Explicit constructions of mutually unbiased bases via Hadamard matrices

TL;DR

This paper provides explicit algebraic and computational methods for constructing mutually unbiased bases in various finite dimensions, emphasizing the special cases of 4 and 6, and connecting these to algebraic structures and symmetries.

Contribution

It introduces explicit analytical conditions for MUBs in dimension 4, explores tensor-product and Fourier-family constructions, and offers a systematic computational framework for testing candidate bases.

Findings

Complete set of 5 MUBs in dimension 4 via tensor-product construction.

Limited to 3 MUBs in dimension 6 due to structural constraints.

Systematic computational framework for testing phase vectors.

Abstract

We present a detailed computational and algebraic study of Mutually Unbiased Bases (MUBs) in finite-dimensional Hilbert spaces, with a particular focus on dimensions 2, 3, 4, and the challenging case of 6. Starting from the Hadamard-phase parametrization, we derive explicit analytical conditions for mutual unbiasedness in dimension 4, providing a tractable system of trigonometric constraints on the phase parameters. We then explore a tensor-product construction via Pauli operators, highlighting the algebraic and group-theoretical origin of MUBs in two-qubit systems, and demonstrating how these constructions yield a complete set of 5 MUBs in dimension 4. Extending our approach, we investigate the Fourier-family method in dimension 6, where the absence of a prime-power structure imposes strong rigidity constraints and limits the known constructions to sets of 3 MUBs. We provide a systematic computational framework for testing candidate phase vectors, bridging the gap between analytical insight and numerical exploration. Finally, we generalize the discussion to arbitrary prime-power dimensions, emphasizing the role of finite-field structures, Heisenberg-Weyl operators, and discrete symmetries in generating complete sets of MUBs. Our work offers a transparent, line-by-line verification methodology, highlighting both the geometric and algebraic richness of MUBs, and clarifying why certain dimensions resist full analytical constructions. This study serves as a comprehensive resource for researchers seeking both theoretical understanding and practical construction of MUBs in quantum information science.