Quantum Statistical Mechanics and Density Operators

A density operator is the common language for quantum mixtures, subsystems, and thermal equilibrium. It predicts every measurement on the chosen system, yet it does not reveal a unique story about how that state was prepared. This page develops that distinction first in a finite-dimensional Hilbert space, where every trace is finite, and then states the extra condition needed for infinite-dimensional thermal systems.

Required background. Calculus and Linear Algebra provides multivariable differentiation, linear maps, inner products, and basis changes; Quantum States, Observables, and Measurement provides Hilbert-space states and observables, while Unitary Dynamics and Pictures of Motion provides unitary evolution. Helpful background. Classical and Variational Mechanics provides Hamiltonians and conserved quantities; States, Processes, and the Laws of Thermodynamics provides equilibrium state variables, temperature, and entropy, Statistical Ensembles and Partition Functions supplies Gibbs weights, and Thermodynamic Potentials and Stability supplies the free-energy and stability language used below.

Density operators and operational equivalence

A density operator on a finite-dimensional Hilbert space obeys

$$\rho=\rho^\dagger, \qquad \rho\geq0, \qquad \operatorname{Tr}\rho=1.$$

For an observable $A$,

$$\langle A\rangle=\operatorname{Tr}(\rho A).$$

A pure state has $\rho=|\psi\rangle\langle\psi|$ and therefore $\rho^2=\rho$ and $\operatorname{Tr}\rho^2=1$. A mixed state has at least two nonzero eigenvalues and $\operatorname{Tr}\rho^2\lt 1$. If preparations $|\psi_a\rangle$ occur with probabilities $p_a$, then

$$\rho=\sum_a p_a|\psi_a\rangle\langle\psi_a|.$$

This ensemble decomposition is generally not unique. For example,

$$\frac{I}{2} =\frac12\bigl(|0\rangle\langle0|+|1\rangle\langle1|\bigr) =\frac12\bigl(|+\rangle\langle+|+|-\rangle\langle-|\bigr).$$

No measurement on the qubit can decide which of these two ensembles was used, because all outcome probabilities depend only on $\rho$. Preskill develops this operational equivalence and the ambiguity of ensemble decompositions in Preskill 2015, §§2.3 and 2.5, pp. 16–22 and 26–35.

Closed-system unitary evolution gives

$$i\frac{d\rho}{dt}=[H,\rho], \qquad \rho(t)=U(t)\rho(0)U^\dagger(t).$$

It preserves the eigenvalues, purity, and von Neumann entropy. Relaxation to a Gibbs state therefore requires an environment, a coarse-grained description, a restriction on the observables or initial states, or a limiting statement. Unitary evolution of the complete closed system cannot change the spectrum of its density operator.

Gibbs states and the free-energy principle

Weak coupling to a large equilibrium reservoir at temperature $T\gt 0$ gives the canonical state

$$\rho_\beta=\frac{e^{-\beta H}}{Z}, \qquad Z=\operatorname{Tr}e^{-\beta H}, \qquad \beta=\frac1T.$$

We set $k_{\mathrm B}=1$, so $T$ has energy units and entropy is dimensionless. In an energy eigenbasis, $\rho_\beta$ has weights $e^{-\beta E_n}/Z$. The restriction of $\rho_\beta$ to each degenerate energy eigenspace is proportional to the identity, so there is no preferred basis within that eigenspace. The operator form and thermal trace rule are derived in Tokmakoff 2009, p. 5, equations (1.20)–(1.24).

In finite dimension $Z$ is automatically finite. On an infinite-dimensional Hilbert space, the same expression defines a density operator only when $e^{-\beta H}$ is trace class, equivalently when $Z\lt \infty$ for the temperature and volume under discussion. The remainder of the derivations returns to finite dimension; an infinite-dimensional extension must additionally ensure that every displayed mean energy and entropy is finite, so no undefined $\infty-\infty$ free-energy difference is formed.

The von Neumann entropy, with $0\ln0=0$ in the spectral definition, is

$$S(\rho)=-\operatorname{Tr}(\rho\ln\rho).$$

For the Gibbs state, $\ln\rho_\beta=-\beta H-\ln Z$, hence

$$S=\beta U+\ln Z, \qquad U=\operatorname{Tr}(\rho_\beta H), \qquad F=-T\ln Z.$$

This is both quantum and statistical: the state space and observables are quantum, while the Gibbs eigenvalues encode thermal weighting.

The equilibrium minimum principle follows from quantum relative entropy. In finite dimension, $\operatorname{supp}\rho=(\ker\rho)^\perp$ is the span of eigenvectors with nonzero eigenvalues. With natural logarithms, define

$$D(\rho\lVert\sigma)= \begin{cases} \operatorname{Tr}\!\left[\rho(\ln\rho-\ln\sigma)\right], & \operatorname{supp}\rho\subseteq\operatorname{supp}\sigma,\\ +\infty,&\text{otherwise}. \end{cases}$$

For density operators, Klein’s inequality states $D(\rho\lVert\sigma)\geq0$, with equality exactly when $\rho=\sigma$. The support condition and positivity are stated in Watrous 2018, Definitions 5.17–5.18 and Proposition 5.22, pp. 265–269. Since a finite-dimensional Gibbs state at $T\gt 0$ has full support, every trial density operator is admissible and

$$D(\rho\lVert\rho_\beta) =\beta\left(\mathcal F_\beta[\rho]-F_\beta\right)\geq0, \qquad \mathcal F_\beta[\rho] =\operatorname{Tr}(\rho H)-T S(\rho).$$

Thus $\rho_\beta$ uniquely minimizes $\mathcal F_\beta$ and the minimum is $F_\beta=-T\ln Z$. This is an equilibrium variational statement, not a real-time account of thermalization.

