A Szegő type theorem and distribution of symplectic eigenvalues

We study the properties of stationary G-chains in terms their generating functions. In particular, we prove an analogue Szeg\H{o} limit theorem for symplectic eigenvalues, derive expression entropy rate quantum Gaussian processes, and distribution eigenvalues truncated block Toeplitz matrices. also introduce a concept numerical range, analogous to that some its basic properties, mainly context operators.