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arXiv:2604.04164 · Journal of Physics A Mathematical and Theoretical

Generalized Numerical Construction of MUBs: A Group Theoretical Investigation

B. Gültekin, S. B. Samuel, Z. Gedik

quant-phSubmitted Apr 5, 2026• Announced Apr 7, 2026• 0 citations• Revised Sep 7, 2026 (v2)
v1 · Apr 5, 2026v2 · Sep 7, 2026
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Abstract

Mutually Unbiased Bases (MUBs) constitute a fundamental geometric structure in quantum theory, known for providing an optimal measurement scheme for quantum state tomography. In prime and prime-power dimensions, analytical constructions of maximal sets of MUBs are well-known and standard construction relies on the Weyl-Heisenberg (WH) group and finite fields. In non-prime-power dimensions, on the other hand, the existence of such maximal sets remains an open question. We present a generalized numerical method of constructing MUBs without any reliance on \textit{a priori} group structure or specific algebraic frameworks. Formulating the problem at the level of Gram matrix, we reduce the search for complete sets of d+1 MUBs to a phase space optimisation problem. We use the fact that the MUB Gram matrix is a projection matrix, and the third- and fourth-order trace constraints are necessary and sufficient conditions for a projection matrix. We further develop a classification framework based on third-order Bargmann invariants and automorphism groups, allowing us to probe the underlying algebraic and geometric structure of the resulting configurations. Numerical applications of this method in dimensions $3$, $4$, and $5$ demonstrate that all numerically constructed solutions are mutually isomorphic, are isolated points in phase space, and possess automorphism groups that coincide exactly with the Clifford group, the normalizer of the WH group. In the lowest non-prime-power dimension $6$ where the existence of a complete set remains open, our numerical search yields no solution. However, for incomplete sets of solutions, our search finds no set of four MUBs, but it yields several inequivalent triples, which we classify and whose neighbourhood we then investigate.

Journal reference
J. Phys. A: Math. Theor. 59, 365303 (2026)
DOI
10.1088/1751-8121/ae99a0
Comments
22 Pages, 4 Figures, 5 Tables

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The structural characterization of high-dimensional mutually unbiased bases (MUBs) by classifying MUBs subsets remains a major open problem. The existing methods not only fail to conclude on the exact classification, but also are severely limited by computational resources and suffer from the numerical precision problem. Here we introduce an operational approach to identify the inequivalence of MUBs subsets, which has less time complexity and entirely avoids the computational precision issues. For arbitrary MUBs subsets of $k$ elements in any prime dimension, this method yields a universal analytical upper bound for the amount of MUBs equivalence classes. By applying this method through simple iterations, we further obtain tighter classification upper bounds for any prime dimension $d\leq 37$. Crucially, the comparison of these upper bounds with existing lower bounds successfully determines the exact classification for all MUBs subsets in any dimension $d \leq 17$. We further extend this method to the case that the dimension is a power of prime number. This general and scalable framework for the classification of MUBs subsets sheds new light on related applications.

