Quantum Entanglement Swapping
Bell measurement creates entanglement between particles that never interacted
Entanglement Swapping: Start with two independent Bell pairs (1,2) and (3,4). A Bell measurement on particles 2 and 3 collapses their joint state into one of four Bell states, simultaneously projecting particles 1 and 4 — which have never interacted — into an entangled state.
|Φ⁺⟩₁₂ ⊗ |Φ⁺⟩₃₄ = ½(|Φ⁺⟩₂₃|Φ⁺⟩₁₄ + |Φ⁻⟩₂₃|Φ⁻⟩₁₄ + |Ψ⁺⟩₂₃|Ψ⁺⟩₁₄ + |Ψ⁻⟩₂₃|Ψ⁻⟩₁₄)
The four Bell states are: Φ± = (|00⟩±|11⟩)/√2, Ψ± = (|01⟩±|10⟩)/√2.
Used in quantum repeaters for long-distance quantum communication.