Monogamy of Entanglement — CKW Tangle Conservation

Three-Qubit State

Amplitude on A — θ 45.0°

B / C balance — φ 45.0°

Quantum State

|ψ⟩ = 0.71|100⟩ + 0.50|010⟩ + 0.50|001⟩

Pairwise Entanglement

Concurrence C(A,B)

0.500

Concurrence C(A,C)

0.500

τ(AB)+τ(AC) = 0.500τ(A|BC) = 0.500

Residual tangle τ(A|BC)

0.500

Monogamy gap

0.000

How it works

Take a single excitation shared by three qubits — the generalized W-class state |ψ⟩ = a|100⟩ + b|010⟩ + c|001⟩ with a² + b² + c² = 1. This is exactly the family Coffman, Kundu and Wootters used in 2000 to prove that entanglement cannot be freely shared.

Pairwise concurrence:  C(A,B) = 2ab      C(A,C) = 2ac
One-vs-rest tangle:    τ(A|BC) = 4a²(1-a²)
CKW monogamy:          τ(AB) + τ(AC) ≤ τ(A|BC)

For this particular state family the inequality is always an exact equality: C(A,B)² + C(A,C)² = 4a²b² + 4a²c² = 4a²(b²+c²) = τ(A|BC). Slide θ (sets a = cos θ) or φ (splits the rest between b = sinθ·cosφ and c = sinθ·sinφ) and the green+orange bar stays glued to the τ(A|BC) marker — pushing more entanglement into the A–B bond leaves strictly less available for A–C, and vice versa. Qubit A simply cannot be maximally entangled with both B and C at once.

θ slider — how much amplitude sits on qubit A versus the B/C pair (a = cos θ).
φ slider — how the remaining amplitude splits between B and C.
Bond glow/thickness — each bond's brightness is its concurrence; node size is that qubit's excitation probability (a², b² or c²).
Monogamy gap — τ(A|BC) − [τ(AB)+τ(AC)], numerically ≈0 for this state class, confirming the CKW equality live.

Real-world relevance: this monogamy bound is why entanglement swapping, quantum repeaters and eavesdropping-detection (e.g. E91 QKD) all work — an eavesdropper who entangles with a qubit necessarily steals entanglement away from the legitimate pair, and that theft is bounded exactly by this inequality.

Qubit A A–B bond (C(A,B)) A–C bond (C(A,C))

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