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Quantum Information and Computation

Families of bipartite states classifiable by the positive partial transposition criterion

Frank E. S. Steinhoff, M. C. de Oliveira

Published May 1, 2010• 2 citations
Publisher

Abstract

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.

DOI
10.5555/2011362.2011372

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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 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 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.
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.
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 parameterized family of n ⊗ n PPT (positive partial transpose) states of corank one for each n ≥ 3. With a suitable choice of parameters, we show that they are n ⊗ n PPT entangled edge states of corank one for 3 ≤ n ≤ 1000. They violate the range criterion for separability in the most extreme way. Note that corank one is the smallest possible corank for such states. The corank of the partial transpose is given by 2n − 3, which is also the smallest possible corank for the partial transposes of PPT entangled edge states of corank one. They provide the first explicit examples of such states for n ≥ 4.
We analyze a class of entangled states for bipartite $d\ensuremath{\bigotimes}d$ systems, with ddd nonprime. The entanglement of such states is revealed by the construction of canonically associated entanglement witnesses. The structure of the states is very simple and similar to the one of isotropic states: they are a mixture of a separable and a pure entangled state whose supports are orthogonal. Despite such a simple structure, in an opportune interval of the mixing parameter their entanglement is not revealed by partial transposition. Moreover, for a restricted set of such states, we prove that there exists an interval of the mixing parameter such that both partial transposition and realignment (i.e., all permutational criteria in the bipartite setting) fail to detect them as entangled. In the range in which the states are positive under partial transposition (PPT), they are not distillable; on the other hand, the states in the considered class are provably distillable as soon as they are nonpositive under partial transposition. The states are associated to any set of more than two pure states. The analysis is extended to the multipartite setting. By an opportune selection of the set of multipartite pure states, it is possible to construct mixed states which are PPT with respect to any choice of bipartite cuts and nevertheless exhibit genuine multipartite entanglement. Finally, we show that every kkk-positive but not completely positive map is associated to a family of nondecomposable maps.