Fidelity for multimode thermal squeezed states

In the theory of quantum transmission of information the concept of fidelity plays a fundamental role. An important class of channels, which can be experimentally realized in quantum optics, is that of Gaussian quantum channels. In this work we present a general formula for fidelity in the case of two arbitrary Gaussian states. From this formula one can get a previous result [H. Scutaru, J. Phys. A 31, 3659 (1998)], for the case of a single mode. The concept of fidelity used in this paper is the standard one [D. Bures, Trans. Am. Math. Soc. 135, 199 (1969); A. Uhlmann, Rep. Math. Phys. 9, 273 (1976); R. Josza, J. Mod. Opt. 41, 2314 (1994)]. It can be defined by $F({{ρ}}₁,{{ρ}}₂{)}^{{=}{def}}{max}_{|{{ψ}}₁〉,|{{ψ}}₂〉}|〈{{ψ}}₁|{{ψ}}₂〉{|}²,$ where $|{{ψ}}ᵢ〉,$ $i=1,2$ are purifications of the density matrices ${{ρ}}ᵢ.$