Graph-Theoretic Approach for Self-Testing in Bell Scenarios

Bharti，Kishor1; Ray，Maharshi2; Xu，Zhen Peng3; Hayashi，Masahito4,5,6,7; Kwek，Leong Chuan1,8,9,10; Cabello，Adán11,12

2022-07-01

10.1103/PRXQuantum.3.030344

PRX Quantum   Impact Factor & Quartile

2691-3399

Self-testing is a technology to certify states and measurements using only the statistics of the experiment. Self-testing is possible if some extremal points in the set BQ of quantum correlations for a Bell experiment are achieved, up to isometries, with specific states and measurements. However, BQ is difficult to characterize, so it is also difficult to prove whether or not a given matrix of quantum correlations allows for self-testing. Here, we show how some tools from graph theory can help to address this problem. We observe that BQ is strictly contained in an easy-to-characterize set associated with a graph, Θ(G). Therefore, whenever the optimum over BQ and the optimum over Θ(G) coincide, self-testing can be demonstrated by simply proving self-testability with Θ(G). Interestingly, these maxima coincide for the quantum correlations that maximally violate many families of Bell-like inequalities. Therefore, we can apply this approach to prove the self-testability of many quantum correlations, including some that are not previously known to allow for self-testing. In addition, this approach connects self-testing to some open problems in discrete mathematics. We use this connection to prove a conjecture [M. Araújo et al., Phys. Rev. A, 88, 022118 (2013)] about the closed-form expression of the Lovász theta number for a family of graphs called the Möbius ladders. Although there are a few remaining issues (e.g., in some cases, the proof requires the assumption that measurements are of rank 1), this approach provides an alternative method to self-testing and draws interesting connections between quantum mechanics and discrete mathematics.

SCI

English

Others

Physics

Quantum Science & Technology ; Physics, Applied ; Physics, Multidisciplinary

WOS:000869014900001

AMER PHYSICAL SOC

2-s2.0-85139213505

Scopus

1.Centre for Quantum Technologies,National University of Singapore,Singapore,117543,Singapore
2.Department of Information Engineering,Graduate School of Engineering,Mie University,Mie,Tsu,514-8507,Japan
3.Naturwissenschaftlich-Technische Fakultät,Universität Siegen,Siegen,Walter-Flex-Straße 3,57068,Germany
4.Shenzhen Institute for Quantum Science and Engineering,Southern University of Science and Technology,Shenzhen,518055,China
5.International Quantum Academy (SIQA),Shenzhen,518048,China
6.Guangdong Provincial Key Laboratory of Quantum Science and Engineering,Southern University of Science and Technology,Shenzhen,518055,China
7.Graduate School of Mathematics,Nagoya University,Nagoya,464-8602,Japan
8.MajuLab,CNRS-UNS-NUS-NTU International Joint Research Unit,Singapore UMI 3654,Singapore,Singapore
9.National Institute of Education,Nanyang Technological University,Singapore,637616,Singapore
10.Quantum Science and Engineering Centre (QSec),Nanyang Technological University,Singapore,Singapore
11.Departamento de Física Aplicada II,Universidad de Sevilla,Sevilla,E-41012,Spain
12.Instituto Carlos i de Física Teórica y Computacional,Universidad de Sevilla,Sevilla,E-41012,Spain

Bharti，Kishor,Ray，Maharshi,Xu，Zhen Peng,et al. Graph-Theoretic Approach for Self-Testing in Bell Scenarios[J]. PRX Quantum,2022,3(3).

Bharti，Kishor,Ray，Maharshi,Xu，Zhen Peng,Hayashi，Masahito,Kwek，Leong Chuan,&Cabello，Adán.(2022).Graph-Theoretic Approach for Self-Testing in Bell Scenarios.PRX Quantum,3(3).

Bharti，Kishor,et al."Graph-Theoretic Approach for Self-Testing in Bell Scenarios".PRX Quantum 3.3(2022).