Reduced Density Matrices

Canonical treatment: Reduced Density Operators maintains the complete theory, examples, exercises, and references.

This bridge is intentionally limited to Core density-matrix calculations and interpretation for subsystem predictions.

Calculation and meaning

For a joint density matrix $\rho_{AB}$,

$$\rho_A=\operatorname{Tr}_B\rho_{AB}.$$

In product bases, the matrix elements are

$$(\rho_A)_{ii'}=\sum_j(\rho_{AB})_{ij,i'j}.$$

The result reproduces every expectation value of an observable local to $A$:

$$\operatorname{Tr}(\rho_A M_A) = \operatorname{Tr}\!\left[\rho_{AB}(M_A\otimes I_B)\right].$$

Check Hermiticity, positivity, and unit trace. Do not interpret $\rho_A$ as a post-measurement conditional state: reduction discards a subsystem without conditioning on an outcome.

Continue to the canonical page for channel structure, shared spectra of pure bipartite states, correlations, dynamics, and infinite-dimensional qualifications.