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arXiv:quant-ph/0610259

Contractions, Matrix Paramatrizations, and Quantum Information

Tseng, M. C.

Published Oct 31, 2006• 0 citations
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Abstract

In this note, we discuss dilation-theoretic matrix parametrizations of contractions and positive matrices. These parametrizations are then applied to some problems in quantum information theory. First we establish some properties of positive maps, or entanglement witnesses. Two further applications, concerning concrete dilations of completely positive maps, in particular quantum operations, are given.

DOI
10.48550/arxiv.quant-ph/0610259

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In this note, we discuss dilation-theoretic matrix parametrizations of contractions and positive matrices. These parametrizations are then applied to some problems in quantum information theory. First we establish some properties of positive maps, or entanglement witnesses. Two further applications, concerning concrete dilations of completely positive maps, in particular quantum operations, are given.
There are various notions of positivity for matrices and linear matrix-valued maps that play important roles in quantum information theory. The cones of positive semidefinite matrices and completely positive linear maps, which represent quantum states and quantum channels respectively, are the most ubiquitous positive cones. There are also many natural cones that can been regarded as "more" or "less" positive than these standard examples. In particular, entanglement theory deals with the cones of separable operators and entanglement witnesses, which satisfy very strong and weak positivity properties respectively. Rather complementary to the various cones that arise in entanglement theory are norms. The trace and operator norms for operators and the diamond and completely bounded norms for superoperators are the typical norms that are seen throughout quantum information theory. In this work our main goal is to develop a family of norms that play a role analogous to the cone of entanglement witnesses. We investigate the basic mathematical properties of these norms, including their relationships with other well-known norms, their isometry groups, and their dual norms. We also make the place of these norms in entanglement theory rigorous by showing that entanglement witnesses arise from minimal operator systems, and analogously our norms arise from minimal operator spaces. Finally, we connect the various cones and norms considered here to several seemingly unrelated problems from other areas. We characterize the problem of whether or not non-positive partial transpose bound entangled states exist in terms of one of our norms, and provide evidence in favour of their existence. We also characterize the minimum gate fidelity of a quantum channel, the maximum output purity and its completely bounded counterpart, and the geometric measure of entanglement in terms of these norms.
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.
The most general evolution of the density matrix of a quantum system with a finite-dimensional state space is by stochastic maps which take a density matrix linearly into the set of density matrices. These dynamical stochastic maps form a linear convex set that may be viewed as supermatrices. The property of Hermiticity of density matrices renders an associated supermatrix Hermitian and hence diagonalizable. The positivity of the density matrix does not make the associated supermatrix positive though. If the map itself is positive, it is called completely positive and it has a simple parametrization. This is extended to all positive (not completely positive) maps. A general dynamical map that does not preserve the norm of the density matrices it acts on can be thought of as the contraction of a norm-preserving map of an extended system. The reconstruction of such extended dynamics is also given.