Random and free positive maps with applications to entanglement detection

Abstract

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, PPT entanglement. As an application, we considerably refine our understanding of PPT states in the case where one of the spaces is large whereas the other one remains small.

References in corpus (5)

• Randomizing quantum states: Constructions and applications
• Random matrix techniques in quantum information theory
• Extensive nonadditivity of privacy
• On superconvergence of sums of free random variables
• Criteria and new classes of k-positive maps

Cited by in corpus (6)

• Random matrix techniques in quantum information theory
• Positive tensor products of qubit maps and n-tensor-stable positive qubit maps
• On the asymptotic distribution of block-modified random matrices
• Products of Complex Rectangular and Hermitian Random Matrices
• On the separability of unitarily invariant random quantum states - the unbalanced regime
• Moment Methods on compact groups: Weingarten calculus and its applications