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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,

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

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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.

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