A New Generalization of von Neumann Relative Entropy

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(rho||sigma) and von Neumann entropy S(rho) to f-von Neumann relative entropy and f-von Neumann entropy induced by a logarithm-like function f, respectively, and explore their properties. We prove that is nonnegative and then prove that has nonnegativity, boundedness, concavity, subadditivity and so on. Later, we show the stability and continuity of the 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.