Von neumann entropy as information rate

Abstract

Recently it has been shown that quantum theory can be viewed as a classical probability theory by treating Hilbert space as a measure space (H, B(H)) of “events” or “hidden states.” Each density operator\(\hat W = \sum _{n = 1}^\infty {\text{ }}w_n \hat \prod _{E_n } \) defines a setℳŵ of probability measures such thatμ(En)=wn (alln). Coding elements ψeH by subspacesEn entails distortion. We show that the von Neumann entropyS(Ŵ) = -trŴInŴequals the effective rate at which the Hilbert space produces information with zero expected distortion, and comment on the meaning of this.