What are Bell states?

What are Bell states?

Bell states are quantum states of two qubits that represent simple examples of quantum entanglement. When one of the two qubits is measured, it takes on a specific value, and the second qubit is forced to also take on a specific value, as the entangled state collapses.

(Learn more about entangled states in the PennyLane Codebook.)

Bell states are also known as EPR states or EPR pairs.

There are four Bell states, which are as below.

$$ |\beta_{00}\rangle = \frac{|00\rangle + |11\rangle}{\sqrt{2}} \\ |\beta_{01}\rangle = \frac{|01\rangle + |10\rangle}{\sqrt{2}} \\ |\beta_{10}\rangle = \frac{|00\rangle - |11\rangle}{\sqrt{2}} \\ |\beta_{11}\rangle = \frac{|01\rangle - |10\rangle}{\sqrt{2}} $$

For notations such as $|01\rangle$ and $|11\rangle$, the first digit in the brackets refers to the first qubit, and the second digit refers to the second qubit.

The Bell states can be produced by a very simple quantum circuit consisting of a Hadamard gate and a CNOT gate, as shown below.

A simple quantum circuit for producing Bell states.

If qubit x starts with a state of $|0\rangle$, the Hadamard gate puts it into a superposition of $|0\rangle$ and $|1\rangle$. The CNOT gate flips qubit y depending on the state of qubit x, but since qubit x is in a superposition of $|0\rangle$ and $|1\rangle$ this makes the final state of qubit y dependent on what the final state of qubit x turns out to be — thus the two qubits are part of an entangled state.