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.