Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres

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More than 50 years ago1, John Bell proved that no theory of nature that obeys locality and realism2 can reproduce all the predictions of quantum theory: in any local-realist theory, the correlations between outcomes of measurements on distant particles satisfy an inequality that can be violated if the particles are entangled. Numerous Bell inequality tests have been reported3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13; however, all experiments reported so far required additional assumptions to obtain a contradiction with local realism, resulting in ‘loopholes’13, 14, 15, 16. Here we report a Bell experiment that is free of any such additional assumption and thus directly tests the principles underlying Bell’s inequality. We use an event-ready scheme17, 18, 19 that enables the generation of robust entanglement between distant electron spins (estimated state fidelity of 0.92 ± 0.03). Efficient spin read-out avoids the fair-sampling assumption (detection loophole14, 15), while the use of fast random-basis selection and spin read-out combined with a spatial separation of 1.3 kilometres ensure the required locality conditions13. We performed 245 trials that tested the CHSH–Bell inequality20 S ≤ 2 and found S = 2.42 ± 0.20 (where S quantifies the correlation between measurement outcomes). A null-hypothesis test yields a probability of at most P = 0.039 that a local-realist model for space-like separated sites could produce data with a violation at least as large as we observe, even when allowing for memory16, 21 in the devices. Our data hence imply statistically significant rejection of the local-realist null hypothesis. This conclusion may be further consolidated in future experiments; for instance, reaching a value of P = 0.001 would require approximately 700 trials for an observed S = 2.4. With improvements, our experiment could be used for testing less-conventional theories, and for implementing device-independent quantum-secure communication22 and randomness certification23, 24.

At a glance


  1. Bell-test schematic and experimental realization.
    Figure 1: Bell-test schematic and experimental realization.

    a, Bell-test set-up: two boxes, A and B, accept binary inputs (a, b) and produce binary outputs (x, y). In an event-ready scenario, an additional box C gives a binary output signalling that A and B were successfully prepared. b, Experimental realization. The set-up consists of three separate laboratories, A, B and C. The boxes at locations A and B each contain a single NV centre in diamond. A quantum random-number generator (RNG) is used to provide the input. The NV electronic spin is read out in a basis that depends on the input bit, and the resultant signal provides the output. A box at location C records the arrival of single photons that were previously emitted by, and entangled with, the spins at A and B. c, Experimental set-up at A and B. The NV centre is located in a low-temperature confocal microscope (Obj.). Depending on the output of the RNG, a fast switch (Sw.) transmits one of two different microwave pulses (P0 and P1) into a gold line deposited on the diamond surface (inset, scanning electron microscope image). Pulsed red and yellow lasers are used to resonantly excite the optical transitions of the NV centre. The emission (dashed arrows) is spectrally separated into an off-resonant part (phonon side band, PSB) and a resonant part (zero-phonon line, ZPL), using a dichroic mirror (DM). The PSB emission is detected with a single-photon counter (APD). The ZPL emission is transmitted through a beam-sampler (BS, reflection ≤4%) and wave plates (λ/2 and λ/4), and sent to location C through a single-mode fibre. d, Set-up at location C. The fibres from A and B are connected to a fibre-based beam splitter (FBS) after passing a fibre-based polarizer (POL). Photons in the output ports are detected and recorded. e, Aerial photograph of the campus of Delft University of Technology indicating the distances between locations A, B and C. The red dotted line marks the path of the fibre connection. Aerial photograph by Slagboom en Peeters Luchtfotografie BV.

  2. Space-time analysis of the experiment.
    Figure 2: Space–time analysis of the experiment.

    a, Space–time diagram of a single repetition of the entanglement generation. The x axis denotes the distance along the lines AC and CB. After spin initialization, spin-photon entanglement is generated, such that the two photons from A and B arrive simultaneously at C where the detection time of the photons is recorded. Successful preparation of the spins is signalled (bell symbol) by a specific coincidence detection pattern. Independent of the event-ready signal, the set-ups at locations A and B choose a random basis (RNG symbol), rotate the spin accordingly and start the optical spin read-out (measurement symbol). Vertical bars indicate durations. The event-ready signal lies outside the future light cone (coloured regions) of the random basis choices of A and B. b, Space–time diagram of the Bell test. The x axis denotes the distance along the line AB. The read-out on each side is completed before any light-speed signal can communicate the basis choice from the other side. The uncertainty in the depicted event times and locations is much smaller than the symbol size. c, Single-shot spin read-out fidelity at location A as a function of read-out duration (set by the latest time that detection events are taken into account). Blue (orange) line, fidelity of outcome +1 (−1) when the spin is prepared in ms = 0 (ms = ±1); green line, average read-out fidelity; dotted line, read-out duration used (3.7 μs). The inset shows the relevant ground and excited-state levels (not to scale).

