Multi-Qubit States

So far, we’ve learned what a quantum state is and its different properties but only in the context of a single qubit. However, a single qubit can’t really do much. What we are really interested in is how we can study multiple qubits and their states.

Two qubits

Let’s start by taking an example of two qubits, both initially in the $|0\rangle$ state.

We can simply represent the combined state of the two qubits as $|00\rangle$. Now, what does this mean in vector form? In vector notation we take the tensor-product or outer product of the vectors as $|0\rangle \otimes|0\rangle$. To keep things simple and avoid writing huge vectors and matrices, we use the bra-ket notation, $|00\rangle$.

Qubits in superposition

Things get interesting when our qubits are in a superposition. Building from the knowledge about a single qubit in superposition being both $|0\rangle$ and $|1\rangle$ simultaneously, two qubits in superposition would be in four different states at the same time. Let’s see how this works.

Our first qubit is in a superposition, so we can represent it as $|q_0\rangle = |0\rangle + |1\rangle$ ...