Coding Theorems for Quantum Communication Channels

Let H be a Hilbert space providing a quantum-mechanical description for the physical carrier of information. A simple model of quantum communication channel consists of the input alphabet A = {1, ..., a} and a mapping i→ Si from the input alphabet to the set of quantum states in H. A quantum state is a density operator, i. e. positive operator S in H with unit trace, TrS = 1. Sending a letter i results in producing the signal state Si of the information carrier. Like in the classical case, the input is described by an apriori probability distribution π = {πi} on A. At the receiving end of the channel a quantum measurement is performed, which mathematically is described by a resolution of identity in H, that is by a family X = {Xj} of positive operators in H satisfying ∑ j Xj = I, where I is the unit operator in H [Holevo 1973]. The probability of the output j conditioned upon the input i by definition is equal to P (j|i) = TrSiXj . The classical case is embedded into this picture by assuming that all operators in question commute, hence are diagonal in some basis labelled by index ω; in fact by taking Si = diag[S(ω|i)], Xj = diag[X(j|ω)], we have a classical channel with transition probabilities S(ω|i) and the classical decision rule X(j|ω), so that P (j|i) = ∑ ωX(j|ω)S(ω|i). We call such channel quasiclassical. The Shannon information is given by the usual formula