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IEEE Conference on Cybernetics and Intelligent Systems

Operational separability criteria and matrix reorderings

Wei Cheng

Published Sep 1, 2008• 0 citations
Publisher

Abstract

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.

DOI
10.1109/iccis.2008.4670886

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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.
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.
The correlation matrix (CM) criterion is a recently derived powerful sufficient condition for the presence of entanglement in bipartite quantum states of arbitrary dimensions. It has been shown that it can be stronger than the positive partial transpose (PPT) criterion, as well as the computable cross norm or realignment (CCNR) criterion in different situations. However, it remained as an open question whether there existed sets of states for which the CM criterion could be stronger than both criteria simultaneously. Here, we give an affirmative answer to this question by providing examples of entangled states that scape detection by both the PPT and CCNR criteria whose entanglement is revealed by the CM condition. We also show that the CM can be used to measure the entanglement of pure states and obtain lower bounds for the entanglement measure known as tangle for general (mixed) states.
Given two bipartite quantum states and the convex linear combination of them, we discuss the relation between the entanglement of the convex linear combination state and the entanglement of states being combined. This is achieved by characterizing quantum states quantitatively via the positive partial transpose (PPT) criterion and the computable cross-norm or realignment (CCNR) criterion. Inspired by the Horodecki's 3 ⊗ 3 quantum states, we also give explicit examples to illustrate all possible cases of convex linear combination. Finally, as an application of this method, we show how to construct new bipartite PPT entangled states from known PPT entangled states by convex linear combination.
Recently, a powerful separability criterion was introduced by O. Rudolf in [5] and by K. Chen et al. in [6] — basing on realignment of elements of density matrix. 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 tri-parite entanglement. We generalize it by introducing a family of so called permutation separability criteria for multipartite states. Namely, any permutation of indices of density matrix written in product basis leads to a separability criterion. Partial transpose and realignment criterion are special cases of permutation criteria.
It is well known that the positive partial transpose (PPT) criterion for determining separability is an operational separability criterion. However, in a high-dimensional (>6) situation, this criterion is not sufficient. How can we judge whether an entangled state in any high-dimensional quantum system is PPT or not? Here we propose a linear algebra method for checking the positivity of the partial transpose of a state and present a set of entangled mixed states with non-positive partial transpose.
Entanglement is a key feature in many quantum technologies, including secure communication protocols and quantum computing. However, detecting it in mixed quantum states remains a challenging task. While the positive partial transposition (PPT) and computable cross-norm and realignment criteria are well-established tools for entanglement detection in general, and are especially effective in Bell-diagonal states, their connection to the underlying group structure of this state family has not been fully explored. In this work, we analyze the PPT and realignment criteria for Bell-diagonal states from a group-theoretic point of view. Our results demonstrate that the group structure of Bell-diagonal states provides a clear framework for analyzing and computing these two entanglement-detection criteria, thereby highlighting the connection between entanglement and group structure. This unified perspective offers insights into the mathematical and physical properties of entanglement in structured quantum systems and ties the PPT and realignment criteria for Bell-diagonal states to experimental procedures.