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

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 $\le \frac{1}{\sqrt{d}}(1+O(d^{-\frac{1}{4}}))<2$, for proper choices of parameters, the class of dimensions $d$ being infinitely large.