The Kullback-Leibler divergence as the logarithm of the quantum bit error rate and the lost of information at the BB84 protocol

We demonstrate that the KullbackLeibler Divergence emerges in a natural manner from a theory of entropy inside of BB84 protocol under the assumption that the sending and receiving of data are governed by polarized photons without any explicit law that defines the coincidence of bits in both parties. More than an analysis that explore the concepts of encryption, in this paper we focus on the possible apparition of events of order and disorder. In this manner we built a formalism entirely based on the entropy from random events that are generated as part of the strategy of security. For a photon linear polarization, the entropy might be high enough that would be a source of errors as reflected on the lost of information by a 10% due to polarization superposition of states.