A Maximum Value for the Kullback–Leibler Divergence between Quantized Distributions

The Kullback–Leibler (KL) divergence is a widely used measure for comparing probability distributions, but it faces limitations such as its unbounded nature and the lack of comparability between distributions with different quantum values (the discrete unit of probability). This study addresses these challenges by introducing the concept of quantized distributions, which are probability distributions formed by distributing a given discrete quantity or quantum. This study establishes an upper bound for the KL divergence between two quantized distributions, enabling the development of a normalized KL divergence that ranges between 0 and 1. The theoretical findings are supported by empirical evaluations, demonstrating the distinct behavior of the normalized KL divergence compared to other commonly used measures. The results highlight the importance of considering the quantum value when applying the KL divergence, offering insights for future advancements in divergence measures.