Uhlmann fidelity between two-mode Gaussian states

TL;DR

This paper derives explicit formulas for the Uhlmann fidelity between two-mode Gaussian states, facilitating the measurement of quantum correlations in quantum physics applications.

Contribution

It provides the first explicit, calculable formula for the fidelity between two arbitrary two-mode Gaussian states, extending previous limited cases.

Findings

Explicit formulas for one- and two-mode cases

Fidelity expressed as a product of exponential and symplectic spectrum factors

Application to quantifying quantum correlations

Abstract

We analyze the Uhlmann fidelity of a pair of $n$-mode Gaussian states of the quantum radiation field. This quantity is shown to be the product of an exponential function depending on the relative average displacement and a factor fully determined by the symplectic spectrum of the covariance matrix of a specific Gaussian state. However, it is difficult to handle our general formula unless the Gaussian states commute or at least one of them is pure. On the contrary, in the simplest cases $n=1$ and $n=2$, it leads to explicit analytic formulae. Our main result is a calculable expression of the fidelity of two arbitrary two-mode Gaussian states. This can be applied to build reliable measures of quantum correlations between modes in various branches of quantum physics.