Appl Math Optim 39:33–59 (1999)
1999 Springer-Verlag New York Inc.
Compound Channels, Transition Expectations, and Liftings
and M. Ohya
Dipartimento di Matematica, Centro Matematico V. Volterra,
Universit´a di Roma II, Rome, Italy
Department of Information Sciences,
Science University of Tokyo,
Noda City, Chiba 278, Japan
Abstract. In Section 1 we introduce the notion of lifting as a generalization of the
notion of compound state introduced in  and  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  and we characterize those transition expectations
which arise from compound states in the sense of .
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  are uniquely determined by a classical generalization of both
the quantum random walks of  and the locally diagonalizable states considered
In Section 4, as a ﬁrst application of the above results, we prove that the at-
tenuation (beam splitting) process for optical communication treated in  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.
Key Words. Compound state, Transition expectation, Lifting, Channel, Quantum
probability, Quantum Markov chain, Beam splitting, Optical communication.
AMS Classiﬁcation. 81Q99, 94A40, 60J27.