Jensen Shannon Divergence as a Measure of the Degree of Entanglement

Discerning possible candidates for measuring distances between quantum states is a subject of perennial interest. Many of these measures were first defined as distances between probability distributions and subsequently employed as distance-measures in Hilbert space. Let H be the Hilbert space associated with a quantum system and let S be the set of all states, i.e. the set of self-adjoint, (semi)positive and trace-one operators. A frequently employed notion of distance between quantum states is the relative entropy, which is a natural extension to the realm of quantum mechanics of the Kullback-Leibler divergence. This quantity, however, is not useful for ascertaining the degree of purification of an arbitrary state with respect to a pure reference state. The relative entropy of an operator ρ, with respect to an operator σ, both belonging to S, is given by S(ρ‖σ) = tr[ρ(log ρ− log σ)], (1)