Lyapunov exponents for Quantum Channels: an entropy formula and generic properties

Abstract

We denote by M k the set of k by k matrices with complex entries. We consider quantum channels φ L of the form: given a measurable function L : M k → M k and a measure µ on M k we define the linear operator φ L :On a previous work the authors show that for a fixed measure µ it is generic on the function L the Φ-Erg property (also irreducibility). Here we will show that the purification property is also generic on L for a fixed µ.Given L and µ there are two related stochastic process: one takes values on the projective space P (C k ) and the other on matrices in M k . The Φ-Erg property and the purification condition are good hypothesis for the discrete time evolution given by the natural transition probability. In this way it will follow that generically onOn the previous work it was presented the concepts of entropy of a channel and of Gibbs channel; and also an example (associated to a stationary Markov chain) where this definition of entropy (for a quantum channel) matches the Kolmogorov-Shanon definition of entropy. We estimate here the larger Lyapunov exponent for the above mentioned example and we show that it is equal to − 1 2 h, where h is the entropy of the associated Markov probability.