A quantitative algebraic analysis of BB’84 with maximal entropy

The paper provides a quantitative algebraic analysis of a BB’84-type quantum key distribution protocol. The analysis is done in an algebraic setting, where classical and quantum variables form a module for the quantale formed from the communication and quantum actions. The module-quantale pair is endowed with sup-maps that encode uncertainties of agents involved in the protocol, about the variables and about the actions. The right adjoint to the action of the quantale on the module provides a dynamic modality, read as “after”. The right adjoints to the uncertainty maps provide epistemic modalities, read as “belief” of agents. Using these and the axioms of the algebra, we can express and verify whether the agents share a secret after running the protocol. The need for probabilities is felt, since in the presence of an intruder, agents cannot fully share their secret. We enter quantities into the analysis via degrees/probabilities of belief. These probabilities are derived from the number of choices that an agent has about actions and propositions involved in the protocol, these include actions of an intruder. For simplicity, we have assumed that the choices have a uniform chance of happening, hence as if assuming that the entropies of agents’ choice sets are maximal. Using these probabilities, we show how the purpose of the actions in the protocol is to increase the agents’ degrees of belief and to decrease the intruder’s degree of belief. We show how a classical version of the protocol, in which the intruder can copy the passing qbit, is less efficient, since the intruder is able to obtain a higher degree of belief there. We also show how security amplification’s role is to decrease the intruder’s degree of belief.