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Bell States

Bell states are the four maximally entangled two-qubit states that form an orthonormal basis for the two-qubit Hilbert space and are the fundamental resource for quantum information protocols.

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What it means

The four Bell states are: |Phi+⟩ = (|00⟩+|11⟩)/sqrt(2), |Phi-⟩ = (|00⟩-|11⟩)/sqrt(2), |Psi+⟩ = (|01⟩+|10⟩)/sqrt(2), |Psi-⟩ = (|01⟩-|10⟩)/sqrt(2).They represent maximal entanglement between two qubits -- measuring both gives perfectly correlated results at any distance, visible once the results are compared; each side alone still sees 50/50, so no signal is sent.Bell states are created by applying a Hadamard gate followed by a CNOT gate.They are the key resource in quantum teleportation, superdense coding, quantum key distribution (BB84/E91), and Bell inequality tests that distinguish quantum mechanics from local hidden variable theories.

Everyday analogy

Bell states are like a pair of dice that each land randomly, yet when the two rolls are compared they always agree (or are always complementary), no matter how far apart. Unlike real dice, the result is not set before the roll, and nothing is sent between them.
Think of Bell states as two pages of a book torn in half -- neither half makes sense alone, but together they carry the complete message.

Common misconceptions

  • Bell state correlation does NOT allow faster-than-light communication -- the individual measurement results are still random; only the correlation is predetermined.
  • Bell states are NOT the only entangled states -- they are the maximally entangled two-qubit states, but partial entanglement also exists.

Key takeaways

  • The four Bell states form a basis of maximally entangled two-qubit states.
  • Created by Hadamard + CNOT circuit.
  • Fundamental resource for teleportation, superdense coding, and quantum cryptography.

Check your understanding

Which circuit creates the Bell state |Phi+⟩ from |00⟩?

  1. A.CNOT then H
  2. B.H on first qubit then CNOT
  3. C.X on both qubits
  4. D.Z then CNOT
Show the answer

Answer: B. H on first qubit then CNOT

Why: Applying H to the first qubit creates superposition, then CNOT entangles the two qubits: H⊗I followed by CNOT on |00⟩ gives (|00⟩+|11⟩)/sqrt(2) = |Phi+⟩.

Builds on

QubitSuperpositionCNOT GateHadamard Gate

Builds toward

Quantum EntanglementQuantum TeleportationGHZ State

Related concepts

Quantum Measurement

Primary source: J. S. Bell, Physics Physique Fizika 1, 195 (1964), doi:10.1103/PhysicsPhysiqueFizika.1.195

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