I am interested to find out if there is any method for computing the trace distance between two arbitrary bosonic quantum Gaussian states. It is simple to compute the trace distance between two thermal states because they are both diagonal in the photon-number-state basis. The same is true for coherent states because they are pure, we can then related to fidelity, and there are known expressions for the fidelity between two arbitrary bosonic quantum Gaussian states. (In fact, we can get the trace distance for any two pure quantum Gaussian states by relating to fidelity.) But in general, it seems like a challenging question.