Loophole-free Bell inequality violation with superconducting circuits

Abstract

Superposition, entanglement and non-locality constitute fundamental features of quantum physics. The fact that quantum physics does not follow the principle of local causality^1-3 can be experimentally demonstrated in Bell tests⁴ performed on pairs of spatially separated, entangled quantum systems. Although Bell tests, which are widely regarded as a litmus test of quantum physics, have been explored using a broad range of quantum systems over the past 50 years, only relatively recently have experiments free of so-called loopholes⁵ succeeded. Such experiments have been performed with spins in nitrogen-vacancy centres⁶, optical photons^7-9 and neutral atoms¹⁰. Here we demonstrate a loophole-free violation of Bell's inequality with superconducting circuits, which are a prime contender for realizing quantum computing technology¹¹. To evaluate a Clauser-Horne-Shimony-Holt-type Bell inequality⁴, we deterministically entangle a pair of qubits¹² and perform fast and high-fidelity measurements¹³ along randomly chosen bases on the qubits connected through a cryogenic link¹⁴ spanning a distance of 30 metres. Evaluating more than 1 million experimental trials, we find an average S value of 2.0747 ± 0.0033, violating Bell's inequality with a P value smaller than 10^-108. Our work demonstrates that non-locality is a viable new resource in quantum information technology realized with superconducting circuits with potential applications in quantum communication, quantum computing and fundamental physics¹⁵.

Fig. 1. Schematic of the Bell test experiment.

Two parties A and B choose random-input bits (a, b) at the space–time locations indicated by stars and perform measurements on a pair of entangled quantum systems (in this work, superconducting circuit qubits) yielding output bits (x, y) at space–time locations indicated by crosses. The shaded areas indicate the forwards light cones originating at the space–time location of the random-input-bit-generation events. The inset in the middle indicates the offset angle θ between the measurement bases of the two qubits (main text).

Fig. 2. S value, entangled state and readout fidelity.

a, Calculated S value for a Bell test performed in the xy basis of the Bloch sphere versus (readout-corrected) Bell state concurrence $\mathcal{C}\left({\rho }_{\mathrm{AB}}\right)$ and average qubit-readout fidelity ${\mathcal{F}}_{\mathrm{r}}=\sqrt{{\mathcal{F}}_{\mathrm{r}}^{\mathrm{A}}{\mathcal{F}}_{\mathrm{r}}^{\mathrm{B}}}$. The blue data point indicates the experimentally achieved readout fidelity and concurrence (with correction for readout errors). b, Real part of the density matrix ρ of the Bell state $∣{\psi }^{+}⟩$ reconstructed using quantum state tomography corrected for readout errors. The blue bars indicate the measured, the grey wireframes the ideal values and the red wireframes the results of a master equation simulation.

Fig. 3. Cryogenic microwave quantum link.

a, Computer-aided design (CAD) model. b–d, Photographs of the 30-m-long cryogenic set-up. Dilution refrigerators at each end host the quantum devices that are connected through a waveguide cooled to below 50 mK over the full distance. A central pulse tube cooler provides additional cooling power to the two outermost radiation shields. The photographs are taken at the position of the corresponding eye pictograms shown in a. A, (b), centre (c) and B (d).

Fig. 4. Space–time diagram of the experiment.

The left, vertical time axis schematically shows the microwave pulses applied to the qubits locally at each node. The right axis indicates the duration of the individual Bell test protocol segments: RNG, signal propagation (prop.), qubit basis rotation and measurement. The space–time location of the start and stop events of a Bell test trial are marked with stars and crosses, respectively. The red and blue regions indicate the future light cones of the start events. The inset on the bottom right indicates the approximate spatial location of the start and stop events in the RNG and ADC, relative to the vertical centre axis of each cryostat.

Fig. 5. Bell inequality violation versus measurement basis offset angle.

a, Quantum correlations ⟨xy⟩_(a,b) of individual Bell tests versus offset angle θ. The 17 data points are results of individual Bell tests with $n_{max}/17=\mathrm{61,680}$ trials each, incremented by θ = π/8. The dashed curves are calculated using a master equation simulation. b, Corresponding S values calculated from the data shown in a. Points are experimental data, and the dashed red line is extracted from a master equation simulation. Error bars are roughly on the order of the marker size, see text for details. c, Measured S values of 13 individual Bell tests, with $n_{max}/13=\mathrm{80,659}$ trials each, and offset angles around the expected optimum value ${\theta }_{S_{\max }}$ incremented by θ = π/32. d, Same as in c but for the expected optimum value  ${\theta }_{S_{\min }}$. The green lines in b–d mark the threshold value |S| = 2, and all points in the green shaded region correspond to Bell tests that violate the CHSH inequality.