On the mathematical foundations of mutually unbiased bases

  • Koen Thas


In order to describe a setting to handle Zauner’s conjecture on mutually unbiased bases (MUBs) (stating that in \(\mathbb {C}^d\), a set of MUBs of the theoretical maximal size \(d + 1\) exists only if d is a prime power), we pose some fundamental questions which naturally arise. Some of these questions have important consequences for the construction theory of (new) sets of maximal MUBs. Partial answers will be provided in particular cases; more specifically, we will analyze MUBs with associated operator groups that have nilpotence class 2, and consider MUBs of height 1. We will also confirm Zauner’s conjecture for MUBs with associated finite nilpotent operator groups.


Mutually unbiased base Zauner’s conjecture Pauli group Operator group Nilpotence 



I want to thank Markus Grassl for many helpful communications on the subject of the present paper. In particular, he provided valuable comments on the definition of “height” in Sect. 5 in a previous version of this note. I also want to thank two anonymous referees for several useful suggestions.


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Authors and Affiliations

  1. 1.Department of MathematicsGhent UniversityGhentBelgium

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