Subsystems and Local Observables

Canonical treatment: Local and Global Observables maintains the classification, measurements, correlations, examples, exercises, and references.

This bridge is intentionally limited to embedding subsystem observables into a composite Hilbert space.

Local embedding

If $M_A$ acts on $\mathcal H_A$, its action on $\mathcal H_A\otimes\mathcal H_B$ is

$$M_A\otimes I_B.$$

Similarly, an observable local to $B$ is $I_A\otimes N_B$. The two embeddings commute:

$$[M_A\otimes I_B, I_A\otimes N_B]=0.$$

A product observable $M_A\otimes N_B$ probes a joint correlation; a sum such as $M_A\otimes I+I\otimes N_B$ contains local contributions; and a general global observable need not have either form.

Local expectation values can be computed from the reduced state,

$$\langle M_A\otimes I_B\rangle=\operatorname{Tr}(\rho_A M_A).$$

Continue to the canonical page for joint measurements, singlet correlations, and genuinely global observables.