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 $|b\rangle$.

$$|ab\rangle=|a\rangle\otimes|b\rangle=a_0 b_0|0\rangle|0\rangle+a_0 b_1|0\rangle|1\rangle+a_1 b_0|1\rangle|0\rangle+a_1 b_1|1\rangle|1\rangle$$

In this equation, $|ab\rangle$ is an arbitrary name. The last term is the four combinations reordered to have the amplitudes at the beginning. But what does $|a\rangle\otimes|b\rangle$ mean?

The term $|a\rangle\otimes|b\rangle$ is the tensor product of the two vectors $|a\rangle$ and $|b\rangle$.

The tensor product (denoted by the symbol $\otimes$) is the mathematical way of calculating the amplitudes. In general, the tensor product of two vectors $v$ and $w$ is a vector of all combinations. Like this:

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