Proof of a Modified Jaynes's Estimation Theory

It is proved that the state of maximum entropy having observed values for the n observables, X1, . . . , Xn, is the same state that minimises the matrix of covariances of any n locally unbiased estimators for n parameters for the probability distribution of X1, . . . , Xn. We sketch how to get a similar result in quantum theory, in which X1, . . . , Xn are (not necessarily commuting) quadratic forms that are bounded relative to a positive self-adjoint operator H such that exp(−βH) is of trace-class for some positive β.