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

TL;DR

This paper investigates the use of orthogonal polynomials on the unit circle to construct quantum states, finding limitations for mutually unbiased bases and providing examples of balanced states, with extensions to infinite dimensions.

Contribution

It introduces a constructive approach using orthogonal polynomials for quantum state phenomena, highlighting its limitations and providing new examples of balanced states.

Findings

Orthogonal polynomials do not produce mutually unbiased bases.

Examples of balanced states with respect to certain bases are provided.

Extensions of these concepts to infinite-dimensional spaces are discussed.

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.