Detection of multipartite entanglement in spin rings by use of exchange energy

We investigate multipartite entanglement in rings of arbitrary spins with antiferromagnetic interactions between nearest neighbors. In particular, we show that the nondegenerate ground state of rings formed by an even number $\left(N\right)$ of spins is $N$-partite entangled, and exchange energy can thus be used as a multipartite-entanglement witness. We develop a general approach to compute the energy minima corresponding to biseparable states, and provide numerical results for a representative set of systems. Despite its global character, exchange energy also allows a spin-selective characterization of entanglement. In particular, in the presence of a magnetic defect, one can derive separability criteria for each individual spin, and use exchange energy for detecting entanglement between this and all the other spins.

Figure 1

(a,b) Relative angle between the boundary spins $\left({\overline{\theta }}_A\right)$ as a function of that between the local fields $\left({\theta }_B\right)$, in 3- and 4-qubit chains. Different symbols refer to different values of the difference in modulus between the fields: $Z_B=0$ (dots), $0.25$ (squares), and $0.45$ (triangles). All the results correspond to $|{\mathbf{z}}_B|=1/2$, which maximize (minimize) ${\overline{\theta }}_A$ for $N_A=3$ $\left(N_A=4\right)$, for any given value of $Z_B$. (c,d) Difference between the moduli of the boundary spins of $A$ $\left({\overline{Z}}_A\right)$, as a function of $Z_B$, for relative angles ${\overline{\theta }}_A=0$ and ${\overline{\theta }}_A=\pi$ (with $N_A=3$ and $N_A=4$, respectively). Different symbols correspond to different values of the moduli: $|{\mathbf{z}}_B|=0.5$ (dots), $0.25$ (squares), and $0.05$ (triangles).

Figure 2

(a) Relative angle between the boundary spins of $A$ $\left({\overline{\theta }}_A\right)$ as a function of ${\theta }_B$ for different lengths $N_A$ of the qubit chain. Green corresponds to odd-numbered chains $\left(N_A=2n+1\right)$, while violet refers to even-numbered ones $\left(N_A=2n\right)$. Different symbols correspond to different values of $n$: 1 (dots), 2 (squares), 3 (triangles). (b) Angle ${\overline{\theta }}_A$ as a function of ${\theta }_B$, for chains formed by spins $s>1/2$. Red and blue correspond to 4- and 3-spin chains, respectively. Different symbols correspond to different spin lengths: $s=1/2$ (dots), 1 (squares), $3/2$ (triangles). All the reported values are obtained for $|{\mathbf{z}}_B|=s$ and $Z_B=0$, which maximize (minimize) ${\overline{\theta }}_A$, for each given value of ${\theta }_B$.

Figure 3

Energies of biseparable states corresponding to different bipartitions, $E_{\text{bs}}(N_A,N_B)$, and referred to the ground state energy, $E_0$. The number of spins in the ring is $N=8$, and their length is given by $s=1/2,1,3/2$.

Figure 4

Energetic cost $E_{\text{bs}}^k-E_0$ of disentangling the $k\text{th}$ spin from the remaining ones, for the different molecules of the ${\mathrm{Cr}}_7\text{M}$ series. The position of the magnetic defect corresponds to $k=5$.