A common experimental strategy for demonstrating nonclassical correlations is to show violation of a Bell inequality by measuring a continuously emitted stream of entangled photon pairs. The measurements involve the detection of photons by two spatially separated parties. The detection times are recorded and compared to quantify the violation. The violation critically depends on determining which detections are coincident. Because the recorded detection times have ``jitter,'' coincidences cannot be inferred perfectly. In the presence of settings-dependent timing errors, this can allow a local-realistic system to show apparent violation, the so-called ``coincidence loophole.'' Here, we introduce a family of Bell inequalities based on signed, directed distances between the parties' sequences of recorded time tags. Given that the time tags are recorded for synchronized, fixed observation periods and that the settings choices are random and independent of the source, violation of these inequalities unambiguously shows nonclassical correlations violating local realism. Distance-based Bell inequalities are generally useful for two-party configurations where the effective size of the measurement outcome space is large or infinite. We show how to systematically modify the underlying Bell functions to improve the signal-to-noise ratio and to quantify the significance of the violation.
A common experimental strategy for demonstrating nonclassical correlations is to show violation of a Bell inequality by measuring a continuously emitted stream of entangled photon pairs. The measurements involve the detection of photons by two spatially separated parties. The detection times are recorded and compared to quantify the violation. The violation critically depends on determining which detections are coincident. Because the recorded detection times have jitter,'' coincidences cannot be inferred perfectly. In the presence of settings-dependent timing errors, this can allow a local-realistic system to show apparent violation, the so-called coincidence loophole.'' Here, we introduce a family of Bell inequalities based on signed, directed distances between the parties' sequences of recorded time tags. Given that the time tags are recorded for synchronized, fixed observation periods and that the settings choices are random and independent of the source, violation of these inequalities unambiguously shows nonclassical correlations violating local realism. Distance-based Bell inequalities are generally useful for two-party configurations where the effective size of the measurement outcome space is large or infinite. We show how to systematically modify the underlying Bell functions to improve the signal-to-noise ratio and to quantify the significance of the violation.
In a local realist model, physical properties are defined prior to and independent of measurement and no physical influence can propagate faster than the speed of light. Proper experimental violation of a Bell inequality would show that the world cannot be described with such a model. Experiments intended to demonstrate a violation usually require additional assumptions that make them vulnerable to a number of ``loopholes.'' In both pulsed and continuously pumped photonic experiments, an experimenter needs to identify which detected photons belong to the same pair, giving rise to the coincidence-time loophole. Here, via two different methods, we derive Clauser-Horne- and Eberhard-type inequalities that are not only free of the fair-sampling assumption (thus not being vulnerable to the detection loophole), but also free of the fair-coincidence assumption (thus not being vulnerable to the coincidence-time loophole). Both approaches can be used for pulsed as well as for continuously pumped experiments. Moreover, as they can also be applied to already existing experimental data, we finally show that a recent experiment [Giustina et al., Nature (London) 497, 227 (2013)] violated local realism without requiring the fair-coincidence assumption.
John Bell’s theorem of 1964 states that local elements of physical reality, existing independent of measurement, are inconsistent with the predictions of quantum mechanics (Bell, J. S. (1964), Physics (College. Park. Md). Specifically, correlations between measurement results from distant entangled systems would be smaller than predicted by quantum physics. This is expressed in Bell’s inequalities. Employing modifications of Bell’s inequalities, many experiments have been performed that convincingly support the quantum predictions. Yet, all experiments rely on assumptions, which provide loopholes for a local realist explanation of the measurement. Here we report an experiment with polarization-entangled photons that simultaneously closes the most significant of these loopholes. We use a highly efficient source of entangled photons, distributed these over a distance of 58.5 meters, and implemented rapid random setting generation and high-efficiency detection to observe a violation of a Bell inequality with high statistical significance. The merely statistical probability of our results to occur under local realism is less than 3.74×10-31, corresponding to an 11.5 standard deviation effect.
