MSc Thesis Defense: Buğra Gültekin, ALGEBRAIC AND GENERALIZED NUMERICAL CONSTRUCTIONS OF MUBS: TOWARDS AN EXPLORATION IN NON-PRIME DIMENSIONS

ALGEBRAIC AND GENERALIZED NUMERICAL CONSTRUCTIONS OF MUBS: TOWARDS AN EXPLORATION IN NON-PRIME DIMENSIONS

Buğra Gültekin
Physics, MSc. Thesis, 2025

Thesis Jury

Prof. Dr. Zafer Gedik (Thesis Advisor)

Doç. Dr. Göktuğ Karpat

Dr. Öğr. Üyesi Onur Pusuluk

Date & Time: December 17th, 2025 – 11:30 AM

Place: FENS G025

Keywords : MUBs, Gram Matrix, Bargmann Invariants, Triple Products

Abstract

Mutually Unbiased Bases (MUBs) are special type of basis and one of the fundamental structures in quantum information theory, providing the optimal measurement scheme for quantum state tomography. While analytical recipes exist for constructing the maximal set of d+1 MUBs in prime and prime-power dimensions, the existence of such maximal sets in non-prime dimensions remains an open question. This thesis explores the algebraic and geometric structures of MUBs to address this existence problem. The study first reviews established analytical methods, including the operator-algebraic construction based on the Weyl-Heisenberg (WH) group and the combinatorial approach utilizing relative difference sets. Following this, a generalized numerical construction method is introduced. Unlike traditional methods, this approach generates MUBs via the Gram matrix without relying on a priori group structure or specific algebraic frameworks. The construction enforces the geometric properties of MUBs through scalar constraints derived from third and fourth-order traces of the Gram matrix. To analyze the resulting structures, a classification framework is employed using Bargmann invariants —specifically the triple product tensor — and their automorphism groups. This method allows for the identification of inequivalent MUB families based on their inherent symmetries. Numerical applications of this method in dimensions 3, 4, and 5 demonstrate that all constructed solutions have the underlying group structure of Weyl-Heisenberg group, possessing symmetry groups isomorphic to the Clifford group. While the investigation in dimension 6 yielded no maximal MUB sets within the limited scope of numerical trials, the developed framework provides a robust tool for future exploration of non-prime dimensions and the classification of MUBs.