Manipulating Complex Hybrid Entanglement and Testing Multipartite Bell Inequalities in a Superconducting Circuit

Quantum correlations in observables of multiple systems not only are of fundamental interest, but also play a key role in quantum information processing. As a signature of these correlations, the violation of Bell inequalities has not been demonstrated with multipartite hybrid entanglement involving both continuous and discrete variables. Here we create a five-partite entangled state with three superconducting transmon qubits and two photonic qubits, each encoded in the mesoscopic field of a microwave cavity. We reveal the quantum correlations among these distinct elements by joint Wigner tomography of the two cavity fields conditional on the detection of the qubits and by test of a five-partite Bell inequality. The measured Bell signal is $8.381\pm 0.038$, surpassing the bound of 8 for a four-partite entanglement imposed by quantum correlations by 10 standard deviations, demonstrating the genuine five-partite entanglement in a hybrid quantum system.

Figure 1

Device sketch and pulse sequence. (a) Device sketch. The device involves three superconducting transmon qubits, labeled from $Q_1$ to $Q_3$, with $Q_1$ ($Q_2$) dispersively coupled to the first (second) storage cavity, denoted as $S_1$ ($S_2$), and $Q_3$ coupled to both storage cavities. Each qubit is independently coupled to a readout cavity. The storage cavities are used for storing the photonic fields, while the readout cavities for detecting the qubit states. (b) Schematic of hybrid entanglement containing three discrete variables and two continuous variables. (c) Experimental pulse sequence to create and characterize the entangled state in (b). The procedure consists of three parts. (1) Generation of the multipartite entangled state for the three transmon qubits and the two storage cavities, realized by a sequence of operations, including initialization of the system to the ground state, a pair of phase-space displacements $D_1({\alpha }_1=1.782)$ and $D_2({\alpha }_2=1.782)$ on the two cavity fields, rotation $R_y^{\pi /2}$ of $Q_3$, $Q_3$-state-dependent cavity phase shifts, a second pair of cavities displacements $D_1({\alpha }_3)$ and $D_2({\alpha }_4)$, and $\pi$ rotations $R_{\mathbf{n},\theta ,0}^{\pi }$ and $R_{y,0}^{\pi }$ to $Q_1$ and $Q_2$ conditional on the vacuum states of $S_1$ and $S_2$, respectively. The angle ($\theta$) between the rotation axis of $R_{\mathbf{n},\theta ,0}^{\pi }$ and the $\mathbf{n}$ axis is variable. (2) Measurement of the qubit observables with the appropriate $\pi /2$ prerotations around the $\mathbf{m}$ axis on the equatorial plane $R_{{\mathbf{m}}_j}^{\pi /2}$ before the state readout in the basis $\{|g_j⟩,|e_j⟩\}$. $R_{{\mathbf{m}}_j}^{\pi /2}=R_{-y}^{\pi /2}$ for the joint Wigner measurement and $R_{{\mathbf{m}}_j}^{\pi /2}=R_{-y}^{\pi /2}$ or $R_{\pm x}^{\pi /2}$ for the Bell signal measurement. (3) Displaced photon-number parity detection for each cavity, realized by performing a phase-space displacement $D_j(-{\gamma }_j)$ and then sandwiching a conditional cavity $\pi$-phase shift between two qubit rotations $R_y^{\pi /2}$ on the corresponding qubit. $D_j(-{\gamma }_j)$ is used for conditional Wigner tomography and for ${\sigma }_x$ measurement of the corresponding cavity cat state qubit. For ${\sigma }_y$ measurement of each cavity, the displacement $D_j(-{\gamma }_j)$ is replaced by the combination of two perpendicular displacements (see the main text).

Figure 2

Conditional two-mode Wigner tomography for $S_1$ and $S_2$ after generation of the GHZ state of Eq. (4) with $\theta =0$. Two-dimensional plane-cuts along the $\mathrm{Re}({\gamma }_1)\text{−}\mathrm{Re}({\gamma }_2)$ axes with $\mathrm{Im}({\gamma }_j)=0$ and along the $\mathrm{Im}({\gamma }_1)\text{−}\mathrm{Im}({\gamma }_2)$ axes with $\mathrm{Re}({\gamma }_j)=0$ are shown in the left and right panels. We displace each mode with $-{\beta }_1/2$ and $-{\beta }_2/2$, respectively, so that the center of the fringes is the origin. We note that the fringe direction relative to the horizontal axis is determined by ${\tan }^{-1}[\mathrm{Im}({\beta }_1)/\mathrm{Im}({\beta }_2)]$ and ${\tan }^{-1}[\mathrm{Re}({\beta }_1)/\mathrm{Re}({\beta }_2)]$ for the left and right panels respectively. (a) Cuts of the measured conditional Wigner function, $W_1({\gamma }_1,{\gamma }_2)$, defined as the joint Wigner functions for the two cavity fields conditional on $X_1{X}_2{X}_3=1$. (b) The expected $W_1({\gamma }_1,{\gamma }_2)$, calculated from the ideal state of Eq. (4) with ${\beta }_1=-2.7-0.2i$ and ${\beta }_2=0.8+2.3i$. (c) Cuts of the measured $W_{-1}({\gamma }_1,{\gamma }_2)$ conditional on $X_1{X}_2{X}_3=-1$. (d) The expected $W_{-1}({\gamma }_1,{\gamma }_2)$, also calculated from the ideal state as (a).

Figure 3

Bell signal measurement. (a) Measured expectation value of the Bell operator $⟨\mathcal{B}⟩$ as a function of the relative phase ($\theta$) of the produced GHZ state. An oscillation is observed as predicted. Experiment data are marked with red. Their standard deviations are obtained from ten repeated measurements when the system is stable and are smaller than marker sizes. Simulated results based on qutip in Python [54, 55] are shown as the blue curve. In the simulation, we neglect the system decoherence in the GHZ state generation and calculate the expectation value of the Bell operator of Eq. (7) with thus obtained GHZ state. The maximum of the measured Bell signals $|⟨\mathcal{B}⟩|=8.381\pm 0.038$ not only exceeds the bound of 4 allowed by local realism (red region), but also above the bound of 8 for a four-partite entangled quantum system (light blue region), demonstrating the genuine five-partite entanglement in a hybrid system. (b) Correlations combining the maximum of the measured Bell signals in (a). Blue bars are experimental data and grey ones represent simulated results. Error bars are from repeated measurements. The measured results are lower due to the imperfections in the experiment (see the main text), but the overall distribution is consistent with simulation.