Orthogonal Polynomials on the Unit Circle, Mutually Unbiased Bases, and Balanced States

Abstract Two interesting phenomena for the construction of quantum states are that of mutually unbiased bases and that of balanced states. We explore a constructive approach to each phenomenon that involves orthogonal polynomials on the unit circle. In the case of mutually unbiased bases, we show that this approach does not produce such bases. In the case of balanced states, we provide examples of pairs of orthonormal bases and states that are balanced with respect to them. We also consider extensions of these ideas to the infinite dimensional setting.