A Szeg\H{o} type theorem and distribution of symplectic eigenvalues

TL;DR

This paper extends Szeg\

Contribution

It introduces a Szeg\

Findings

Proves a Szeg\

Derives an entropy rate expression for quantum Gaussian processes.

Analyzes symplectic eigenvalue distribution of block Toeplitz matrices.

Abstract

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