Quantum mechanical retrodiction through an extended mean King problem

The mean King problem is a conditional retrodiction problem. In this problem Alice prepares a two prime-dimensional particles state and avails one of the particles to the King who measures its state in one of mutually unbiased bases of his choice. The King tells Alice his choice of basis after she completes a control measurement on his particle. Conditioned on this knowledge, she now infers the state observed by the King by utilizing the outcome of her control measurement. In the extended mean King problem, studied in this paper, the King does not tell Alice his measurement basis, but instead both the King and Alice repeat their measurements. Proper ordering of these allows Alice to deduce both the basis used by the King and the outcome of his first measurement, with the King reticent throughout, i.e., this protocol effects a complete (almost) retrodiction of the King's first measurement.