Fidelity, as a measure of the distinguishability of states, is an important concept in quantum mechanics, quantum optics and quantum information theory. Recently, the explicit expressions of fidelity for single-mode squeezed states have been given. However, in experimental studies, especially in non-degenerate parametric down-conversion, two photons are generated and one studies two-or more-mode systems. In this paper we study the Bures fidelity for thermal states of a diagonalizable quadratic Hamiltonian in multi-mode Fock space. To the best of our knowledge, no one has yet attempted to give an explicit general formula of fidelity of mixed states in multi-mode systems.

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