Mutually Unbiased Quantum Observables

We begin by defining mutually unbiased (MU) observables on a finite dimensional Hilbert space. We also consider the more general concept of parts of MU observables. The relationships between MU observables, value-complementary observables and two other conditions involving sequential products of observables are discussed. We next present a special motivating case of MU observables called finite position and momentum observables. These are atomic observables related by a finite Fourier transform. Finite position and momentum observables are employed to give examples of parts of MU observables that are value-complementary and those that are not value-complementary. Various open problems involving these concepts are presented. These problems mainly involve extending this work from sharp observables to unsharp observables.