The Operational Meaning of Min- and Max-Entropy

In this paper, we show that the conditional min-entropy <i>H</i> _min(<i>A</i> |<i>B</i>) of a bipartite state <i>rhoAB</i> is directly related to the maximum achievable overlap with a maximally entangled state if only local actions on the <i>B</i>-part of <i>rhoAB</i> are allowed. In the special case where <i>A</i> is classical, this overlap corresponds to the probability of guessing <i>A</i> given <i>B</i>. In a similar vein, we connect the conditional max-entropy <i>H</i> _max(<i>A</i> |<i>B</i>) to the maximum fidelity of <i>rhoAB</i> with a product state that is completely mixed on <i>A</i>. In the case where <i>A</i> is classical, this corresponds to the security of <i>A</i> when used as a secret key in the presence of an adversary holding <i>B</i>. 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 <i>A</i> given <i>B</i> is a lower bound on the number of uniform secret bits that can be extracted from <i>A</i> relative to an adversary holding <i>B</i>.