Bell State and GHZ State

What is Bell states?

The Bell states are four specific maximally entangled quantum states of two qubits.
$$\begin{equation} \begin{aligned} \vert \Psi^\pm \rangle = \frac{1}{\sqrt{2}}(\vert 0\rangle\vert 1\rangle\pm\vert 1\rangle\vert 0\rangle)\\ \vert \Phi^\pm \rangle = \frac{1}{\sqrt{2}}(\vert 0\rangle\vert 0\rangle\pm\vert 1\rangle\vert 1\rangle) \end{aligned} \end{equation}$$

How to generate and measure Bell states?

It is easy to generate and measure Bell states with Hadamard gate and Controled-Not gate.

The Bell measurement: the gates on the left hand side allow us to generate the four Bell states from the four possible different inputs. Reversing the order of the gates (right-hand side of the diagram) corresponds to a Bell measurement.

Hadamard gate

It is equivalent to the following unitary transformation:
$$\begin{equation} \begin{aligned} \vert 0\rangle \rightarrow \frac{1}{\sqrt{2}}(\vert 0\rangle+\vert 1\rangle)\\ \vert 1\rangle \rightarrow \frac{1}{\sqrt{2}}(\vert 0\rangle-\vert 1\rangle) \end{aligned} \end{equation}$$

Controled-Not gate

It flips the second of two qubits if and only if the first is $\vert 1\rangle$, namely

$$\begin{equation} \begin{aligned} \vert 0\rangle\vert 0\rangle \rightarrow \vert 0\rangle\vert 0\rangle\\ \vert 0\rangle\vert 1\rangle \rightarrow \vert 0\rangle\vert 1\rangle\\ \vert 1\rangle\vert 0\rangle \rightarrow \vert 1\rangle\vert 1\rangle\\ \vert 1\rangle\vert 1\rangle \rightarrow \vert 1\rangle\vert 0\rangle \end{aligned} \end{equation}$$
The first qubit control whether apply ‘NOT’ on the other qubit.

Greenberger–Horne–Zeilinger state (GHZ state)

Maximally entangled three-particle states, that is,
$$\begin{equation} \begin{aligned} \frac{1}{\sqrt{2}}(\vert a\rangle\vert b\rangle\vert c\rangle\pm\vert \overline{a}\rangle\vert \overline{b}\rangle\vert \overline{c}\rangle) \end{aligned} \end{equation}$$
where a, b and c can each take the values 0 and 1 and $\overline{a}$ and $\overline{b}$and $\overline{c}$ denote NOT-a and NOT-b
I feel confused in this place. Wikipedia says GHZ state is only ${\displaystyle |\mathrm {GHZ} \rangle ={\frac {|0\rangle ^{\otimes M}+|1\rangle ^{\otimes M}}{\sqrt {2}}}.}$ which is different with this paper “Quantum Teleportation and Multi-photon Entanglement, Jian-Wei Pan”

Reference

1. https://en.wikipedia.org/wiki/Bell_state
2. https://en.wikipedia.org/wiki/Greenberger%E2%80%93Horne%E2%80%93Zeilinger_state
3. Quantum Teleportation and Multi-photon Entanglement, Jian-Wei Pan