Rényi multi-entropy of graph states and the $n -> 1$ limit

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 $n\to1$ 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 $n \to 1$ 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 $2r + 1 \geq 7$ qubits raises it from $(r+1)\log 2$ to $(2r-1) \log 2$. Thus the continuation is not an entanglement monotone.