  We consider the extraction of shared secret key from correlations that are
generated by either a classical or quantum source. In the classical setting,
two honest parties (Alice and Bob) use public discussion and local randomness
to distill secret key from some distribution $p_{XYZ}$ that is shared with an
unwanted eavesdropper (Eve). In the quantum settings, the correlations
$p_{XYZ}$ are delivered to the parties as either an \textit{incoherent} mixture
of orthogonal quantum states or as \textit{coherent} superposition of such
states; in both cases, Alice and Bob use public discussion and local quantum
operations to distill secret key. While the power of quantum mechanics
increases Alice and Bob's ability to generate shared randomness, it also equips
Eve with a greater arsenal of eavesdropping attacks. Therefore, it is not
obvious who gains the greatest advantage for distilling secret key when
replacing a classical source with a quantum one.
  In this paper we first demonstrate that the classical key rate is equivalent
to the quantum key rate when the correlations are generated incoherently in the
quantum setting. For coherent sources, we next show that the rates are
incomparable, and in fact, their difference can be arbitrarily large in either
direction. However, we identify a large class of non-trivial distributions
$p_{XYZ}$ that possess the following properties: (i) Eve's advantage is always
greater in the quantum source than in its classical counterpart, and (ii) for
the quantum entanglement shared between Alice and Bob in the coherent source,
the so-called entanglement cost/squashed entanglement/relative entropy of
entanglement can all be computed. With property (ii), we thus present a rare
instance in which the various entropic entanglement measures of a quantum state
can be explicitly calculated.
