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 dot or scalar product $|\braket{v_1|v_2}| = \frac{1}{\sqrt{d}}$. The upper bound on the number of such bases is $d+1$. Construction methods to achieve this bound are known for cases 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 Resolvable Block Designs (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 of the order of $\mathcal{O}(d^{3/2})$ in numbers, having high angular distances among them.

翻译：在维度 $d$ 中，互无偏基（MUBs）是 $\mathbb{C}^d$ 上的一组正交基，使得对于任意两个属于不同基的向量 $v_1, v_2$，其点积或标量积满足 $|\braket{v_1|v_2}| = \frac{1}{\sqrt{d}}$。此类基数量的上界为 $d+1$。当 $d$ 为素数幂时，已知有构造方法能达到该上界。但在其他情况，以及考虑实数域而非复数域的结果时，限制更为严格。因此，文献中考虑对该模型的某些松弛，进而研究了近似互无偏基（AMUB）。这使得在 $\mathbb{C}^d$ 和 $\mathbb{R}^d$ 中都能构造出大量此类对象。据此，我们提出近乎完美互无偏基（APMUB）的概念，其中将内积绝对值 $|\braket{v_1|v_2}|$ 限制为双值，即 0 和 $\leq \frac{1+\mathcal{O}(d^{-\lambda})}{\sqrt{d}}$，且 $\lambda > 0$，分子 $1 + \mathcal{O}(d^{-\lambda}) \leq 2$。所构造的每个向量具有重要特征：其大部分分量为零，而非零分量大小相等。我们的方法基于与可分解区组设计（RBDs）相关的组合结构。我们证明，对于若干复合维度 $d$，可构造 $\mathcal{O}(\sqrt{d})$ 个 APMUB，而在这些情况下 MUB 的数量显著较小。具体而言，该结果适用于形如 $(q-e)(q+f)$ 的 $d$，其中 $q, e, f \in \mathbb{N}$，满足 $0 \leq f \leq e$（$e, f$ 为常数），且 $q$ 为素数幂。我们还证明，此类 APMUB 可提供数量级为 $\mathcal{O}(d^{3/2})$ 的双角向量集，且这些向量间具有高角距离。