Energy and multipartite entanglement in multidimensional and frustrated spin models

We investigate the relation between the entanglement properties of a quantum state and its energy for macroscopic spin models. To this aim, we develop a general method to compute energy bounds for states without certain forms of multipartite entanglement. Violation of these bounds implies the presence of these types of multipartite entanglement. As examples, we investigate the Heisenberg model in different dimensions, the Ising model and the $XX$ model in the presence of a magnetic field. Finally, by studying the Heisenberg model on a triangular lattice, we demonstrate that our techniques can be applied also to frustrated systems.

Figure 1

(Color online) Schematic view of a four-producible state in a spin model of 24 qubits. Dashed lines correspond to interactions between disentangled qubits and solid lines represent the interactions between qubits which are allowed to be entangled.

Figure 2

(Color online) A possible grouping for a pure two-producible state in 12-qubit spin system. See text for details.

Figure 3

(Color online) Entanglement gap $E_g(k,B)$ for two-partite, three-partite, four-partite, and five-partite entanglement for the Ising model in a transverse magnetic field. See text for details.

Figure 4

(Color online) Derivative of the entanglement gap $F(k,B)=\partial E_g(k,B)∕\partial B$ for $k=1,2$, and $4$.

Figure 5

(Color online) Entanglement in thermal states of the $XX$ model in a magnetic field. The regions in the $T$-$B$ plane are shown where the different types of multipartite entanglement can be detected with our method.

Figure 6

(Color online) A two-producible state on a triangular lattice. Solid lines represent possible entanglement between the qubits. The triangle $(2,5,3)$ is type A, i.e., it does not have entanglement between its qubits. The triangle $(1,2,3)$ is of type B since qubits 1 and 2 may be entangled. When estimating $C_i$ for the block of the qubits $1$ and $2$, the two triangles $(1,2,3)$ and $(1,2,4)$ are estimated via Eq. (31), since there are definitely of type B. See text for further details.