# A Pedagogical Note on the Computation of Relative Entropy of Two n-Mode Gaussian States

## Document Type

Conference Article

## Publication Title

Springer Proceedings in Mathematics and Statistics

## Abstract

A formula for the relative entropy S(ρ||σ)=Trρ(logρ-logσ) of two gaussian states ρ, σ in the boson Fock space Γ(Cn) is presented. It is shown that the relative entropy has a classical and a quantum part: The classical part consists of a weighted linear combination of relative Shannon entropies of n pairs of Bernouli trials arising from the thermal state composition of the gaussian states ρ and σ. The quantum part has a sum of n terms, that are functions of the annihilation means and the covariance matrices of 1-mode marginals of the gaussian state ρ′, which is equivalent to ρ under a disentangling unitary gaussian symmetry operation of the state σ. A generalized formula for the Petz-Rényi relative entropy Sα(ρ||σ)=-1α-1logTrρασ1-α,0<α<1 for gaussian states ρ, σ is also presented. Furthermore it is shown that Sα(ρ| | σ) converges to the limit S(ρ| | σ) as α increases to 1.

## First Page

55

## Last Page

72

## DOI

10.1007/978-3-031-06170-7_2

## Publication Date

1-1-2022

## Recommended Citation

Parthasarathy, K. R., "A Pedagogical Note on the Computation of Relative Entropy of Two n-Mode Gaussian States" (2022). *Conference Articles*. 451.

https://digitalcommons.isical.ac.in/conf-articles/451

## Comments

Open Access, Green