  3. Characterization of the set-up and the entangled state.
    Figure 3: Characterization of the set-up and the entangled state.

    a, The probability to obtain spin state |↑right fence at location A (left panel) or B (right panel) when a single photon is detected in the early or late time bin at location C. In the left (right) panel, only emission from A (B) was recorded. Dotted bars are corrected for finite spin read-out fidelity and yield remaining errors of 1.4% ± 0.2% (1.6% ± 0.2%) and 0.8% ± 0.4% (0.7% ± 0.4%) for early and late detection events, respectively, from set-up A (B). These errors include imperfect rejection of the excitation laser pulses, detector dark counts, microwave-pulse errors and off-resonant excitation of the NV. b, Two-photon quantum interference signal, with dt the time between the two photo-detection events. When the NV centres at A and B emit indistinguishable photons, coincident detections of two photons, one in each output arm of the beam-splitter at C, are expected to vanish. The observed contrast between the cases of indistinguishable (orange) and distinguishable (grey) photons (3 versus 28 events in the central peak) yields a visibility of (90 ± 6)% (Supplementary Information). c, Characterization of the Bell set-up using (anti-)parallel read-out angles. The spins at A (left arrows on the x axis) and B (right arrows on the x axis) are read out along the ±Z axis (left panels) or the ±X axis (right panels). The numbers in brackets are the raw number of events. The dotted lines represent the expected correlations on the basis of the characterization measurements presented in a and b (Supplementary Information). The data yield a strict lower bound29 on the state fidelity to |ψright fence of 0.83 ± 0.05. Error bars are 1 s.d.

  4. Loophole-free Bell inequality violation.
    Figure 4: Loophole-free Bell inequality violation.

    a, Summary of the data and the CHSH correlations. The read-out bases corresponding to the input values are indicated by the green (for A) and blue (for B) arrows. Dotted lines indicate the expected correlation on the basis of the spin read-out fidelities and the characterization measurements presented in Fig. 3 (Supplementary Information). Numbers above the bars represent the number of correlated and anti-correlated outcomes, respectively. Error bars shown are , with n(a,b) the number of events with inputs (a, b). b, Statistical analysis for n = 245 trials. For the null-hypothesis test performed (Supplementary Information), the dependence of the P value on the I value is shown (complete analysis, red). Here , with k the number of times (−1)(ab)xy = 1. (For equal n(a,b), I = S with S defined in equation (1).) A small P value indicates strong evidence against the null hypothesis. We find k = 196, which results in a rejection of the null hypothesis with a P ≤ 0.039. For comparison, we also plot the P value for an analysis (conventional analysis, orange) assuming independent and identically distributed (i.i.d.) trials, Gaussian statistics, no memory and perfect random-number generators.


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Author information

  1. Present address: Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA.

    • H. Bernien


  1. QuTech, Delft University of Technology, PO Box 5046, 2600 GA Delft, The Netherlands

    • B. Hensen,
    • H. Bernien,
    • A. E. Dréau,
    • A. Reiserer,
    • N. Kalb,
    • M. S. Blok,
    • J. Ruitenberg,
    • R. F. L. Vermeulen,
    • R. N. Schouten,
    • D. Elkouss,
    • S. Wehner,
    • T. H. Taminiau &
    • R. Hanson
  2. Kavli Institute of Nanoscience Delft, Delft University of Technology, PO Box 5046, 2600 GA Delft, The Netherlands

    • B. Hensen,
    • H. Bernien,
    • A. E. Dréau,
    • A. Reiserer,
    • N. Kalb,
    • M. S. Blok,
    • J. Ruitenberg,
    • R. F. L. Vermeulen,
    • R. N. Schouten,
    • T. H. Taminiau &
    • R. Hanson
  3. ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain

    • C. Abellán,
    • W. Amaya,
    • V. Pruneri &
    • M. W. Mitchell
  4. ICREA-Institució Catalana de Recerca i Estudis Avançats, Lluis Companys 23, 08010 Barcelona, Spain

    • V. Pruneri &
    • M. W. Mitchell
  5. Element Six Innovation, Fermi Avenue, Harwell Oxford, Didcot, Oxfordshire OX11 0QR, UK

    • M. Markham &
    • D. J. Twitchen


B.H., H.B. and R.H. devised the experiment. B.H., H.B., A.E.D., A.R., M.S.B., J.R., R.F.L.V. and R.N.S. built and characterized the experimental set-up. M.W.M., C.A. and V.P. designed the quantum random-number generators (QRNGs), M.W.M. and C.A. designed the randomness extractors, and W.A. and C.A. built the interface electronics and the QRNG optics, the latter with advice from V.P. C.A. and M.W.M. designed and implemented the QRNG statistical metrology. C.A. designed and implemented the QRNG output tests. M.M. and D.J.T. grew and prepared the diamond device substrates. H.B. and M.S.B. fabricated the devices. B.H., H.B., A.E.D., A.R. and N.K. collected and analysed the data, with support from T.H.T. and R.H. D.E. and S.W. performed the theoretical analysis. B.H., A.R., T.H.T., D.E., S.W. and R.H. wrote the manuscript. R.H. supervised the project.

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  1. Supplementary Information (2.1 MB)

    This file contains Supplementary Text and Data, Supplementary Figures 1-7, Supplementary Table 1 and additional references.

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