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Prove the proposed upper and lower bounds for GM_1^{(3)} in general finite-dimensional systems

Prove that for arbitrary finite-dimensional tripartite quantum systems the genuine tri-entropy satisfies 0 ≤ GM_1^{(3)}(A;B;C) ≤ (1/2)·log(min{dim 𝓗_A, dim 𝓗_B, dim 𝓗_C}).

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Background

Using stabilizer-state decompositions, the authors show that GM_1{(3)} counts the total GHZ content and is bounded by half the logarithm of the smallest local Hilbert-space dimension within that class, motivating a broader conjecture.

They verify the bound in several examples (including SU(2)k links and generalized GHZ states) and suggest it holds generally, but no proof is provided.

References

Therefore, it is temping to conjecture that the bound eq-bound of $\mathrm{GM}_1{(3)}(A;B;C)$ also holds for general finite systems.

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)\ ,

— 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)