A maximum value for the Kullback-Leibler divergence between quantum discrete distributions

This work presents an upper-bound for the maximum value that the Kullback-Leibler (KL) divergence can reach for a newly introduced class of probability distributions, called quantum distributions (QD). In particular, the aim is to find a discrete distribution $U$ which maximize the finite KL divergence from a given $P$ under the assumption that $P$ and $U$ have been generated by distributing a fixed discretized quantity. The theoretical findings are used for proposing a notion of normalized KL divergence that is empirically shown to behave differently from already known measures.