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Generalized Numerical Construction of MUBs: A Group Theoretical Investigation

Buğra Gültekin, Solomon B. Samuel, Zafer Gedik

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 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 in dimension $d$ 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 valid 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. Though the scope of the search was limited, in dimension $d = 6$ our numerical search yielded no MUBs within explored parameter space.

Generalized Numerical Construction of MUBs: A Group Theoretical Investigation

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 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 MUBs in dimension 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 valid 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 , , and 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. Though the scope of the search was limited, in dimension our numerical search yielded no MUBs within explored parameter space.

Paper Structure

This paper contains 18 sections, 40 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Analytical Construction
  3. Operator Algebraic Method
  4. Weyl-Heisenberg group
  5. Prime dimensions
  6. Prime-power dimensions
  7. Generalized Numerical Construction
  8. Gram Matrix
  9. Trace Functions
  10. Construction
  11. Isolated and Continuous Families of Solutions
  12. Group Theoretical Classification: Symmetries and Triple Products
  13. Bargmann Invariants and Triple Product Tensor
  14. Permutation Symmetry
  15. Symmetry Group of the Known Solutions
  16. ...and 3 more sections

Figures (3)

  • Figure 1: Commuting classes in dimension 3: points $(p_1,p_2)\in\mathbb{Z}_3^2$ and lines through the origin representing $\mathcal{C}_1,\dots,\mathcal{C}_4$. This geometric structure is a form of finite affine plane.
  • Figure 2: Commuting classes in dimension 4. The intersection points are marked with $\times$.
  • Figure 3: Generating set elements of the triple product tensor plotted on the unit circle (left column) and their distributions in the histogram form (right column), for dimensions $d=3$ (top), $d=4$ (middle), and $d=5$ (bottom). Red dots mark the distinct phases.