If a number operator $N$ is conserved, so $[H,N]=0$, and the reservoir also exchanges particles, then

$$\rho_{\beta,\mu}=\frac{e^{-\beta(H-\mu N)}}{\Xi}, \qquad \Xi=\operatorname{Tr}e^{-\beta(H-\mu N)}.$$

The state exists only when $\Xi\lt \infty$. For several noncommuting charges, an exponential ansatz can still be studied, but it requires a separately stated constraint and equilibrium principle; it is not the ordinary grand-canonical construction by default.

Reduced states and entanglement

For a bipartite state $\rho_{AB}$, the reduced state $\rho_A$ is the unique operator satisfying

$$\operatorname{Tr}_A(\rho_A A) =\operatorname{Tr}_{AB}\!\left[\rho_{AB}(A\otimes I_B)\right]$$

for every observable $A$ on subsystem $A$. In any orthonormal basis $\{|b\rangle\}$ of $B$,

$$\rho_A=\operatorname{Tr}_B\rho_{AB} =\sum_b (I_A\otimes\langle b|)\rho_{AB}(I_A\otimes|b\rangle).$$

The result is independent of the chosen basis. Preskill derives the expectation identity, basis formula, and reduced-state properties in Preskill 2015, §2.3.1, pp. 18–21.

For the Bell state

$$|\Phi^+\rangle_{AB}=\frac{|00\rangle+|11\rangle}{\sqrt2}, \qquad \rho_A=\rho_B=\frac{I}{2}.$$

The joint state is pure, so $S(\rho_{AB})=0$, while each reduced state has $S(\rho_A)=S(\rho_B)=\ln2$. More generally, a bipartite pure state’s two reduced density operators have the same nonzero eigenvalues, as follows from the Schmidt decomposition in Preskill 2015, §2.4, pp. 23–25.

This local mixedness is not an unknown classical label that a better measurement on $A$ could reveal; it records correlations with $B$. Conversely, a reduced state can happen to equal a Gibbs state, but mixedness alone supplies neither a Hamiltonian nor a temperature. Reduced-state entropy becomes thermodynamic entropy only after an additional physical argument identifies the subsystem, observables, equilibrium state, and thermodynamic limit or coarse graining.

Exercise: a thermal qubit

Let a two-level system have energies $0$ and $\epsilon\gt 0$. Compute its thermal state, mean energy, and entropy, and find the limits $T\to0$ and $T\to\infty$.

Solution

The partition function and density operator are

$$Z=1+e^{-\beta\epsilon}, \qquad \rho_\beta=\frac{1}{Z} \begin{pmatrix} 1&0\\ 0&e^{-\beta\epsilon} \end{pmatrix}.$$

Writing $q=e^{-\beta\epsilon}/Z=1/(e^{\beta\epsilon}+1)$, the energy and entropy are

$$U=q\epsilon, \qquad S=-(1-q)\ln(1-q)-q\ln q.$$

As $T\to0$, $q\to0$, the state approaches the pure ground state and $S\to0$. As $T\to\infty$, $q\to1/2$, so $\rho_\beta\to I/2$ and $S\to\ln2$. If the lowest energy instead had degeneracy $g$, the Gibbs limit would be the equal mixture on that eigenspace and would retain entropy $\ln g$.

Common pitfalls

A mixture is not a coherent superposition. The state $(|0\rangle+|1\rangle)/\sqrt2$ has off-diagonal coherence in the energy basis and is pure. The equal probabilistic mixture of $|0\rangle$ and $|1\rangle$ is $I/2$ and is mixed.

Diagonal does not mean classical. Every density operator is diagonal in its eigenbasis. What is physically relevant is which observables are accessible, which basis is selected by the Hamiltonian or environment, and whether coherence is present relative to those observables.

Gibbs form is not a thermalization proof. Showing that $\rho_\beta$ minimizes a free-energy functional identifies equilibrium. It does not show that a closed finite system dynamically converges to that state.

Scope toward gravity

Finite-dimensional Gibbs states do not by themselves explain irreversible equilibration. Infinite systems can admit inequivalent phases, while quantum fields introduce local algebras, regulator dependence, and ultraviolet-divergent subsystem entropies absent from the finite-dimensional examples here.

No formula on this page identifies von Neumann entropy with Bekenstein–Hawking, Wald, generalized, or fine-grained radiation entropy. Bekenstein–Hawking and Wald entropy are gravitational horizon constructions. In semiclassical continuum settings, generalized and radiation entropies additionally require a specified state and region or algebra, a renormalization prescription, and a controlled regime developed in Black-Hole Thermodynamics and Semiclassical Gravity. A reduced density operator may look thermal without having been prepared by a heat bath; thermality remains relative to a Hamiltonian, temperature, and chosen observable algebra.

The chapter sequence now continues to Fluids, Plasmas, and Kinetic Theory, where distribution functions evolve rather than merely label equilibrium.

References

• Preskill, John. Lecture Notes for Ph219/CS219: Quantum Information, Chapter 2, “Foundations I: States and Ensembles,” updated July 2015, §§2.3–2.5, pp. 16–35. California Institute of Technology. Official PDF.
• Tokmakoff, Andrei. “The Density Matrix,” MIT 5.74 Introductory Quantum Mechanics II (Spring 2009), pp. 2–6, equations (1.8)–(1.33). Massachusetts Institute of Technology OpenCourseWare. Official PDF.
• Watrous, John. The Theory of Quantum Information. Cambridge University Press, 2018, §5.2.1, pp. 265–269. Author’s PDF. doi:10.1017/9781316848142.