Quantum relative entropy

Let $\rho$ and $\sigma$ be density-operators on a finite-dimensional Hilbert space $H$ . The quantum relative entropy (also called Umegaki relative entropy) is defined by

D(\rho\|\sigma) := \begin{cases} \operatorname{Tr}\!\big(\rho(\log\rho-\log\sigma)\big), & \text{if }\operatorname{supp}(\rho)\subseteq \operatorname{supp}(\sigma),\\[4pt] +\infty, & \text{otherwise.} \end{cases}

The support condition is needed because $\log\sigma$ is not finite on the kernel of $\sigma$ . The operators $\log\rho$ and $\log\sigma$ are defined by spectral calculus.

Classical case

If $\rho$ and $\sigma$ commute, they are simultaneously diagonalizable and the formula reduces to the Kullback–Leibler divergence of their eigenvalue distributions.

Properties

Nonnegativity (Klein inequality): $D(\rho\|\sigma)\ge 0$ , with equality iff $\rho=\sigma$ .
Not symmetric: typically $D(\rho\|\sigma)\ne D(\sigma\|\rho)$ ; it is not a metric.
Data processing inequality: for any completely positive trace-preserving map $\Phi$ , $D(\rho\|\sigma)\ \ge\ D(\Phi(\rho)\|\Phi(\sigma)).$ In particular, taking $\Phi=\operatorname{Tr}_B$ (see partial trace) gives $D(\rho_{AB}\|\sigma_{AB})\ \ge\ D(\rho_A\|\sigma_A), \quad \rho_A=\operatorname{Tr}_B(\rho_{AB}),\ \sigma_A=\operatorname{Tr}_B(\sigma_{AB}).$

Relation to von Neumann entropy

If $d=\dim H$ and $\tau=I/d$ is the maximally mixed state, then

D(\rho\|\tau)=\log d - S(\rho),

where $S(\rho)$ is the von Neumann entropy.