Almost Perfect Mutually Unbiased Bases that are Sparse

Article Type

Research Article

Publication Title

Journal of Statistical Theory and Practice

Abstract

Selected ideas of statistical designs are exploited in this paper in constructions related to Mutually Unbiased Bases (MUBs). In dimension d, MUBs are a collection of orthonormal bases over Cd such that for any two vectors v1,v2 belonging to different bases, the dot or scalar product |⟨v1|v2⟩|=1d. 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 Cd as well as in Rd. In this regard, we propose the concept of Almost Perfect MUBs (APMUB), where we restrict the absolute value of inner product |⟨v1|v2⟩| to be two-valued, one being 0 and the other ≤1+O(d-λ)d, such that λ>0 and the numerator 1+O(d-λ)≤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), that are used extensively in statistical designs. We show that for several composite dimensions d, one can construct O(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∈N, with the conditions 0≤f≤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 O(d32) 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.

DOI

10.1007/s42519-024-00414-2

Publication Date

12-1-2024

Comments

Open Access; Green Open Access

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