Mutual information between thermo-field doubles and disconnected holographic boundaries

We use mutual information as a measure of the entanglement between 'physical' and thermo-field double degrees of freedom in field theories at finite temperature. We compute this "thermo-mutual information" in simple toy models: a quantum mechanics two-site spin chain, a two dimensional massless fermion, and a two dimensional holographic system. In holographic systems, the thermo-mutual information is related to minimal surfaces connecting the two disconnected boundaries of an eternal black hole. We derive a number of salient features of this thermo-mutual information, including that it is UV finite, positive definite and bounded from above by the standard mutual information for the thermal ensemble. We relate the construction of the reduced density matrices used to define the thermo-mutual information to the Schwinger-Keldysh formalism, ensuring that all our objects are well defined in Euclidean and Lorentzian signature.