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Journal of Physics A Mathematical and General

On the cross norm criterion for separability

Oliver Rudolph

Published May 12, 2003• 36 citations
Publisher

Abstract

The computable cross norm (CCN) criterion for separability is neither weaker nor stronger than the positive partial transpose (PPT) criterion.
DOI
10.1088/0305-4470/36/21/311

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So far the positive partial transpose (PPT) criterion and the computable cross norm (CCN) criterion are two strongest operational separability criteria of bipartite mixed quantum states. In this paper, we exactly connect the PPT criterion and the CCN criterion with the matrix reorderings found by Poluikis and Hill [J. A. Poluikis and R. D. Hill, Linear Algebra and Its Applications 35: 1-10 (1981)] and Oxenrider and Hill [C. J. Oxenrider and R. D. Hill, Linear Algebra and Its Applications 69: 205-212 (1985)]. Motivated by work by Chen etc. [K. Chen, S. Albeverio, and S.-M. Fei, Physical Review Letters 95, 040504 (2005), Physical Review Letters 95, 210501 (2005)], we compute the trace norm of several bound entangled states and illuminate the equivalence among the elements of group generated by the eight matrix reorderings.

IEEE Conference on Cybernetics and Intelligent Systems•Sep 1, 2008•0 citations
The computable cross-norm (CCN) criterion is a powerful analytical and computable separability criterion for bipartite quantum states, which is also known to systematically detect bound entanglement. In certain aspects this criterion complements the well-known Peres positive partial transpose (PPT) criterion. In the present paper we study important analytical properties of the CCN criterion. We show that in contrast to the PPT criterion it is not sufficient in dimension $2\ifmmode\times\else\texttimes\fi{}2.$ In higher dimensions, theorems connecting the fidelity of a quantum state with the CCN criterion are proved. We also analyze the behavior of the CCN criterion under local operations and identify the operations that leave it invariant. It turns out that the CCN criterion is in general not invariant under local operations.
So far the positive partial transpose (PPT) criterion and the computable cross norm (CCN) criterion are two strongest operational separability criteria of bipartite mixed quantum states. In this paper, we exactly connect the PPT criterion and the CCN criterion with the matrix reorderings found by Poluikis and Hill [J. A. Poluikis and R. D. Hill, Linear Algebra and Its Applications 35: 1-10 (1981)] and Oxenrider and Hill [C. J. Oxenrider and R. D. Hill, Linear Algebra and Its Applications 69: 205-212 (1985)]. Motivated by work by Chen etc. [K. Chen, S. Albeverio, and S.-M. Fei, Physical Review Letters 95, 040504 (2005), Physical Review Letters 95, 210501 (2005)], we compute the trace norm of several bound entangled states and illuminate the equivalence among the elements of group generated by the eight matrix reorderings.