Quantum sources and a quantum coding theorem

We define a large class of quantum sources and prove a quantum analog of the asymptotic equipartition property. Our proof relies on using local measurements on the quantum source to obtain an associated classical source. The classical source provides an upper bound for the dimension of the relevant subspace of the quantum source, via the Shannon–McMillan noiseless coding theorem. Along the way we derive a bound for the von Neumann entropy of the quantum source in terms of the Shannon entropy of the classical source, and we provide a definition of ergodicity of the quantum source. Several explicit models of quantum sources are also presented.