Quantum metrics based upon classical Jensen–Shannon divergence

Abstract

Jensen–Shannon divergence is an important distinguishability measure between probability distributions that finds interesting applications within the context of Information Theory. In particular, this classical divergence belongs to a remarkable class of divergences known as Csiszár or f-divergences. In this paper we analyze the problem of obtaining a distance measure between two quantum states starting from the classical Jensen–Shannon divergence between two probability distributions. Considering the Jensen–Shannon divergence as a Csiszár divergence, we first focus on the problem of distinguishability between two pure quantum states. We find a quantum version of the classical Jensen–Shannon divergence that differs from the previously introduced Quantum Jensen–Shannon Divergence. The two quantum versions of Jensen–Shannon divergence have different interpretations within the framework of Quantum Information Theory. Whereas the former quantum version of Jensen–Shannon divergence can be interpreted as the Holevo bound, the alternative quantum version obtained in this work equals the accessible information. Furthermore, we obtain a monoparametric family of metrics between two quantum pure states. Finally, we extend this family of metrics to the case of mixed quantum states by means of the concept of purification.