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The generalized partial transposition criterion for infinite-dimensional quantum systems

Siqing YanYu GuoJinchuan Hou

Published Jan 1, 2014• 0 citations
OpenAlex

Abstract

The positive partial transposition (PPT) criterion and the realignment criterion constitute two of the most important criteria for detecting entanglement. The general- ized partial transposition (GPT) criterion which contains the PPT criterion and the realignment criterion as special cases, provides a necessary condition for a multipartite state to be separable. Here we extend the GPT criterion to the infinite- dimensional multipartite (bipartite) case and show that it includes multipartite (bipartite) PPT criterion and multipartite (bipartite) realignment criterion as special cases as well.

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The positive partial transposition (PPT) criterion and the realignment criterion constitute two of the most important criteria for detecting entanglement. The general- ized partial transposition (GPT) criterion which contains the PPT criterion and the realignment criterion as special cases, provides a necessary condition for a multipartite state to be separable. Here we extend the GPT criterion to the infinite- dimensional multipartite (bipartite) case and show that it includes multipartite (bipartite) PPT criterion and multipartite (bipartite) realignment criterion as special cases as well.
We construct a family of bipartite states of arbitrary dimension whose eigenvalues of the partially transposed matrix can be inferred directly from the block structure of the global density matrix. We identify from this several subfamilies in which the PPT criterion is both necessary and sufficient. A sufficient criterion of separability is obtained, which is fundamental for the discussion. We show how several examples of states known to be classifiable by the PPT criterion indeed belong to this general set. Possible uses of these states in numerical analysis of entanglement and in the search of PPT bound entangled states are briefly discussed.
It is challenging task to detect and measure genuine multipartite entanglement. We investigate the problem by considering the average based positive partial transposition(PPT) criterion and the realignment criterion. Sufficient conditions for detecting genuine tripartite entanglement are presented. We also derive lower bounds for the genuine tripartite entanglement concurrence with respect to the conditions. While the PPT criterion and the realignment criterion are powerful for detecting bipartite entanglement and for providing lower bounds of bipartite concurrences, our results give an effective operational way to detect and measure the genuine tripartite entanglement.
We construct a family of bipartite states of arbitrary dimension whose eigenvalues of thepartially transposed matrix can be inferred directly from the block structure of the globaldensity matrix. We identify from this several subfamilies in which the PPT criterion isboth necessary and sufficient. A sufficient criterion of separability is obtained, which isfundamental for the discussion. We show how several examples of states known to beclassifiable by the PPT criterion indeed belong to this general set. Possible uses of thesestates in numerical analysis of entanglement and in the search of PPT bound entangledstates are briefly discussed.
We begin by describing our study of the singular value decomposition. We move on describing the separability/entanglement detection problem for a two-qubit system. After describing a few other criterions used for detection of entangled states, we divert our focus to higher dimensional systems, where we find a necessary condition on the minimum singular value for a d ⊗ d dimensional state to be separable. The condition is based on realignment criterion and structural physical approximation of partial transposition (SPA-PT) criterion. We also discuss the geometrical interpretation of the necessary condition. We have also included some examples that show the situations where other criterion might fail but our condition is useful to detect entangled states. The condition we found is useful in the fact that only the smallest singular value of the realigned matrix is required instead of all the singular values that are required in the other criterion. This makes our criterion have a small computational advantage. After this, we also discussed some ways to approximate the smallest singular value which may in turn help finding entangled states when used along with the above criterion. Index Terms—entanglement detection, quantum information theory, realignment criterion, singular value, structural physical approximation of partial transposition.
Based on the realignment moments of density matrix, we study parameterized entanglement criteria for bipartite and multipartite states. By adjusting the different parameter values, our criterion can detect not only bound entangled states, but also non-positive partial transpose entangled states for bipartite quantum systems. Moreover, we propose the definition of multipartite realignment moments and generalize the result of bipartite systems to obtain a sufficient criterion to detect entanglement for multipartite quantum states in arbitrary dimensions. And we further improve the conclusion to obtain another new entanglement criterion. The new method can detect more entangled states than previous methods as backed by detailed examples.