Characterizing maximal families of mutually unbiased bases

TL;DR

This paper characterizes maximal families of mutually unbiased bases using partitioned unitary error bases and introduces a new, simpler construction from finite fields, with novel tensor diagrammatic methods.

Contribution

It provides a comprehensive characterization of maximal MUBs via partitioned unitary error bases and offers a new finite field-based construction method.

Findings

Maximal MUBs are characterized by partitioned unitary error bases.

A new, simpler construction of MUBs from finite fields is proposed.

Tensor diagrammatic characterizations are introduced for these structures.

Abstract

We show that maximal families of mutually unbiased bases are characterized in all dimensions by partitioned unitary error bases, up to a choice of a family of Hadamards. Furthermore, we give a new construction of partitioned unitary error bases, and thus maximal families of mutually unbiased bases, from a finite field, which is simpler and more direct than previous proposals. We introduce new tensor diagrammatic characterizations of maximal families of mutually unbiased bases, partitioned unitary error bases, and finite fields as algebraic structures defined over Hilbert spaces.