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Mutually Unbiased Bases (MUBs) constitute a fundamental geometric structure in quantum theory, known for providing an optimal measurement scheme for quantum state tomography. In prime and prime-power dimensions, analytical constructions of maximal sets of MUBs are well-known and standard construction relies on the Weyl-Heisenberg (WH) group and finite fields. In non-prime-power dimensions, on the other hand, the existence of such maximal sets remains an open question. We present a generalized numerical method of constructing MUBs without any reliance on \textit{a priori} group structure or specific algebraic frameworks. Formulating the problem at the level of Gram matrix, we reduce the search for complete sets of d+1 MUBs to a phase space optimisation problem. We use the fact that the MUB Gram matrix is a projection matrix, and the third- and fourth-order trace constraints are necessary and sufficient conditions for a projection matrix. We further develop a classification framework based on third-order Bargmann invariants and automorphism groups, allowing us to probe the underlying algebraic and geometric structure of the resulting configurations. Numerical applications of this method in dimensions 333, 444, and 555 demonstrate that all numerically constructed solutions are mutually isomorphic, are isolated points in phase space, and possess automorphism groups that coincide exactly with the Clifford group, the normalizer of the WH group. In the lowest non-prime-power dimension 666 where the existence of a complete set remains open, our numerical search yields no solution. However, for incomplete sets of solutions, our search finds no set of four MUBs, but it yields several inequivalent triples, which we classify and whose neighbourhood we then investigate.
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.
Mutually Unbiased bases has various application in quantum information procession and coding theory. There can be maximum d + 1 MUBs in C^d and d/2 +1 MUBs in R^d. But , over R^d MUBs are known to be non existent when d is odd and for most of the other even d there are mostly 3 Real MUBs. In case of C^d the construction for complete set of MUBs are known for only Prime Power dimension. Thus in general large set of MUBs are not known, particularly for composite dimensions which are not of the form of prime powers. Because of this, there are many constructions of Approximate version of MUBs. In this paper we make an attempt to define certain measures to characterise the AMUBs. Our construction of measures derives its inspiration from the applications of MUBs, and based on them, we define certain quantifiable measures, which are can be computed and gives estimates of how close the Approximate MUBs are to the MUBs. We use geometric interpretation, projective design features of MUBs and applications like Optimal State determination and Entropic Uncertainty of MUBs. We show generic relationship between these measures and show that it can be evaluated for APMUBs without known the exact construction details, thereby showing that definition of APMUB is sufficient completely characterise it. We also evaluate these measure for an interesting class of AMUBs called Weak MUBs and certain AMUBs constructed using RBDs.
The connection between maximal sets of mutually unbiased bases (MUBs) in a prime-power dimensional Hilbert space and finite phase-space geometries is well known. In this article, we classify MUBs according to their degree of covariance with respect to the natural symmetries of a finite phase-space, which are the group of its affine symplectic transformations. We prove that there exist maximal sets of MUBs that are covariant with respect to the full group only in odd prime-power dimensional spaces, and in this case, their equivalence class is actually unique. Despite this limitation, we show that in dimension [Formula: see text] covariance can still be achieved by restricting to proper subgroups of the symplectic group, that constitute the finite analogues of the oscillator group. For these subgroups, we explicitly construct the unitary operators yielding the covariance.
Discrete structures in product Hilbert spaces are investigated. For monopartite systems of size ddd one relies on the Weyl--Heisenberg group WH(d)WH(d)WH(d), while in the case of composite Hilbert spaces with the local dimensions did_idi​ we identify designs covariant with respect to the product group, ⨂iWH(di)\bigotimes_i WH(d_i)⨂i​WH(di​). In analogy with magic -- a quantity attaining its maximum for states fiducial with respect to WH(d)WH(d)WH(d) -- we introduce a similar quantifier of {\sl product magic}, defined with respect to the product group. The maximum of this quantity over all equimodular vectors yields fiducial states that generate d=∏idid = \prod_i d_id=∏i​di​ \textit{a priori} isoentangled mutually unbiased bases (MUBs), which, when supplemented by the identity, form their complete set. Such fiducial states are explicitly constructed in all prime-power dimensions d=pnd = p^nd=pn with p≥3p\ge 3p≥3. The result for p≥5p\ge 5p≥5 extends the construction of Klappenecker and R{ö}tteler, whereas for p=3p=3p=3 it is mathematically distinct and is based on Galois rings. The global maximum of the quantifier of product magic for d=23d=2^3d=23 yields fiducial states corresponding to the symmetric informationally complete (SIC) generalized measurement of Hoggar. Our approach feeds into a unifying perspective in which highly symmetric quantum designs emerge from fiducial states with extremal properties via structured group-orbit constructions.
One way to construct a maximal set of mutually unbiased bases (MUBs) in a prime-power dimensional Hilbert space is by means of finite phase-space methods. MUBs obtained in this way are covariant with respect to some subgroup of the group of all affine symplectic phase-space transformations. However, this construction is not canonical: as a consequence, many different choices of covariance subgroups are possible. In particular, when the Hilbert space is 2n dimensional, it is known that covariance with respect to the full group of affine symplectic phase-space transformations can never be achieved. Here we show that in this case there exist two essentially different choices of maximal subgroups admitting covariant MUBs. For both of them, we explicitly construct a family of 2n covariant MUBs. We thus prove that, contrary to the odd dimensional case, maximally covariant MUBs are very far from being unique in even prime-power dimensions.
The structural characterization of high-dimensional mutually unbiased bases (MUBs) by classifying MUBs subsets remains a major open problem. The existing methods not only fail to conclude on the exact classification, but also are severely limited by computational resources and suffer from the numerical precision problem. Here we introduce an operational approach to identify the inequivalence of MUBs subsets, which has less time complexity and entirely avoids the computational precision issues. For arbitrary MUBs subsets of kkk elements in any prime dimension, this method yields a universal analytical upper bound for the amount of MUBs equivalence classes. By applying this method through simple iterations, we further obtain tighter classification upper bounds for any prime dimension d≤37d\leq 37d≤37. Crucially, the comparison of these upper bounds with existing lower bounds successfully determines the exact classification for all MUBs subsets in any dimension d≤17d \leq 17d≤17. We further extend this method to the case that the dimension is a power of prime number. This general and scalable framework for the classification of MUBs subsets sheds new light on related applications.