Entanglement measures in embedding quantum simulators with nuclear spins

We implement an embedding quantum simulator (EQS) in nuclear spin systems. The experiment consists of a simulator of up to three qubits, plus a single ancillary qubit, where we are able to efficiently measure the concurrence and the three-tangle of two-qubit and three-qubit systems as they undergo entangling dynamics. The EQS framework allows us to drastically reduce the number of measurements needed for this task, which otherwise would require full-state reconstruction of the qubit system. Our simulator is built of the nuclear spins of four ${}^{13}\mathrm{C}$ atoms in a molecule of trans-crotonic acid manipulated with NMR techniques.

Figure 1

Molecular structure and Hamiltonian parameters of ${}^{13}\mathrm{C}$-labeled trans-crotonic acid. In experiments, $\mathrm{C}1, \mathrm{C}2, \mathrm{C}3$, and $\mathrm{C}4$ are used as a four-qubit simulator. In the table, the chemical shifts and $J$-couplings (in Hz) are presented by the diagonal and off-diagonal elements, respectively. The last row of the table shows $T_2$ (in seconds).

Figure 2

Quantum circuit and corresponding NMR pulse sequence. (a) Quantum circuit consisting of four controlled-not gates and one local rotation $R_y\left(\theta \right)$, which implements the evolution associated with the Hamiltonian $H=-\omega {\sigma }_y\otimes {\sigma }_x\otimes {\sigma }_x$. The upper (red) line represents the ancillary qubit in the EQS, which is held by the nuclear spin of atom C3. Black solid and dashed lines represent the work and idle qubits, respectively. The dotted controlled-not gates can be avoided for initial states of the form $|0000\rangle$. (b) NMR pulse sequence corresponding to the circuit in panel (a). The orange and blue rectangles represent, respectively, $\pi /2$ and $\pi$ pulses around the directions indicated on top of them. Parameters ${\tau }_1$ and ${\tau }_2$ take values ${\tau }_1=1/2J_{\text{C}3,\text{C}4}$ and ${\tau }_2=1/2J_{\text{C}3,\text{C}2}$. (c) Quantum circuit for the implementation of the Hamiltonian $H=-\omega {\sigma }_y\otimes {\sigma }_x\otimes {\sigma }_x\otimes {\sigma }_x$, consisting of six controlled-not gates and one local rotation $R_y\left(\theta \right)$.

Figure 3

Experimental results for the evolution of concurrence, $\mathcal{C}\left(t\right)$. Panels (a) and (b) show the time evolution of the expectation values of the EQS observables ${\sigma }_z{\sigma }_y{\sigma }_y$ and ${\sigma }_x{\sigma }_y{\sigma }_y$, respectively. (c) Reconstructed concurrence $\mathcal{C}\left(t\right)$ of the simulated model from the values of the measured $\langle {\sigma }_z{\sigma }_y{\sigma }_y\rangle$ and $\langle {\sigma }_x{\sigma }_y{\sigma }_y\rangle$. The dots represent experimental data and the lines stem from theory predictions. The error bars are calculated from the estimated imperfections of the GRAPE pulses, PPS preparation, and $T_2$-decoherence effects.

Figure 4

Experimental results for the evolution of the three-tangle ${\mathcal{E}}_3$. Panels (a)–(f) show the expectation values of observables ${\sigma }_z{\sigma }_0{\sigma }_y{\sigma }_y, {\sigma }_x{\sigma }_0{\sigma }_y{\sigma }_y, {\sigma }_z{\sigma }_x{\sigma }_y{\sigma }_y, {\sigma }_x{\sigma }_x{\sigma }_y{\sigma }_y, {\sigma }_z{\sigma }_z{\sigma }_y{\sigma }_y$, and ${\sigma }_x{\sigma }_z{\sigma }_y{\sigma }_y$, respectively. Panel (g) provides the result of the time evolution of the three-tangle ${\mathcal{E}}_3\left(t\right)$ computed from the measurements of the previous six observables. The dots are experimental points and the lines are theory predictions. The error bars are estimated from the noise introduced by the GRAPE pulses, imperfect PPS preparation, and the $T_2$-decoherence effect.

Figure 5

Real and imaginary parts of the reconstructed PPS matrix. Panels (a) and (b) respectively show the real and imaginary elements of the PPS matrix reconstructed in the experiments. The $x$ and $y$ axes represent the index number of the row and columns of the PPS matrix from 1 to 16. The $z$ axis shows the value of each element of the PPS matrix.