Summary form only given. In a Bell experiment, one prepares pairs of entangled particles and sends them to two observers, Alice and Bob, for measurement and detection [1]. For some choices of their measurement settings, Alice and Bob may observe strong correlations between their results in accordance with the predictions of quantum mechanical theory. Conversely, any physical theory that assumes no physical influence can be faster than the speed of light and that properties of physical systems are elements of reality [2] - a local realistic theory - can predict only a limited amount of correlation between Alice's and Bob's measurement outcomes. Upon observing correlations sufficient to violate Bell's inequality, Alice and Bob must abandon local realism.It is common that in an experiment, some particles emitted by the source will not be detected. Then the subset of detected particles might display correlations that violate the Bell inequality although the entire ensemble can be described by a local realistic theory [3]. To achieve a conclusive Bell violation without assuming that the detected particles are a “fair” sample, a highly efficient setup is necessary. Due to experimental limitations, fair sampling has been assumed in most Bell experiments performed to date, and it has never been possible to avoid this assumption with photons due to the absence of efficient sources and detectors. Here we report the first Bell experiment with photons that does not rely on any fair-sampling assumption. In our experiment we employ the Eberhard inequality, a Bell inequality that by construction does not rely on the fair-sampling assumption [4]. Our source of entangled photon pairs utilizes bulk-crystal spontaneous parametric downconversion in a Sagnac configuration, which has shown to be very efficient [5, 6]. For photon detection, we employ superconducting transition-edge sensors, which not only offer high efficiency but also are intrinsically free of dark counts [7]. This combination of features is imperative for an experiment in which no correction of count rates can be tolerated. We achieve uncorrected coupling efficiencies over 73% in each arm of our source, facilitating a violation of the inequality by nearly 70 standard deviations after five minutes of measurement. Since the first experimental Bell test, a satisfactory laboratory realization of the motivating gedankenexperiment has remained an outstanding challenge. The two other main assumptions include “locality” [8, 9] and “freedom of choice” [10]. Invoking any of these renders an experiment vulnerable to explanation by a local realistic theory. The realization of an experiment that is free of all three assumptions-a so-called loopholefree Bell test-remains an important outstanding goal for the physics community with strong implications for quantum technologies. We note that with our experiment, photons are the first physical system for which each of these three assumptions has been successfully addressed, albeit in different experiments. This represents promise for practical applications like one-sided device-independent quantum key distribution [11], and in an additional test, we also demonstrate that our experimental apparatus is capable of implementing this protocol on both sides.
It is theoretically and experimentally shown that photons emitted by statistically independent incoherent classical light sources and measured in the far field in spatially separated modes may display spatial correlations akin to path-entanglement of photons produced by quantum sources. By measuring higher order photon-correlations at different locations, i.e., m photons in one mode and one photon in another mode, we experimentally demonstrate for m≥6 a violation of Bell-type inequalities for spatial degrees of freedom. The spatial correlations among the photons can be understood from state projection where the detection of the first m photons projects the sources onto a state which emits the subsequent photon in a strongly correlated manner. From this perspective the entanglement and violation of Bell's inequalities appears as a consequence of nonvanishing cross correlations between noncommuting quadrature phase components of the two spatially separated fields after m photons have been recorded. In this way we show that classical systems may produce spatial field correlations violating local realistic theories.
We show that correlations inconsistent with any locally causal description can be a generic feature of measurements on entangled quantum states. Specifically, spatially separated parties who perform local measurements on a maximally entangled state using randomly chosen measurement bases can, with significant probability, generate nonclassical correlations that violate a Bell inequality. For n parties using a Greenberger-Horne-Zeilinger state, this probability of violation rapidly tends to unity as the number of parties increases. We also show that, even with both a randomly chosen two-qubit pure state and randomly chosen measurement bases, a violation can be found about 10% of the time. Among other applications, our work provides a feasible alternative for the demonstration of Bell inequality violation without a shared reference frame.
We analyze nonclassical correlations between outcomes of measurements conducted on two spatial radiation modes. These correlations cannot be simulated with statistical mixtures of coherent states or, more generally, with non-negative phase-space functions of quantum states and measurements. We argue that nonclassical correlations are naturally related to Bell nonlocality, the former being a more general class of quantum correlations. Indeed, it is known that local realistic as well as noncontextual models correspond to non-negative solutions to a system of linear equations for the joint probability distributions of all observables. We demonstrate that nonclassical correlations correspond to a particular solution to this system, which may have negative values even if local realism is not violated. A modification of Bell inequalities enables us to test such correlations. At the same time, our approach leads to a formulation of Bell inequalities applicable also to continuous variables. The results are illustrated with two-mode squeezed vacuum states and with hybrid entangled states (Schr\"odinger-Cat states), one mode being analyzed by balanced and the other one by unbalanced homodyne detection.