Repeated application of <superadditivity of quantum relative entropy> gives
$$
D(\rho_{1\ldots n}\|\sigma_1\otimes\cdots\otimes\sigma_n)
\geq\sum_{j=1}^nD(\rho_j\|\sigma_j).
$$

Multipartite superadditivity of quantum relative entropy

Apply bipartite <superadditivity of quantum relative entropy> first to system $1$ and systems $2,\ldots,n$, and then repeat on the remaining joint state. <Mathematical induction> gives
$$
\boxed{
D(\rho_{1\ldots n}\|
\sigma_1\otimes\cdots\otimes\sigma_n)
\geq\sum_{j=1}^nD(\rho_j\|\sigma_j)}.
$$
This is <multipartite superadditivity of quantum relative entropy>. Repeated application of bipartite additivity similarly gives
$$
\boxed{
D(\rho_1\otimes\cdots\otimes\rho_n\|
\sigma_1\otimes\cdots\otimes\sigma_n)
=\sum_{j=1}^nD(\rho_j\|\sigma_j)}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 323 / 4 / ii / Solution

The <Umegaki relative entropy> is
$$
D(\rho\|\sigma)
=\operatorname{Tr}\rho(\log\rho-\log\sigma)
$$
when the support of $\rho$ is contained in that of $\sigma$, and $+\infty$ otherwise. Its <additivity of quantum relative entropy> is
$$
D(\rho_A\otimes\rho_B\|\sigma_A\otimes\sigma_B)
=D(\rho_A\|\sigma_A)+D(\rho_B\|\sigma_B),
$$
its <superadditivity of quantum relative entropy> is
$$
D(\rho_{AB}\|\sigma_A\otimes\sigma_B)
\geq D(\rho_A\|\sigma_A)+D(\rho_B\|\sigma_B),
$$
and its <data-processing inequality for quantum relative entropy> is $D(T(\rho)\|T(\sigma))\leq D(\rho\|\sigma)$ for every <quantum channel> $T$.

Additivity follows from the logarithm of a <tensor product>,
$$
\log(\rho_A\otimes\rho_B)
=\log\rho_A\otimes I+I\otimes\log\rho_B,
$$
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, leaving
$$
S(\rho_A)+S(\rho_B)-S(\rho_{AB})
=I(A:B)_\rho\geq0.
$$
This is the nonnegativity of <quantum mutual information>, equivalently <Subadditivity of Von Neumann entropy>.

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 323 / 4 / i / Solution

For a bipartite <density operator> $\rho_{AB}$ and a product reference state,
$$
D(\rho_{AB}\|\sigma_A\otimes\sigma_B)
\geq D(\rho_A\|\sigma_A)+D(\rho_B\|\sigma_B).
$$
The difference between the two sides is the nonnegative <quantum mutual information> of $\rho_{AB}$.

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 Incoming links: Superadditivity of quantum relative entropy