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arXiv:0811.3604 · New Journal of Physics

Positive maps, majorization, entropic inequalities and detection of entanglement

Remigiusz Augusiak, Julia Stasińska

Published May 27, 2009• 13 citations

Abstract

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.

DOI: 10.1088/1367-2630/11/5/053018

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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.

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.

Recently, a toolkit of highly symmetric techniques employing matrix inequalities has been developed to detect entanglement in various ways. Here we unifiedly explain in detail these methods, and expand them to a new family of positive maps with further detection capabilities. In the simplest case, we generalize the reduction map to detect more generic states using both multiple copies and local filters. Through the Choi-Jamiołkowski isomorphism, this family of maps leads to a construction of multipartite entanglement witnesses. Discussions and examples are provided regarding the detection of states with local positive partial transposition and the use of multiple copies.

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.

We apply random matrix and free probability techniques to the study of linear maps of interest in quantum information theory. Random quantum channels have already been widely investigated with spectacular success. Here, we are interested in more general maps, asking only for k-positivity instead of the complete positivity required of quantum channels. Unlike the theory of completely positive maps, the theory of k-positive maps is far from being completely understood, and our techniques give many new parametrized families of such maps. We also establish a conceptual link with free probability theory and show that our constructions can be obtained to some extent without random techniques in the setup of free products of von Neumann algebras. Finally, we study the properties of our examples and show that for some parameters, they are indecomposable. In particular, they can be used to detect the presence of entanglement missed by the partial transposition test, that is, positive partial transposition entanglement. As an application, we considerably refine our understanding of positive partial transposition states in the case where one of the spaces is large, whereas the other one remains small.