Factorization and dilation problems for completely positive maps on von Neumann algebras

TL;DR

This paper investigates the factorization and dilation properties of Markov maps on von Neumann algebras, providing new examples of non-factorizable maps and addressing open problems in quantum information theory and noncommutative inequalities.

Contribution

It introduces simple examples of non-factorizable Markov maps on matrix algebras and solves open problems related to the quantum Birkhoff conjecture and noncommutative inequalities.

Findings

Existence of non-factorizable Markov maps on M_n(C) for all n≥3

A semigroup of Markov maps on M_4(C) with non-factorizable elements for small t

Negative resolution of an open problem in quantum information theory

Abstract

We study factorization and dilation properties of Markov maps between von Neumann algebras equipped with normal faithful states, i.e., completely positive unital maps which preserve the given states and also intertwine their automorphism groups. The starting point for our investigation has been the question of existence of non-factorizable Markov maps, as formulated by C. Anantharaman-Delaroche. We provide simple examples of non-factorizable Markov maps on M_n(C) for all n\geq 3, as well as an example of a one-parameter semigroup (T(t))_{t\geq 0} of Markov maps on M_4(C) such that T(t) fails to be factorizable for all small values of t > 0. As applications, we solve in the negative an open problem in quantum information theory concerning an asymptotic version of the quantum Birkhoff conjecture, as well as we sharpen the existing lower bound estimate for the best constant in the noncommutative little Grothendieck inequality.