Finite quantum statistical system - Imported Knowlpedia Content

Positivity: ρ ≥ 0 \rho \ge 0 ρ ≥ 0 , meaning ⟨ ψ , ρ ψ ⟩ ≥ 0 \langle \psi,\rho\psi\rangle \ge 0 ⟨ ψ , ρ ψ ⟩ ≥ 0 for all ψ ∈ H \psi\in H ψ ∈ H .
Unit trace: Tr ⁡ ( ρ ) = 1 \operatorname{Tr}(\rho)=1 Tr ( ρ ) = 1 , where Tr ⁡ \operatorname{Tr} Tr is the operator trace (<a class="knowl" href="/quantum-foundations/trace-operator/" data-knowl="/fragments/quantum-foundations/trace-operator/core.html" aria-expanded="false">Trace Operator ).

A finite quantum statistical system consists of a triple $(\mathcal H,\mathcal A,H)$ , where $\mathcal H$ is a finite-dimensional complex Hilbert space, $\mathcal A\subseteq\mathcal B(\mathcal H)$ is a unital observable algebra, and $H=H^*\in\mathcal A$ is the Hamiltonian. A state is a positive normalized functional on $\mathcal A$ ; on $\mathcal B(\mathcal H)$ it is represented uniquely by a density operator.

Dynamics and equilibrium

The Hamiltonian generates the dynamics

\tau_t(A)=e^{itH/\hbar}Ae^{-itH/\hbar}.

For each inverse temperature $\beta>0$ , it also determines the partition function and Gibbs state.

Scope

This finite-dimensional model avoids the domain and trace-class issues that arise for unbounded Hamiltonians or infinite systems. Those settings require additional analytic hypotheses and, for equilibrium, the operator-algebraic KMS formulation.