Hardy's formulation of the Einstein-Podolsky-Roser paradox and the usual formulation based on Bell-type inequalities are compared. The role of the supplementary assumptions in an experimental test based on Hardy's approach is discussed. It is proved that even in Hardy's approach the supplementary assumptions play a fundamental role in refusing Einstein locality on the basis of the experimental results. The lowest value of quantum efficiency for a loophole-free experiment based on Hardy's approach is deduced.
Hardy's formulation of the Einstein-Podolsky-Roser paradox and the usual formulation based on Bell-type inequalities are compared. The role of the supplementary assumptions in an experimental test based on Hardy's approach is discussed. It is proved that even in Hardy's approach the supplementary assumptions play a fundamental role in refusing Einstein locality on the basis of the experimental results. The lowest value of quantum efficiency for a loophole-free experiment based on Hardy's approach is deduced.
A recent discovery by Hardy [Phys. Rev. Lett. 71, 1665 (1993)] is developed and extended. It shows that the inconsistency between quantum mechanics and Einstein-Podolsky-Rosen assumptions about locality and reality can be demonstrated for two photons or particles with spin 1/2, as in experiments that test Bell inequalities, but without inequalities, as in what Greenberger, Horne, and Zeilinger discovered for three or four particles [in Bell's Theorem, Quantum Theory, and Conceptions of the Universe, edited by M. Kafatos (Kluwer, Dordrecht, 1989), p. 74]. Here all the possibilities are found. The measurement choices can be different for the two particles. They are described directly in terms of angles for spin measurements. Hardy's result and its converse are both proved: almost any state will do (with measurement choices made to accommodate) and any choice of two different measurement possibilities for each particle will do (with a state made to accommodate). A simple example is given; it allows the calculations to be done quickly with Pauli matrices.
Local realistic models cannot completely describe all predictions of quantum mechanics. This is known as Bell's theorem that can be revealed either by violations of Bell inequality, or all-versus-nothing proof of nonlocality. Hardy's paradox is an important all-versus-nothing proof and is considered as ``the simplest form of Bell's theorem.'' In this work, we advance the study of Hardy's paradox based on Bell's inequality. Our formalism is essentially equivalent to the ``logical Bell inequality'' formalism developed by Abramsky and Hardy, but ours is more easily applied to experimental tests. Remarkably, we find that not only for a two-qubit two-setting case but also for a two-qubit four-setting case, we can construct stronger Hardy-type paradoxes based on Bell's inequality, whose successful probabilities can attain four times and seven times larger than the original one, respectively, thus providing more friendly tests for experiment. Meanwhile, we experimentally test the stronger Hardy-type paradoxes in a two-qubit system. Within the experimental errors, the experimental results coincide with the theoretical predictions.
Bell's theorem states that the quantum mechanical description of physical quantities cannot be fully explained by local realistic theories, laying a solid basis for various quantum information applications. Hardy's paradox is celebrated as the simplest form of Bell's theorem concerning its "All versus Nothing" approach to test local realism. However, due to experimental imperfections, existing tests of Hardy's paradox require additional assumptions of the experimental systems, and these assumptions constitute potential loopholes for faithfully testing local realistic theories. Here, we experimentally demonstrate Hardy's nonlocality through a photonic entanglement source. By achieving a detection efficiency of 82.2%, a quantum state fidelity of 99.10%, and applying high-speed quantum random number generators for the measurement setting switching, the experiment is implemented in a loophole-free manner. During 6 h of running, a strong violation of P_{Hardy}=4.646×10^{-4} up to 5 standard deviations is observed with 4.32×10^{9} trials. A null hypothesis test shows that the results can be explained by local realistic theories with an upper bound probability of 10^{-16348}. These testing results provide affirmative evidence against local realism, and establish an advancing benchmark for quantum information applications based on Hardy's paradox.
Abstract This chapter presents a detailed discussion of Bell inequalities and their experimental tests. Starting from the Aharonov-Bohm version of the Einstein-Podolsky-Rosen (EPR) paradox, we review EPR’s criteria for realism and completeness and Bell’s locality condition, before deriving the Clauser-Horne-Shimony-Holt (CHSH) inequality. This is followed by an exposition on Bell’s original inequality and other Bell inequalities by Wigner, as well as Clauser and Horne. We then discuss the milestones in the experimental violation of Bell inequalities, from the experiment of Freedman and Clauser in the 1970s, to the efforts of Aspect et al. in the 1980s, the experiments by the teams of Zeilinger and Gisin in the 1990s, and culminating the loophole-free tests in 2015. Finally we turn to a brief discussion of the interpretations of quantum mechanics, focusing on contrasting the realist views held by, e.g., Einstein and Bell, and the informational views with proponents such as Brukner and Zeilinger
This chapter begins with a review of the ideas of locality and realism which provide the basis for the Einstein–Podolsky–Rosen paradox. The general conditions defining a local, realistic theory are stated, and then used to prove Bell's theorem, which yields — for any local, realistic theory — an inequality satisfied by certain linear combinations of correlations. It is shown that this inequality is violated by quantum theory and by experiment. The chapter ends with a discussion of possible experimental loopholes and the alternative Greenberger–Horne–Zeilinger (GHZ) state, which would, in principle, make the use of statistical correlations unnecessary.