A New Generalization of von Neumann Relative Entropy

Abstract

In quantum information, von Neumann relative entropy has a great applications and operational interpretations in diverse fields, and von Neumann entropy is an important tool for describing the uncertainty of a quantum state. In this paper, we generalize the classical von Neumann relative entropy S(ρ||σ) and von Neumann entropy S(ρ) to f-von Neumann relative entropy $\widetilde {S}_{f}(\rho ||\sigma )$ and f-von Neumann entropy $\widetilde {S}_{f}(\rho )$ induced by a logarithm-like function f, respectively, and explore their properties. We prove that $\widetilde {S}_{f}(\rho ||\sigma )$ is nonnegative and then prove that $\widetilde {S}_{f}(\rho )$ has nonnegativity, boundedness, concavity, subadditivity and so on. Later, we show the stability and continuity of the $\widetilde {S}_{f}(\rho )$ with respect to the trace distance. In the case that f(x) = −log x, the resulted entropies reduce the classical von Neumann relative entropy and von Neumann entropy, respectively. This means that our results extend the usual results to a more general setting and then have some potential applications in quantum information.