Consider the circuit below, where a Hadamard gate is connected to qubit $q_0$ and a controlled-NOT (CNOT) gate is connected to $|q_0q_1\rangle$ after the Hadamard gate:

The Hadamard gate in Fig. 1 effects the transformations: $|0\rangle\to|+\rangle$ and $|1\rangle\to|-\rangle$.

The CNOT gate in Fig. 1 has its control qubit connected to the $q_0$ line, and its target qubit connected to the $q_1$ line.

The unitary matrix representing the CNOT gate is

$\text{CNOT} = \begin{bmatrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0\end{bmatrix}.$

If the input to the CNOT gate is the state $|q_0q_1\rangle = a|00\rangle + b|01\rangle + c|10\rangle + d|11\rangle = \begin{bmatrix}a & b & c & d\end{bmatrix}^\top$, then the output is

$\text{CNOT}|q_0q_1\rangle = \begin{bmatrix}a & b & d & c\end{bmatrix}^\top = a|00\rangle + b|01\rangle + d|10\rangle + c|11\rangle.$

Thus, as an example, if the input is $|q_0q_1\rangle = |10\rangle$, then the Hadamard gate transforms it to $|-,0\rangle = \frac{1}{\sqrt{2}}(|00\rangle - |10\rangle)$, and the CNOT gate further transforms it to

$\text{CNOT}\frac{1}{\sqrt{2}}\begin{bmatrix}1 & 0 & -1 & 0\end{bmatrix}^\top = \frac{1}{\sqrt{2}}\begin{bmatrix}1 & 0 & 0 & -1\end{bmatrix}^\top = \frac{1}{\sqrt{2}}(|00\rangle - |11\rangle).$

Table 1 is the truth table summarising the outputs corresponding to basis-state inputs.

Table 1: Truth table for quantum circuit in Fig. 1.

Input $|q_0q_1\rangle$	Output
$|00\rangle$	$|\Phi^+\rangle = |\beta_{00}\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$
$|01\rangle$	$|\Psi^+\rangle = |\beta_{01}\rangle = \frac{1}{\sqrt{2}}(|01\rangle + |10\rangle)$
$|10\rangle$	$|\Phi^-\rangle = |\beta_{10}\rangle = \frac{1}{\sqrt{2}}(|00\rangle - |11\rangle)$
$|11\rangle$	$|\Psi^-\rangle = |\beta_{11}\rangle = \frac{1}{\sqrt{2}}(|01\rangle - |10\rangle)$

The output states in Table 1 are called the Bell states or Einstein-Podolsky-Rosen (EPR) pairs [KLM07, p. 75; NC10, Sec. 1.3.6], and can be represented concisely as

$|\beta_{xy}\rangle = \frac{1}{\sqrt{2}}\left[|0,y\rangle + (-1)^x|1,\bar{y}\rangle \right].$

When the Bell states are used as an orthonormal basis, they are called the Bell basis [Wil17, pp. 91-93].