Computation of sandwiched relative alpha-entropy of two n-mode gaussian states

TL;DR

This paper derives a formula for the sandwiched relative alpha-entropy between two n-mode Gaussian states, using a specific parametrization to facilitate the computation.

Contribution

It provides a closed-form expression for the sandwiched relative alpha-entropy of Gaussian states, extending previous parametrization techniques.

Findings

Derived a formula for sandwiched relative alpha-entropy of Gaussian states

Utilized the $ ext{E}_2$-parametrization for computation

Applicable to n-mode Gaussian states in boson Fock space

Abstract

A formula for the sandwiched relative $\alpha$-entropy $\widetilde{D}_\alpha(\rho\vert\vert\sigma)=\frac{1}{\alpha-1}\, \ln\,{\rm Tr}\, \left(\sigma^{\frac{1-\alpha}{2\alpha}}\,\rho\,\sigma^{\frac{1-\alpha}{2\alpha}}\right)^\alpha$ for $0 < \alpha < 1$, of two $n$ mode gaussian states $\rho$, $\sigma$ in the boson Fock space $\Gamma(\mathbb{C}^n)$ is presented. This computation extensively employs the $\mathcal{E}_2$-parametrization of gaussian states in $\Gamma(\mathbb{C}^n)$ introduced in J. Math. Phys. {\bf 62} (2021), 022102.