  Relative entropy is a measure of distinguishability for quantum states, and
plays a central role in quantum information theory. The family of Renyi
entropies generalizes to Renyi relative entropies that include as special cases
most entropy measures used in quantum information theory. We construct a
Euclidean path-integral approach to Renyi relative entropies in conformal field
theory, then compute the fidelity and the relative entropy of states in one
spatial dimension at zero and finite temperature using a replica trick. In
contrast to the entanglement entropy, the relative entropy is free of
ultraviolet divergences, and is obtained as a limit of certain correlation
functions. The relative entropy of two states provides an upper bound on their
trace distance.
