Theoretical Statistics and Mathematics Unit, ISI Delhi

November 28, 2018 (Wednesday) ,
3:30 PM at Webinar

Speaker:
Tiju Cherian John,
ISI Bangalore

Title:
Williamson's Normal Form: A passage through linear algebra to quantum theory

Abstract of Talk

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.