Entangled States

Canonical treatment: Entangled States contains the maintained derivations, mixed-state qualifications, examples, exercises, and references.

This bridge is intentionally limited to the minimal definition and operational distinction needed in Core Formalism.

Minimal definition

After specifying a bipartition $A|B$, a pure state is a product state if

$$\lvert\Psi\rangle_{AB}=\lvert\psi\rangle_A\otimes\lvert\phi\rangle_B.$$

It is entangled across $A|B$ when no such factorization exists. Entanglement is therefore relative to a subsystem decomposition; the same vector can have different status under a different split.

Operational distinction

An entangled pure joint state can give mixed reduced states to both subsystems. It can also exhibit correlations that no assignment of independent pure subsystem states reproduces. These facts do not make entanglement a force or a faster-than-light signaling mechanism.

As a quick contrast,

$$\lvert+\rangle_A\lvert-\rangle_B$$

is product despite having several product-basis amplitudes, whereas the Bell state $(\lvert00\rangle+\lvert11\rangle)/\sqrt2$ is entangled.

Continue to the canonical page for coefficient-matrix tests, Schmidt criteria, reduced-state calculations, and mixed-state separability.