bibkey: camargo2026gaussianmultientropy authors: H. A. Camargo, M. Nishida year: 2026 title: “Genuine Multi-Entropy of Fully Symmetric Gaussian States” doi: 10.48550/arXiv.2609.30754 url: https://arxiv.org/abs/2609.30754v1 claim: “For fully symmetric pure Gaussian states with diagonal a ≥ 1, the tripartite genuine Rényi multi-entropy is conjectured to satisfy GM^{(3)}_n ∼ ((2−n)/(2n)) log a as a → ∞ for every replica index n and every number of modes N; GM^{(3)}_2 = 0 is proved for every pure bosonic Gaussian state.” strata_touched:

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Camargo and Nishida, genuine multi-entropy of fully symmetric Gaussian states

H. A. Camargo and M. Nishida, Genuine Multi-Entropy of Fully Symmetric Gaussian States, arXiv:2609.30754v1 (25 September 2026; hep-th, cross-listed quant-ph and cond-mat.stat-mech).

Verified locator

DOI: 10.48550/arXiv.2609.30754. Primary version: https://arxiv.org/abs/2609.30754v1 (the only arXiv version). The TeX source draftv1.tex of v1 supplies §2.1 (the multi-entropy, the twists $g_A,g_B,g_C$, the replica contraction and the genuine multi-entropy), §2.3 (the pure Gaussian state and its density kernel), §3.1 (the fully symmetric state and its off-diagonal entry), §3.2 (Table 1 and the conjecture, Eq. (3.8), label AsymB) and §4 (the proof of ${\mathrm{GM}}_2^{(3)}=0$).

Source statements

Multi-entropy (§2.1): “$S_n^{(\mathtt{q})}(A_1:A_2:\cdots :A_{\mathtt{q}}):=\frac{1}{1-n}\frac{1}{n^{\mathtt{q}-2}}\log \frac{Z_n^{(\mathtt{q})}}{(Z_1^{(\mathtt{q})}{)}^{n^{\mathtt{q}-1}}}$”, with, for $\mathtt{q}=3$, “$g_A=(1,2,\dots ,n)(n+1,n+2,\dots ,2n)\cdots (n^2-n+1,n^2-n+2,\dots ,n^2)$”, “$g_B=(1,n+1,\dots ,n^2-n+1)(2,n+2,\dots ,n^2-n+2)\cdots (n,2n,\dots ,n^2)$”, “$g_C=(1)(2)\cdots (n^2)$”, and “$Z_n^{(3)}={\prod }_{i=1}^{n^2}{\delta }_{{\alpha }_i^{\prime }}^{{\alpha }_{g_A(i)}}{\delta }_{{\beta }_i^{\prime }}^{{\beta }_{g_B(i)}}{\delta }_{{\gamma }_i^{\prime }}^{{\gamma }_{g_C(i)}}{\prod }_{j=1}^{n^2}{\rho }_{{\alpha }_j{\beta }_j{\gamma }_j}^{{\alpha }_j^{\prime }{\beta }_j^{\prime }{\gamma }_j^{\prime }}$”.

Genuine multi-entropy (§2.1): “${\text{GM}}_n^{(3)}(A:B:C):=S_n^{(3)}(A:B:C)-\frac{1}{2}\left(S_n^{(2)}(AB:C)+S_n^{(2)}(BC:A)+S_n^{(2)}(CA:B)\right)$”.

Gaussian kernel (§2.3): “$\rho (\mathbf{q},{\mathbf{q}}^{\prime })={\left(\det \left(\frac{\mathbf{R}}{\pi }\right)\right)}^{1/2}\exp \left[-\frac{1}{2}\left({\mathbf{q}}^{\top }\cdot \mathbf{W}\cdot \mathbf{q}+{{\mathbf{q}}^{\prime }}^{\top }\cdot {\mathbf{W}}^{\ast }\cdot {\mathbf{q}}^{\prime }\right)\right]$”.

Fully symmetric state (§3.1): “all diagonal elements are $a\ge 1$ and all off-diagonal elements are $e^{-}$, where $e^{-}:=\frac{(a^2-1)(N-2)-\sqrt{a^2-1}\sqrt{(a^2-1)N^2+4(N-1)}}{2a(N-1)}$”.

Conjecture (§3.2, Eq. (3.8)): “${\text{GM}}_n^{(3)}\sim \frac{2-n}{2n}\log a(a\to \infty )$ … we \emph{conjecture} that the asymptotic behavior \eqref{AsymB} is valid for the fully symmetric Gaussian states with any $n$ and $N$.”

Scope

The paper derives Eq. (3.8) from exact formulas for $N\le 8$ and $n\le 4$ (Table 1) and proves ${\mathrm{GM}}_2^{(3)}=0$ for every pure bosonic Gaussian state (§4). The fully symmetric states have real $\mathbf{W}$.