Entanglement
Entanglement is the quantum possibility that a composite system has a definite state even though its parts do not each have their own pure states.
Plain-language picture
Classical correlation can come from ignorance. Charlie gives one coin to Alice and one to Bob; if Alice later sees a penny, she can infer what Bob has. Nothing mysterious is required because the coins had definite identities all along.
Entanglement is different. The joint state can be pure and fully specified while the local subsystems are not described by their own kets.
Two-spin example
A singlet-style state:
(|ud⟩ - |du⟩)/√2This state predicts strong correlations between Alice’s and Bob’s spin measurements. But Alice cannot choose her outcome, and Bob’s local statistics do not become a controllable message.
Why density matrices enter
When the total state is entangled, the best description of one part alone is usually a reduced Density matrix, not a pure ket. This is the technical way to say that the part does not have its own complete pure-state description.
Common pitfalls
- Do not reduce entanglement to classical ignorance.
- Do not use entanglement as a faster-than-light signaling mechanism.
- Do not assume that knowing the whole implies knowing pure states for the parts.