Entanglement detection on an NMR quantum-information processor using random local measurements

Random local measurements have recently been proposed to construct entanglement witnesses and thereby detect the presence of bipartite entanglement. We experimentally demonstrate the efficacy of one such scheme on a two-qubit NMR quantum-information processor. We show that a set of three random local measurements suffices to detect the entanglement of a general two-qubit state. We experimentally generate states with different amounts of entanglement and show that the scheme is able to clearly witness entanglement. We perform complete quantum state tomography for each state and compute state fidelity to validate our results. Further, we extend previous results and perform a simulation using random local measurements to optimally detect bipartite entanglement in a hybrid system of $2\otimes 3$ dimensionality.

Figure 1

(a) Structure of the ${}^{13}\mathrm{C}$ enriched chloroform molecule with the two qubits labeled as ${}^1\mathrm{H}$ and ${}^{13}\mathrm{C}$. Tabulated experimental NMR parameters with chemical shifts ${\nu }_i$ and spin-spin coupling $J_{ij}$ in hertz and relaxation times $T_1$ and $T_2$ in seconds and (b) ${}^1\mathrm{H}$ and ${}^{13}\mathrm{C}$ NMR spectra obtained at thermal equilibrium after a $\frac{\pi }{2}$ readout pulse. The spectral resonances of each qubit are labeled by the logical state $\left\{\right|0\rangle ,|1\rangle \}$ of the other qubit.

Figure 2

(a) Quantum circuit to implement the entanglement detection protocol. The first red box creates states with different amounts of entanglement. The second red box maps the observables $O_i$ to the $z$ magnetization of either qubit. Only one $z$ magnetization is finally measured in an experiment (inner green box). (b) NMR pulse sequence for the quantum circuit. Unfilled rectangles represent $\frac{\pi }{2}$ pulses, while solid rectangles represent $\pi$ pulses. Tuning of the interaction between qubits is controlled by varying the ${\tau }_i$ time period and the $z$-pulse rotation angle (gray rectangle). Pulse phases are written above each pulse, with a bar indicating negative phase. The $\tau$ evolution period was fixed at $\frac{1}{4J_{CH}}$, where $J_{\mathrm{CH}}$ is the strength of the scalar coupling.

Figure 3

NMR spectra of ${}^1\mathrm{H}$ and ${}^{13}\mathrm{C}$ nuclei, showing the experimentally measured expectation values of (a) $\langle O_1\rangle$, (b) $\langle O_2\rangle$, and (c) $\langle O_3\rangle$ of the Bell state $|{\varphi }^{-}\rangle =\frac{1}{2}\left(\right|00\rangle -|11\rangle )$. The expectation values have been measured by the NMR pulse sequences given in Table 1.

Figure 4

Real part of the tomographed density matrix for the states described in Table 2. (a)–(d) Maximally entangled Bell states and (e)–(f) separable states. (g)–(t) The tomographs represent states of different degrees of entanglement. The state fidelity is written above each tomograph. The arrangement of the rows and columns of the bar graphs is per the computational basis of the two-qubit system $\left\{\right|00\rangle ,|01\rangle ,|10\rangle ,|11\rangle \}$.

Figure 5

(a) Bar graph showing the fraction of the detected entangled states plotted as a function of the number of local measurements from the simulation on the qubit-qutrit system. Plots of (a) $\mathrm{Tr}\left(W_I{\rho }_{AB}\right)$, (b) $\mathrm{Tr}\left(W_{II}{\rho }_{AB}\right)$, and (c) $\mathrm{Tr}\left(W_{III}{\rho }_{AB}\right)$ for $0\le \alpha \le 0.5$ and $0\le \gamma \le 1$. The entanglement witness operators $W_I,W_{II},W_{III}$ are constructed by choosing sets of one, two, ad three random local measurements at a time, respectively. A reference plane (gray shaded) at vanishing trace is also plotted to better differentiate positive and negative values.