Representation of superoperators in double phase space

TL;DR

This paper extends phase space methods to represent superoperators in quantum mechanics using double phase space, revealing new integral relationships and connecting the Choi-Jamiolkowsky isomorphism with double Wigner or chord transforms.

Contribution

It introduces a novel phase space representation of superoperators via double phase space, linking the Choi-Jamiolkowsky isomorphism to double Wigner or chord transforms.

Findings

Choi-Jamiolkowsky isomorphism as double Wigner/chord transform

New integral relationships between Wigner and chord distributions

Representation of superoperators in double phase space

Abstract

Operators in quantum mechanics - either observables, density or evolution operators, unitary or not - can be represented by c-numbers in operator bases. The position and momentum bases are in one to one correspondence with lagrangian planes in double phase space, but this is also true for the well known Wigner-Weyl correspondence based on translation and reflection operators. These phase space methods are here extended to the representation of superoperators. We show that the Choi-Jamiolkowsky isomorphism between the dynamical matrix and the linear action of the superoperator constitutes a "double" Wigner or chord transform when represented in double phase space. As a byproduct several previously unknown integral relationships between products of Wigner and chord distributions for pure states are derived.