The strength of a nonlocality proof is examined in terms of the amount of evidence that the corresponding experiment provides for the nonlocality of Nature. An experimental implementation of such a proof gives data whose statistics will differ from the statistics that are possible under a local description of Nature. The strength of the experiment is quantified by the expected deviation between the observed frequencies, which are given by the laws of quantum mechanics, and the closest possible local theory. Varying the frequencies of the measurement settings gives different experimental implementations of a nonlocality proof, giving each implementation its own strength. The statistical strength of a nonlocality proof is thus determined by the experimental implementation that maximizes its statistical deviation from all possible local theories. It is shown that the deviation between quantum mechanics and a local theory is best expressed by the Kullback-Leibler distance between the probability distributions over the measurement outcomes that the respective theories predict. Specifically, it is proven that the Kullback-Leibler distance is optimal for three methods of hypothesis testing: frequentist, Bayesian, and information theoretic hypothesis testing. The nonlocality proofs that are analyzed in this article are: Bell’s original proof, an improved version
The strength of a nonlocality proof is examined in terms of the amount of evidence that the corresponding experiment provides for the nonlocality of Nature. An experimental implementation of such a proof gives data whose statistics will differ from the statistics that are possible under a local description of Nature. The strength of the experiment is quantified by the expected deviation between the observed frequencies, which are given by the laws of quantum mechanics, and the closest possible local theory. Varying the frequencies of the measurement settings gives different experimental implementations of a nonlocality proof, giving each implementation its own strength. The statistical strength of a nonlocality proof is thus determined by the experimental implementation that maximizes its statistical deviation from all possible local theories. It is shown that the deviation between quantum mechanics and a local theory is best expressed by the Kullback-Leibler distance between the probability distributions over the measurement outcomes that the respective theories predict. Specifically, it is proven that the Kullback-Leibler distance is optimal for three methods of hypothesis testing: frequentist, Bayesian, and information theoretic hypothesis testing. The nonlocality proofs that are analyzed in this article are: Bell’s original proof, an improved version
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.
This master thesis is centered around the the mathematical framework of quantum\nprobability and quantum statistics. Quantum statistics is the axiomatization of the\nphysical theory of quantum mechanics pioneered by von Neumann in the 1930s.\nIn contrast to many other physical theories which are described by deterministic\nmodels quantum mechanics is a stochastic theory. In fact, quantum probability\ncan be viewed as an extension to Kolomogorov’s “classical” probability theory.\nEspecially in the last decade, due to first physical realizations of quantum\ncomputational systems and the rise of quantum information theory, the subject of\nquantum statistics became more and more important and many applications for\nmathematical statisticians and probabilists opened up.\nIn this thesis we will be concerned with the optimization of so-called nonlocality\nproofs which are methods to show the “non-classicality” of certain probability\ndistributions within the framework of quantum statistics. In particular,\none is interested in measuring the statistical strength of such nonlocality proofs,\nsometimes called Bell tests. One of the main results of this thesis is the analysis\nof certain Bell tests where the corresponding measure space is described by\nan infinite dimensional separable Hilbert space, corresponding to infinitely many\npossible outcomes. In particular, this gives numerical evidence for a new quantum\nBell inequality describing the boundary of the space of quantum probability\ndistributions for the considered setting. These results have been published in the\nfollowing letter, S. Zohren and R. D. Gill, “Maximal violation of the Collins-\nGisin-Linden-Massar-Popescu inequality for infinite dimensional states” submitted\nto Phys. Rev. Lett..
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.
Statistical tests are needed to determine experimentally whether a hypothetical theory based on local realism can be an acceptable alternative to quantum mechanics. It is impossible to rule out local realism by a single test, as often claimed erroneously. The “strength” of a particular Bell inequality is measured by the number of trials that are needed to invalidate local realism at a given confidence level. Various versions of Bell's inequality are compared from this point of view. It is shown that Mermin's inequality for Greenberger-Horne-Zeilinger states requires fewer tests than the Clauser-Horne-Shimony-Holt inequality or than its chained variants applied to a singlet state, and also than Hardy's proof of nonlocality.
Statistical tests are needed to determine experimentally whether a hypothetical theory based on local realism can be an acceptable alternative to quantum mechanics. It is impossible to rule out local realism by a single test, as often claimed erroneously. The “strength” of a particular Bell inequality is measured by the number of trials that are needed to invalidate local realism at a given confidence level. Various versions of Bell's inequality are compared from this point of view. It is shown that Mermin's inequality for Greenberger-Horne-Zeilinger states requires fewer tests than the Clauser-Horne-Shimony-Holt inequality or than its chained variants applied to a singlet state, and also than Hardy's proof of nonlocality.
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.