Location of the minimum in the free-scalar multi-entropy curve

Determine whether the minimum of the excess multi-entropy for three adjacent intervals in the free scalar theory occurs at the equal-subsystem configuration ℓ=L/3, corresponding to the proposed multi-entropy time.

Background

The paper compares its Gaussian multi-entropy curve with results for three adjacent intervals on a circle in a free scalar conformal field theory. The cited plot indicates a minimum of the excess multi-entropy, but does not resolve its precise location.

The proposed multi-entropy time in that setup is the equal-subsystem point, where the interval lengths satisfy ℓ=L−2ℓ and hence ℓ=L/3. The authors state that the available plot does not establish whether the observed minimum coincides with this point.

References

Whether such a minimum coincides with the corresponding ``multi-entropy time'' $t_{\Delta S}=\ell|_{\ell=L-2\ell}=L/3$, is not evident from the plot alone, but cannot be discarded.

— Genuine Multi-Entropy of Fully Symmetric Gaussian States  (2609.30754 - Camargo et al., 25 Sep 2026) in Section 5, final paragraph