Reformulation for arbitrary mixed states of Jones' Bayes estimation of pure states

Jones has cast the problem of estimating the pure state |ψ〉 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 φk, k = 1,,…,N, from an N-trial analyzer, that is a collection of N bases of the Hilbert space Cd. The desired posterior/inferred distribution is then simply proportional to the likelihood of Πk = 1N |〈ψ|φk〉|2. Here, Jones' approach is extended to “the more realistic experimental case of mixed input states.” As the (unnormalized) prior over the d × d density matrices (ϱ), the recently-developed reparameterization and unitarily-invariant measure, |ϱ|2d + 1, is utilized. The likelihood is then taken to be Πk = 1N 〈φk|ϱ|φk〉, reducing to that of Jones when ϱ corresponds to a pure state. 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.