  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.
