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Chinese Physics B

Experimentally testing Hardy’s theorem on nonlocality with entangled mixed states

Dai-He Fan, Maochun Dai, Wei-Jie Guo, Lian-Fu Wei

Published Apr 1, 2017• 2 citations
Publisher

Abstract

Hardy’s theorem on nonlocality has been verified by a series of experiments with two-qubit entangled pure states. However, in this paper we demonstrate the experimental test of the theorem by using the two-photon entangled mixed states. We first investigate the generic logic in Hardy’s proof of nonlocality, which can be applied for arbitrary two-qubit mixed polarization entangled states and can be reduced naturally to the well-known logic tested successfully by the previous pure state experiments. Then, the optimized violations of locality for various experimental parameters are delivered by the numerical method. Finally, the logic argued above for testing Hardy’s theorem on nonlocality is demonstrated experimentally by using the mixed entangled-photon pairs generated via pumping two type-I BBO crystals. Our experimental results shows that Hardy’s proof of nonlocality can also be verified with two-qubit polarization entangled mixed states, with a violation of about 3.4 standard deviations.

DOI
10.1088/1674-1056/26/4/040302

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Hardy’s theorem on nonlocality has been verified by a series of experiments with two-qubit entangled pure states. However, in this paper we demonstrate the experimental test of the theorem by using the two-photon entangled mixed states. We first investigate the generic logic in Hardy’s proof of nonlocality, which can be applied for arbitrary two-qubit mixed polarization entangled states and can be reduced naturally to the well-known logic tested successfully by the previous pure state experiments. Then, the optimized violations of locality for various experimental parameters are delivered by the numerical method. Finally, the logic argued above for testing Hardy’s theorem on nonlocality is demonstrated experimentally by using the mixed entangled-photon pairs generated via pumping two type-I BBO crystals. Our experimental results shows that Hardy’s proof of nonlocality can also be verified with two-qubit polarization entangled mixed states, with a violation of about 3.4 standard deviations.
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.
We present the results of some experimental tests of quantum nonlocality performed by two-photon states, entangled both in polarization and momentum, namely hyperentangled states and two-photon four-qubit linear cluster states. These states, which double the number of available qubits with respect to the standard two-photon entangled states, are engineered by a simple experimental method, which adopts linear optics and a single type I nonlinear crystal. The tests of local realism performed with these states represent a generalization of the Greenberger, Home, and Zeilinger (GHZ) theorem to the case of two entangled particles.
Hardy’s paradox (equivalently, Hardy’s non-locality or Hardy’s test) [Phys. Rev. Lett. 68, 2981 (1992)] is used to show non-locality without inequalities, and it has been tested several times using optical circuits. We, for the first time, experimentally test Hardy’s paradox of non-locality in superconducting qubits. For practical verification of Hardy’s paradox, we argue that the error-modeling used in optical circuits is not useful for superconducting qubits. So, we propose a new error-modeling for Hardy’s paradox and a new method to estimate the lower bound on Hardy’s probability (i.e., the probability of a specific event in Hardy’s test) for superconducting qubits. Our results confirmed the theory that any non-maximally entangled state of two qubits violates Hardy’s equations; whereas, any maximally entangled state and product state of two qubits do not exhibit Hardy’s non-locality. Further, we point out the difficulties associated with the practical implementation of quantum protocols based on Hardy’s paradox and propose possible remedies. We also propose two performance measures for any two qubits of any quantum computer based on superconducting qubits.
Recent years have witnessed a rapidly growing interest in high-dimensional quantum entanglement for fundamental studies as well as towards novel applications. Therefore, the ability to verify entanglement between physical qudits, ddd-dimensional quantum systems, is of crucial importance. To show nonclassicality, Hardy's paradox represents ``the best version of Bell's theorem'' without using inequalities. However, so far it has only been tested experimentally for bidimensional vector spaces. Here, we formulate a theoretical framework to demonstrate the ladder proof of Hardy's paradox for arbitrary high-dimensional systems. Furthermore, we experimentally demonstrate the ladder proof by taking advantage of the orbital angular momentum of high-dimensionally entangled photon pairs. We perform the ladder proof of Hardy's paradox for dimensions 3 and 4, both with the ladder up to the third step. Our paper paves the way towards a deeper understanding of the nature of high-dimensionally entangled quantum states and may find applications in quantum information science.