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arXiv:0709.0027 · Physical Review A

Robustness of entangled states that are positive under partial transposition

Somshubhro Bandyopadhyay, Sibasish Ghosh A, Vwani Roychowdhury

Published Mar 12, 2008• 15 citations
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Abstract

We study robustness of bipartite entangled states that are positive under partial transposition (PPT). It is shown that almost all PPT entangled states are unconditionally robust, in the sense, both inseparability and positivity are preserved under sufficiently small perturbations in its immediate neighborhood. Such unconditionally robust PPT entangled states lie inside an open PPT entangled ball. We construct examples of such balls whose radii are shown to be finite and can be explicitly calculated. This provides a lower bound on the volume of all PPT entangled states. Multipartite generalization of our constructions is also outlined.

DOI
10.1103/physreva.77.032318

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We study robustness of bipartite entangled states that are positive under partial transposition (PPT). It is shown that almost all PPT entangled states are unconditionally robust, in the sense, both inseparability and positivity are preserved under sufficiently small perturbations in its immediate neighborhood. Such unconditionally robust PPT entangled states lie inside an open PPT entangled ball. We construct examples of such balls whose radii are shown to be finite and can be explicitly calculated. This provides a lower bound on the volume of all PPT entangled states. Multipartite generalization of our constructions is also outlined.
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In this paper we show an explicit construction of multipartite class of entangled states with the PPT (Positive Partial Transposition) property in every cut. We investigate properties of this class of states focusing on the trace distance from the set of separable states. We provide an explicit sub-class of the multipartite entangled PPT states which are arbitrary far from the set of separable states. We argue, that in the multipartite case the mentioned distance increases with dimension of the local Hilbert space. In our construction is we do not have to use many copies of initial state living on the smaller space to boost the trace distance as in the previous attempts to this problem.
From both theoretical and experimental points of view symmetric states constitute an important class of multipartite states. Still, entanglement properties of these states, in particular those with positive partial transposition (PPT), lack a systematic study. Aiming at filling in this gap, we have recently affirmatively answered the open question of existence of four-qubit entangled symmetric states with PPT and thoroughly characterized entanglement properties of such states [J. Tura et al., Phys. Rev. A 85, 060302(R) (2012)]. With the present contribution we continue on characterizing PPT entangled symmetric states. On the one hand, we present all the results of our previous work in a detailed way. On the other hand, we generalize them to systems consisting of an arbitrary number of qubits. In particular, we provide criteria for separability of such states formulated in terms of their ranks. Interestingly, for most of the cases, the symmetric states are either separable or typically separable. Then, edge states in these systems are studied, showing in particular that to characterize generic PPT entangled states with four and five qubits, it is enough to study only those that assume few (respectively, two and three) specific configurations of ranks. Finally, we numerically search for extremal PPT entangled states in such systems consisting of up to 23 qubits. One can clearly notice regularity behind the ranks of such extremal states, and, in particular, for systems composed of odd numbers of qubits we find a single configuration of ranks for which there are extremal states.
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.
The discovery of entangled quantum states from which one cannot distill pure entanglement constitutes a fundamental recent advance in the field of quantum information. Such bipartite bound-entangled (BE) quantum states could fall into two distinct categories: (1) Inseparable states with positive partial transposition (PPT), and (2) states with negative partial transposition (NPT). While the existence of PPT BE states has been confirmed, only one class of conjectured NPT BE states has been discovered so far. We provide explicit constructions of a variety of multicopy undistillable NPT states, and conjecture that they constitute families of NPT BE states. For example, we show that for every pure state of Schmidt rank greater than or equal to 3, one can construct n-copy undistillable NPT states, for any n> 1.n>~1.n> 1. The abundance of such conjectured NPT BE states, we believe, considerably strengthens the notion that being NPT is only a necessary condition for a state to be distillable.
We construct a class of quantum bipartite $d\ensuremath{\bigotimes}d$ states which are positive under partial transposition (PPT states). This class is invariant under the maximal commutative subgroup of U(d)\mathrm{U}(d)U(d) and contains as special cases many well-known examples of PPT states. States from our class provide criteria for testing the indecomposability of positive maps. Such maps are crucial for constructing entanglement witnesses.