What is a Bell State?
Bell states are the simplest examples of quantum entanglement, and the foundation of quantum teleportation, superdense coding, and quantum cryptography. Here's how to build one.
The intuition
A Bell state is a 2-qubit quantum state where the qubits are maximally entangled. This means you can't describe each qubit independently. Their fates are linked.
Measure the first qubit: you get 0 or 1 with equal probability (50/50). But the moment you measure it, the second qubit's outcome is instantly determined, even if it's light-years away.
This isn't a trick. It's quantum mechanics. Einstein called it “spooky action at a distance.” Bell's theorem (1964) proved the correlation is genuinely quantum, ruling out hidden pre-set values.
The Bell state Φ⁺ written out:
|Φ⁺⟩ = (|00⟩ + |11⟩) / √2
50% probability of measuring 00, 50% of measuring 11
How to build a Bell state in Qiskit
Two gates. That's it. Here's the complete circuit:
from qiskit import QuantumCircuit
from qiskit_aer import AerSimulator
# Create a 2-qubit, 2-classical-bit circuit
qc = QuantumCircuit(2, 2)
# Step 1: Put qubit 0 in superposition
qc.h(0)
# State: (|00⟩ + |10⟩) / √2
# Step 2: Entangle qubit 1 with qubit 0
qc.cx(0, 1)
# State: (|00⟩ + |11⟩) / √2 ← Bell state Φ+
# Step 3: Measure
qc.measure([0, 1], [0, 1])
# Run simulation
sim = AerSimulator()
result = sim.run(qc, shots=2048).result()
print(result.get_counts())
# Output: {'00': ~1024, '11': ~1024}
# Note: '01' and '10' NEVER appear. That's entanglement.Start: |00⟩
Both qubits start in |0⟩. The full 2-qubit state is |00⟩: a definite, classical state.
H gate → superposition
Hadamard on qubit 0. Now the state is (|00⟩ + |10⟩)/√2. Qubit 0 is in superposition, qubit 1 is still |0⟩.
CNOT → entanglement
CNOT flips qubit 1 when qubit 0 is |1⟩. Result: (|00⟩ + |11⟩)/√2. The Bell state. Maximum entanglement.
The four Bell states
There are four maximally entangled 2-qubit states, forming an orthonormal basis (the Bell basis).
(|00⟩ + |11⟩) / √2Circuit: H on q0, then CNOT(0,1)
Default Bell pair, used in teleportation and superdense coding
(|00⟩ − |11⟩) / √2Circuit: H, CNOT, then Z on q0
Relative phase flip, used in error correction
(|01⟩ + |10⟩) / √2Circuit: H, X on q1, then CNOT(0,1)
Anti-correlated: each qubit gives the opposite result
(|01⟩ − |10⟩) / √2Circuit: H, X on q1, CNOT, Z on q0
Singlet state, used in Bell inequality experiments
What are Bell states used for?
Quantum Teleportation
Transmit an arbitrary qubit state from Alice to Bob using a shared Bell pair + 2 classical bits. No physical qubit is transported.
Superdense Coding
Send 2 classical bits of information using only 1 qubit. Encode information into which Bell state you create.
Quantum Cryptography (BB84)
Create provably secure encryption keys. Any eavesdropping disturbs the Bell pairs and is immediately detectable.
Bell Test Experiments
Prove quantum mechanics is fundamentally non-classical by violating Bell inequalities, ruling out local hidden variables.
Build it yourself, right now
The Playground has the Bell state pre-loaded. Run it, modify it, break it, fix it. That's how you actually learn quantum computing.
Free forever on Track 1. No credit card.