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<p>The structural characterization of high-dimensional mutually unbiased bases (MUBs) by classifying MUB 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 MUB subsets, which has less time complexity and entirely avoids the computational precision issues. For arbitrary MUB subsets of <i>k</i> 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 <i>d</i> ⩽ 37. Crucially, by combining these upper bounds with lower bounds obtained from the Shannon entropy criterion, one obtains the exact classification for all MUB subsets in any dimension <i>d</i> ⩽ 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 MUB subsets sheds new light on related applications.</p>

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Efficient identification of the inequivalence of mutually unbiased bases via finite operators

  • 期刊论文

  • 发表日期：

    2026年5月19日

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  • Jianxin Song,
  • Zhen-Peng Xu,
  • Changliang Ren

摘要

The structural characterization of high-dimensional mutually unbiased bases (MUBs) by classifying MUB 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 MUB subsets, which has less time complexity and entirely avoids the computational precision issues. For arbitrary MUB 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 ⩽ 37. Crucially, by combining these upper bounds with lower bounds obtained from the Shannon entropy criterion, one obtains the exact classification for all MUB subsets in any dimension d ⩽ 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 MUB subsets sheds new light on related applications.