Graph-Theoretic Approach for Self-Testing in Bell Scenarios
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 B_{Q} of quantum correlations for a Bell experiment are achieved, up to isometries, with specific states and measurements. However,...
| Main Authors: | , , , , , |
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| Format: | Article |
| Language: | English |
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American Physical Society
2022-09-01
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| Series: | PRX Quantum |
| Online Access: | http://doi.org/10.1103/PRXQuantum.3.030344 |
| _version_ | 1798001708547178496 |
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| author | Kishor Bharti Maharshi Ray Zhen-Peng Xu Masahito Hayashi Leong-Chuan Kwek Adán Cabello |
| author_facet | Kishor Bharti Maharshi Ray Zhen-Peng Xu Masahito Hayashi Leong-Chuan Kwek Adán Cabello |
| author_sort | Kishor Bharti |
| collection | DOAJ |
| description | 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 B_{Q} of quantum correlations for a Bell experiment are achieved, up to isometries, with specific states and measurements. However, B_{Q} 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 B_{Q} is strictly contained in an easy-to-characterize set associated with a graph, Θ(G). Therefore, whenever the optimum over B_{Q} 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. |
| first_indexed | 2024-04-11T11:40:35Z |
| format | Article |
| id | doaj.art-e5a9d3b5b1454c7e9a22d2cd79165cea |
| institution | Directory Open Access Journal |
| issn | 2691-3399 |
| language | English |
| last_indexed | 2024-04-11T11:40:35Z |
| publishDate | 2022-09-01 |
| publisher | American Physical Society |
| record_format | Article |
| series | PRX Quantum |
| spelling | doaj.art-e5a9d3b5b1454c7e9a22d2cd79165cea2022-12-22T04:25:50ZengAmerican Physical SocietyPRX Quantum2691-33992022-09-013303034410.1103/PRXQuantum.3.030344Graph-Theoretic Approach for Self-Testing in Bell ScenariosKishor BhartiMaharshi RayZhen-Peng XuMasahito HayashiLeong-Chuan KwekAdán CabelloSelf-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 B_{Q} of quantum correlations for a Bell experiment are achieved, up to isometries, with specific states and measurements. However, B_{Q} 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 B_{Q} is strictly contained in an easy-to-characterize set associated with a graph, Θ(G). Therefore, whenever the optimum over B_{Q} 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.http://doi.org/10.1103/PRXQuantum.3.030344 |
| spellingShingle | Kishor Bharti Maharshi Ray Zhen-Peng Xu Masahito Hayashi Leong-Chuan Kwek Adán Cabello Graph-Theoretic Approach for Self-Testing in Bell Scenarios PRX Quantum |
| title | Graph-Theoretic Approach for Self-Testing in Bell Scenarios |
| title_full | Graph-Theoretic Approach for Self-Testing in Bell Scenarios |
| title_fullStr | Graph-Theoretic Approach for Self-Testing in Bell Scenarios |
| title_full_unstemmed | Graph-Theoretic Approach for Self-Testing in Bell Scenarios |
| title_short | Graph-Theoretic Approach for Self-Testing in Bell Scenarios |
| title_sort | graph theoretic approach for self testing in bell scenarios |
| url | http://doi.org/10.1103/PRXQuantum.3.030344 |
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