Entanglement Negativity on Random Spin Networks

We investigate multipartite entanglement for quantum states of 3 d space geometry, described via generalised random spin networks with ﬁxed areas, in the context of background independent approaches to quantum gravity. We focus on entanglement negativity as a well deﬁned witness of quantum correlations for mixed states, in our setting describing generic subregions of the boundary of a quantum 3 d region of space. In particular, we consider a generic tripartition of the boundary of an open spin network state and we compute the typical R´enyi negativity of two boundary subregions A and B immersed in the environment C , explicitly for a set of simple open random spin network states. We use the random character of the spin network to exploit replica and random average techniques to derive the typical R´enyi negativty via a classical generalised Ising model correspondence generally used for random tensor networks in the large bond regime. For trivially correlated random spin network states, with only local entanglement between spins located on the network edges, we ﬁnd that typical log negativity displays a holographic character, in agreement with the results for random tensor networks, in large spin limit. When non-local bulk entanglement between intertwiners at the vertices is considered the negativity increases, while at the same time the holographic scaling is generally perturbed by the bulk contribution.