Explicit constructions of mutually unbiased bases via Hadamard matrices

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.

• The paper introduces analytic and computational methods for constructing mutually unbiased bases using Hadamard matrices and phase parameters.
• It demonstrates explicit constructions in dimensions 2, 3, and 4 while highlighting structural barriers in the non-prime-power dimension 6.
• The study leverages algebraic techniques and group theory to provide practical criteria for verifying mutual unbiasedness in quantum systems.

Explicit Constructions of Mutually Unbiased Bases via Hadamard Matrices

Introduction

This work addresses the explicit construction of mutually unbiased bases (MUBs) in finite-dimensional Hilbert spaces, focusing on analytic and computational frameworks that leverage Hadamard matrices and phase parametrizations. Particular attention is given to the tractable dimensions $d=2,3,4$ and the analytically intransigent case $d=6$. The study characterizes the origin of MUBs both via direct matrix computation and through their algebraic structure (finite fields, Pauli groups, Weyl-Heisenberg operators), and provides constructive criteria for verifying MUB conditions, elucidating why the construction of maximal sets is dimension-sensitive.

Analytical and Computational Construction of MUBs in Low Dimensions

Dimension 2

For $d=2$, the full set of three MUBs is constructed via the computational basis and normalized columns of the standard Hadamard matrix $H_2$ and its variants under diagonal phase operation. A line-by-line computation confirms that all basis pairs satisfy $|\langle e|f\rangle|^2 = 1/2$, confirming mutual unbiasedness. This serves as the canonical archetype, exploiting the isomorphism between orthonormal vectors and Hadamard structures.

Dimension 3

In $d=3$, the Weyl--Heisenberg construction is employed, utilizing the operators $X$ and $Z$ with the relation $ZX = \omega XZ$ for $\omega = e^{2\pi i /3}$. The resulting set features the computational basis (eigenbasis of $d=6$0) and three further bases as eigenbases of $d=6$1, $d=6$2, and $d=6$3. The explicit calculation of overlaps demonstrates uniform modulus-squared entries $d=6$4 for all off-diagonal basis pairs, with the group structure manifesting the full MUB set.

Dimension 4

For $d=6$5, the construction proceeds by tensor-product extension $d=6$6, yielding a four-dimensional Hadamard matrix whose normalized columns define MUBs. The core innovation here is the explicit introduction of a diagonal phase matrix $d=6$7: the resulting basis $d=6$8 can be continuously parametrized by $d=6$9. The paper derives analytical trigonometric conditions under which two such parametrized bases are mutually unbiased. This parametric family highlights the surplus degrees of freedom in composite, prime-power Hilbert spaces, and the conditions are reduced to polynomial constraints in $d=2$0, and their pairwise differences.

Further, the Pauli group structure on $d=2$1 is shown to result in a classification of commuting sets, each corresponding to a unique MUB. This algebraic perspective is mapped to the Hadamard-phase formalism, evidencing the underlying group-theoretical unity.

The Structure and Barriers in Dimension 6

In $d=2$2, the construction attempts using the Fourier matrix $d=2$3 and phase parametrization fail to generalize the $d=2$4 flexibility. The mutual unbiasedness condition between candidate bases leads to a nonlinear system for the six phases, and only three mutually unbiased bases are constructible analytically. The rigidity is attributed to the non-prime-power nature of 6, as $d=2$5 does not possess a product structure akin to lower dimensions. The existence of isolated complex Hadamard matrices and the lack of continuous families in $d=2$6 is discussed, corroborating established results about the maximal set problem.

Generalization to Prime-Power Dimensions

The work revisits the finite field construction for $d=2$7, where $d=2$8 MUBs can always be obtained using eigenstructures of the generalized Pauli group and trace maps over $d=2$9. For prime $H_2$0, the classical $H_2$1 construction is used; for higher prime powers, the methodology exploits the existence of $H_2$2 maximal abelian subgroups, whose common eigenbases yield the full MUB set. The methodology is infeasible for dimensions lacking a finite field structure, explaining the obstruction in $H_2$3, and the algebraic gap is rigorously formalized.

Connections are established with the representation theory of the Heisenberg-Weyl group and the structure of $H_2$4 Lie algebras, with MUBs corresponding to maximal commuting operator sets or discrete phase space striations.

Theoretical and Practical Implications

The explicit constructions and verification protocols developed provide transparent tools for analyzing MUBs and underscore the structural reasons for their existence or absence in specific dimensions. The analytic conditions on phase parameters in dimension 4 serve as a concrete method for classifying or enumerating MUBs, and the computational approach offers a platform for systematic exploration beyond analytically solvable cases.

The insights into the algebraic and geometric obstructions at non-prime-power dimensions sharpen the theoretical understanding of the maximal MUB set problem and inform future numerical and group-theoretical investigations. The explicit demonstration of these obstructions—both via matrix computation and symmetry analysis—renders the study a valuable resource for foundational and applied research, including quantum state tomography and quantum cryptographic protocol design.

Conclusion

This study constitutes a technically detailed resource for researchers investigating mutually unbiased bases and their explicit construction. By detailing both computational and algebraic strategies, it clarifies why certain dimensions yield a maximal set of MUBs while others do not, with particular focus on the analytic parametrization in dimension 4 and the structural barriers in dimension 6. The explicit algebraic and computational formulations presented provide effective means of constructing, classifying, and verifying MUBs and expose the deep interplay between Hilbert space geometry, group representations, and operator theory. Future work will likely continue to leverage and generalize these frameworks in both foundational studies and quantum technology implementations.

Reference: "Explicit constructions of mutually unbiased bases via Hadamard matrices" (2604.02234).