On Construction of Approximate Real Mutually Unbiased Bases for an infinite class of dimensions $d otequiv 0 mod 4$

2025-07-09

AI Summary

Real mutually unbiased bases (MUBs) are known to not exist in dimensions $d > 2$ where $d otequiv 0 pmod{4}$, imposing a fundamental limitation on exact constructions. Method: We introduce a novel construction of approximate real MUBs (ARMUBs) using real Hadamard matrices to generate families of approximately orthogonal matrices, bounding the absolute inner product between any two vectors from distinct bases by $frac{1}{sqrt{d}}(1 + O(d^{-1/2}))$. Contribution/Results: For infinitely many odd dimensions—specifically those of the form $d = (4n - t)s$—we construct more than $lceil sqrt{d} ceil$ ARMUBs efficiently and with tunable parameters. The achieved inner-product bound asymptotically approaches the theoretical optimum. This work overcomes the strict dimensional constraints of classical MUB theory and substantially extends the feasibility of constructing high-dimensional approximately unbiased structures in the real setting.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Multiagent Systems: Adversarial Agents
• Intelligent Robots: Other Foundations and Applications

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

It is known that real Mutually Unbiased Bases (MUBs) do not exist for any dimension $d > 2$ which is not divisible by 4. Thus, the next combinatorial question is how one can construct Approximate Real MUBs (ARMUBs) in this direction with encouraging parameters. In this paper, for the first time, we show that it is possible to construct $> lceil sqrt{d} ceil$ many ARMUBs for certain odd dimensions $d$ of the form $d = (4n-t)s$, $t = 1, 2, 3$, where $n$ is a natural number and $s$ is an odd prime power. Our method exploits any available $4n imes 4n$ real Hadamard matrix $H_{4n}$ (conjectured to be true) and uses this to construct an orthogonal matrix ${Y}_{4n-t}$ of size $(4n - t) imes (4n - t)$, such that the absolute value of each entry varies a little from $frac{1}{4n-t}$. In our construction, the absolute value of the inner product between any pair of basis vectors from two different ARMUBs will be $leq frac{1}{sqrt{d}}(1 + O(d^{-frac{1}{2}})) < 2$, for proper choices of parameters, the class of dimensions $d$ being infinitely large.

Problem

• Construct Approximate Real MUBs for non-divisible-by-4 dimensions
• Develop ARMUBs with >√d bases for specific odd dimensions
• Ensure inner product bounds between ARMUB vectors are minimized

Innovation

• Constructs ARMUBs for odd dimensions d
• Uses real Hadamard matrices H_4n
• Ensures inner product bounds < 2