Multipartite quantum correlations reveal frustration in a quantum Ising spin system

We report a nuclear magnetic resonance experiment, which simulates the quantum transverse Ising spin system in a triangular configuration, and further demonstrate that multipartite quantum correlations can be used to distinguish between the frustrated and the nonfrustrated regimes in the ground state of this system. Adiabatic state preparation methods are used to prepare the ground states of the spin system. We employ two different multipartite quantum correlation measures to analyze the experimental ground state of the system in both the frustrated and the nonfrustrated regimes. As expected from theoretical predictions, the experimental data confirm that the nonfrustrated regime shows higher multipartite quantum correlations compared to the frustrated one.

Figure 1

Energy level diagram. Solid (red) curves, which are labeled $E_0$ and $E_1$, represent, respectively, the energy levels of the ground state and the excited state, which is relevant in the calculation of the adiabatic evolution rate [Eq. (6)]. Dashed (blue) curves represent the energy levels corresponding to the other excited states.

Figure 2

Chemical structure of the molecule (left) and table of Hamiltonian parameters (right). In the table, the diagonal elements are the chemical shifts ${\nu }_i$ (in Hz) of the fluorine spins and the off-diagonal elements are the scalar coupling constants $J_{ij}$ (in Hz) between them.

Figure 3

(a) Negativity and (b) discord scores. Dashed (red) curves correspond to the theoretically expected results, obtained by applying ideal unitary operations to the ideal initial state. Open (blue) circles were obtained by applying the ideal unitary operations to the experimental initial state. The (blue) asterisks correspond to experimental results. In (b) dash-dotted (blue) curves correspond to the theoretically expected results for the mixed states of the NMR system, and the $y$ axis for the dashed (red) curve is given at the right. For simulating all the theoretically expected density matrices, we used Trotter's approximation of Eq. (12) and fixed the total number of steps for the adiabatic evolution as 21 for each, in both regimes. The real parts of the experimental density matrices are also shown for the initial state $|---\rangle \langle ---|$ (middle) and the final states in the nonfrustrated (left) and frustrated (right) regimes, respectively.

Figure 4

Bipartite quantum correlations. (a) Negativity, $N_{12}$; (b) quantum discord, $D_{12}$. Dashed (red) curves correspond to theoretically expected results, obtained by applying ideal unitary operations to the ideal initial state. Open (blue) circles were obtained by applying ideal unitary operations to the experimental initial state. The (blue) asterisks correspond to the experimental results. In (b) the experimental results were calculated using the full mixed-state density matrices of the NMR system, the (blue) dash-dotted curve represents the theoretically expected results for the same, and the $y$ axis for the dashed (red) curve is given at the right.