# Entropic bounds between two thermal equilibrium states. (arXiv:1802.00839v1 [quant-ph])

The positivity conditions of the relative entropy between two thermal

equilibrium states $\hat{\rho}_1$ and $\hat{\rho}_2$ are used to obtain upper

and lower bounds for the subtraction of their entropies, the Helmholtz

potential and the Gibbs potential of the two systems. These limits are

expressed in terms of the mean values of the Hamiltonians, number operator, and

temperature of the different systems. In particular, we discuss these limits

for molecules which can be represented in terms of the Franck--Condon

coefficients. We emphasize the case where the Hamiltonians belong to the same

system at two different times $t$ and $t'$. Finally, these bounds are obtained

for a general qubit system and for the harmonic oscillator with a time

dependent frequency at two different times.