The Conditional Common Information in Classical and Quantum Secret Key Distillation

In this paper, we consider two extensions of the Gács–Körner common information to three variables, the <italic>conditional common information</italic> (cCI) and the <italic>coarse-grained conditional common information</italic> (ccCI). Both quantities are shown to be useful technical tools in the study of classical and quantum resource transformations. In particular, the ccCI is shown to have an operational interpretation as the optimal rate of secret key extraction from an eavesdropped classical source <inline-formula> <tex-math notation="LaTeX">$p_{XYZ}$ </tex-math></inline-formula> when Alice (<inline-formula> <tex-math notation="LaTeX">$X$ </tex-math></inline-formula>) and Bob (<inline-formula> <tex-math notation="LaTeX">$Y$ </tex-math></inline-formula>) are unable to communicate but share common randomness with the eavesdropper Eve (<inline-formula> <tex-math notation="LaTeX">$Z$ </tex-math></inline-formula>). Moving to the quantum setting, we consider two different ways of generating a tripartite quantum state from classical correlations <inline-formula> <tex-math notation="LaTeX">$p_{XYZ}$ </tex-math></inline-formula>: 1) coherent encodings <inline-formula> <tex-math notation="LaTeX">$\sum _{xyz}\sqrt {p_{xyz}} |xyz\rangle $ </tex-math></inline-formula> and 2) incoherent encodings <inline-formula> <tex-math notation="LaTeX">$\sum _{xyz}p_{xyz} |xyz\rangle \langle xyz|$ </tex-math></inline-formula>. We study how well can Alice and Bob extract secret key from these quantum sources using quantum operations compared with the extraction of key from the underlying classical sources <inline-formula> <tex-math notation="LaTeX">$p_{XYZ}$ </tex-math></inline-formula> using classical operations. 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. We first demonstrate that the classical key rate of <inline-formula> <tex-math notation="LaTeX">$p_{XYZ}$ </tex-math></inline-formula> is equivalent to the quantum key rate for an incoherent quantum encoding of the distribution. For coherent encodings, we next show that the classical and quantum rates are generally incomparable, and in fact, their difference can be arbitrarily large in either direction. Finally, we introduce a “zoo” of entangled tripartite states all characterized by the conditional common information of their encoded probability distributions. Remarkably, for these states almost all entanglement measures, such as Alice and Bob’s entanglement cost, squashed entanglement, and relative entropy of entanglement, can be sharply bounded or even exactly expressed in terms of the conditional common information. In the latter case, we thus present a rare instance in which the various entropic entanglement measures of a quantum state can be explicitly calculated.