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arXiv:1001.1750 · Physical Review A

Statistical strength of experiments to reject local realism with photon pairs and inefficient detectors

Yanbao Zhang, Emanuel H. Knill, Scott Glancy

Published Mar 18, 2010• 16 citations
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Abstract

Because of the fundamental importance of Bell's theorem, a loophole-free demonstration of a violation of local realism (LR) is highly desirable. Here, we study violations of LR involving photon pairs. We quantify the experimental evidence against LR by using measures of statistical strength related to the Kullback-Leibler (KL) divergence, as suggested by van Dam et al. [W. van Dam, R. D. Gill, and P. D. Grunwald, IEEE Trans. Inf. Theory. 51, 2812 (2005)]. Specifically, we analyze a test of LR with entangled states created from two independent polarized photons passing through a polarizing beam splitter. We numerically study the detection efficiency required to achieve a specified statistical strength for the rejection of LR depending on whether photon counters or detectors are used. Based on our results, we find that a test of LR free of the detection loophole requires photon counters with efficiencies of at least $89.71%$, or photon detectors with efficiencies of at least $91.11%$. For comparison, we also perform this analysis with ideal unbalanced Bell states, which are known to allow rejection of LR with detector efficiencies above $2/3$.

DOI
10.1103/physreva.81.032117

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Because of the fundamental importance of Bell's theorem, a loophole-free demonstration of a violation of local realism (LR) is highly desirable. Here, we study violations of LR involving photon pairs. We quantify the experimental evidence against LR by using measures of statistical strength related to the Kullback-Leibler (KL) divergence, as suggested by van Dam et al. [W. van Dam, R. D. Gill, and P. D. Grunwald, IEEE Trans. Inf. Theory. 51, 2812 (2005)]. Specifically, we analyze a test of LR with entangled states created from two independent polarized photons passing through a polarizing beam splitter. We numerically study the detection efficiency required to achieve a specified statistical strength for the rejection of LR depending on whether photon counters or detectors are used. Based on our results, we find that a test of LR free of the detection loophole requires photon counters with efficiencies of at least 89.7189.71%89.71, or photon detectors with efficiencies of at least 91.1191.11%91.11. For comparison, we also perform this analysis with ideal unbalanced Bell states, which are known to allow rejection of LR with detector efficiencies above 2/32/32/3.
Reliable and loophole-free demonstrations of the violation of local realism (LR) are highly desirable not only for understanding the foundation of quantum mechanics but also for facilitating quantum information processing, such as quantum key distribution and randomness expansion. To date, LR has been experimentally violated, but with loopholes, by testing predetermined Bell inequalities. This thesis presents a framework for verifying and quantifying the violation of LR without relying on a particular Bell inequality.\nFirst, the experimental resources, such as the quantum state, measurement settings, detection effciency, and visibility required for a violation of LR, are studied via a measure called the statistical strength. The higher the statistical strength, the more confidence in a violation of LR one has after a suffciently large number of experimental data. Particularly, we study the minimum detection effciency required to achieve any given statistical strength level in tests of LR with entangled states created from two independent polarized photons passing through a polarizing beam splitter. It is shown that, compared with photon detectors, photon counters make violations of LR easier to detect for any nonzero probability of multiple photons in an output beam of the polarizing beam splitter.\nSecond, to quantify the statistical evidence against LR obtained from a finite number of experimental data, one can choose a test statistic, such as a Bell-inequality violation, to measure the amount of violation of LR. It is desirable to bound the probability, according to LR, of obtaining a test statistic at least as extreme as that observed. This probability is known as a p-value for the hypothesis test of LR. We propose a protocol to bound such a p-value. The bound provided is asymptotically tight, if the prepared quantum state and measurement settings are stable during an experiment. Therefore, the proposed protocol is asymptotically optimal, and the bound provided is a standardized measure of success for experimental tests of LR. One can quantitatively compare different experimental tests based on this bound. Moreover, the bound provided is valid even if the quantum state varies arbitrarily and local realistic models depend on previous measurement settings and outcomes. Hence, this bound facilitates device-independent and nonlocality-based quantum information processing. For comparison, bounds of p-values derived from Bell-inequality violations using the number of standard deviations of violation of a Bell inequality or using martingale theory are studied. It is found that putative bounds derived from the number of standard deviations of violation are not valid and bounds from martingale theory are not tight.\nFinally, a simplified and effcient data analysis protocol using a set of Bell inequalities is proposed and compared with the above optimal and martingale-based protocols. The simplified protocol provides as good as and typically tighter p-value bounds than the martingale-based protocol, and the bounds provided can even be asymptotically tight. Moreover, the simplified protocol can be applied to any test with linear witnesses, such as tests for verifying entanglement, system dimensionality, or steering.
We present a loophole-free violation of local realism using entangled photon pairs. We ensure that all relevant events in our Bell test are spacelike separated by placing the parties far enough apart and by using fast random number generators and high-speed polarization measurements. A high-quality polarization-entangled source of photons, combined with high-efficiency, low-noise, single-photon detectors, allows us to make measurements without requiring any fair-sampling assumptions. Using a hypothesis test, we compute p values as small as 5.9×10^{-9} for our Bell violation while maintaining the spacelike separation of our events. We estimate the degree to which a local realistic system could predict our measurement choices. Accounting for this predictability, our smallest adjusted p value is 2.3×10^{-7}. We therefore reject the hypothesis that local realism governs our experiment.
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.
Local realism is the worldview in which physical properties of objects exist independently of measurement and where physical influences cannot travel faster than the speed of light. Bell's theorem states that this worldview is incompatible with the predictions of quantum mechanics, as is expressed in Bell's inequalities. Previous experiments convincingly supported the quantum predictions. Yet, every experiment requires assumptions that provide loopholes for a local realist explanation. Here, we report a Bell test that closes the most significant of these loopholes simultaneously. Using a well-optimized source of entangled photons, rapid setting generation, and highly efficient superconducting detectors, we observe a violation of a Bell inequality with high statistical significance. The purely statistical probability of our results to occur under local realism does not exceed 3.74×10^{-31}, corresponding to an 11.5 standard deviation effect.
There exist numerous proofs of Bell's theorem, stating that quantum mechanics is incompatible with local realistic theories of nature. Here the strength of such nonlocality proofs is defined in terms of the amount of evidence against local realism provided by the corresponding experiments. Statistical considerations show that the amount of evidence should be measured by the Kullback-Leibler (KL) or relative entropy divergence. The statistical strength of the following proofs is determined: Bell's original proof and Peres' optimized variant of it, and proofs by Clauser, Horne, Shimony, and Holt (CHSH), Hardy, Mermin, and Greenberger, Horne, and Zeilinger (GHZ). The GHZ proof is at least four and a half times stronger than all other proofs, while of the two-party proofs, the one of CHSH is the strongest.