The state is a genuine 8-amplitude 3-qubit pure state |ψ⟩ ∈ ℂ⁸, mixing a GHZ component (|000⟩+e^{iδ}|111⟩) with a W-class component (a|100⟩+b|010⟩+c|001⟩). Every quantity below is computed from scratch each frame: real partial traces ρAB=TrC|ψ⟩⟨ψ|, ρAC=TrB|ψ⟩⟨ψ|, ρA=TrBC|ψ⟩⟨ψ| built by summing outer products of complex amplitudes, then real Wootters concurrence C=max(0, √λ₁−√λ₂−√λ₃−√λ₄) from the eigenvalues of R=ρ(σy⊗σy)ρ*(σy⊗σy) — obtained via a from-scratch Faddeev–LeVerrier characteristic polynomial and a Durand–Kerner complex root solver, no shortcuts.
Pairwise tangle: τ(AB)=C(A,B)² τ(AC)=C(A,C)²
One-vs-rest tangle: τ(A|BC) = 4·det(ρ_A) (exact for a pure global state)
CKW monogamy: τ(AB) + τ(AC) ≤ τ(A|BC)
Pure GHZ (α=0°) makes both pairwise concurrences exactly zero while τ(A|BC)=1 — a strict inequality, entanglement is entirely non-local across the pair split. Pure W (α=90°) saturates the inequality to an exact equality, just like the 3D flagship version of this sim. Sliding α between them shows the transition; "Verify 2000 random states" draws fully random 8-dimensional complex states (not restricted to this family) and confirms the CKW margin never dips below zero.
- α — blends GHZ-type entanglement (monogamous, non-distillable pairwise) with W-type (freely shareable pairwise, saturating equality).
- θ, φ — redistribute the W component's amplitude across A, B, C, exactly as in the 3D version.
- δ — relative phase between |000⟩ and |111⟩; changes interference terms in the reduced density matrices without changing the marginal probabilities.
Real-world relevance: this exact bound is why quantum repeaters, entanglement swapping and eavesdropping-detection protocols (e.g. E91 QKD) work — an eavesdropper who entangles with a qubit necessarily steals entanglement away from the legitimate pair, bounded exactly by this inequality.