Tensor product in quantum computing

We don't need to understand a super generalized version of tensor products to know what they mean in basic quantum computing!

Intuitively, taking a tensor product of two qubits simply means putting them together on the same quantum system/computer.

When we write the bra-ket notation: $|00⟩$ that is the same as $|0⟩\otimes |0⟩$.

The tensor product is called a "product" because it distributes over addition.

E.g. consider:

Intuitively, in this operation we just put a Hadamard gate qubit together with a second pure $|0⟩$ qubit.

And the outcome still has the second qubit as always 0, because we haven't made them interact.

The quantum state $\frac{|00⟩+|10⟩}{\sqrt{2}}$ is called a separable state, because it can be written as a single product of two different qubits. We have simply brought two qubits together, without making them interact.

If we then add a CNOT gate to make a Bell state:we can now see that the Bell state is non-separable: we've made the two qubits interact, and there is no way to write this state with a single tensor product. The qubits are fundamentally entangled.