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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.

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 one qubit instantly determines the state of the other, regardless of distance.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 magic dice that always show correlated results -- if one shows 6, the other instantly shows 6 too (or its complement), no matter how far apart they are.
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

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

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