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Journal of Physics A Mathematical and General

Separability criterion induced from cross norm is not equivalent to positive partial transpose

Seyed Javad Akhtarshenas, M. A. Jafarizadeh

Published Jan 22, 2003• 3 citations
Publisher

Abstract

Recently, a computational criterion of separability induced by the greatest cross norm was proposed by Rudolph (2002 Preprint quant-ph/0202121). There, Rudolph conjectured that the new criterion is neither weaker nor stronger than the positive partial-transpose criterion for separability. We show that there exists a counterexample to the this claim, that is, the proposed criterion is not equivalent to the positive partial-transpose criterion.

DOI
10.1088/0305-4470/36/5/322

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Recently, a computational criterion of separability induced by the greatest cross norm was proposed by Rudolph (2002 Preprint quant-ph/0202121). There, Rudolph conjectured that the new criterion is neither weaker nor stronger than the positive partial-transpose criterion for separability. We show that there exists a counterexample to the this claim, that is, the proposed criterion is not equivalent to the positive partial-transpose criterion.
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.
We present a framework for deciding whether a quantum state is separable or entangled using covariance matrices of locally measurable observables. This leads to the covariance matrix criterion as a general separability criterion. We demonstrate that this criterion allows us to detect many states where the familiar criterion of the positivity of the partial transpose fails. It turns out that a large number of criteria that have been proposed to complement the positive partial transpose criterion---the computable cross norm or realignment criterion, the criterion based on local uncertainty relations, criteria derived from extensions of the realignment map, and others---are in fact a corollary of the covariance matrix criterion.
A block matrix [AXX∗B]\left[ \begin{smallmatrix}A & X \\{{X}^{*}} & B \\\end{smallmatrix} \right][AX∗​XB​] is positive partial transpose (PPT) if both [AXX∗B]\left[ \begin{smallmatrix}A & X \\{{X}^{*}} & B \\\end{smallmatrix} \right][AX∗​XB​] and [AX∗XB]\left[ \begin{smallmatrix}A & {{X}^{*}} \\X & B \\\end{smallmatrix} \right][AX​X∗B​] are positive semi-definite. This class is significant in studying the separability criterion for density matrices. The current paper presents new relations for such matrices. This includes some equivalent forms and new related inequalities that extend some results from the literature. In the end of the paper, we present some related results for positive semi-definite block matrices, which have similar forms as those presented for PPT matrices, with applications that include significant improvement of numerical radius inequalities.
Recently, a new and powerful separability criterion was introduced in [O. Rudolph, quant-ph/0202121] and [Chen {\it et al.}, quant-ph/0205017]. Composing the main idea behind the above criterion and the necessary and sufficient condition in terms of positive maps, we provide a characterization of separable states by means of linear contractions. The latter need not be positive maps. We extend the idea to multipartite systems, and find that, somewhat suprisingly, partial realigment (unlike partial transposition) can detect genuinely triparite entanglement. We also show that for multipartite system any permutation of indices of density matrix leads to separability criterion.