Reduced density matrix (construction)

Subsystem state obtained by tracing out degrees of freedom; it encodes all local expectations and correlations of a quantum state.

Reduced density matrix (construction)

A reduced density matrix is the quantum analogue of a marginal distribution: it describes the state of a subsystem when the rest of the degrees of freedom are ignored. In quantum statistical mechanics, it is the object that turns a global thermal state into a locally testable state on a finite region.

Definition via partial trace

Let the system Hilbert space factorize as $\mathcal H=\mathcal H_A\otimes \mathcal H_B$ , where $A$ is the subsystem of interest and $B$ is its complement (environment).

A (mixed) quantum state is represented by a density operator $\rho$ on $\mathcal H$ , which in finite dimensions is a positive semidefinite matrix with trace equal to $1$ :

\rho \ge 0, \qquad \operatorname{Tr}_{\mathcal H}(\rho)=1.

The reduced density matrix on $A$ is defined by the partial trace over $B$ :

\rho_A := \operatorname{Tr}_B(\rho).

It is characterized by the universal property: for every observable $O_A$ acting only on $A$ ,

\operatorname{Tr}_{\mathcal H_A}(\rho_A\, O_A) ={} \operatorname{Tr}_{\mathcal H}\big(\rho\, (O_A\otimes I_B)\big).

Equivalently, choosing any orthonormal basis $\{|b\rangle\}$ of $\mathcal H_B$ (compatible with the inner product ), one has the concrete formula

\rho_A = \sum_b \langle b|\rho|b\rangle,

where each $\langle b|\rho|b\rangle$ is an operator on $\mathcal H_A$ .

This definition guarantees that local expectations computed using the reduced state agree with the global ensemble average .

Reduced thermal state in the canonical ensemble

For a quantum Hamiltonian $H$ at inverse temperature $\beta$ , the canonical ensemble uses the thermal density matrix

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

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

Given a region $A$ (e.g. a finite subset of a lattice), the reduced thermal density matrix is

\rho_{A,\beta} ={} \operatorname{Tr}_{A^c}\!\left(\rho_\beta\right) ={} \frac{\operatorname{Tr}_{A^c}\!\left(e^{-\beta H}\right)}{\operatorname{Tr}\!\left(e^{-\beta H}\right)}.

All local thermodynamic quantities and local correlation functions supported in $A$ can be computed from $\rho_{A,\beta}$ .

Consistency across regions and infinite-volume viewpoint

For nested finite regions $A\subset B$ , reduced states satisfy a consistency (marginalization) property:

\rho_A = \operatorname{Tr}_{B\setminus A}(\rho_B).

This mirrors the way finite-volume marginals are consistent in classical Gibbs measures; compare the role played by a DLR specification in the classical setting. In many constructions of infinite systems, one proves existence of an infinite-volume state by controlling these finite-region reduced density matrices and then taking a suitable weak limit .

Physical interpretation

Local physics: $\rho_A$ is the complete description of outcomes of measurements confined to $A$ .
Open systems viewpoint: tracing out $B$ represents ignorance of (or intentional coarse-graining over) the environment.
Entropy: the von Neumann entropy $S(\rho_A) = -\operatorname{Tr}(\rho_A \log \rho_A)$ is the quantum analogue of the entropy concepts discussed in Gibbs/Shannon entropy and thermodynamic entropy (with important quantum-specific features due to entanglement).

Reduced density matrices are also the natural inputs for defining quantum thermal correlation functions restricted to a subsystem.