The Two Qubit States and Their Transformation

Two qubit states

Let’s say we have two qubits. We can call them $|a\rangle$ and $|b\rangle$. Each of the two qubits has its own probability amplitudes: $|a\rangle=a_0|0\rangle+a_1|1\rangle=\begin{bmatrix}a_0 \\ a_1\end{bmatrix}$ and $|b\rangle=b_0|0\rangle+b_1|1\rangle=\begin{bmatrix}b_0 \\ b_1\end{bmatrix}$. When we look at these two qubits concurrently, there are four different combinations of the basis states. Each of these combinations has its probability amplitude. These are the products of the probability amplitudes of the two corresponding states.

• $a_0|0\rangle b_0 |0\rangle$
• $a_0|0\rangle b_1 |1\rangle$
• $a_1|1\rangle b_0 |0\rangle$
• $a_1|1\rangle b_1 |1\rangle$

These four states form a quantum system on their own. Therefore, we can represent them in a single equation. While we are free to choose an arbitrary name for the state, we use $|ab\rangle$ because this state is the collective quantum state of $|a\rangle$ and ...