A Lambda Calculus for Density Matrices with Classical and Probabilistic Controls

Abstract

In this paper we present two flavors of a quantum extension to the lambda calculus. The first one, \(\lambda _\rho \), follows the approach of classical control/quantum data, where the quantum data is represented by density matrices. We provide an interpretation for programs as density matrices and functions upon them. The second one, \(\lambda _\rho ^\circ \), takes advantage of the density matrices presentation in order to follow the mixed trace of programs in a kind of generalised density matrix. Such a control can be seen as a weaker form of the quantum control and data approach.