REFORMULATION FOR ARBITRARY MIXED STATES OF JONES BAYES ESTIMATION OFPURE STATES

Jones has cast the problem of estimating the pure state \psi] of a d-d imensional quantum system into a Bayesian framework. The normalized un iform ray measure over such states is employed as the prior distributi on. The data consist of observed eigenvectors phi k, k = 1,,..., N, fr om an N-trial analyzer, that is a collection of N bases of the Hilbert space Cd. Th, desired posterior/inferred distribution is then simply proportional to the likelihood of Pi(k=1)(N) \[psi\phi(k)\(2). Here, J ones' approach is extended to ''the more realistic experimental case o f mixed input states.'' As the (unnormalized) prior over the d x d den sity matrices (rho), the recently-developed reparameterization and uni tarily-invariant measure, \rho\(2d-1), is utilized. The likelihood is then taken to be Pi(k=1)(N) [phi(k)\rho\phi(k)], reducing to that of J ones when rho corresponds to a pure state. In the case of a pure state , however, the associated prior and posterior probabilities are then z ero. Some analytical results for the case d = 2 are presented.