Experimental classification of entanglement in arbitrary three-qubit pure states on an NMR quantum information processor

We undertake experimental detection of the entanglement present in arbitrary three-qubit pure quantum states on an NMR quantum information processor. Measurements of only four observables suffice to experimentally differentiate between the six classes of states which are inequivalent under stochastic local operation and classical communication. The experimental realization is achieved by mapping the desired observables onto Pauli $z$ operators of a single qubit, which is directly amenable to measurement. The detection scheme is applied to known entangled states as well as to states randomly generated using a generic scheme that can construct all possible three-qubit states. The results are substantiated via direct full quantum state tomography as well as via negativity calculations and the comparison suggests that the protocol is indeed successful in detecting tripartite entanglement without requiring any a priori information about the states.

Figure 1

(a) Quantum circuit to achieve mapping of the state $\rho$ to either of the states ${\rho }_{21}^{}$, ${\rho }_{23}^{}$, ${\rho }_{29}^{}$, or ${\rho }_{53}^{}$ followed by measurement of qubit three in the computational basis. (b) NMR pulse sequence of the quantum circuit given in panel (a). All the unfilled rectangles denote $\frac{\pi }{2}$ spin-selective rf pulses while filled rectangles denote $\pi$ pulses. Pulse phases are written above the respective pulse and a bar over a phase represents negative phase. Delays are given by ${\tau }_{ij}^{}=1/\left(8J_{ij}\right)$; $i,j$ label the qubit and $J$ is the coupling constant.

Figure 2

(a) Molecular structure of ${}^{13}\mathrm{C}$-labeled diethyl fluoromalonate and NMR parameters. NMR spectra of (b) thermal equilibrium state and (c) pseudopure state. Each peak is labeled with the logical state of the qubit which is passive during the transition. Horizontal scale represents the chemical shifts in ppm.

Figure 3

Bar plots of the expectation values of the observables $O$, $O_1$, $O_2$, and $O_3$ for states numbered from 1–27 (Table 3). The horizontal axes denote the state number while the vertical axes represent the values of the respective observable. Black, cross-hatched, and unfilled bars represent the theoretical (The.), directly (Dir.) measured from experiment, and QST-derived expectation values, respectively.