On the Wigner distribution of the reduced density matrix

Consider a bipartite quantum system consisting of two subsystems $A$ and $B$. The reduced density matrix of $A$ 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 the 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 from the Wigner formalism point of view.

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Maurice de Gosson has been financed by the Grant P 33447-N of the Austrian Science Fund FWF.

Published 30 October 2023