Average of uncertainty-product for observables

Lin Zhang, Jiamei Wang

Published 2016-08-26Version 1

The goal of this paper is to calculate exactly the average of uncertainty-product of observables and to establish its typicality over the whole set of finite dimensional quantum pure states. Firstly, we investigate the average uncertainty of an observable over isospectral density matrices with a fixed spectrum. By letting the isospectral density matrices be of rank-one, we get the average uncertainty of an observable restricted to pure quantum states. Physically, ensemble of particles of large number as a closed system is represented by mixed state $\rho$. When we measure observable $A$ at a mixed state $\rho$ with many repetitions, we suggest that each time we measure $A$ at a point within the isospectral density matrices, i.e. the unitary orbit $\mathcal{U}_\rho$ of $\rho$. Thus it is suitable for taking average of uncertainty of observable $A$ over the whole unitary orbit $\mathcal{U}_\rho$. Based on this result, we finally get the calculation of the average of uncertainty-product over the whole set of mixed quantum states.