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Ascertain whether the three-qubit W-state satisfies the proposed GM_1^{(3)} bound

Determine whether the three-qubit W-state |W⟩ = (|001⟩ + |010⟩ + |100⟩)/√3 satisfies the conjectured bound 0 ≤ GM_1^{(3)}(A;B;C) ≤ (1/2)·log(min{dim 𝓗_A, dim 𝓗_B, dim 𝓗_C}) exactly, and compute its precise GM_1^{(3)} value in the n→1 limit.

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Background

The authors discuss data for the Rényi multi-entropy of the W-state from prior work and infer a likely range for the coefficient c_W in S_1{(3)}[W] = c_W·log 3, suggesting the bound probably holds.

They explicitly note that it remains uncertain whether the W-state exactly satisfies the proposed bound, making its exact GM_1{(3)} value and compliance with the bound a concrete unresolved question.

References

Although it remains uncertain whether the W-state exactly satisfies the bound eq-bound, the data in eq-W-RME strongly suggest that it most likely does.

eq-bound:

0≤GM1(3)(A;B;C)≤12log⁡ ⁣( ⁣min⁡ ⁣{dim⁡HA,dim⁡HB,dim⁡HC} ⁣) ,0 \le \mathrm{GM}_1^{(3)}(A;B;C) \le \frac{1}{2}\log\! \Big(\!\min\!\left\{\dim \mathcal{H}_A, \dim \mathcal{H}_B, \dim \mathcal{H}_C\right\}\!\Big)\ ,

eq-W-RME:

S2(3)[W]=log⁡3 ,S3(3)[W]≈0.93log⁡3 ,S4(3)[W]≈0.88log⁡3 ,S5(3)[W]≈0.84log⁡3 ,S6(3)[W]≈0.82log⁡3 .\begin{split} &S_2^{(3)}[\text{W}] = \log 3\ , \quad S_3^{(3)}[\text{W}] \approx 0.93\log 3\ , \quad S_4^{(3)}[\text{W}] \approx 0.88 \log 3\ , \\ &S_5^{(3)}[\text{W}] \approx 0.84 \log 3\ , \quad S_6^{(3)}[\text{W}] \approx 0.82\log 3\ . \end{split}

— Multi-entropy from Linking in Chern-Simons Theory (2510.18408 - Yuan et al., 21 Oct 2025) in Section 5.3 (Possible lower and upper bounds of the genuine tri-entropy)