Solution to the King’s Problem in Prime Power Dimensions

The King’s Problem [1] is the following: A physicist is trapped on an island ruled by a mean king who promises to set her free if she can give him the answer to the following puzzle. The physicist is asked to prepare a d-state quantum system in any state of her choosing and give it to the king, who measures one of several sets of mutually unbiased observables (this term will be defined below) on it. Following this, the physicist is allowed to make a control measurement on the system, as well as any other systems it may have been coupled to in the preparation phase. The king then reveals which set of observables he measured and the physicist is required to predict correctly all the eigenvalues he found. A special case of this problem was first introduced and solved in [2] for the case of d = 2: there the king is given a spin-1/2 particle (or qubit) and allowed to measure one of the spin components σ x,σy, or σz on it. Some variants of this basic problem were discussed in [3 – 5]. Then a solution to the problem for d = 3 was presented in [6], following which a solution for arbitraryan prime d was presented in [1]. The purpose of the present paper is to generalize the solution in [1] to arbitrary prime power dimensions. Let A and B be two observables of a d-state quantum system with orthonormal eigenstates {|αi〉} and {|βi〉}, respectively. These observables (and their eigenstates) are said to be mutually unbiased [7] if the inner products of all pairs of eigenstates across the two bases have the same magnitude, i. e. if |〈βi|α j〉| = 1/ √ d for all i and j. The observables are also sometimes spoken of as being “mutually complementary” or “maximally noncommutative” [8] because, given any eigenstate of one,