Convergence of the quantum Boltzmann map

by R. F. Streater; Commun. Math. Phys., 98, 177-185, 1985.

ABSTRACT

We consider a non-linear map on the space of density matrices, which we call the Boltzmann map tau. It is the composition of a doubly stochastic map T on the space of n-body states, and the conditional expectation onto the one-body space. When T is ergodic, then the iterates of tau take any initial state to the uniform distribution. If the energy-levels are equally spaced, and T conserves energy and is ergodic on each energy shell, then the iterates of tau take any initial state of finite energy to a canonical distribution.

During the proof we prove an inequality which bounds the relative entropy between two states by half the square of the Hilbert-Schmidt norm of their difference. This might be called a quantum Kullback inequality.

A result replacing the Hilbert-Schmidt norm by the trace-norm had already been obtained by Ohya, Hiai et al. in 1981.