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

We denote by $M_k$ the set of $k$ by $k$ matrices with complex entries. We consider quantum channels $\phi_L$ of the form: given a measurable function $L:M_k\to M_k$ and a measure $\mu$ on $M_k$ we define the linear operator $\phi_L:M_k \to M_k$, by the law $\rho \,\to\,\phi_L(\rho) = \int_{M_k} L(v) \rho L(v)^\dagger \, \dm(v).$ On a previous work the authors show that for a fixed measure $\mu$ it is generic on the function $L$ the $\Phi$-Erg property (also irreducibility). Here we will show that the purification property is also generic on $L$ for a fixed $\mu$. Given $L$ and $\mu$ 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 $\Phi$-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 $\int |L(v)|^2 \log |L(v)|\, \ d\mu(v)<\infty$, then the Lyapunov exponents $\infty > \gamma_1\geq \gamma_2\geq ...\geq \gamma_k\geq -\infty$ 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.