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 $L:M_k\to M_k$ and a measure $μ$ on $M_k$ we define the linear operator $ϕ_L:M_k \to M_k$, by the law $ρ\,\to\,ϕ_L(ρ) = \int_{M_k} L(v) ρL(v)^\dagger \, \dm(v).$ 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 on $L$, if $\int |L(v)|^2 \log |L(v)|\, \ dμ(v)<\infty$, then the Lyapunov exponents $\infty > γ_1\geq γ_2\geq ...\geq γ_k\geq -\infty$ are well defined. On 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 $-\frac{1}{2} \,h$, where $h$ is the entropy of the associated Markov probability.