Partial Trace and Purification

Corresponds to Lectures 11–12. The partial trace $\mathrm{Tr}_B$ takes a density matrix on a composite system $AB$ and produces the unique density matrix on $A$ alone consistent with every measurement outcome statistic a local observer of $A$ could ever see — the rigorous resolution of the steering question from the previous note (no information actually travels; the reduced state on $A$ is unaffected by what’s done to $B$, only which decomposition of it is natural changes). Purification runs this in reverse: any mixed state $\rho_A$ can be written as the reduced state of some pure state $|\psi\rangle_{AR}$ on a larger system including a fictitious “reference” $R$ — so mixedness always has an interpretation as entanglement with something outside the system being described, a recurring trick for converting mixed-state questions into (often easier) pure-state ones.

Outline (TODO — flesh out each)

• Partial trace — definition and uniqueness
• Reduced density matrices
• Purifications
• Non-uniqueness of purification; the role of isometries