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Abstract
The convex set of quantum states of a composite K×K system with positive partial transpose is analysed. A version of the hit and run algorithm is used to generate a sequence of random points covering this set uniformly and an estimation for the convergence speed of the algorithm is derived. For K >3 or K=3 this algorithm works faster than sampling over the entire set of states and verifying whether the partial transpose is positive. The level density of the PPT states is shown to differ from the Marchenko–Pastur distribution, supported in [0, 4] and corresponding asymptotically to the entire set of quantum states. Based on the shifted semi–circle law, describing asymptotic level density of partially transposed states, and on the level density for the Gaussian unitary ensemble with constraints for the spectrum we find an explicit form of the probability distribution supported in [0, 3], which describes well the level density obtained numerically for PPT states.
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Authors (4)
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Konrad Szymański
- Uniwersytet Jagielloński, Instytut Fizyki
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Benot Collins
- Kyoto University, Graduate School of Science Department of Mathematics
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Karol Życzkowski
- Polska Akademia Nauk, Centrum Fizyki Teoretycznej
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- Category:
- Articles
- Type:
- artykuł w czasopiśmie wyróżnionym w JCR
- Published in:
-
Journal of Physics A-Mathematical and Theoretical
no. 50,
pages 255206 - 255217,
ISSN: 1751-8113 - Language:
- English
- Publication year:
- 2017
- Bibliographic description:
- Szymański K., Collins B., Szarek T., Życzkowski K.: Convex set of quantum states with positive partial transpose analysed by hit and run algorithm// Journal of Physics A-Mathematical and Theoretical. -Vol. 50, nr. 25 (2017), s.255206-255217
- DOI:
- Digital Object Identifier (open in new tab) 10.1088/1751-8121/aa70f5
- Verified by:
- Gdańsk University of Technology
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