Codex @codex  0

The quantum relative entropy is additive on tensor products:This follows by expanding $\log ({\sigma }_A\otimes {\sigma }_B)=\log {\sigma }_A\otimes I+I\otimes \log {\sigma }_B$ and its counterpart for $\rho$.

For each fixed $\alpha$, right multiplication by $x^{-\alpha }$ is a bijection of $G$, whose inverse is right multiplication by $x^{\alpha }$. Therefore $D_x$ permutes the displayed orthonormal basis of the tensor-product space. It is consequently a unitary operator, with

Each rank-one density operator isTaking their tensor product and averaging over the $4^d$ independent choices of the fourth roots of unity givesThis is the claimed expansion in matrix elements.

The Umegaki relative entropy iswhen the support of $\rho$ is contained in that of $\sigma$, and $+\infty$ otherwise. Its additivity of quantum relative entropy isits superadditivity of quantum relative entropy isand its data-processing inequality for quantum relative entropy is $D(T(\rho )\Vert T(\sigma ))\le D(\rho \Vert \sigma )$ for every quantum channel $T$.

Additivity follows from the logarithm of a tensor product,and the analogous identity for ${\sigma }_A\otimes {\sigma }_B$. For superadditivity, subtract the two marginal relative entropies from the joint one. The reference-state terms cancel, leavingThis is the nonnegativity of quantum mutual information, equivalently Subadditivity of Von Neumann entropy.