The volume of Gaussian states and of two-qubit states by information geometry.

The issue of the volume of sets of states in the phase space framework can be addressed in order to distinguish classical from quantum states as well as to find separable states within all quantum states. In finite dimensional systems, several metrics are introduced in order to compute the volume of physical states. However, when going to infinite dimensional systems, problems arise also from the non-compactness of the support of states.

Thus, on the one hand, we have the difficulties in analysing infinite dimensional systems, while on the other hand we still lack a unifying approach for evaluating volumes of classical and quantum states.

To deal with these problems, we propose to exploit information geometry. In so doing, we associate a Riemannian manifold to a generic Gaussian system. Then, we consider a volume measure as the volume of the manifold associated with a set of states of the system. We are able to overcome the difficulty of an unbounded volume by introducing a regularizing function stemming from energy bounds, which acts as a form of compactification of the support of Gaussian states. We then proceed to consider a different regularizing function which satisfies some nice properties of canonical invariance. Finally, we find the volumes of classical, quantum, and quantum entangled states for two-mode Gaussian systems, showing chains of strict inclusions.

This approach is extended to two-qubit systems by resorting to the Husimi $Q$-function that is a truly probability distribution function. Above all we address the question of whether such an approach gives results similar to other approaches based on quantum version of the Fisher metric, as Helstrom quantum Fisher metric and as Wigner-Yanase-like quantum Fisher metric. We focus on states having maximally disordered subsystems and analyze the behavior of the volume of sub-manifolds of separable and entangled states with fixed purity. We show that the all above mentioned approaches give the same qualitative results.