Realization of an entanglement-assisted quantum delayed-choice experiment

The puzzling properties of quantum mechanics, i.e., wave-particle duality, entanglement, and superposition, have been dissected experimentally over the past decades. However, hidden-variable (HV) models based on three classical assumptions of wave-particle objectivity, determinism, and independence strive to explain the same experiments or even defeat quantum mechanics. Meanwhile, the steady development of quantum technologies continues to enable us to test experimentally the predictions of quantum mechanics and HV theories. We report an experimental demonstration of an entanglement-assisted quantum delayed-choice scheme using a liquid nuclear magnetic resonance quantum-information processor. This experiment is based on the recently proposed scheme [R. Ionicioiu et al., Nat. Commun. 5. 3997 (2014)], which predicted different results for quantum mechanics and HV theories. In our experiments, the intensities and the visibilities of the interference are consistent with the theoretical predictions of quantum mechanics. The results show that a contradiction with the experiments is indeed appearing when all three assumptions of the HV models are combined, though any two of those assumptions are compatible with the experiments.

Figure 1

Wheeler delayed-choice experiment. (a) The classical WDC setup. If BS2 is present, it is a closed interferometer. If BS2 is absent, it is an open interferometer. (b)The quantum delayed choice. First qubit is initially prepared in $|0\rangle$. Second qubit is an auxiliary quantum system initially prepared in the superposition state $cos\alpha |0\rangle +sin\alpha |1\rangle$. $R\left(\alpha \right)=exp\left(-i\alpha {\sigma }_y\right)$ is used to prepare the state of the ancillary qubit initially in the $|0\rangle$ state. (c) EAQDC experiment. EPR is an Einstein-Podolsky-Rosen pair, $H$ is the Hadamard gate, $\phi$ is a phase shifter, and MZI is a Mach-Zehnder interferometer.

Figure 2

Quantum circuit designs. (a) Equivalent circuit of the control-Hadamard gate. There is no operation on qubit $A$ if qubit $B$ is in the $|0\rangle$ state before two-qubit operator $U_{12}$ on qubits $AB$. Qubit $A$ undergoes a rotation $R_y(\pi /2)=e^{-i\pi {\sigma }_y/4}$ if qubit $B$ is in the $|1\rangle$ state, followed by two-qubit operator $U_{12}$ on qubits $AB$. (b) Feasible quantum circuit of the EAQDC experiment. It can be performed directly in NMR. $U_{12}$ and $U_{23}$ are two-qubit gates. Others are one-qubit gates.

Figure 3

Molecular structure and properties of NMR sample. ${}^{13}\mathrm{C}, {}^1\mathrm{H},$ and ${}^{19}\mathrm{F}$ comprise a three-qubit NMR quantum processor. The chemical shifts and $J_{ij}$-coupling constants of this molecule are given as diagonal elements and off-diagonal elements in the table. $T_1$ and $T_2$ are the longitudinal and transversal relaxation times, respectively, which can be obtained by the standard inversion recovery and Hahn echo sequences [29].

Figure 4

Entire pulse sequence of the entanglement-assisted quantum delayed-choice experiment. Light-orange and deep-orange pulses are hard pulses. Deep-orange pulses represent the $X$ pulse: $R_x\left(\theta \right)=e^{-i\theta {\sigma }_x/2}$. Light-orange pulses represent the $Y$ pulse: $R_y\left(\theta \right)=e^{-i\theta {\sigma }_y/2}$, where the rotation angle $\theta$ is on the top of each pulse. Ladder pulses are shaped pulses which are strongly modulated pulses [30, 31]. The first and second ladder pulses, respectively, act as a phase shifter $\phi$ on qubit $A$ and a rotation around the $Y$ axis $Y\left(\alpha \right)=exp\left(i\alpha {\sigma }_y\right)$ on qubit $C$. $J_{ij}$ is the $J$-coupling constant between the $i\mathrm{th}$ and the $j\mathrm{th}$ nuclear spin. In our experiments, $J_{ab}=160.8$ Hz, and $J_{bc}=47.6$ Hz. The $J$-coupling evolution times are ${\tau }_1=1/\left(4J_{bc}\right), {\tau }_2=1/\left(8J_{ab}\right)$, and ${\tau }_3=1/\left(4J_{ab}\right)$. Each shaded rectangle is for realizing two-qubit gates. First, second, and third dotted rectangles, respectively, realize the evolutions $e^{-i\pi {\sigma }_z^b{\sigma }_z^c/4}, e^{-i\pi {\sigma }_z^a{\sigma }_z^b/8}$, and $e^{-i\pi {\sigma }_z^a{\sigma }_z^b/4}$. The $\pi /2$ detection pulse is shown in black.

Figure 5

Experimental spectra of ${}^{13}\mathrm{C}$. States of the other two qubits decide the position of the peaks, which are labeled beside each peak.

Figure 6

Experimental intensities and visibilities. (a) Intensities $I_{A|c=0}$ and $I_{A|c=1}$ versus phase $\phi$ for different values of $\alpha$. The quantum mechanical predictions are shown by solid lines. This experiment was repeated four times. The average values of experimental results are shown by symbols. Almost all error bars are covered by the symbols because the experimental values have a small standard deviation of less than 0.03. (b) Visibilities $V_{A|c=0}$ and $V_{A|c=1}$ versus rotation angle $\alpha$. QM predicts that $V_{A|c=0}$ (blue, left column) and $V_{A|c=1}$ (cyan, right column), respectively, vary as ${sin}^2\alpha$ and ${cos}^2\alpha$. In HV theories, the visibilities do not distinguish $c=0,1$ cases. The dotted line $V^{\mathrm{HV}}=0.5$ illustrates this.