  The symplectic eigenvalues play a significant role in finite mode quantum
information theory, and Williamson normal form proves to be a valuable tool in
this area. Understanding the symplectic spectrum of a Gaussian Covariance
Operator is a crucial task. Recently, an infinite-dimensional analogue of
Williamson Normal form was discovered, which has been instrumental in studying
infinite mode Gaussian quantum states. However, most existing results pertain
to finite-dimensional operators, leaving a dearth of literature in the
infinite-dimensional context. The focus of this article is on employing
approximation techniques to estimate the symplectic spectrum of certain
infinite-dimensional operators. These techniques are well-suited for a
particular class of operators, including specific types of infinite mode
Gaussian Covariance Operators. Our approach involves computing the Williamson
Normal form and deriving bounds for the symplectic spectrum of these operators.
As a practical application, we explicitly compute the symplectic spectrum of
Gaussian Covariance Operators. Through this research, we aim to contribute to
the understanding of symplectic eigenvalues in the context of
infinite-dimensional operators, opening new avenues for exploration in quantum
information theory and related fields.
