  Assume that two distant parties, Alice and Bob, as well as an adversary, Eve,
have access to (quantum) systems prepared jointly according to a tripartite
state. 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 classical
correlations and (ii) key distillation from pure tripartite quantum states.
  Firstly, we obtain generalisations of fundamental results related to
scenarios (i) and (ii), including upper bounds on the key rate. Moreover, based
on an embedding of classical distributions into quantum states, we are able to
find new connections between protocols and quantities in the standard scenarios
(i) and (ii).
  Secondly, we study specific properties of key distillation protocols. In
particular, we show that every protocol that makes use of pre-shared key can be
transformed into an equally efficient protocol which needs no pre-shared key.
This result is of practical significance as it applies to quantum key
distribution (QKD) protocols, but it also implies that the key rate cannot be
locked with information on Eve's side. Finally, we exhibit an arbitrarily large
separation between the key rate in the standard setting where Eve is equipped
with quantum memory and the key rate in a setting where Eve is only given
classical memory. This shows that assumptions on the nature of Eve's memory are
important in order to determine the correct security threshold in QKD.
