The seed of the notion of symplectic eigenvalues was implanted by John Williamson in his 1936 seminal paper on the normal forms of linear dynamical systems. His main result, which concerns diagonalization of positive definite matrices by symplectic matrices, attracted the attention of only pure mathematicians working in the fields of symplectic geometry and linear system theory. Its recognition among physicists came much later in 1970s and 1980s when the concept of Williamson's normal form became relevant in quantum optics and Hamiltonian mechanics research. The theory of symplectic eigenvalues gained substantial popularity within the physics community in 1990s and 2000s as the symplectic techniques became central to continuous-variable quantum information theory, particularly in the study of quantum Gaussian states.
Much of the initial theory of symplectic eigenvalues was developed by quantum information theorists till the early 2010s. Arguably, the first matrix analysis treatment of the topic was done by Bhatia and Jain in 2015 (arXiv:1803.04647). Since the last decade, symplectic eigenvalue has become a topic of intense research by physicists and mathematicians due to its established importance in various fields of sciences. Remarkably, several results on symplectic eigenvalues have been found to be analogous to those of eigenvalues of Hermitian matrices with appropriate interpretations. In particular, symplectic analogs of famous eigenvalue inequalities are known today such as Sing-Thompson theorem, Cauchy's interlacing principle, Weyl's inequalities, Lidskii's inequalities, and Schur--Horn theorem.
In this talk, we will discuss the basic theory of symplectic eigenvalues, including some of the beautiful results, and its significance in quantum information theory. In the end, we will discuss some interesting future research work.
Dr. Hemant Mishra is a faculty member in the Dept of Mathematics at IIT (ISM) Dhanbad. He obtained his M.Sc. from IIT Guwahati and his Ph.D. from ISI Delhi. Subsequently he served as a Postdoctoral Fellow at the Cornell University, USA. His research interest are in Matrix Analysis, Quantum Information Theory, Linear Algebra, Functional Analysis.
