Von Neumann entropy as information rate (Q1057577)

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 \(\hat W=\sum^{\infty}_{n=1}w_ n{\hat \Pi}_{E_ n}\) defines a set \({\mathcal M}_{\hat W}\) of probability measures such that \(\mu (E_ n)=w_ n\) (all n). Coding elements \(\psi\in H\) by subspaces \(E_ n\) entails distortion. We show that the von Neumann entropy \(S(\hat W)=-tr \hat W \ln \hat W\) equals the effective rate at which the Hilbert space produces information with zero expected distortion, and comment on the meaning of this.