# Multipartite entanglement in spin chains and the Hyperdeterminant. (arXiv:1802.02596v1 [quant-ph])

A way to characterize multipartite entanglement in pure states of a spin

chain with $n$ sites and local dimension $d$ is by means of the Cayley

hyperdeterminant. The latter quantity is a polynomial constructed with the

components of the wave function $\psi_{i_1, \dots, i_n}$ which is invariant

under local unitary transformation. For spin 1/2 chains (i.e. $d=2$) with $n=2$

and $n=3$ sites, the hyperdeterminant coincides with the concurrence and the

tangle respectively. In this paper we consider spin chains with $n=4$ sites

where the hyperdeterminant is a polynomial of degree 24 containing around $2.8

\times 10^6$ terms. This huge object can be written in terms of more simple

polynomials $S$ and $T$ of degrees 8 and 12 respectively. In this paper we

compute $S$, $T$ and the hyperdeterminant for eigenstates of the following spin

chain Hamiltonians: the transverse Ising model, the XXZ Heisenberg model and

the Haldane-Shastry model. Those invariants are also computed for random

states, the ground states of random matrix Hamiltonians in the Wigner-Dyson

Gaussian ensembles and the quadripartite entangled states defined by Verstraete

et al. in 2002. Finally, we propose a generalization of the hyperdeterminant to

thermal density matrices.