Reduced States

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

This bridge is intentionally limited to subsystem-state motivation and the minimum reduced-state dictionary.

Subsystem state

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

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

The result is not an approximation. It is the unique operator that reproduces every prediction available to measurements on $A$:

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

Reduction discards information accessible only through $B$ or joint measurements, especially correlations. A pure joint state may therefore have a mixed reduced state, and different joint states can share the same marginal.

Use Partial Trace: First Encounter for one mechanical calculation, then continue to the canonical page for uniqueness, dynamics, and detailed examples.