Quantum uncertainty of mixed states based on skew information

Abstract

The uncertainty of a mixed state has two quite different origins: classical mixing and quantum randomness. While the classical aspect (mixedness) is significantly quantified by the von Neumann entropy, it seems that we still do not have a well accepted measure of quantum uncertainty. In terms of the skew information introduced by Wigner and Yanase in 1963 in the context of quantum measurements, we will propose an intrinsic measure for synthesizing quantum uncertainty of a mixed state and investigate its fundamental properties. We illustrate how it arises naturally from a naive hidden-variable approach to entanglement and how it exhibits a simple relation to the notion of negativity, which is an entanglement monotone introduced quite recently. We further show that it has a dramatic nonextensive feature resembling the probability law relating operations of two events. This measure of quantum uncertainty provides an alternative quantity complementary to the von Neumann entropy for studying mixedness and quantum correlations.