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We consider the extraction of shared secret key from correlations that aregenerated by either a classical or quantum source. In the classical setting,two honest parties (Alice and Bob) use public discussion and local randomnessto distill secret key from some distribution $p_{XYZ}$ that is shared with anunwanted eavesdropper (Eve). In the quantum settings, the correlations$p_{XYZ}$ are delivered to the parties as either an \textit{incoherent} mixtureof orthogonal quantum states or as \textit{coherent} superposition of suchstates; in both cases, Alice and Bob use public discussion and local quantumoperations to distill secret key. While the power of quantum mechanicsincreases Alice and Bob's ability to generate shared randomness, it also equipsEve with a greater arsenal of eavesdropping attacks. Therefore, it is notobvious who gains the greatest advantage for distilling secret key whenreplacing a classical source with a quantum one.  In this paper we first demonstrate that the classical key rate is equivalentto the quantum key rate when the correlations are generated incoherently in thequantum setting. For coherent sources, we next show that the rates areincomparable, and in fact, their difference can be arbitrarily large in eitherdirection. However, we identify a large class of non-trivial distributions$p_{XYZ}$ that possess the following properties: (i) Eve's advantage is alwaysgreater in the quantum source than in its classical counterpart, and (ii) forthe quantum entanglement shared between Alice and Bob in the coherent source,the so-called entanglement cost/squashed entanglement/relative entropy ofentanglement can all be computed. With property (ii), we thus present a rareinstance in which the various entropic entanglement measures of a quantum statecan be explicitly calculated.

Quantum Versus Classical Advantages in Secret Key Distillation (and Their Links to Quantum Entanglement)