Entropic bounds between two thermal equilibrium states

The positivity conditions of the relative entropy between two thermal equilibrium states ${\widehat{\rho }}_1$ and ${\widehat{\rho }}_2$ 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 that 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 $t^{\prime }$. Finally, these bounds are obtained for a general qubit system and for the harmonic oscillator with a time-dependent frequency at two different times.

Figure 1

Upper (blue) and lower (red) bounds for the difference of entropies $\mathrm{\Delta }S$ (a) as a function of the angle between the Bloch vectors $\theta$ given two thermal equilibrium states with temperature $T_1=10$ and (b) as a function of temperature $T_1$ with fixed $\theta =\pi /4$. In both cases, we assume $T_2=15$, $\left|h\right|=\sqrt{61}$, and $\left|\mathfrak{h}\right|=\sqrt{17}$. The gray curves correspond to the analytic result.

Figure 2

(a) Upper (blue), lower (red) bounds and analytical expression (gray) for the difference of entropies $S_2(T_2,t^{\prime })-S_1(T_1,t)$ for a harmonic oscillator with time-dependent frequency $\omega \left(t\right)=\sqrt{t}$, with equal temperatures $T_1=T_2=10$. (b) a 2D cut of the previous plot for $t^{\prime }=1$.

Figure 3

Upper (blue) and lower (red) bounds for the difference of the entropies $\mathrm{\Delta }S(t,t^{\prime })$ as a function of $t^{\prime }$ for a time-dependent frequency harmonic oscillator $[\omega \left(t\right)=\sqrt{1+cos\left(2t\right)/2}]$ between two thermal equilibrium states with temperatures (a) $T_1=T_2=10$ at the fixed time $t=0.1$ and (b) $T_1=15$, $T_2=10$ at the fixed time $t=3$. The dashed curve corresponds to the plot of the frequency.

Figure 4

Upper (blue) and lower (red) bounds for the difference of entropies $S(T_2,t^{\prime })-S(T_1,t)$ (gray) for a harmonic oscillator with time-dependent frequency $\omega \left(t\right)=\sqrt{t}$. Here, (a) $T_1=T_2=10$ and (b) $t=5$ and $t^{\prime }=10$.