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Reduction criterion for separability

Cerf, N. J. and Adami, C. and Gingrich, R. M. (1999) Reduction criterion for separability. Physical Review A, 60 (2). pp. 898-909. ISSN 1050-2947. doi:10.1103/PhysRevA.60.898.

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We introduce a separability criterion based on the positive map Γ:ρ→(Tr ρ)-ρ, where ρ is a trace-class Hermitian operator. Any separable state is mapped by the tensor product of Γ and the identity into a non-negative operator, which provides a simple necessary condition for separability. This condition is generally not sufficient because it is vulnerable to the dilution of entanglement. In the special case where one subsystem is a quantum bit, Γ reduces to time reversal, so that this separability condition is equivalent to partial transposition. It is therefore also sufficient for 2×2 and 2×3 systems. Finally, a simple connection between this map for two qubits and complex conjugation in the “magic” basis [Phys. Rev. Lett. 78, 5022 (1997)] is displayed.

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Adami, C.0000-0002-2915-9504
Additional Information:©1999 The American Physical Society Received 31 October 1997; revised 14 December 1998 We acknowledge useful discussions with Michal Horodecki. We are also grateful to Chris Fuchs for communicating to us unpublished results of Ref. [13], especially the connection between the map M and the "magic" basis for two qubits. This work was supported in part by NSF Grant Nos. PHY 94-12818 and PHY 94-20470, and by a grant from DARPA/ARO through the QUIC Program (No. DAAH04-96-1-3086).
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Deposited On:25 May 2006
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