On the Wigner Distribution of the Reduced Density Matrix

TL;DR

This paper rigorously derives how the Wigner distribution of a reduced density matrix in bipartite quantum systems can be obtained by integrating the total system's Wigner distribution over one subsystem's phase space, with applications to Gaussian states.

Contribution

It provides a rigorous proof for the relation between the Wigner distributions of a bipartite system and its reduced state, using the Weyl--Wigner--Moyal formalism, and applies it to Gaussian states.

Findings

Wigner distribution of reduced state obtained by phase space integration

Explicit description for Gaussian mixed states

Discussion on purification in Wigner formalism

Abstract

CConsider a bipartite quantum system consisting of two subsystems A and B. The reduced density matrix ofA a is obtained by taking the partial trace with respect to B. In this work, we will show that the Wigner distribution of this reduced density matrix is obtained by integrating the total Wigner distribution with respect to the phase space variables corresponding to subsystem B. The proof we give is rigorous (as opposed to those found in the literature) and makes use of the Weyl--Wigner--Moyal phase space formalism. Our main result is applied to general Gaussian mixed states, of which it gives a particularly simple and precise description. We also briefly discuss the purification of a mixed state from the Wigner formalism point of view.