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

Standard symmetrized variance with applications to coherence, uncertainty and entanglement