REFORMULATION FOR ARBITRARY MIXED STATES OF JONES BAYES ESTIMATION OF PURE STATES

摘要

Jones has cast the problem of estimating the pure state psi] of a d-dimensional quantum system into a Bayesian framework. The normalized uniform ray measure over such states is employed as the prior distribution. The data consist of observed eigenvectors phi k, k = 1,,..., N, from 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) [psiphi(k)(2). Here, Jones' approach is extended to ''the more realistic experimental case of mixed input states.'' As the (unnormalized) prior over the d x d density matrices (rho), the recently-developed reparameterization and unitarily-invariant measure, ho(2d-1), is utilized. The likelihood is then taken to be Pi(k=1)(N) [phi(k)hophi(k)], reducing to that of Jones when rho corresponds to a pure state. In the case of a pure state, however, the associated prior and posterior probabilities are then zero. Some analytical results for the case d = 2 are presented. [References: 67]