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ON THE STRUCTURE OF POSITIVE MAPS; FINITE DIMENSIONAL CASE

W Ladys, Law A. Majewski

Published Aug 16, 2016• 15 citations

Abstract

A natural and intrinsic characterization of the structure of the set $\mathfrak{C}$ of positive unital maps is given, i.e. it is shown that $\mathfrak{C}$ is isometrically isomorphic to the subset $\gD$ of bp-positive density matrices endowed with the geometry given by the norm $\alpha$ dual to the Grothendieck projective norm $\pi$, the structure of $\gD$ is determined by the set of its exposed points, and finally a characterization of exposed points of $\gD$ in terms of convex analysis is presented. This seems to be an answer to an old open problem, characterization of the structure of the set of positive maps, studied both in Quantum Information and Operator Algebras. Our arguments are based on the concept of exposed points and links between tensor products and mapping spaces. It should be emphasized that the concept of exposed point depends strongly on the geometry, hence the role of the norm $\alpha$ is crucial.

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A natural and intrinsic characterization of the structure of the set

\mathfrak{C}

of positive unital maps is given, i.e. it is shown that

\mathfrak{C}

is isometrically isomorphic to the subset

\gD

of bp-positive density matrices endowed with the geometry given by the norm

\alpha

dual to the Grothendieck projective norm

\pi

, the structure of

\gD

is determined by the set of its exposed points, and finally a characterization of exposed points of

\gD

in terms of convex analysis is presented. This seems to be an answer to an old open problem, characterization of the structure of the set of positive maps, studied both in Quantum Information and Operator Algebras. Our arguments are based on the concept of exposed points and links between tensor products and mapping spaces. It should be emphasized that the concept of exposed point depends strongly on the geometry, hence the role of the norm

\alpha

is crucial.

Abstract Very recently, Karder and Petek completely described maps on density matrices (positive semidefinite matrices with unit trace) preserving certain entropy-like convex functionals of any convex combination. As a result, maps could be characterized that preserve von Neumann entropy or Schatten

p

-norm of any convex combination of quantum states (whose mathematical representatives are the density matrices). In this note we consider these latter two problems on the set of invertible density operators, in a much more general setting, on the set of positive invertible elements with unit trace in a

C^{\ast }

-algebra.

A new method of analysing positive bistochastic maps on the algebra of complex matrices [Formula: see text] has been proposed. By identifying the set of such maps with a convex set of linear operators on [Formula: see text], one can employ techniques from the theory of compact semigroups to obtain results concerning asymptotic properties of positive maps. It turns out that the idempotent elements play a crucial role in classifying the convex set into subsets, in which representations of extremal positive maps are to be found. It has been shown that all positive bistochastic maps, extremal in the set of all positive maps of [Formula: see text], that are not Jordan isomorphisms of [Formula: see text] are represented by matrices that fall into two possible categories, determined by the simplest idempotent matrices: one by the zero matrix, and the other by a one-dimensional orthogonal projection. Some norm conditions for matrices representing possible extremal maps have been specified and examples of maps from both categories have been brought up, based on the results published previously.

We investigate the set (a) of positive, trace preserving maps acting on density matrices of size N and a sequence of its nested subsets: the sets of maps which are (b) decomposable, (c) completely positive, and (d) extended by identity impose positive partial transpose and (e) are superpositive. Working with the Hilbert–Schmidt (Euclidean) measure, we derive tight explicit two-sided bounds for the volumes of all five sets. A sample consequence is the fact that, as N increases, a generic positive map becomes not decomposable and, a fortiori, not completely positive. Due to the Jamiołkowski isomorphism, the results obtained for quantum maps are closely connected to similar relations between the volume of the set of quantum states and the volumes of its subsets (such as states with positive partial transpose or separable states) or supersets. Our approach depends on the systematic use of duality to derive quantitative estimates and on various tools of classical convexity, high-dimensional probability, and geometry of Banach spaces, some of which are not standard.