Tensor product states

Tensor products are how quantum mechanics builds the state space of a composite system from the state spaces of its parts.

Definition by basis

If Alice has basis states |a⟩ and Bob has basis states |b⟩, the combined system has basis states:

|ab⟩ = |a⟩ ⊗ |b⟩

If Alice’s space has dimension N_A and Bob’s has dimension N_B, the combined space has dimension N_A N_B.

For two spins:

|uu⟩, |ud⟩, |du⟩, |dd⟩

A product state has the form:

|Ψ_AB⟩ = |A⟩ ⊗ |B⟩

A general state is a sum over product-basis states:

|Ψ_AB⟩ = Σ_ab ψ_ab |ab⟩

If ψ_ab cannot be factored into Alice-only amplitudes times Bob-only amplitudes, the state is entangled.

Operators can act on one subsystem while leaving the other alone:

M_A ⊗ I_B
I_A ⊗ M_B

This is why Alice and Bob can have compatible measurements on separated subsystems.

Do not confuse |ab⟩ with two separate states; it is one state of the combined system.
Do not add vectors from different subsystem Hilbert spaces.
Do not assume every state in the tensor product factors.