Bell States

The four Bell states are the maximally entangled two-qubit states and form an orthonormal basis for two qubits.

The four states

The Bell states, also called EPR pairs, are the four maximally entangled states of two qubits:

• |Phi+> = (|00> + |11>)/sqrt(2)
• |Phi-> = (|00> - |11>)/sqrt(2)
• |Psi+> = (|01> + |10>)/sqrt(2)
• |Psi-> = (|01> - |10>)/sqrt(2)

They are mutually orthogonal and normalised, so they form a complete basis for the four-dimensional two-qubit space. Any two-qubit state can be expanded in this Bell basis instead of the computational basis.

How to make one

A Bell state is produced by a simple circuit: apply a Hadamard to the first qubit, then a CNOT with the first as control and second as target. Starting from |00> this yields |Phi+>. Changing the input among |00>, |01>, |10>, |11> selects which of the four Bell states appears.

python

# Bell state |Phi+> with a small statevector simulation
import numpy as np
H = np.array([[1,1],[1,-1]])/np.sqrt(2)
CNOT = np.array([[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]])
state = np.kron(H@np.array([1,0]), np.array([1,0]))
state = CNOT @ state
print(np.round(state,3))  # [0.707 0 0 0.707]

Why they matter

Bell states are the workhorses of quantum information. Teleportation and superdense coding both consume a shared Bell pair. A joint measurement in the Bell basis (a Bell measurement) distinguishes the four states and is the key operation in those protocols.

Maximal entanglement

Each Bell state has a maximally mixed reduced state: trace out either qubit and you get the identity divided by two, meaning a single qubit of the pair carries no local information at all. Its entanglement entropy is exactly one bit, the maximum for two qubits.