What is a Bell state?
A Bell state is a maximally entangled two-qubit state.
The most commonly used Bell state is:
It has perfect quantum correlations and cannot be factored into two independent qubit states.
🧩 What is a Bell state circuit?
A Bell state circuit is the smallest quantum circuit that generates entanglement from an initially separable state.
It uses only two gates:
- Hadamard (H)
- Controlled-NOT (CNOT)
🔄 Step-by-Step Bell State Circuit
Initial state
We always start with two qubits initialized to:
Step 1: Apply (H \otimes I)
Apply a Hadamard gate to the first qubit:
This creates superposition, but the qubits are not yet entangled.
Step 2: Apply CNOT
Use the first qubit as control, second as target:
🎉 This final state is entangled.
🧠 Why this circuit creates entanglement
| Gate | Role |
|---|---|
| Hadamard | Creates superposition |
| CNOT | Correlates qubits conditionally |
👉 Neither gate alone is sufficient:
- H alone → superposition, no entanglement
- CNOT alone on |00⟩ → no effect
Together → entanglement
📐 Circuit diagram interpretation
|0⟩ ──H──●──
│
|0⟩ ─────⊕──- ● = control qubit
- ⊕ = target (NOT applied if control = 1)
🔔 All four Bell states (generated variants)
By changing inputs or adding Pauli gates, we get:
🔗 Where Bell state circuits are used
| Application | Why Bell states matter |
|---|---|
| Quantum teleportation | Shared entanglement |
| Superdense coding | 2 classical bits via 1 qubit |
| Quantum cryptography | Security via entanglement |
| Bell inequality tests | Non-classical correlations |
🧠 One-line takeaway
A Bell state circuit is the minimal quantum circuit (H + CNOT) that converts superposition into entanglement.