Complementary Decompositions and Unextendible Mutually Unbiased Bases

TL;DR

This paper investigates the structure of unextendible mutually unbiased bases (MUBs) through the lens of complementary subalgebras, revealing new decompositions and establishing strong unextendibility results.

Contribution

It introduces new complementary decompositions of matrix algebras and demonstrates their role in proving the unextendibility of certain sets of MUBs.

Findings

Linear span of fewer than d+1 factors lacks pure states

Some complementary decompositions lead to unextendible MUB sets

Established strong unextendibility for specific MUB configurations

Abstract

Unextendible sets of Mutually Unbiased Bases (MUBs) are examined from the point of view of complementary subalgebras. We show, that the linear span of less than $d+1$ factors of $M_d \otimes M_d$ does not contain pure states, and therefore some complementary decompositions give rise to undextendable sets of MUBs. We provide some new complementary decompositions, and thus prove strong unextendibility of some set of MUBs.