Bell State

The four maximally entangled two-qubit states, a key resource for quantum communication protocols.

Bell states (also called EPR pairs or Bell pairs) are the four maximally entangled states of two qubits. They’re named after physicist John Bell and are fundamental to quantum information.

The Four Bell States

$$|{\Phi }^{+}⟩=\frac{1}{\sqrt{2}}(|00⟩+|11⟩)$$ $$|{\Phi }^{-}⟩=\frac{1}{\sqrt{2}}(|00⟩-|11⟩)$$ $$|{\Psi }^{+}⟩=\frac{1}{\sqrt{2}}(|01⟩+|10⟩)$$ $$|{\Psi }^{-}⟩=\frac{1}{\sqrt{2}}(|01⟩-|10⟩)$$

Alternative notation: $|{\beta }_{00}⟩,|{\beta }_{01}⟩,|{\beta }_{10}⟩,|{\beta }_{11}⟩$.

Properties

Maximally Entangled

Each Bell state has maximum entanglement. Tracing out either qubit gives the maximally mixed state: $${\text{Tr}}_B(|{\Phi }^{+}⟩⟨{\Phi }^{+}|)=\frac{I}{2}$$

Perfect Correlations

Measuring both qubits in the same basis always gives correlated results:

Orthonormal Basis

The four Bell states form a complete orthonormal basis for two qubits. Any two-qubit state can be written as a superposition of Bell states.

Creating Bell States

The standard circuit (Bell circuit):

|0⟩ ──H────●──── |Φ+⟩
           │
|0⟩ ───────⊕────

1. Hadamard on first qubit
2. CNOT with first as control

Different input states produce different Bell states.

Bell Measurement

A Bell measurement projects onto the Bell basis, distinguishing all four states.

Circuit (reverse of creation):

     ●────H──── Measure → tells which Bell state
     │
     ⊕──────── Measure →

Applications

Bell State as a Resource

Think of a Bell pair as a resource that can be “spent”:

• Teleportation consumes one Bell pair to send one qubit
• Superdense coding consumes one Bell pair to send two classical bits

Entanglement is a currency in quantum information!

See also: Entanglement, Bell Inequality, Quantum Teleportation, CNOT Gate