Sfb 288 Differential Geometry and Quantum Physics

Abstract for Sfb Preprint No. 466

On a quantum version of Shannon's conditional entropy

R. Schrader

In this article we propose a quantum version of Shannon's conditional entropy. Given two density matrices $ ho$ and $sigma$ on a finite dimensional Hilbert space and with $S( ho)=- r holn ho$ being the usual von Neumann entropy, this quantity $S( ho|sigma)$ is concave in $ ho$ and satisfies $0le $ ( ho|sigma)le S( ho)$, a quantum analogue of Shannon's famous inequality. Thus we view $ ( ho|sigma)$ as the entropy of $ ho$ conditioned by $sigma$. The second inequality is an equality if $sigma$ is a multiple of the identity. In contrast to the classical case, however, $S( ho| ho)=0$ if and only if the non-vanishing eigenvalues of $ ho$ are all non-degenerate. Also in general and again in contrast to the corresponding classical situation $S( ho,sigma)=S(sigma)+S( ho|sigma)$ is not symmetric in $ ho$ and $sigma$ even if they commute. We also show that there is no quantum version of conditional entropy in terms of two density matrices, which shares more properties with the classical case and which in particular reduces to the classical case when the two density matrices commute. As an alternative we propose to use spectral resolutions of the unit matrix instead of density matrices. We briefly compare this with the algebraic approach of Connes and St{o}rmer and Connes, Narnhofer and Thirring.

---

Get a gzip-compressed PostScript copy of this preprint
preprint466.ps.gz (100 kB)

---