In dimension $d$, Mutually Unbiased Bases (MUBs) are a collection of orthonormal bases over $\mathbb{C}^d$ such that for any two vectors $v_1, v_2$ belonging to different bases, the scalar product $|\braket{v_1|v_2}| = \frac{1}{\sqrt{d}}$. The upper bound on the number of such bases is $d+1$. Constructions to achieve this bound are known when $d$ is some power of prime. The situation is more restrictive in other cases and also when we consider the results over real rather than complex. Thus, certain relaxations of this model are considered in literature and consequently Approximate MUBs (AMUB) are studied. This enables one to construct potentially large number of such objects for $\mathbb{C}^d$ as well as in $\mathbb{R}^d$. In this regard, we propose the concept of Almost Perfect MUBs (APMUB), where we restrict the absolute value of inner product $|\braket{v_1|v_2}|$ to be two-valued, one being 0 and the other $ \leq \frac{1+\mathcal{O}(d^{-\lambda})}{\sqrt{d}}$, such that $\lambda > 0$ and the numerator $1 + \mathcal{O}(d^{-\lambda}) \leq 2$. Each such vector constructed, has an important feature that large number of its components are zero and the non-zero components are of equal magnitude. Our techniques are based on combinatorial structures related to RBDs. We show that for several composite dimensions $d$, one can construct $\mathcal{O}(\sqrt{d})$ many APMUBs, in which cases the number of MUBs are significantly small. To be specific, this result works for $d$ of the form $(q-e)(q+f), \ q, e, f \in \mathbb{N}$, with the conditions $0 \leq f \leq e$ for constant $e, f$ and $q$ some power of prime. We also show that such APMUBs provide sets of Bi-angular vectors which are $\mathcal{O}(d^{\frac{3}{2}})$ in numbers, having high angular distances among them. Finally, as the MUBs are equivalent to a set of Hadamard matrices, we show that the APMUBs are so with the set of Weighing matrices.

翻译：在维数 $d$ 中，无偏基（MUBs）是 $\mathbb{C}^d$ 上一组标准正交基的集合，使得对于任意两个属于不同基的向量 $v_1, v_2$，其内积的绝对值 $|\braket{v_1|v_2}| = \frac{1}{\sqrt{d}}$。此类基的数量上界为 $d+1$。当 $d$ 是素数的幂时，已有构造达到此上界。在其他情况下，以及考虑实数域而非复数域时，条件更为严格。因此，文献中考虑了该模型的某些松弛形式，并研究了近似无偏基（AMUB）。这使得我们能在 $\mathbb{C}^d$ 和 $\mathbb{R}^d$ 中构造可能大量存在的此类对象。在此背景下，我们提出几乎完美无偏基（APMUB）的概念，其中内积绝对值 $|\braket{v_1|v_2}|$ 被限制为两个值：0 和 $\leq \frac{1+\mathcal{O}(d^{-\lambda})}{\sqrt{d}}$，其中 $\lambda > 0$ 且分子 $1 + \mathcal{O}(d^{-\lambda}) \leq 2$。每个构造的向量具有重要特征：其大量分量为零，而非零分量幅值相等。我们的技术基于与可分解平衡不完全区组设计（RBD）相关的组合结构。我们证明，对于若干复合维数 $d$，可构造 $\mathcal{O}(\sqrt{d})$ 个 APMUBs，而在这些情况下 MUBs 的数量显著较少。具体而言，该结果适用于形式为 $(q-e)(q+f)$ 的 $d$，其中 $q, e, f \in \mathbb{N}$，条件 $0 \leq f \leq e$ 且 $e, f$ 为常数，$q$ 为某素数的幂。我们还证明此类 APMUBs 可提供数量为 $\mathcal{O}(d^{\frac{3}{2}})$ 的双角向量集，且它们之间具有高角距离。最后，由于 MUBs 等价于一组 Hadamard 矩阵，我们证明 APMUBs 等价于一组称量矩阵。