Loophole-free Bell inequality violation with superconducting circuits

The study investigates whether superconducting circuits can exhibit non-local correlations that violate Bell’s inequality without relying on additional assumptions (closing all major loopholes). Building on Bell’s 1964 proposal, the experiment tests local causality by performing space-like separated, randomly chosen measurements on entangled qubits, and evaluating a CHSH Bell parameter. The motivation is twofold: foundational—probing the limits of local hidden-variable theories—and practical—enabling device-independent quantum information protocols. Prior loophole-free tests have used NV centres, optical photons and neutral atoms. Demonstrating a loophole-free violation with macroscopic, microwave-controlled superconducting circuits addresses a long-standing challenge and opens the door to certified non-locality as a resource for quantum technologies. The key research goal is to generate high-concurrence remote entanglement over large cryogenic distances and perform fast, high-fidelity, random-basis measurements within a strict relativistic timing budget to simultaneously close locality, fair-sampling, and memory loopholes, while supporting measurement independence.

Early Bell tests (1970s–1990s) demonstrated violations but left loopholes open, spurring decades of refinements. Loophole-free violations were only achieved recently with NV centres, optical photons, and neutral atoms (2015–2018). In superconducting circuits, prior experiments addressed individual aspects: closing the detection loophole with Josephson phase qubits; using human-generated randomness to support measurement independence; and violating Bell’s inequality with qubits coupled via on-chip or metre-scale microwave links. Remote entanglement in superconducting platforms had been achieved within a single cryostat and between two cryostats over 5 m. High-fidelity, rapid single-shot readout techniques (≈50 ns, >98% fidelity) were established, crucial for Bell tests requiring space-like separation. Device-independent applications—self-testing, certification of quantum processors, DI-QKD, randomness generation/expansion, and randomness amplification—provide strong motivation to realize loophole-free tests in superconducting systems, which are leading candidates for scalable quantum computing.

• System architecture: Two dilution refrigerators (sites A and B) each host a transmon qubit with local readout and remote entanglement circuitry at ≈15 mK. The nodes are connected by a 30 m superconducting aluminium waveguide forming a cryogenic quantum microwave link cooled to below 50 mK along its full length. The waveguide exhibits negligible thermal occupation and loss <1 dB/km at cryogenic temperatures. The large-scale cryogenic infrastructure includes high-reflectance radiation shields, superinsulation, low thermal conductivity supports, flexible thermal connections, and a central pulse tube cooler for intermediate stages; total shield mass >1.3 tons (<80 K), ≈90 kg below 50 mK.
• Qubit control and readout: Each qubit is driven with nanosecond-scale, amplitude- and phase-shaped microwave pulses and flux-bias pulses. State readout uses a dedicated resonator with a Purcell filter; a separate photon-transfer resonator (also Purcell-filtered) couples via coax to the waveguide for entanglement generation.
• Entanglement protocol: Deterministic entanglement of the stationary qubits into |φ+⟩=(|ge⟩+|eg⟩)/√2 is achieved via direct photon exchange through the waveguide using shaped microwave photon techniques. Quantum state tomography (corrected for readout errors) yields Bell-state fidelity Fr=80.4% and concurrence C=0.765. Without readout-error correction, Fr=78.9% and C=0.689.
• Random basis choice and timing: Independent random number generators (RNGs) at each node, located ≈2 m from the qubits, produce bits a and b that select one of two measurement bases. From the space-time ‘start’ event (random bit creation), the RNG output becomes a voltage pulse at the switch output after 17.10±0.14 ns. Basis-selection microwave pulses are routed via side-access cryostat ports to minimize propagation delay to ~14 ns. Basis rotation duration is 12 ns. The timing sequence per trial comprises: RNG (~17 ns), propagation (~14 ns), basis rotation (12 ns), and readout (50 ns).
• Readout and data acquisition: Single-shot readout is integrated over 50 ns with fidelities FA=99.05% and FB=97.60%. Readout signals are similarly routed through side ports, with ~14 ns propagation to the ADC/FPGA located ~1 m from the qubits. The measurement result is defined at the time the last portion of the readout pulse reaches the ADC input. All outcomes are recorded to avoid fair-sampling bias.
• Space-time separation and locality closure: The shortest inter-site distance between start and stop space-time points is measured as d=32.824 m ± 4.6 mm, defining a light-travel time budget τ=109.489±0.015 ns. Independent timing measurements give a total trial duration t2−t1=107.40±0.26 ns, which is within τ by ~8 standard deviations, ensuring space-like separation of basis choices and outcomes.
• CHSH evaluation: From outcomes x,y∈{±1} for settings (a,b)∈{0,1}^2, compute correlators ⟨xy⟩(a,b) and the CHSH parameter S=⟨xy⟩(0,0)+⟨xy⟩(0,1)−⟨xy⟩(1,0)+⟨xy⟩(1,1). The maximum achievable S for given concurrence C and average readout fidelity F=√(FA FB) follows Smax=2√2 F C. Basis choices are selected to maximize violation (optimal relative angle near −π/4 after calibrating a relative phase offset θ≈160.0° between sites). Statistical analysis uses methods robust to memory effects, avoiding IID assumptions.
• Experimental runs: Four main experiments with nmax=2^20=1,048,576 trials each (≈20 min per run). Additional scans vary the measurement-basis offset θ in steps (π/8 and π/32) with ~60,000–80,000 trials per angle to map correlations and S(θ). Master-equation simulations model imperfections (notably photon loss dominated by a circulator used for waveguide photon extraction during protocol characterization).

