Seminar at SMU Delhi

Williamson's normal form in finite dimensions states that, if $A \in M_{2n}(\mathbb{R})$ is strictly positive, then there exists a symplectic matrix $L \in M_{2n}(\mathbb{R})$ and an $n \times n$ diagonal matrix $D$ % $= \diag{(d_1,d_2, \dots, d_n)}$, with $d_1\geq d_2\geq\cdots\geq d_n >0$ such that $$A= L^T \begin{bmatrix} D &0\\ 0& D \end{bmatrix}L. $$ The matrix $D$ is uniquely determined up to permutation of its entries and the diagonal entries are known as \emph{symplectic eigenvalues} of $A$. We prove an infinite dimensional analogue of the above theorem using elementary techniques in Hilbert space theory. Also, we will discuss a few applications of this theorem in the study of quantum Gaussian states, which are a non-commutative analogue of classical Gaussian distributions.