Searching for Mutually Unbiased Observables

Abstract

We call two non-degenerate observables “mutually unbiased” if every eigenstate of one of them is a state of complete ignorance relative to the other one. For example, for a spin-1/2 particle, the spin components in the x, y, and z directions are all unbiased with respect to each other, since a knowledge of any of these components entails a complete lack of knowledge about the others. It turns out that if, for a given system, there exist a sufficient number of mutually unbiased observables, then measurements of these observables can be used to determine as efficiently as possible the state of an ensemble of such systems. For a system with a finite number N of orthogonal states, the number of mutually unbiased observables one needs is N+1. We have shown that precisely N+1 such observables can indeed be found if N is a power of a prime. We do not know whether this number can be found if N is not a power of a prime.