# Is the quantum mutual information variance bounded from above?

The relative entropy variance between two quantum states $$\rho$$ and $$\sigma$$ is defined to be

$$V(\rho\|\sigma) = \text{Tr}(\rho(\log\rho - \log\sigma)^2) - D(\rho\|\sigma)^2,$$

where $$D(\rho\|\sigma)$$ is the quantum relative entropy.

Analogous to the mutual information, one can define the mutual information variance for a bipartite state $$\rho_{AB}$$ by choosing the second argument of the relative entropy variance to be the product of the marginals $$\rho_A\otimes\rho_B$$. The mutual information is bounded by $$\log d^2$$, where $$d$$ is the dimension of the registers $$A$$ and $$B$$. My question is if the mutual information variance is also bounded? That is,

$$V(\rho_{AB}\|\rho_A\otimes\rho_B) \leq \ ?$$

Yes, for any bipartite state $$\rho_{AB}$$ we have $$V(\rho_{AB}\| \rho_A \otimes \rho_B) \leq 4 \left( \log (2 d_A^2 + 1) \right)^2$$ where $$d_A$$ is the dimension of the Hilbert space of $$A$$. For a proof I refer you to Corollary III.5 in Entropy accumulation with improved second-order term.