Quantum systems, duality and separable morphisms of operator Hilbert systems

The separable morphisms between operator systems play a fundamental role in many aspects of quantum information theory. A key result proven by Paulsen, Todorov and Tomforde (2011) asserts that a separability of a linear mapping between finite dimensional matrix algebras is equivalent to its property to be an entanglement breaking mapping. The latter in turn is equivalent to max-matrix (or min-max-matrix) positive mapping of the related operator system structures. Thus a separable channel can be thought as a max-matrix positive mapping between finite-dimensional matrix algebras preserving the related traces. Whether the separable morphisms characterize the max-matrix positive maps of operator systems is an open problem. How to be with the min-max-matrix positive maps? On this concern a possible characterization of separable morphisms between some operator systems is of great importance. In the first part of the present talk we classify quantum systems among the quantum spaces. In the normed case we obtain a complete solution to the problem when an operator space turns out to be an operator system. The min and max quantizations of a local order are described in terms of the min and max envelopes of the related state spaces. In particular, the operator Hilbert space of Pisier turns out to be an operator system, which possesses the self-duality property, and we obtain a solution to the problem on the max-matrix positive maps between operator systems. It is established a link between unital positive maps and Pietch factorizations, which allows us to describe all separable morphisms from an abelian C*-algebra to an operator Hilbert system. Finally, we provide a key property of entanglement breaking maps that involves operator Hilbert systems.