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?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.