Efficient Identification the Inequivalence of Mutually Unbiased Bases via Finite Operators

Abstract: The structural characterization of high-dimensional mutually unbiased bases (MUBs) by classifying MUBs subsets remains a major open problem. The existing methods not only fail to conclude on the exact classification, but also are severely limited by computational resources and suffer from the numerical precision problem. Here we introduce an operational approach to identify the inequivalence of MUBs subsets, which has less time complexity and entirely avoids the computational precision issues. For arbitrary MUBs subsets of $k$ elements in any prime dimension, this method yields a universal analytical upper bound for the amount of MUBs equivalence classes. By applying this method through simple iterations, we further obtain tighter classification upper bounds for any prime dimension $d\leq 37$. Crucially, the comparison of these upper bounds with existing lower bounds successfully determines the exact classification for all MUBs subsets in any dimension $d \leq 17$. We further extend this method to the case that the dimension is a power of prime number. This general and scalable framework for the classification of MUBs subsets sheds new light on related applications.