An Ergodic Theorem for Quantum Diagonal Measures

Abstract

We extend the Nonconventional Ergodic Theorem for generic measures by Furstenberg, to several situations of interest arising from quantum dynamical systems. We deal with the diagonal state canonically associated to the product state (i.e. quantum "diagonal measures"), or to convex combinations of diagonal measures for non ergodic cases. For the sake of completeness, we treat also the Nonconventional Ergodic Theorem for compact dynamical systems, that is when the unitary generating the dynamics in the GNS representation is almost periodic. The Nonconventional Ergodic Theorem allows in a natural way to determine the limit of the three-point correlations, naturally relevant for the knowledge of the ergodic properties of a dynamical system. Mathematics Subject Classification: 46L55, 47A35, 37A55.

Diagonal couplings of quantum Markov chains

“…For a state ϕ on a noncommutative algebra M, however, the straightforward definition via ϕ ∆ (a⊗ b) := ϕ(a·b) for all a, b ∈ M fails to be positive in general. A workaround to deal with this issue was proposed by R. Duvenhage [Duv08] and F. Fidaleo [Fid09]. In the following we repeat this construction and derive some elementary properties.…”

Balance Between Quantum Markov Semigroups

“…The modular conjugation J associated to µ (and to ζ) is then obtained as the conjugate linear operator J : H → H given by J(e p ⊗ e q ) = e q ⊗ e p for all p, q = 1, 2, 3, .... Furthermore, (33) j(π(a)) := Jπ(a) * J = π ′ (a T ) for all a ∈ B(H), where a T denotes the transpose of a in the basis e 1 , e 2 , e 3 , .... This allows us to apply the general notions from the earlier sections explicitly to this specific case.…”

Nonconventional ergodic theorems for quantum dynamical systems

“…13 and Theorem 4.2 of Ref. 9. Indeed, let (A, α, ω) be a C * -dynamical system based on the unital C * -algebra A, the automorphism α, and the invariant state ω.…”