Symplectic Dilations, Gaussian States and Gaussian Channels

TL;DR

This paper demonstrates that any real matrix can be dilated to a symplectic matrix, explores properties of Gaussian states, and investigates implications for quantum Gaussian channels, highlighting open problems in the field.

Contribution

It provides a simple matrix algebra method for symplectic dilation of real matrices and analyzes their impact on Gaussian states and channels.

Findings

Every real 2n×2n matrix admits a symplectic dilation.

Properties of Gaussian states in L^2(ℝ^n) are summarized.

Implications for quantum Gaussian channels and open problems are discussed.

Abstract

By elementary matrix algebra we show that every real $2n \times 2n$ matrix admits a dilation to an element of the real symplectic group $Sp (2(n+m))$ for some nonnegative integer $m.$ Our methods do not yield the minimum value of $m,$ for which such a dilation is possible. After listing some of the main properties of Gaussian states in $L^2 (\mathbb{R}^n),$ we analyse the implications of symplectic dilations in the study of quantum Gaussian channels which lead to some interesting open problems, particularly, in the context of the work of Heinosaari, Holevo and Wolf \cite{3}.