Quantum Gibbs state

Let $\mathcal H$ be a finite-dimensional complex Hilbert space (see finite-dimensional complex Hilbert space ) and let $H$ be the system Hamiltonian (quantum Hamiltonian ), a self-adjoint operator on $\mathcal H$.

Fix an inverse temperature $\beta>0$ (see inverse temperature β ). Define the quantum partition function as

using the operator trace (trace ). The quantum Gibbs state at inverse temperature $\beta$ is the density operator

This is a state on the observable algebra (observable algebra ) given by the expectation functional

In the language of quantum states, $\rho_\beta$ is a thermal density-operator state (compare density operator ).

Physical interpretation

The Gibbs state describes thermal equilibrium (thermodynamic equilibrium ) for a system weakly coupled to a heat bath at temperature $T = (k_B\beta)^{-1}$ (see Boltzmann constant and temperature ).

If $H$ has spectral decomposition $H=\sum_n E_n \,|n\rangle\langle n|$, then $\rho_\beta$ is diagonal in the energy eigenbasis with weights

so $\rho_\beta$ assigns the familiar Boltzmann probabilities to energy levels.

Key properties

1. Positivity and normalization.
   $\rho_\beta$ is positive semidefinite and $\operatorname{Tr}(\rho_\beta)=1$, hence it is a valid quantum state (density operator state ).

2. Stationarity under time evolution.
   Since $\rho_\beta$ is a function of $H$, it commutes with $H$, so it is invariant under the Heisenberg dynamics generated by $H$. This is one precise sense in which it represents equilibrium.

3. Variational (free-energy) principle.
   Let $S(\rho)=-\operatorname{Tr}(\rho\log\rho)$ be the von Neumann entropy (von Neumann entropy ). Define the Helmholtz free-energy functional

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

   which matches the thermodynamic Helmholtz free energy (Helmholtz free energy ) in equilibrium. Then $\rho_\beta$ uniquely minimizes $\mathcal F_\beta(\rho)$ over all density operators on $\mathcal H$.

   Equivalently, for any state $\rho$,

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

   where $D(\cdot\|\cdot)$ is quantum relative entropy (quantum relative entropy ).

4. Equilibrium free energy from the partition function.
   The equilibrium Helmholtz free energy is

   $$F(\beta)\;=\;-\beta^{-1}\log Z(\beta),$$

   connecting quantum partition function to statistical free energy .

5. High- and low-temperature limits.

   • As $\beta\to 0^+$ (high temperature), $\rho_\beta \to I/\dim\mathcal H$, the maximally mixed state.
   • As $\beta\to\infty$ (low temperature), $\rho_\beta$ concentrates on the ground-state subspace (ground-state projector normalized by its degeneracy).

6. KMS characterization (finite systems).
   The Gibbs state is exactly the $\beta$-KMS equilibrium state for the dynamics generated by $H$ (see KMS condition (finite) ). This property controls the analytic/imaginary-time structure of thermal correlation functions (quantum correlation function ).

7. Useful inequalities and bounds.
   Bounds on $Z(\beta)$ and on equilibrium free energies often use trace inequalities such as the Golden–Thompson inequality .