Quantum information and phase transitions: Fidelity and state distinguishability

We review the definition and properties of fidelity as a measure of distinguishability of quantum states, presenting some of its geometric aspects. We apply then fidelity to the study of phase transitions in condensed matter systems, both at zero and finite temperatures. It does not rely on any already known order parameter, being useful not only in the usual Landau type phase transitions but also in phase transitions displaying topological order. The so called fidelity susceptibility and its relation to the usual thermodynamic susceptibilities and response functions are discussed. Finally, we consider the fidelity between different reduced density matrices for a subsystem of a larger system.