Uncertainty relations for positive-operator-valued measures

How much unavoidable randomness is generated by a positive-operator-valued measure (POVM)? We address this question using two complementary approaches. First, we study the variance of a real variable associated with the POVM outcomes. In this context we introduce an uncertainty operator which measures how much additional noise is introduced by carrying out a POVM rather than a von Neumann measurement. We illustrate this first approach by studying the variances of joint estimates of sigma(x) and sigma(z) for spin-1/2 particles. We show that for unbiased measurements the sum of these variances is lower bounded by 1. In our second approach we study the entropy of the POVM outcomes. In particular, we try to establish lower bounds on the entropy of the POVM outcomes. We illustrate this second approach by examples.