A Meaner King uses Biased Bases

Abstract

The mean king problem is a quantum mechanical retrodiction problem, in which Alice has to name the outcome of an ideal measurement on a d-dimensional quantum system, made in one of (d+1) orthonormal bases, unknown to Alice at the time of the measurement. Alice has to make this retrodiction on the basis of the classical outcomes of a suitable control measurement including an entangled copy. We show that the existence of a strategy for Alice is equivalent to the existence of an overall joint probability distribution for (d+1) random variables, whose marginal pair distributions are fixed as the transition probability matrices of the given bases. In particular, for d=2 the problem is decided by John Bell's classic inequality for three dichotomic variables. For mutually unbiased bases in any dimension Alice has a strategy, but for randomly chosen bases the probability for that goes rapidly to zero with increasing d.