Susskind QM Lecture 6 - Combining Systems Entanglement
One-line takeaway
The state space of a composite quantum system is a tensor product, and that construction permits states that are not states of the parts separately.
Plain-language map
Lecture 6 introduces Alice and Bob as labels for subsystems. If Alice has state space S_A and Bob has state space S_B, the combined system lives in S_A ⊗ S_B. The dimension multiplies.
dim(S_A ⊗ S_B) = dim(S_A) dim(S_B)For two spins, the product basis is:
|uu⟩, |ud⟩, |du⟩, |dd⟩A product state factors into Alice’s state and Bob’s state. But a general state in the four-dimensional space does not have to factor. That is where entanglement enters.
Classical correlation versus entanglement
Susskind first separates ordinary correlation from quantum entanglement. Classical correlation can come from shared history plus ignorance. Entanglement is not just ignorance about pre-existing labels; it is a property of the composite quantum state.
A singlet-style example is:
(|ud⟩ - |du⟩)/√2The pair has a well-defined joint state, while the individual spins do not have their own pure spin states.
Core links
Common pitfalls
- Do not add Alice’s vector to Bob’s vector; they live in different spaces.
- Do not treat |ab⟩ as two states. It is one basis state of the combined system.
- Do not mistake classical correlation for entanglement.