Rényi multi-entropy of graph states and the n −> 1 limit
Abstract
Rényi multi-entropy extends the Rényi entropy to pure states of more than two parties. It is defined by a replica contraction at integer replica number n, and the multi-entropy is its n→1 limit. In holography this limit is conjectured to compute the area of a minimal soap-film web in the bulk. Understanding multi-entropy for general states, however, is harder. Unlike the bipartite case, the replica contraction is not a moment of a density matrix, and the continuation to n=1 has to be found by other means. We study this problem for qubit graph states. We show that at every odd n the contraction equals 2−k/2, where k is a sum of ranks of matrices over a finite field, one row and one column per qubit. We define a class of graphs that we call collision-controlled, for which the rank sum is a polynomial in n. This class includes graphs without even cycles and all complete bipartite graphs. The natural polynomial n → 1 continuation is a finite sum over partitions of the parties. While the Rényi multi-entropy is non-increasing under single-qubit Pauli measurements at every odd integer n>1, the continuation to n=1 is not. A Z measurement on one qubit of an odd ring with 2r + 1 ≥ 7 qubits raises it from (r+1)log 2 to (2r−1) log 2. Thus the continuation is not an entanglement monotone.
Related
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BibTeX
@misc{akella2026rényi,
title = {{Rényi multi-entropy of graph states and the $n -> 1$ limit}},
author = {Sriram Akella and Abhijit Gadde},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27080},
eprint = {2610.27080},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27080},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Rényi multi-entropy of graph states and the n −> 1 limit
AU - Sriram Akella
AU - Abhijit Gadde
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.27080
UR - https://arxiv.org/abs/2610.27080
AB - Rényi multi-entropy extends the Rényi entropy to pure states of more than two parties. It is defined by a replica contraction at integer replica number n, and the multi-entropy is its n→1 limit. In holography this limit is conjectured to compute the area of a minimal soap-film web in the bulk. Understanding multi-entropy for general states, however, is harder. Unlike the bipartite case, the replica contraction is not a moment of a density matrix, and the continuation to n=1 has to be found by other means. We study this problem for qubit graph states. We show that at every odd n the contraction equals 2^−k/2, where k is a sum of ranks of matrices over a finite field, one row and one column per qubit. We define a class of graphs that we call collision-controlled, for which the rank sum is a polynomial in n. This class includes graphs without even cycles and all complete bipartite graphs. The natural polynomial n → 1 continuation is a finite sum over partitions of the parties. While the Rényi multi-entropy is non-increasing under single-qubit Pauli measurements at every odd integer n>1, the continuation to n=1 is not. A Z measurement on one qubit of an odd ring with 2r + 1 ≥ 7 qubits raises it from (r+1)log 2 to (2r−1) log 2. Thus the continuation is not an entanglement monotone.
ER -