# Connection between smooth max-relative entropy and smooth max-information

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?