The entangled ergodic theorem and an ergodic theorem for quantum "diagonal measures"

Abstract

Let $U$ be a unitary operator acting on the Hilbert space $H$, $\a:\{1,..., 2k\}\mapsto\{1,..., k\}$ a pair--partition, and finally $A_{1},...,A_{2k-1}\in B(H)$. We show that the ergodic average $$ \frac{1}{N^{k}}\sum_{n_{1},...,n_{k}=0}^{N-1} U^{n_{\a(1)}}A_{1}U^{n_{\a(2)}}... U^{n_{\a(2k-1)}}A_{2k-1}U^{n_{\a(2k)}} $$ converges in the strong operator topology when $H$ is generated by the eigenvectors of $U$, that is when the dynamics induced by the unitary $U$ on $H$ is almost periodic. This result improves the known ones relative to the entangled ergodic theorem. We also prove the noncommutative version of the ergodic result of H. Furstenberg relative to diagonal measures. This implies that ${\displaystyle \frac{1}{N}\sum_{n=0}^{N-1} U^{n}AU^{n}}$ converges in the strong operator topology for other interesting situations where the involved unitary operator does not generate an almost periodic dynamics, and the operator $A$ is noncompact.