Partial Trace: First Encounter

Canonical treatment: Partial Trace maintains the basis-independence proof, matrix rules, algorithms, exercises, and references.

This bridge is intentionally limited to the minimal partial-trace calculation needed before the full reduced-state treatment.

Working rule

For a bipartite operator, tracing out $B$ leaves an operator on $A$:

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

On product dyads,

$$\operatorname{Tr}_B\!\left( \lvert a\rangle\langle a'\rvert\otimes \lvert b\rangle\langle b'\rvert \right) = \langle b'\vert b\rangle\, \lvert a\rangle\langle a'\rvert.$$

Apply this rule term by term and check that the result is Hermitian, positive, and has unit trace when the input is a density operator.

Minimal example

For $\lvert\Phi^+\rangle=(\lvert00\rangle+\lvert11\rangle)/\sqrt2$, the cross terms vanish under $\operatorname{Tr}_B$ and

$$\rho_A=\frac12\left(\lvert0\rangle\langle0\rvert+\lvert1\rangle\langle1\rvert\right)=\frac{I}{2}.$$

The canonical page develops why this answer is basis independent and how to compute it from blocks, indices, or software.