Explicit constructions of mutually unbiased bases via Hadamard matrices

• URL: http://arxiv.org/abs/2604.02234v1
• Date: Thu, 02 Apr 2026 16:28:23 GMT
• Title: Explicit constructions of mutually unbiased bases via Hadamard matrices
• Abstract summary: We present a detailed study of Mutually Unbiased Bases (MUBs) in finite-dimensional Hilbert spaces.<n>We focus on dimensions 2, 3, 4, and the challenging case of 6.<n>Our work offers a transparent, line-by-line verification methodology.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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