Executive Summary : | Williamson's normal form for positive definite matrices is a symplectic analogue of the spectral theorem in linear algebra. With the advent of continuous variable quantum information (QI) theory and quantum computing, interest in the symplectic spectrum of positive matrices has soared. This theory is practically sound, mathematically elegant, and challenging. Quantum computers and quantum communication systems transform and transmit information using systems like electrons and photons, whose behavior is intrinsically quantum mechanical. Gaussian states and transformations are primary tools for analyzing continuous-variable quantum information processing. In 2019, B. V. Rajarama Bhat and T. C. John extended Williamson's normal form to bounded positive invertible operators on infinite-dimensional separable real Hilbert spaces, leading to the notion of symplectic spectrum for a positive invertible operator on infinite-dimensional real Hilbert spaces. This project focuses on developing a Symplectic Spectral Theory analogous to the spectral theory of normal operators on complex Hilbert spaces. The project will consider three types of problems: generalizing various inequalities and results concerning the symplectic eigenvalues of a finite-dimensional operator into the infinite-dimensional setting, approximating the symplectic spectrum of an infinite-dimensional operator using finite-dimensional truncations, and investigating whether analogues of results in spectral theory, such as spectral representation theorem, spectral mapping theorem, and spectral inclusion theorems, are available for the symplectic spectrum. |