Unifying classical and quantum key distillation

Assume that three parties, Alice, Bob, and Eve, are presented with a sequence of identical tripartite quantum states. In addition, Alice and Bob can use local operations and authenticated public classical communication. Their goal is to establish a key which is unknown to Eve. We initiate the study of this scenario as a unification of two standard scenarios: (i) key distillation (agreement) from random variables and (ii) key distillation from pure tripartite quantum states. We obtain generalisations of fundamental results of scenarios (i) and (ii), including upper bounds given by the intrinsic information, squashed entanglement, and the relative entropy of entanglement. Based on an embedding of classical distributions into quantum states, we also find new connections between protocols and quantities in the standard scenarios (i) and (ii). In addition, we study properties of key distillation protocols. In particular, we show that every protocol for key distillation that makes use of a pre-shared key can be transformed into a protocol having identical (or better) rate without making use of the pre-shared key. This result is of practical significance as it applies to quantum key distribution protocols, but it also implies that the key rate cannot be locked with information on Eve's side. Finally, we show that an arbitrarily large separation can occur between the key-distillation rate when Eve is equipped with quantum memory and the key-distillation rate when Eve is only given classical memory