  We show that the conditional min-entropy Hmin(A|B) of a bipartite state
rho_AB is directly related to the maximum achievable overlap with a maximally
entangled state if only local actions on the B-part of rho_AB 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 Hmax(A|B) to the maximum fidelity of rho_AB 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.
