Quantum entanglement is perhaps the most counterintuitive phenomenon in all of physics. When two particles become entangled, measuring one instantaneously determines the state of the other, regardless of the distance separating them.

Bell States: The Maximally Entangled Pairs

The four Bell states form a complete basis for two-qubit entangled states:

$$|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$$

$$|\Phi^-\rangle = \frac{1}{\sqrt{2}}(|00\rangle - |11\rangle)$$

$$|\Psi^+\rangle = \frac{1}{\sqrt{2}}(|01\rangle + |10\rangle)$$

$$|\Psi^-\rangle = \frac{1}{\sqrt{2}}(|01\rangle - |10\rangle)$$

In the state $|\Phi^+\rangle$, if we measure the first qubit and find $|0\rangle$, the second qubit is guaranteed to be $|0\rangle$ as well — and vice versa. There is no classical explanation for this correlation.

The EPR Paradox

In 1935, Einstein, Podolsky, and Rosen published their famous paper arguing that quantum mechanics must be incomplete. They believed that “hidden variables” must predetermine measurement outcomes. Einstein famously called entanglement “spooky action at a distance.”

Bell’s Inequality

In 1964, John Bell derived an inequality that any local hidden variable theory must satisfy:

$$|E(a, b) - E(a, b')| + |E(a', b) + E(a', b')| \leq 2$$

where $E(a, b)$ is the correlation function for measurements along directions $a$ and $b$. Quantum mechanics predicts violations of this inequality, reaching a maximum of $2\sqrt{2}$ (the Tsirelson bound).

Experiments — from Alain Aspect’s pioneering 1982 tests to the 2022 Nobel Prize-winning loophole-free experiments — have consistently confirmed that nature violates Bell’s inequality.

Creating Entanglement with Qiskit

Here’s how to create and verify a Bell state on a quantum computer:

from qiskit import QuantumCircuit
from qiskit.quantum_info import Statevector
from qiskit.visualization import plot_histogram

# Create a Bell state |Φ+⟩
qc = QuantumCircuit(2, 2)
qc.h(0)        # Hadamard on qubit 0
qc.cx(0, 1)    # CNOT: entangle qubits 0 and 1
qc.measure([0, 1], [0, 1])

# Visualize the circuit
print(qc.draw())

# Simulate the state vector
sv = Statevector.from_instruction(
    QuantumCircuit(2).compose(qc.remove_final_measurements(inplace=False))
)
print("State vector:", sv)

The output should show the state $\frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$ — the $|\Phi^+\rangle$ Bell state.

Applications

Entanglement is the foundation for:

  • Quantum Key Distribution (QKD): Secure communication guaranteed by physics
  • Quantum Teleportation: Transferring quantum states across distances
  • Quantum Computing: The resource that gives quantum computers their power
  • Quantum Sensing: Enhanced precision measurements beyond classical limits

The 2022 Nobel Prize in Physics was awarded to Alain Aspect, John Clauser, and Anton Zeilinger for their experiments establishing the violation of Bell inequalities and pioneering quantum information science.