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

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Conference Article

Publication Title

Springer Proceedings in Mathematics and Statistics


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.

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Open Access, Green

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