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arXiv:1312.4732 · Journal of Physics Conference Series

Detection methods to rule out completely co-positive and bi-entangling operations

Chiara Macchiavello, Matteo A. C. Rossi

Published Dec 6, 2013• 3 citations
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Abstract

In this work we extend the quantum channel detection method developed in Refs. [1, 2] in order to detect other interesting convex sets of quantum channels. First we work out a procedure to detect non completely co-positive maps. Then we focus on the set of so-called bi-entangling operations and show how a map outside this set can be revealed. In both cases we provide explicit examples showing the theoretical technique and the corresponding experimental procedure.

DOI
10.1088/1742-6596/470/1/012005

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In this work we extend the quantum channel detection method developed in Refs. [1, 2] in order to detect other interesting convex sets of quantum channels. First we work out a procedure to detect non completely co-positive maps. Then we focus on the set of so-called bi-entangling operations and show how a map outside this set can be revealed. In both cases we provide explicit examples showing the theoretical technique and the corresponding experimental procedure.
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For any $N\ensuremath{\bigotimes}N$-type bipartite system with even N>2N>2N>2, two-parameter continuous families of unital block-dephasing quantum channels and of weakly optimal entanglement witnesses are constructed from special forms of a parametrized positive map. The parameter domains of the families are complementary subsets of a convex set and this fact reveals unusual combination rules of quantum channels and entanglement witnesses. These rules make it possible to develop many physically implementable direct entanglement-detection methods that provide a unified framework to accomplish both information-processing and entanglement-detection tasks by means of smooth quantum channels, without any reference to entanglement witnesses.
Positive maps applied to a subsystem of a bipartite quantum state constitute a central tool in characterising entanglement. In the multipartite case, however, the direct application of a positive but not completely positive map cannot distinguish if a state is genuinely multipartite entangled or just entangled across some bipartition. We thus generalise this bipartite concept to the multipartite setting by introducing non-positive maps that are positive on the subset of biseparable states but can map to a non-positive element if applied to a genuine multipartite entangled state. We explicitly construct examples of multipartite non-positive maps, obtained from positive maps via a lifting procedure, that in this fashion can reveal genuine multipartite entanglement in a robust way.
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.
One of the most challenging open problems in quantum information theory is to clarify and quantify how entanglement behaves when part of an entangled state is sent through a quantum channel. Of central importance in the description of a quantum channel or completely positive map (CP-map) is the dual state associated to it. The present paper is a collection of well-known, less known and new results on quantum channels, presented in a unified way. We will show how this dual state induces nice characterizations of the extremal maps of the convex set of CP-maps, and how normal forms for states defined on a Hilbert space with a tensor product structure lead to interesting parameterizations of quantum channels.
Higher-dimensional entanglement is a valuable resource for several quantum information processing tasks, and is often characterized by the Schmidt number and specific classes of entangled states beyond qubit-qubit and qubit-qutrit systems. We propose a criterion to detect higher-dimensional entanglement, focusing on determining the Schmidt number of quantum states and identifying significant classes of positive partial transposition and negative partial transposition entangled states. Our approach relies on evaluating moments of generalized positive maps which can be efficiently simulated in real experiments without the requirement of full-state tomography. We demonstrate the effectiveness of our detection scheme through various illustrative examples. As a direct application, we explore the implications of our moment-based detection schemes in identifying useful quantum channels such as non-Schmidt-number breaking channels and non-entanglement breaking channels. Finally, we present an operational implication of our proposed moment criterion through its manifestation in channel discrimination tasks.