Lyapunov Exponents for Quantum Channels: An Entropy Formula and Generic Properties
Abstract We denote by Mk the set of k by k matrices with complex entries. We consider quantum channels φL of the form: given a measurable function L: Mk → Mk and a measure µ on Mk we define the linear operator φL : Mk → Mk , by the law ρ → φL (ρ) = ∫ Mk L(v)ρL(v)† dµ(v). In a previous work, the authors show that for a fixed measure µ the Φ-Erg property is generic on the function L (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 processes: one takes values on the projective space P (ℂ k ) and the other on matrices in Mk . The Φ-Erg property and the purification condition are the nice hypothesis for the discrete time evolution given by the natural transition probability. In this way it will follow that generically on L, if ∫ |L(v)|2 log |L(v)| dµ(v) < ∞, the Lyapunov exponents ∞ > γ 1 ≥ γ 2 ≥ … ≥ γk ≥ −∞ are well defined. In a previous work, the concepts of entropy of a channel and Gibbs channel were presented; and also an example (associated to a stationary Markov chain) in which this definition of entropy (for a quantum channel) matches the Kolmogorov-Shanon definition of entropy. We estimate here the larger Lyapunov exponent for the mentioned example and we show that it is equal to −1/2 h, where h is the entropy of the associated Markov invariant probability.