On duality between quantum maps and quantum states

TL;DR

This paper explores the mathematical relationship between quantum operations and quantum states, introducing criteria for decomposability and analyzing special classes of quantum maps using the Jamiolkowski isomorphism.

Contribution

It establishes a duality framework linking quantum maps and states, and defines a class of unistochastic operations based on extended unitaries.

Findings

Constructive criterion for decomposability of quantum maps

Analysis of unistochastic operations via extended unitaries

Relation between quantum dynamics and kinematics through Jamiolkowski isomorphism

Abstract

We investigate the space of quantum operations, as well as the larger space of maps which are positive, but not completely positive. A constructive criterion for decomposability is presented. A certain class of unistochastic operations, determined by unitary matrices of extended dimensionality, is defined and analyzed. Using the concept of the dynamical matrix and the Jamiolkowski isomorphism we explore the relation between the set of quantum operations (dynamics) and the set of density matrices acting on an extended Hilbert space (kinematics). An analogous relation is established between the classical maps and an extended space of the discrete probability distributions.