bibkey: yadavalliandrejickunjwal2024qualitative
authors: Shiv Akshar Yadavalli; Nikola Andrejic; Ravi Kunjwal
year: 2024
title: "Qualitative equivalence between incompatibility and Bell nonlocality"
doi: 10.1103/PhysRevA.110.L060201
url: https://arxiv.org/abs/2008.10100v3
claim: "The paper shows that every non-trivial joint measurability structure with finitely many vertices has a quantum realization by measurements that enable a Bell violation, so a non-trivial structure is necessary and sufficient for a Bell violation. Its Supplemental Material studies trine qubit measurements with a purity parameter and a common bias. For the first family it states the pairwise joint-measurability condition from earlier work, computes the triplewise threshold numerically by a semidefinite program, and conjectures that threshold to be the piecewise function 1+b, 2/3, 1-b."
strata_touched:
  - D5/S3/Quantum/Measurement/BiasedTrineJointMeasurabilityThreshold
license: citation-only
triage: anchor

Yadavalli, Andrejic and Kunjwal, incompatibility and Bell nonlocality

S. A. Yadavalli, N. Andrejic and R. Kunjwal, arXiv:2008.10100v3 [quant-ph]; Phys. Rev. A 110, L060201 (2024). Quotations are from the v3 Supplemental Material TeX.

The trine directions are ${\vec{n}}_k=\cos \frac{2k\pi }{3}{\vec{e}}_x+\sin \frac{2k\pi }{3}{\vec{e}}_z$, $k\in \{1,2,3\}$. With a global purity parameter $\eta \in [0,1]$ and a bias $b\in [-1,1]$ such that $|b|\le 1-\eta$, the first family is

$$E_k(x_k)=\frac{1}{2}\left((1+x_{k}b)I+\eta x_k{\vec{n}}_k\cdot \vec{\sigma }\right),x_k\in \{-1,+1\}.$$

Citing earlier work, the source states that the set is pairwise jointly measurable if and only if $\eta \le 1+b$ for $b<-\frac{1}{3}$, $\eta \le \sqrt{3-2b^2}-1$ for $|b|\le \frac{1}{3}$, and $\eta \le 1-b$ for $b>\frac{1}{3}$. The triplewise threshold ${\eta }_1^{(3)}(b)$, defined by “the set is triplewise jointly measurable iff $\eta \le {\eta }_1^{(3)}(b)$”, is obtained by a semidefinite program on a discretization of $b$. The authors write:

Based on our numerical evaluation and visual inspection, we conjecture the exact form of ${\eta }_1^{(3)}(b)$ to be $${\eta }_1^{(3)}(b)=\{\begin{array}{ll}1+b,& -1\le b<-\frac{1}{3},\\ \frac{2}{3},& b\in \left[-\frac{1}{3},\frac{1}{3}\right],\\ 1-b,& \frac{1}{3}<b\le 1.\end{array}$$

Verified locator

Source: https://arxiv.org/abs/2008.10100v3

• DOI: https://doi.org/10.1103/PhysRevA.110.L060201 (Phys. Rev. A 110, L060201 (2024); author list, journal, volume, issue and article number from Crossref).
• URL: https://arxiv.org/abs/2008.10100v3. The e-print is a gzipped tar archive; its member qualequiv_supp_arxiv.tex has 258 lines and md5 99ad6f21ee4fb05ae64c577ed2d1efef. Locations:
  • the trine directions, the domain and the two biased families, l. 112–121;
  • the pairwise condition, the triplewise threshold definition and the conjecture (Example 1), l. 133–164.
• The arXiv versions (v1 23 August 2020, v3 20 November 2024) are from the arXiv metadata.