bibkey: murshidguptarussobandyopadhyay2026nonlocality
authors: Shayeef Murshid; Tathagata Gupta; Vincent Russo; Somshubhro Bandyopadhyay
year: 2026
title: "Quantum nonlocality without entanglement and state discrimination measures"
doi: 10.22331/q-2026-07-23-2174
url: https://arxiv.org/abs/2506.20560v3
claim: "The paper constructs product ensembles that LOCC discriminates optimally for unambiguous but not for minimum-error discrimination, and conjectures for the no-repetition ensembles E_{N,k} that the local and global unambiguous optima coincide with value (1-s)^k for k < N and (1-s)^(N-1)(1+(N-1)s) for k = N."
strata_touched:
  - D5/S3/Quantum/PureState/NoRepetitionSequenceUnambiguousDiscrimination
license: citation-only
triage: anchor

Quantum nonlocality without entanglement and state discrimination measures

S. Murshid, T. Gupta, V. Russo and S. Bandyopadhyay, arXiv:2506.20560v3 [quant-ph]; Quantum 10, 2174 (2026). Quotations are from the v3 TeX source.

The global and local optima:

Let $f_{{}_G}\left({\text{E}}_{\psi }\right)$ and $f_{{}_L}\left({\text{E}}_{\psi }\right)$ be the respective global and local optimum, defined as $f_{{}_G}\left({\text{E}}_{\psi }\right)={\sup }_{\text{M}}f\left({\text{E}}_{\psi };\text{M}\right)$ and $f_{{}_L}\left({\text{E}}_{\psi }\right)={\sup }_{\text{M}\in \text{LOCC}}f\left({\text{E}}_{\psi };\text{M}\right).$

The states and the ensemble:

Consider a set of $N$ linearly independent states in ${\mathbb{C}}^N$ with equal and real pairwise overlaps, ${\S }_N=\left\{|{\psi }_i⟩\in {\mathbb{C}}^N:i=1,2,\dots ,N;⟨{\psi }_i|{\psi }_j⟩=s\in (0,1)\text{for }i\ne j\right\}$.

${\mathcal{T}}_{N,k}=\left\{|{\psi }_{i_1}⟩\otimes |{\psi }_{i_2}⟩\otimes \cdots \otimes |{\psi }_{i_k}⟩:i_1,i_2,\dots ,i_k\in \{1,\dots ,N\};i_a\ne i_b\forall a\ne b\right\}$

${\mathcal{E}}_{N,k}=\left\{\left(\frac{1}{{}^N{P}_k},|{\psi }_{i_1}⟩\otimes \cdots \otimes |{\psi }_{i_k}⟩\right):|{\psi }_{i_j}⟩\in {\S }_N\right\}$

The conjecture:

For the family of ensembles of multipartite pure product states ${\text{E}}_{N,k}$ given by Equation \eqref{eq:gensemble} the following holds for all $N\ge 3$ and $2\le k\le N$

followed by the displays $p_{{}_L}^{me}\left({\text{E}}_{N,k}\right)<p_{{}_G}^{me}\left({\text{E}}_{N,k}\right)$ and $p_{{}_L}^{ud}\left({\text{E}}_{N,k}\right)=p_{{}_G}^{ud}\left({\text{E}}_{N,k}\right)$, and

Moreover, the optimal unambiguous discrimination probability of ${\text{E}}_{N,k}$ is given by

followed by the display $p_{{}_G}^{ud}\left({\text{E}}_{N,k}\right)=(1-s{)}^k$ when $k<N$ and $(1-s{)}^{N-1}(1+\left(N-1\right)s)$ when $k=N$ (a cases environment in the source).

(the displayed formulas are written inline here).

Verified locator

Source: https://arxiv.org/abs/2506.20560v3

• URL: https://arxiv.org/abs/2506.20560v3 (source revised_02.tex, md5 1f552f21910a72e61d78246fe6c76ee9, 1571 lines):
  • the global and local optima, l. 208–221;
  • unambiguous discrimination, l. 649–672;
  • the states, products and ensemble, l. 822–840;
  • the conjecture, l. 970–985.
• DOI: 10.22331/q-2026-07-23-2174 (https://doi.org/10.22331/q-2026-07-23-2174; Crossref lists Quantum, volume 10, article 2174, published 23 July 2026).