Total, classical, and quantum uncertainty matrices via operator monotone functions

It is important to distinguish between classical information and quantum information in quantum information theory. In this paper, we first extend the concept of metric-adjusted correlation measure and some related measures to non-Hermitian operators, and establish several relations between the metric-adjusted skew information with different operator monotone functions. By employing operator monotone functions, we next introduce three uncertainty matrices generated by channels: the total uncertainty matrix, the classical uncertainty matrix, and the quantum uncertainty matrix. We establish a decomposition of the total uncertainty matrix into classical and quantum parts and further investigate their basic properties. As applications, we employ uncertainty matrices to quantify the decoherence caused by the action of quantum channels on quantum states, and calculate the uncertainty matrices of some typical channels to reveal certain intrinsic features of the corresponding channels. Moreover, we establish several uncertainty relations that improve the traditional Heisenberg uncertainty relations involving variance.