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arXiv:0709.3779 · Physical Review A

General scheme for construction of scalar separability criteria from positive maps

Remigiusz Augusiak, Julia Stasińska

Published Jan 28, 2008• 8 citations
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Abstract

We present a general scheme allowing for construction of scalar separability criteria from positive, but not completely positive, maps. The concept is based on a decomposition of every positive map $\ensuremath{\Lambda}$ acting on ${M}_{d}(\mathbb{C})$ into a difference of two completely positive maps ${\ensuremath{\Lambda}}_{1}$, ${\ensuremath{\Lambda}}_{2}$, i.e., $\ensuremath{\Lambda}={\ensuremath{\Lambda}}_{1}\ensuremath{-}{\ensuremath{\Lambda}}_{2}$. The scheme may also be treated as a generalization of the known entropic inequalities, which are obtained from the reduction map. Analyses performed on a few classes of states show that the scalar criteria are stronger than the entropic inequalities and when derived from indecomposable maps allow for detection of bound entanglement.

DOI
10.1103/physreva.77.010303

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We present a general scheme allowing for construction of scalar separability criteria from positive, but not completely positive, maps. The concept is based on a decomposition of every positive map \ensuremathΛ\ensuremath{\Lambda}\ensuremathΛ acting on Md(C){M}_{d}(\mathbb{C})Md​(C) into a difference of two completely positive maps \ensuremathΛ1{\ensuremath{\Lambda}}_{1}\ensuremathΛ1​, \ensuremathΛ2{\ensuremath{\Lambda}}_{2}\ensuremathΛ2​, i.e., \ensuremathΛ=\ensuremathΛ1\ensuremath−\ensuremathΛ2\ensuremath{\Lambda}={\ensuremath{\Lambda}}_{1}\ensuremath{-}{\ensuremath{\Lambda}}_{2}\ensuremathΛ=\ensuremathΛ1​\ensuremath−\ensuremathΛ2​. The scheme may also be treated as a generalization of the known entropic inequalities, which are obtained from the reduction map. Analyses performed on a few classes of states show that the scalar criteria are stronger than the entropic inequalities and when derived from indecomposable maps allow for detection of bound entanglement.
Recently it was shown that if a given state fulfils the reduction criterion, it must also satisfy the known entropic inequalities. The natural question arises as to whether it is possible to derive some scalar inequalities stronger than the entropic ones, assuming that stronger criteria based on positive but not completely positive maps are satisfied. In the present paper we show that if certain conditions hold, the extended reduction criterion [H.-P. Breuer, Phys. Rev. Lett 97, 080501 (2006); W. Hall, J. Phys. A 40, 6183 (2007)] leads to some entropiclike inequalities, much stronger than their entropic counterparts. The comparison of the derived inequalities with other separability criteria shows that such an approach might lead to strong scalar criteria detecting both distillable and bound entanglement. In particular, in the case of SO(3)-invariant states it is shown that the present inequalities detect entanglement in regions, in which linear entanglement witnesses based on the extended reduction map fail. It should also be emphasized that in the case of $2\ensuremath{\bigotimes}d$ states the derived inequalities detect entanglement efficiently, while the extended reduction maps are useless, when acting on the qubit subsystem. Moreover, there is a natural way to construct a many-copy entanglement witnesses based on the derived inequalities so, in principle, there is a possibility of experimental realization. Some open problems and possibilities for further research are outlined.
We study families of positive and completely positive maps acting on a bipartite system ${\mathbb{C}}^{M}\ensuremath{\bigotimes}{\mathbb{C}}^{N}$ (with $M\ensuremath{\le}N$). The maps have a property that, when applied to any state (of a given entanglement class), result in a separable state or, more generally, a state of another certain entanglement class (e.g., Schmidt number $\ensuremath{\le}\phantom{\rule{0.16em}{0ex}}k$). This allows us to derive useful families of sufficient separability criteria. Explicit examples of such criteria have been constructed for arbitrary M,NM,NM,N, with a special emphasis on M=2M=2M=2. Our results can be viewed as generalizations of the known facts that in the sufficiently close vicinity of the completely depolarized state (the normalized identity matrix), all states are separable (belong to ``weakly'' entangled classes). Alternatively, some of our results can be viewed as an entanglement classification for a certain family of states, corresponding to mixtures of the completely polarized state with pure states, partial transposes, and/or local transformations thereof.
In this paper, we discuss some general connections between the notions of positive map, weak majorization and entropic inequalities in the context of detection of entanglement among bipartite quantum systems. First, basing on the fact that any positive map can be written as the difference between two completely positive maps Λ=Λ 1 −Λ 2 , we propose a possible way to generalize the Nielsen–Kempe majorization criterion. Then, we present two methods of derivation of some general classes of entropic inequalities useful for the detection of entanglement. While the first one follows from the aforementioned generalized majorization relation and the concept of Schur-concave decreasing functions, the second is based on some functional inequalities. What is important is that, contrary to the Nielsen–Kempe majorization criterion and entropic inequalities, our criteria allow for the detection of entangled states with positive partial transposition when using indecomposable positive maps. We also point out that if a state with at least one maximally mixed subsystem is detected by some necessary criterion based on the positive map Λ, then there exist entropic inequalities derived from Λ (by both procedures) that also detect this state. In this sense, they are equivalent to the necessary criterion [ I ⊗Λ](ϱ AB )⩾0. Moreover, our inequalities provide a way of constructing multi-copy entanglement witnesses and therefore are promising from the experimental point of view. Finally, we discuss some of the derived inequalities in the context of the recently introduced protocol of state merging and the possibility of approximating the mean value of a linear entanglement witness.
The reduction criterion is a well-known necessary condition for separable states, and states violating this condition are entangled and also 1-distillable. In this paper we introduce a set of necessary conditions for separability of multipartite states, obtained from a set of positive but not completely positive maps. These conditions can be thought of as generalizations of the reduction criterion to multipartite systems. We use tripartite Werner states as an example to investigate the entanglement detecting powers of some of these conditions, and we also look at what these conditions mean in terms of distillation. Finally, we show that these maps can be used to give a partial solution to the subsystem problem, as described by Butterley et al.