Thermofield double state (TFD state)

For two copies of a quantum system,Tracing out either copy produces the thermal density operator $e^{-\beta H}/Z$.

The modular Hamiltonian of a density operator is $K=-\log \rho$. For a Gibbs state, $K=\beta H+\log Z$.

Linearizing the von Neumann entropy around a reference state gives $\delta S=\mathrm{Tr}(\delta \rho K)$. Around a thermal state this becomes $\delta S=\beta \delta ⟨H⟩$.