The Operational Meaning of Min- and Max-Entropy

In this paper, we show that the conditional min-entropyHmin(A|B) of a bipartite staterhoABis directly related to the maximum achievable overlap with a maximally entangled state if only local actions on theB-part ofrhoABare allowed. In the special case whereAis classical, this overlap corresponds to the probability of guessingAgivenB. In a similar vein, we connect the conditional max-entropyHmax(A|B) to the maximum fidelity ofrhoABwith a product state that is completely mixed onA. In the case whereAis classical, this corresponds to the security ofAwhen used as a secret key in the presence of an adversary holdingB. 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 guessingAgivenBis a lower bound on the number of uniform secret bits that can be extracted fromArelative to an adversary holdingB.