The Operational Meaning of Min- and Max-Entropy

In this paper, we show that the conditional min-entropy H _min(A |B) of a bipartite state rhoAB is directly related to the maximum achievable overlap with a maximally entangled state if only local actions on the B-part of rhoAB are allowed. In the special case where A is classical, this overlap corresponds to the probability of guessing A given B. In a similar vein, we connect the conditional max-entropy H _max(A |B) to the maximum fidelity of rhoAB with a product state that is completely mixed on A. In the case where A is classical, this corresponds to the security of A when used as a secret key in the presence of an adversary holding B. Because min- and max-entropies are known to characterize information-processing tasks such as randomness extraction and state merging, our results establish a direct connection between these tasks and basic operational problems. For example, they imply that the (logarithm of the) probability of guessing A given B is a lower bound on the number of uniform secret bits that can be extracted from A relative to an adversary holding B.