Purification Overview

Canonical treatment: Purification maintains the constructions, equivalence results, applications, exercises, and references.

This bridge is intentionally limited to density-operator motivation and a first encounter with a purifying reference system.

Definition and construction

A purification of $\rho_A$ is a pure state $\lvert\Psi\rangle_{AR}$ satisfying

$$\operatorname{Tr}_R\lvert\Psi\rangle\langle\Psi\rvert=\rho_A.$$

Given $\rho_A=\sum_k p_k\lvert k\rangle\langle k\rvert$, one choice is

$$\lvert\Psi\rangle_{AR}=\sum_k\sqrt{p_k}\,\lvert k\rangle_A\lvert k\rangle_R.$$

The reference system is an auxiliary mathematical system unless a physical model identifies it with an environment. Purifications are not unique, and no purification selects a privileged ensemble interpretation of $\rho_A$.

The minimum reference dimension equals the rank of $\rho_A$. Continue to the canonical page for nonuniqueness theorems, channel dilation, thermofield connections, and infinite-dimensional qualifications.