Further Constructions of AMUBs for Non-prime power Composite Dimensions
Ajeet Kumar, Subhamoy Maitra
TL;DR
This work addresses the challenge of constructing large mutually unbiased base sets in non-prime-power composite dimensions by leveraging Resolvable Block Designs (RBD). It develops two construction regimes—non-constant and constant block sizes—to produce beta-AMUBs (and ARMUBs when Hadamard matrices exist) with beta typically close to 1 and sparsity near 1; in particular, it shows there exist $\mathcal{O}(\sqrt{d})$ such bases for dimensions with $d=k\,s$ and $|s-k|<\sqrt{d}$, when block sizes are around $\sqrt{d}$. The results provide explicit beta and sparsity bounds in terms of $\mu$ (intersection number) and block sizes, and extend previous real-case ARMUBs to complex settings, aided by new RBD constructions. Overall, the paper advances AMUB/ARMUB theory for broad composite dimensions and clarifies when Hadamard-order availability yields additional ARMUBs, with bases that remain highly sparse and near-MUB in quality.
Abstract
Construction of a large class of Mutually Unbiased Bases (MUBs) for non-prime power composite dimensions ($d = k\times s$) is a long standing open problem, which leads to different construction methods for the class Approximate MUBs (AMUBs) by relaxing the criterion that the absolute value of the dot product between two vectors chosen from different bases should be $\leq \fracβ{\sqrt{d}}$. In this chapter, we consider a more general class of AMUBs (ARMUBs, considering the real ones too), compared to our earlier work in [Cryptography and Communications, 14(3): 527--549, 2022]. We note that the quality of AMUBs (ARMUBs) constructed using RBD$(X,A)$ with $|X|= d$, critically depends on the parameters, $|s-k|$, $μ$ (maximum number of elements common between any pair of blocks), and the set of block sizes. We present the construction of $\mathcal{O}(\sqrt{d})$ many $β$-AMUBs for composite $d$ when $|s-k|< \sqrt{d}$, using RBDs having block sizes approximately $\sqrt{d}$, such that $|\braket{ψ^l_i|ψ^m_j}| \leq \fracβ{\sqrt{d}}$ where $β= 1 + \frac{|s-k|}{2\sqrt{d}}+ \mathcal{O}(d^{-1}) \leq 2$. Moreover, if real Hadamard matrix of order $k$ or $s$ exists, then one can construct at least $N(k)+1$ (or $N(s)+1$) many $β$-ARMUBs for dimension $d$, with $β\leq 2 - \frac{|s-k|}{2\sqrt{d}}+ \mathcal{O}(d^{-1})< 2$, where $N(w)$ is the number of MOLS$(w)$. This improves and generalizes some of our previous results for ARMUBs from two points, viz., the real cases are now extended to complex ones too. The earlier efforts use some existing RBDs, whereas here we consider new instances of RBDs that provide better results. Similar to the earlier cases, the AMUBs (ARMUBs) constructed using RBDs are in general very sparse, where the sparsity $(ε)$ is $1 - \mathcal{O}(d^{-\frac{1}{2}})$.