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Approximating the set of separable states using the positive partial transpose test

Beigi, Salman and Shor, Peter W. (2010) Approximating the set of separable states using the positive partial transpose test. Journal of Mathematical Physics, 51 (4). Art. No. 042202. ISSN 0022-2488.

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The positive partial transpose test is one of the main criteria for detecting entanglement, and the set of states with positive partial transpose is considered as an approximation of the set of separable states. However, we do not know to what extent this criterion, as well as the approximation, is efficient. In this paper, we show that the positive partial transpose test gives no bound on the distance of a density matrix from separable states. More precisely, we prove that, as the dimension of the space tends to infinity, the maximum trace distance of a positive partial transpose state from separable states tends to 1. Using similar techniques, we show that the same result holds for other well-known separability criteria such as reduction criterion, majorization criterion, and symmetric extension criterion. We also bring in evidence that the sets of positive partial transpose states and separable states have totally different shapes.

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Additional Information:© 2010 American Institute of Physics. Received 9 November 2009; accepted 22 February 2010; published online 27 April 2010. Authors are grateful to Karol Życzkowski and Stanisław J. Szarek for providing some background about the comparison of the volume of the sets of separable states and PPT states. S.B. is also thankful to Barbara Terhal for useful discussions.
Subject Keywords:quantum entanglement
Issue or Number:4
Classification Code:PACS: 03.67.Bg; 03.65.Ud
Record Number:CaltechAUTHORS:20100524-134930766
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:18407
Deposited By: Ruth Sustaita
Deposited On:25 May 2010 15:45
Last Modified:03 Oct 2019 01:42

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