Higher-party replica factorization for pure bosonic Gaussian states

Determine whether a special factorization of replica partition functions analogous to the n=2 factorization proved for three-party pure bosonic Gaussian states holds for q-partite general pure bosonic Gaussian states with q>3.

Background

For three parties and replica index n=2, the paper establishes that the replica partition function factorizes into products of bipartite purity-like quantities, implying that the genuine multi-entropy vanishes for arbitrary pure bosonic Gaussian states.

The authors note that n=2 has an analogous special status in random tensor-network states and ask whether the Gaussian factorization extends beyond three parties. Such an extension would clarify the structure of n=2 multi-entropy for arbitrary numbers of parties.