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The max-relative entropy between two states is defined as

$$D_{\max }(\rho \| \sigma):=\log \min \{\lambda: \rho \leq \lambda \sigma\},$$

where $\rho\leq \sigma$ should be read as $\sigma - \rho$ is positive semidefinite. There is also a smoothed version of this quantity and this is given by taking the infimum of $D_{\max}(\rho\|\sigma)$ over all states $\bar{\rho}$ which are within an $\varepsilon$ ball of $\rho$ according to some metric. For example, one may require that the trace distance between $\rho$ and $\bar{\rho}$ is at most $\varepsilon$ and define a ball this way. So we have

$$D^{\varepsilon}_{\max}(\rho\|\sigma) = \inf\limits_{\bar{\rho}\in\mathcal{B}^{\varepsilon}(\rho)}D_{\max}(\bar{\rho}\|\sigma)$$

Now consider the case where $\rho = \rho_{AB}$ (some bipartite state) and $\sigma = \rho_A\otimes\rho_B$. A quantity known as the max-information that $B$ has about $A$ is given by

$$I_{\max}(A:B)_\rho = D_{\max}(\rho_{AB}||\rho_A\otimes\rho_B)$$

Note that this is not the only definition of the max-information (there are several as shown here but they are all equivalent after smoothing). The smoothed max-information for our definition of the max-information is

$$I^{\varepsilon}_{\max}(A:B)_{\rho_{AB}} = \inf\limits_{\bar{\rho}_{AB}\in\mathcal{B}^{\varepsilon}(\rho_{AB})} I(A:B)_{\bar{\rho}} = \inf\limits_{\bar{\rho}_{AB}\in\mathcal{B}^{\varepsilon}(\rho_{AB})}D_{\max}(\bar{\rho}_{AB}\|\bar{\rho}_A\otimes\bar{\rho}_B)$$

In contrast, the smoothed max-relative entropy is

$$D^{\varepsilon}_{\max}(\rho_{AB}\|\rho_A\otimes\rho_B) = \inf\limits_{\bar{\rho}_{AB}\in\mathcal{B}^{\varepsilon}(\rho_{AB})} D_{\max}(\bar{\rho}_{AB}\|\rho_A\otimes\rho_B)$$

Are the two quantities $D^{\varepsilon}_{\max}(\rho_{AB}\|\rho_A\otimes\rho_B)$ and $I^{\varepsilon}_{\max}(A:B)_{\rho_{AB}}$ close to each other (e.g. the difference is some function of $\varepsilon$) such that they are qualitatively equivalent?

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