Standard symmetrized variance with applications to coherence, uncertainty and entanglement

Variance is a ubiquitous quantity in quantum information theory. Given a basis, we consider the averaged variances of a fixed diagonal observable in a pure state under all possible permutations on the components of the pure state and call it the symmetrized variance. Moreover we work out the analytical expression of the symmetrized variance and find that such expression is in the factorized form where two factors separately depends on the diagonal observable and quantum state. By shifting the factor corresponding to the diagonal observable, we introduce the notion named the standard symmetrized variance for the pure state which is independent of the diagonal observable. We then extend the standard symmetrized variance to mixed states in three different ways, which characterize the uncertainty, the coherence and the coherence of assistance, respectively. These quantities are evaluated analytically and the relations among them are established. In addition, we show that the standard symmetrized variance is also an entanglement measure for bipartite systems. In this way, these different quantumness of quantum states are unified by the variance.