Classical Analog to Entanglement Reversibility

In this Letter we study the problem of secrecy reversibility. This asks when two honest parties can distill secret bits from some tripartite distribution $p_{XYZ}$ and transform secret bits back into $p_{XYZ}$ at equal rates using local operation and public communication. This is the classical analog to the well-studied problem of reversibly concentrating and diluting entanglement in a quantum state. We identify the structure of distributions possessing reversible secrecy when one of the honest parties holds a binary distribution, and it is possible that all reversible distributions have this form. These distributions are more general than what is obtained by simply constructing a classical analog to the family of quantum states known to have reversible entanglement. An indispensable tool used in our analysis is a conditional form of the Gács-Körner common information.

Figure 1

A $\mathrm{UBI}\text{−}\mathrm{PD}\downarrow$ distribution where $p_Z(0)=1/5$, $p_Z(1)=4/5$, and $\overline{Z}$ is a full coarse graining of $Z$. In this simplified example, no communication is needed for Alice and Bob to both generate $J_{XY|\overline{Z}}$. Note that $p_{XYZ}$ itself is not BI.