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