R'enyi 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 at least 7 qubits raises it from (r+1) 2 to (2r-1) 2. Thus the continuation is not an entanglement monotone.