Covariance matrix entanglement criterion for an arbitrary set of operators
A criterion for entanglement detection based on covariance matrices for an arbitrary set of observables is formulated. The method generalizes the covariance matrix entanglement criterion by Simon (2000 Phys. Rev. Lett. 84 2726) to a more general set of operators using the positive partial transpose...
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IOP Publishing
2020-01-01
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| Series: | New Journal of Physics |
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| Online Access: | https://doi.org/10.1088/1367-2630/ab9ce7 |
| _version_ | 1797750395479523328 |
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| author | Vinay Tripathi Chandrashekar Radhakrishnan Tim Byrnes |
| author_facet | Vinay Tripathi Chandrashekar Radhakrishnan Tim Byrnes |
| author_sort | Vinay Tripathi |
| collection | DOAJ |
| description | A criterion for entanglement detection based on covariance matrices for an arbitrary set of observables is formulated. The method generalizes the covariance matrix entanglement criterion by Simon (2000 Phys. Rev. Lett. 84 2726) to a more general set of operators using the positive partial transpose test for the covariance matrix. The relation is found by starting from the generalized uncertainty relation for multiple operators, and taking the partial transpose on the bipartition. The method is highly efficient and versatile in the sense that the set of measurement operators can be freely chosen, and there is no constraint on the commutation relations. The main restriction on the chosen set of measurement operators is that the correlators and expectation values of the partially transposed observable operators can be evaluated. The method is particularly suited for systems with higher dimensionality since the computations do not scale with the dimension of the Hilbert space—rather they scale with the number of chosen observables. We illustrate the approach by examining the entanglement between two spin ensembles, and show that it detects entanglement in a basis independent way. |
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| format | Article |
| id | doaj.art-1ac01496927d45f2ae4ae7e49d3055cd |
| institution | Directory Open Access Journal |
| issn | 1367-2630 |
| language | English |
| last_indexed | 2024-03-12T16:32:09Z |
| publishDate | 2020-01-01 |
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| series | New Journal of Physics |
| spelling | doaj.art-1ac01496927d45f2ae4ae7e49d3055cd2023-08-08T15:24:52ZengIOP PublishingNew Journal of Physics1367-26302020-01-0122707305510.1088/1367-2630/ab9ce7Covariance matrix entanglement criterion for an arbitrary set of operatorsVinay Tripathi0Chandrashekar Radhakrishnan1Tim Byrnes2State Key Laboratory of Precision Spectroscopy, School of Physical and Material Sciences, East China Normal University , Shanghai 200062, People’s Republic of China; New York University Shanghai , 1555 Century Ave, Pudong, Shanghai 200122, People’s Republic of China; Department of Electrical and Computer Engineering, University of California , Riverside, CA 92521, United States of AmericaNew York University Shanghai , 1555 Century Ave, Pudong, Shanghai 200122, People’s Republic of China; NYU-ECNU Institute of Physics at NYU Shanghai , 3663 Zhongshan Road North, Shanghai 200062, People’s Republic of ChinaState Key Laboratory of Precision Spectroscopy, School of Physical and Material Sciences, East China Normal University , Shanghai 200062, People’s Republic of China; New York University Shanghai , 1555 Century Ave, Pudong, Shanghai 200122, People’s Republic of China; NYU-ECNU Institute of Physics at NYU Shanghai , 3663 Zhongshan Road North, Shanghai 200062, People’s Republic of China; National Institute of Informatics , 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan; Department of Physics, New York University , New York, NY 10003, United States of AmericaA criterion for entanglement detection based on covariance matrices for an arbitrary set of observables is formulated. The method generalizes the covariance matrix entanglement criterion by Simon (2000 Phys. Rev. Lett. 84 2726) to a more general set of operators using the positive partial transpose test for the covariance matrix. The relation is found by starting from the generalized uncertainty relation for multiple operators, and taking the partial transpose on the bipartition. The method is highly efficient and versatile in the sense that the set of measurement operators can be freely chosen, and there is no constraint on the commutation relations. The main restriction on the chosen set of measurement operators is that the correlators and expectation values of the partially transposed observable operators can be evaluated. The method is particularly suited for systems with higher dimensionality since the computations do not scale with the dimension of the Hilbert space—rather they scale with the number of chosen observables. We illustrate the approach by examining the entanglement between two spin ensembles, and show that it detects entanglement in a basis independent way.https://doi.org/10.1088/1367-2630/ab9ce7entanglement criterionquantum informationcovariance matrix |
| spellingShingle | Vinay Tripathi Chandrashekar Radhakrishnan Tim Byrnes Covariance matrix entanglement criterion for an arbitrary set of operators New Journal of Physics entanglement criterion quantum information covariance matrix |
| title | Covariance matrix entanglement criterion for an arbitrary set of operators |
| title_full | Covariance matrix entanglement criterion for an arbitrary set of operators |
| title_fullStr | Covariance matrix entanglement criterion for an arbitrary set of operators |
| title_full_unstemmed | Covariance matrix entanglement criterion for an arbitrary set of operators |
| title_short | Covariance matrix entanglement criterion for an arbitrary set of operators |
| title_sort | covariance matrix entanglement criterion for an arbitrary set of operators |
| topic | entanglement criterion quantum information covariance matrix |
| url | https://doi.org/10.1088/1367-2630/ab9ce7 |
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