite quantum states, Phys. Rev. Lett.89, 097905 (2002)

    work page 2002

  63. [63]

    Chattopadhyay and D

    I. Chattopadhyay and D. Sarkar, Local indistinguishability and possibility of hiding cbits in activable bound entangled states, Physics Letters A365, 273 (2007)

    work page 2007

  64. [64]

    Markham and B

    D. Markham and B. C. Sanders, Graph states for quantum secret sharing, Phys. Rev. A78, 042309 (2008)

    work page 2008

  65. [65]

    Matthews, S

    W. Matthews, S. Wehner, and A. Winter, Distinguishability of quantum states under restricted families of measurements with an application to quantum data hiding, Communications in Mathematical Physics291, 813 (2009)

    work page 2009

  66. [66]

    Halder, M

    S. Halder, M. Banik, S. Agrawal, and S. Bandyopadhyay, Strong quantum nonlocality without entanglement, Phys. Rev. Lett.122, 040403 (2019)

    work page 2019

  67. [67]

    Zhang and X

    Z.-C. Zhang and X. Zhang, Strong quantum nonlocality in mul- tipartite quantum systems, Phys. Rev. A99, 062108 (2019)

    work page 2019

  68. [68]

    P. Yuan, G. Tian, and X. Sun, Strong quantum nonlocality with- out entanglement in multipartite quantum systems, Phys. Rev. A102, 042228 (2020)

    work page 2020

  69. [69]

    Bhunia, S

    A. Bhunia, S. Bera, I. Biswas, I. Chattopadhyay, and D. Sarkar, Strong quantum nonlocality: Unextendible biseparability be- yond unextendible product basis, Phys. Rev. A109, 052211 (2024)

    work page 2024

  70. [70]

    S. Bera, A. Bhunia, I. Biswas, I. Chattopadhyay, and D. Sarkar, Genuinely entangled subspaces beyond strongly nonlocal un- extendible biseparable bases (2025), arXiv:2508.17154 [quant- ph]

    work page internal anchor Pith review arXiv 2025

  71. [71]

    Agrawal, S

    S. Agrawal, S. Halder, and M. Banik, Genuinely entangled sub- space with all-encompassing distillable entanglement across ev- ery bipartition, Phys. Rev. A99, 032335 (2019)

    work page 2019

  72. [72]

    Halder and R

    S. Halder and R. Augusiak, Local unambiguous unidentifiabil- ity, entanglement generation, and hilbert space splitting (2025), arXiv:2503.05673 [quant-ph]

    work page arXiv 2025

  73. [73]

    Sengupta and A

    R. Sengupta and A. I. Singh, Useful variants and perturbations of completely entangled subspaces and spans of unextendible product bases, Infinite Dimensional Analysis, Quantum Proba- bility and Related Topics27, 2440011 (2024)

    work page 2024

  74. [74]

    Halder and A

    S. Halder and A. Streltsov, Unextendibility, uncompletability, and many-copy indistinguishable ensembles, Quantum Infor- mation and Computation24, 1081–1098 (2024)

    work page 2024

  75. [75]

    The superposition(a u j +b ¯uj )N i̸=j |ui⟩ then produces in- finitely many product vectors

    If a subspaceV⊂H= Nk i=1 Hi contain any nonzero product vector of the form Nk i=1 |ui⟩, then for anyj=1,· · ·,k, one can construct another product vector in extended Hilbertspace ¯H= (H j L ¯H j)N i̸=j Hi by replacing u j with ¯uj ∈ ¯H j. The superposition(a u j +b ¯uj )N i̸=j |ui⟩ then produces in- finitely many product vectors

    work page

  76. [76]

    K. R. Parthasarathy, On the maximal dimension of a completely entangled subspace for finite level quantum systems, Proceed- ings Mathematical Sciences114, 365 (2004)

    work page 2004

  77. [77]

    R. Duan, Y . Xin, and M. Ying, Locally indistinguishable sub- spaces spanned by three-qubit unextendible product bases, Phys. Rev. A81, 032329 (2010). APPENDIX The appendix includes explicit calculations and results that support the actual paper to be understood. I. Prelemenaries Consider a set ofnpure states{ψ i}n−1 i=0 onC dA N CdB, each of the form |ψi...

    work page 2010

  78. [78]

    Consequently, the state a0 a3 |ψ0⟩ + a1 a3 |ψ0⟩ + a2 a3 |ψ0⟩ + |φ ⟩ (52) is also a product state. This implies that the system above admits a nonzero solution of the form (a0 a3 , a1 a3 , a2 a3 ,1) t (53) Equivalently, the nonhomogeneous quadratic system Xt 3PX3 =0,∀P∈Λ A|BC ∪Λ B|CA ∪Λ C|AB (54) withX 3 = (x0,x 1,x 2,1) t, would admit a solution. Case 1:L...

    work page arXiv