Compound Channels, Transition Expectations, and Liftings

Abstract. In Section 1 we introduce the notion of lifting as a generalization of the notion of compound state introduced in [21] and [22] and we show that this notion allows a unified approach to the problems of quantum measurement and of signal transmission through quantum channels. The dual of a linear lifting is a transition expectation in the sense of [3] and we characterize those transition expectations which arise from compound states in the sense of [22]. In Section 2 we characterize those liftings whose range is contained in the closed convex hull of product states and we prove that the corresponding quantum Markov chains [2] are uniquely determined by a classical generalization of both the quantum random walks of [4] and the locally diagonalizable states considered in [3]. In Section 4, as a first application of the above results, we prove that the attenuation (beam splitting) process for optical communication treated in [21] can be described in a simpler and more general way in terms of liftings and of transition expectations. The error probabilty of information transmission in the attenuation process is rederived from our new description. We also obtain some new results concerning the explicit computation of error probabilities in the squeezing case.