On the Impossibility to Extend Triples of Mutually Unbiased Product Bases in Dimension Six

TL;DR

This paper proves analytically that in dimension six, it is impossible to extend certain sets of mutually unbiased product bases, providing strong results without relying on computer algebra.

Contribution

It introduces a novel analytical approach to demonstrate the impossibility of extending MU product bases in dimension six, strengthening previous results.

Findings

Impossible to extend MU product bases in dimension six by a single vector

Removing two states from a MU triple prevents forming a complete set of seven MU bases

Results are among the strongest analytical proofs without computer algebra in this context

Abstract

An analytic proof is given which shows that it is impossible to extend any triple of mutually unbiased (MU) product bases in dimension six by a single MU vector. Furthermore, the 16 states obtained by removing two orthogonal states from any MU product triple cannot figure in a (hypothetical) complete set of seven MU bases. These results follow from exploiting the structure of MU product bases in a novel fashion, and they are among the strongest ones obtained for MU bases in dimension six without recourse to computer algebra.