Quantum correlations on the no-signaling boundary: self-testing and more

Kai-Siang Chen; Gelo Noel M. Tabia; Chellasamy Jebarathinam; Shiladitya Mal; Jun-Yi Wu; Yeong-Cherng Liang

doi:10.22331/q-2023-07-11-1054

Quantum correlations on the no-signaling boundary: self-testing and more

Kai-Siang Chen¹, Gelo Noel M. Tabia^1,2,3, Chellasamy Jebarathinam^4,5, Shiladitya Mal^1,2, Jun-Yi Wu⁶, and Yeong-Cherng Liang^1,2

¹Department of Physics and Center for Quantum Frontiers of Research & Technology (QFort), National Cheng Kung University, Tainan 701, Taiwan
²Physics Division, National Center for Theoretical Sciences, Taipei 10617, Taiwan
³Center for Quantum Technology, National Tsing Hua University, Hsinchu 300, Taiwan
⁴Center for Theoretical Physics, Polish Academy of Sciences, Aleja Lotników 32/46, 02-668 Warsaw, Poland
⁵Department of Physics and Center for Quantum Information Science, National Cheng Kung University, Tainan 70101, Taiwan
⁶Department of Physics, Tamkang University, Tamsui, New Taipei 251301, Taiwan

Published:	2023-07-11, volume 7, page 1054
Eprint:	arXiv:2207.13850v3
Doi:	https://doi.org/10.22331/q-2023-07-11-1054
Citation:	Quantum 7, 1054 (2023).

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Abstract

In device-independent quantum information, correlations between local measurement outcomes observed by spatially separated parties in a Bell test play a fundamental role. Even though it is long-known that the set of correlations allowed in quantum theory lies strictly between the Bell-local set and the no-signaling set, many questions concerning the geometry of the quantum set remain unanswered. Here, we revisit the problem of when the boundary of the quantum set coincides with the no-signaling set in the simplest Bell scenario. In particular, for each Class of these common boundaries containing $k$ zero probabilities, we provide a $(5-k)$-parameter family of quantum strategies realizing these (extremal) correlations. We further prove that self-testing is possible in all nontrivial Classes beyond the known examples of Hardy-type correlations, and provide numerical evidence supporting the robustness of these self-testing results. Candidates of one-parameter families of self-testing correlations from some of these Classes are identified. As a byproduct of our investigation, if the qubit strategies leading to an extremal nonlocal correlation are local-unitarily equivalent, a self-testing statement provably follows. Interestingly, all these self-testing correlations found on the no-signaling boundary are provably non-exposed. An analogous characterization for the set $\mathcal{M}$ of quantum correlations arising from finite-dimensional maximally entangled states is also provided. En route to establishing this last result, we show that all correlations of $\mathcal{M}$ in the simplest Bell scenario are attainable as convex combinations of those achievable using a Bell pair and projective measurements. In turn, we obtain the maximal Clauser-Horne-Shimony-Holt Bell inequality violation by any maximally entangled two-qudit state and a no-go theorem regarding the self-testing of such states.

Featured image: A two-dimensional projection of the no-signaling set $\mathcal{NS}$, an outer approximation $\tilde{Q}$ of the quantum set, and the local set $\mathcal{L}$ of correlations. All (quantum) correlations having the three zero entries $P(0,0|0,0)=P(1,1|0,0)=P(1,0|1,1)=0$ lie on the vertical line where the horizontal value is zero. Among them, $\vec{P}_{\mathcal{Q}}$ gives the maximal $\mathcal{S}_{\text{CHSH}}$ value and self-tests a quantum strategy.

► BibTeX data

@article{Chen2023quantumcorrelations,
  doi = {10.22331/q-2023-07-11-1054},
  url = {https://doi.org/10.22331/q-2023-07-11-1054},
  title = {Quantum correlations on the no-signaling boundary: self-testing and more},
  author = {Chen, Kai-Siang and Tabia, Gelo Noel M. and Chellasamy Jebarathinam and Mal, Shiladitya and Wu, Jun-Yi and Liang, Yeong-Cherng},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {7},
  pages = {1054},
  month = jul,
  year = {2023}
}

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[2] Michael Janas and Michel Janssen, "Broken arrows: Hardy-Unruh chains and quantum contextuality", arXiv:2308.14151, (2023).

[3] Antoni Mikos-Nuszkiewicz and Jędrzej Kaniewski, "Extremal points of the quantum set in the CHSH scenario: conjectured analytical solution", arXiv:2302.10658, (2023).

[4] Yuan Liu, Ho Yiu Chung, and Ravishankar Ramanathan, "Investigations of the boundary of quantum correlations and device-independent applications", arXiv:2309.06304, (2023).

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