• Loophole-free violation: At the optimal basis offset θS, measured S=2.0747±0.0033 (>2 by >22σ) over n=1,048,576 trials, rejecting local realism with P<10^−108.
• Angle dependence: S(θ) shows sinusoidal behavior with maxima near θS=−π/4 and π−π/4. Additional measurements near θS yield S=2.082±0.012. Correlators ⟨xy⟩(a,b) exhibit the expected π/2 phase offsets and reduced contrast due to finite concurrence and readout errors, matching master-equation simulations.
• Entanglement and readout performance: Readout fidelities FA=99.05%, FB=97.60% (50 ns integration). Entangled state tomography: corrected Fr=80.4%, C=0.765; uncorrected Fr=78.9%, C=0.689. These values exceed the thresholds needed to violate CHSH when combined (Smax≈2√2 F C).
• Locality closure: Measured inter-site distance d=32.824±0.0046 m gives τ=109.489±0.015 ns; total trial duration 107.40±0.26 ns, closing the locality loophole with ~8σ margin. All outcomes included (closing fair-sampling); statistical test robust to memory effects.
• Throughput: Repetition rate 12.5 kHz enables highly significant violations within minutes, competitive with or exceeding rates in atomic/solid-state loophole-free tests.

The results directly address whether superconducting circuits—macroscopic quantum devices controlled and measured via microwaves—can realize a loophole-free Bell test. By achieving high concurrence and fast, high-fidelity measurements within a stringent relativistic timing budget over a 30 m cryogenic link, the experiment simultaneously closes locality, fair-sampling, and memory loopholes and supports measurement independence. The observed CHSH violation (S≈2.075) demonstrates non-local correlations incompatible with local-hidden-variable models under standard assumptions, establishing non-locality as a resource available in superconducting platforms. This has significant implications: it enables exploration and eventual deployment of device-independent protocols (self-testing, DI-QKD, certified randomness generation/expansion/amplification) in a leading quantum computing architecture. The strong agreement with master-equation simulations identifies photon loss—dominated by a circulator used during characterization—as the main limitation to higher S. The demonstrated 12.5 kHz repetition rate provides a favorable balance between violation magnitude and statistical throughput compared with other loophole-free platforms. The ability to transmit quantum information reliably over tens of metres at millikelvin temperatures supports visions of cryogenic microwave quantum local area networks interconnecting quantum processors within a facility.

This work demonstrates the first loophole-free Bell inequality violation using superconducting circuits linked by a 30 m cryogenic microwave channel. With deterministic remote entanglement, rapid random-basis selection, high-fidelity 50 ns readout, and precise space-time control, the experiment achieves S=2.0747±0.0033 over more than one million trials, closing major loopholes and yielding an extremely small P value (<10^−108). These results establish non-locality as a practical resource in superconducting systems and open avenues for device-independent quantum information tasks. Future improvements—reducing photon loss by removing the circulator, employing low-loss boards and superconducting cables, or adopting heralded entanglement methods—could raise S beyond 2.4 while maintaining loophole closure. The system’s high repetition rate and robust performance suggest feasibility of device-independent protocols and provide a pathway to cryogenic microwave quantum networks interconnecting superconducting quantum processors.

The conclusions rely on standard yet unavoidable assumptions: accurate space-time characterization of events and measurement independence (RNG outputs are free and independent of hidden variables). The Bell violation magnitude is limited by entanglement infidelity primarily caused by photon loss (notably from a circulator used for photon extraction during protocol characterization). Readout errors, while small, also reduce contrast. Although the statistical analysis is robust to memory effects and all outcomes are included (closing fair-sampling), any residual systematic timing or calibration uncertainties could, in principle, affect margins, though measured timing budgets exceed requirements by ~8 standard deviations.