Quantum relative entropy

Remarks

Example: for $\sigma = I/d$ with $d = \dim H$, one has $\log \sigma = -(\log d) I$. So $S(\rho \| I/d) = \log d - S(\rho)$, where $S$ is the von Neumann entropy.

Non-example of finiteness: for $\rho = |0\rangle\langle 0|$ and $\sigma = |1\rangle\langle 1|$ on $\C^2$, the support of $\rho$ is not contained in that of $\sigma$, so $S(\rho \| \sigma) = +\infty$.

If $\rho$ and $\sigma$ commute, they are diagonal in a common basis with eigenvalues $p_i$ and $q_i$. Then $S(\rho\|\sigma) = \sum_i p_i \log(p_i / q_i)$, the Kullback–Leibler divergence of the two distributions.

Klein's inequality gives $S(\rho \| \sigma) \ge 0$, with equality exactly when $\rho = \sigma$. For every quantum channel $\Phi$, $S(\Phi(\rho) \| \Phi(\sigma)) \le S(\rho \| \sigma)$; this is the monotonicity, or data processing inequality.