On Construction of Approximate Real Mutually Unbiased Bases for an infinite class of dimensions $d \not\equiv 0 \bmod 4$

TL;DR

This paper demonstrates the construction of more than lb1 lb7 lb7 lb7 ARMUBs in certain infinite classes of dimensions where real MUBs do not exist, using Hadamard matrices and orthogonal matrices.

Contribution

It introduces a novel method to construct approximate real MUBs in dimensions not divisible by 4, expanding possibilities for quantum information applications.

Findings

Constructed > lb1 lb7 lb7 lb7 ARMUBs for specific odd dimensions.

Utilized real Hadamard matrices to build orthogonal matrices with near-uniform entries.

Achieved basis inner product bounds below 2, applicable to infinitely many dimensions.

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 $> \lceil \sqrt{d} \rceil$ 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 $4n \times 4n$ real Hadamard matrix $H_{4n}$ (conjectured to be true) and uses this to construct an orthogonal matrix ${Y}_{4n-t}$ of size $(4n - t) \times (4n - t)$, such that the absolute value of each entry varies a little from $\frac{1}{\sqrt{4n-t}}$. In our construction, the absolute value of the inner product between any pair of basis vectors from two different ARMUBs will be $\leq \frac{1}{\sqrt{d}}(1 + O(d^{-\frac{1}{4}})) < 2$, for proper choices of parameters, the class of dimensions $d$ being infinitely large.