Quantum relative entropy data-processing inequality

Let $\rho$ be a quantum state, let $\sigma$ be a positive semi-definite operator, and let $\mathcal{N}$ be a quantum channel. With $D(\rho\|\sigma)$ understood as the quantum relative entropy defined by $D(\rho\|\sigma)=\operatorname{Tr}\!\left[\rho(\log_2 \rho-\log_2 \sigma)\right]$ when the support of $\rho$ is contained in the support of $\sigma$, and as $+\infty$ otherwise, one has $D(\rho\|\sigma) \ge D(\mathcal{N}(\rho)\|\mathcal{N}(\sigma))$.