Asymptotic genuine multi-entropy for arbitrary replica index and mode number

Prove that the asymptotic behavior of the genuine Rényi tri-partite multi-entropy of fully symmetric Gaussian states is GM⁽³⁾ₙ ∼ (2−n)/(2n) log a as the squeezing parameter a tends to infinity for every replica index n and every total number of bosonic modes N.

Background

The paper studies genuine Rényi multi-entropy for fully symmetric Gaussian states, which are parameterized by the squeezing parameter a, the inverse of the single-mode purity. Exact calculations for several replica indices, mode numbers, and subsystem partitions show that the large-squeezing behavior agrees with the genuine multi-entropy of a continuous-variable GHZ state.

The authors conjecture that this leading asymptotic term is independent of the total number of modes and of the subsystem partition. Establishing it for arbitrary n and N would extend the finite examples and provide a general analytic connection between fully symmetric Gaussian states and the continuous-variable GHZ limit.

References

Based on the relationship between the fully symmetric Gaussian states and the proper CV GHZ state, we conjecture that the asymptotic behavior AsymB is valid for the fully symmetric Gaussian states with any $n$ and $N$.

— Genuine Multi-Entropy of Fully Symmetric Gaussian States  (2609.30754 - Camargo et al., 25 Sep 2026) in Section 3.2, following Eq. (3.10); reiterated in Section 6

It would be very interesting to obtain similar analytic formulas for the subsystem configurations corresponding to the Page and Multi-entropy times in the large $N$ regime.

— Genuine Multi-Entropy of Fully Symmetric Gaussian States  (2609.30754 - Camargo et al., 25 Sep 2026) in Section 6, final paragraph before the future-directions list