Bell states and measurements

Figure 1: John Bell

The Bell states (named after physicist John Bell) are four maximally entangled quantum states of two qubits. This post describes the bell states and the quantum circuits for implementing them. Click on the circuit label to open in IBM Quantum Composer.

The four Bell states are:

Circuit 1:\(|\Phi^+\rangle\)

\(\begin{aligned}\hspace{20 mm}|\Phi^+\rangle = \frac{|00\rangle + |11\rangle}{\sqrt2}\end{aligned}\)

Circuit 2: \(|\Phi^-\rangle\)

\(\begin{aligned}\hspace{20 mm}|\Phi^-\rangle = \frac{|00\rangle - |11\rangle}{\sqrt2}\end{aligned}\)

Circuit 3: \(|\Psi^+\rangle\)

\(\begin{aligned}\hspace{20 mm}|\Psi^+\rangle = \frac{|01\rangle + |10\rangle}{\sqrt2}\end{aligned}\)

Circuit 4: \(|\Psi^-\rangle\)

\(\begin{aligned}\hspace{20 mm}|\Psi^-\rangle = \frac{|01\rangle - |10\rangle}{\sqrt2}\end{aligned}\)

All circuits are implemented with a Hadamard gate followed by a controlled NOT gate. Some circuits also utilize the Pauli-X (NOT) gate and Pauli-Z gate.[1]

Note that in some circuit editors, the Pauli-X gate may be rendered with a squared-X symbol rather than a circled-plus symbol as rendered here. Also, this is just one of many possible implementation of the Bell states.[2]

Measurements

By default, measurements are in the Z-basis.

To measure in the X-basis, use a Hadamard gate which rotates the state onto the X-axis.

Figure 2:  The Hadamard gate interchanges
the X- and Z-axes, and inverts the Y-axis

To measure in the Y-basis use an inverse S gate (S^†) followed by a Hadamard gate which rotates the state onto the Y-axis.

Figure 3: The S^† gate performs a quarter turn
clockwise about the Z-axis

The quantum circuits for measuring in the three Pauli bases:

Figure 4: Measuring in Pauli bases

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[1] Description of circuits (where Z_n and X_n act on qubit q[n]):

\(\begin{aligned}\hspace{5 mm}|\Phi^+\rangle = CNOT \cdot H_0 |00 \rangle\end{aligned}\)

\(\begin{aligned}\hspace{5 mm}|\Phi^-\rangle = Z_1 \cdot CNOT \cdot H_0 |00 \rangle\end{aligned}\)

\(\begin{aligned}\hspace{5 mm}|\Psi^+\rangle = X_0 \cdot CNOT \cdot H_0 |00 \rangle\end{aligned}\)

\(\begin{aligned}\hspace{5 mm}|\Psi^-\rangle = X_0 \cdot Z_1 \cdot CNOT \cdot H_0 |00 \rangle\end{aligned}\)

[2] An alternative set of circuits for creating Bell states.

Figure 5: Alternative Bell state circuits