Kullback-Leibler divergence between quantum distributions, and its upper-bound
This work addresses a theoretical limitation in divergence measures for quantum distributions, which are used in practical applications, but it is incremental as it builds on existing concepts.
The paper tackles the problem of bounding the Kullback-Leibler divergence for quantum distributions, presenting an upper-bound under the condition that distributions are comparable over the same quantum value, and proposes a normalized KL divergence that behaves differently from existing measures.
This work presents an upper-bound to value that the Kullback-Leibler (KL) divergence can reach for a class of probability distributions called quantum distributions (QD). The aim is to find a distribution $U$ which maximizes the KL divergence from a given distribution $P$ under the assumption that $P$ and $U$ have been generated by distributing a given discrete quantity, a quantum. Quantum distributions naturally represent a wide range of probability distributions that are used in practical applications. Moreover, such a class of distributions can be obtained as an approximation of any probability distribution. The retrieving of an upper-bound for the entropic divergence is here shown to be possible under the condition that the compared distributions are quantum distributions over the same quantum value, thus they become comparable. Thus, entropic divergence acquires a more powerful meaning when it is applied to comparable distributions. This aspect should be taken into account in future developments of divergences. The theoretical findings are used for proposing a notion of normalized KL divergence that is empirically shown to behave differently from already known measures.