When two quantum systems are considered together, the state space of the composite system is built from the tensor product of the individual state spaces.

If system $A$ has basis states $|0\rangle_A,|1\rangle_A$ and system $B$ has $|0\rangle_B,|1\rangle_B$, their joint basis contains

$$|00\rangle,\ |01\rangle,\ |10\rangle,\ |11\rangle.$$

A general joint state is a superposition

$$|\psi\rangle=\sum_{i,j}c_{ij}|i\rangle_A|j\rangle_B,$$

with

$$\sum_{i,j}|c_{ij}|^2=1.$$

Product states

If the coefficients can be written as

$$c_{ij}=a_i b_j,$$

then

$$|\psi\rangle=|\phi\rangle_A\otimes|\chi\rangle_B$$

is a product state: each subsystem has its own state vector and the joint amplitudes factorize.

Joint measurements

The Born rule now assigns probabilities to pairs of outcomes. Measurements on one subsystem can be correlated with measurements on the other because the probabilities come from one joint state.

The tensor-product construction is what allows a composite quantum system to contain states that cannot be reduced to independent states of its parts.