TFAE: quantum equilibrium at inverse temperature β

Fix a finite quantum system with Hamiltonian $H$ and states given by a density operator $\rho$ (positive, trace $1$). Let $\beta = 1/(k_B T)$ where $T$ is the temperature .

Then the following are equivalent (TFAE):

1. Gibbs form (canonical state).
   $\rho$ equals the Gibbs state

   $$\rho_\beta = \frac{e^{-\beta H}}{Z(\beta)}, \qquad Z(\beta)=\mathrm{Tr}\,e^{-\beta H},$$

   where $Z(\beta)$ is the quantum partition function .

2. Finite-system KMS condition.
   With Heisenberg dynamics $\tau_t(A)=e^{itH}Ae^{-itH}$, the state $\omega(A)=\mathrm{Tr}(\rho A)$ satisfies the KMS condition (finite) at inverse temperature $\beta$: for all observables $A,B$, the function

   $$F_{A,B}(t)=\omega\!\big(A\,\tau_t(B)\big)$$

   extends to a function analytic in the strip $0<\mathrm{Im}\,z<\beta$ with boundary values

   $$F_{A,B}(t+i\beta)=\omega\!\big(\tau_t(B)\,A\big).$$

3. Free-energy minimizer (variational principle).
   $\rho$ minimizes the (Helmholtz) free-energy functional

   $$\mathcal F_\beta(\rho)=\mathrm{Tr}(\rho H)-\beta^{-1}S(\rho),$$

   where $S(\rho)=-\mathrm{Tr}(\rho\log\rho)$ is the von Neumann entropy . Equivalently,

   $$\mathcal F_\beta(\rho)\ge \mathcal F_\beta(\rho_\beta) \quad\text{for all density operators }\rho.$$

   (Compare with statistical free energy in the classical setting.)

4. Maximum entropy subject to mean energy (Lagrange multiplier form).
   For fixed mean energy $U=\mathrm{Tr}(\rho H)$, the state $\rho$ maximizes $S(\rho)$ among all density operators with that same $U$; the maximizer is of Gibbs form with some $\beta$.

5. Relative-entropy gap identity.
   For all density operators $\rho$,

   $$\mathcal F_\beta(\rho)-\mathcal F_\beta(\rho_\beta)=\beta^{-1}D(\rho\|\rho_\beta)\ge 0,$$

   where $D(\rho\|\sigma)=\mathrm{Tr}(\rho(\log\rho-\log\sigma))$ is the quantum analogue of relative entropy (KL divergence) .

Prerequisites: density operators , quantum Gibbs states , KMS condition , and temperature .