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Physical Review A

Orbital-angular-momentum-based experimental test of Hardy's paradox for multisetting and multidimensional systems

Dongkai Zhang, Xiaodong Qiu, Tianlong Ma, Wuhong Zhang, Lixiang Chen

Published May 11, 2020• 13 citations
Publisher

Abstract

Characterizing high-dimensional entangled states is of crucial importance in quantum information science and technology. Recent theoretical progress has been made to extend Hardy's paradox into a general scenario with multisetting and multidimensional systems, which enables the maximum probability of nonlocal events to surpass the bound that was limited in Hardy's original test. Hitherto, no experimental verification has been conducted to verify Hardy's paradox, as most of the previous experimental efforts were restricted to two-dimensional systems. Here, based on two-photon high-dimensional orbital angular momentum (OAM) entanglement, we report an experiment to demonstrate Hardy's paradox for multiple settings and multiple outcomes. We demonstrate the paradox for two-setting higher-dimensional OAM subspaces up to $d=7$, which reveals that the nonlocal events increase with dimension. Furthermore, we showcase the nonlocality with an experimentally recorded probability of 36.77% for a five-setting three-dimensional OAM subspace via entanglement concentration, thus showing a sharper contradiction between quantum mechanics and classical theory.

DOI
10.1103/physreva.101.053821

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Characterizing high-dimensional entangled states is of crucial importance in quantum information science and technology. Recent theoretical progress has been made to extend Hardy's paradox into a general scenario with multisetting and multidimensional systems, which enables the maximum probability of nonlocal events to surpass the bound that was limited in Hardy's original test. Hitherto, no experimental verification has been conducted to verify Hardy's paradox, as most of the previous experimental efforts were restricted to two-dimensional systems. Here, based on two-photon high-dimensional orbital angular momentum (OAM) entanglement, we report an experiment to demonstrate Hardy's paradox for multiple settings and multiple outcomes. We demonstrate the paradox for two-setting higher-dimensional OAM subspaces up to d=7d=7d=7, which reveals that the nonlocal events increase with dimension. Furthermore, we showcase the nonlocality with an experimentally recorded probability of 36.77% for a five-setting three-dimensional OAM subspace via entanglement concentration, thus showing a sharper contradiction between quantum mechanics and classical theory.
Characterizing high-dimensional entangled states holds pivotal significance in quantum information science and technology. Recent theoretical progress has been made to extend the Collins-Gisin-Linden-Massar-Popescu (CGLMP) inequality into a general scenario with multisetting and multidimensional systems. Here, by employing two-photon orbital angular momentum entanglement, we conduct an experiment to demonstrate the CGLMP inequality across multiple settings and outcomes. Our experimental results violate the CGLMP inequality by more than 12 standard deviations, demonstrating that the maximum violation increases with both dimension and setting. Furthermore, we reveal, both theoretically and experimentally, the logical connection between the CGLMP inequality and the general Hardy's paradox. Intriguingly, our analysis establishes that, even in high-dimensional systems, Hardy's paradox can be regarded as a special instance of Bell inequality, with their respective bounds aligning under the no-signaling principle and the information causality principle. Our work may advance the foundational understanding of quantum correlations in high-dimensional quantum systems.
High-dimensional entangled states are of crucial importance, as they provide higher channels for quantum information. Angular position, as a continuous variable, provides an infinite Hilbert space theoretically. It is usually represented in discrete bases experimentally because of the convenient modulation of the width of the angular aperture. Thus the combination of the angular apertures will conveniently shape the two photon states. And the entanglement of the two photon states can be naturally demonstrated by Hardy's paradox. Here, by testing Hardy's paradox for two-setting high-dimensional angular subspaces with dimension ranging from 2 to 7 and for multisetting three-dimensional subspaces with setting ranging from 3 to 5, we reveal the high-dimensional entanglement in the angular-position degree of freedom. We show that the high-dimensional angular-position entanglement can yield a much sharper contradiction between the quantum mechanics and classical theories. Our work shows that the angular variable can be considered as an alternative discrete base to provide the high-dimensional entangled states. Thus it may be used as a new platform for quantum information.
Recent advancements have expanded Hardy's nonlocality arguments into multisetting and multidimensional systems to enhance quantum correlations. In comparison with Hardy's nonlocality argument, Cabello's nonlocality argument (CNA) emerges as a superior choice for illustrating nonlocal features. An open question persists regarding the potential extension of CNA to arbitrary (k,dk, dk,d) scenarios. Here, we answer this question both in theory and experiment. Theoretically, by utilizing compatibility graphs, we construct a new logical framework for multisetting and multidimensional CNA, demonstrating an increase in the maximum successful probability with setting kkk and dimension ddd. Experimentally, by employing controllable photonic orbital angular momentum entanglement, we exhibit nonlocality with an experimentally recorded probability of 20.29% in the (2, 4) scenario and 28.72% in the (6, 2) scenario. Our work showcases a sharper contradiction between quantum mechanics and classical theory, surpassing the bound limited by the original version.
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.
Entanglement in higher dimensions is an attractive concept that is a chal- lenge to realise experimentally. To this end, the entanglement of the orbital angular momentum (OAM) of photons holds promise. The OAM state-space is discrete and theoretically unbounded. In the work that follows, we investi- gate various aspects of OAM entanglement. We show how the correlations in OAM and its conjugate variable, angular position, are determined by phase- matching and the shape of the pump beam in spontaneous parametric down- conversion. We implement tests of quantum mechanics which have been previously done for other variables. We show the Einstein-Podolsky-Rosen paradox for OAM and angle, supporting the incompatibility of quantum me- chanics with locality and realism. We demonstrate violations of Bell-type inequalities, thereby discounting local hidden variables for describing the correlations we observe. We show the Hardy paradox using OAM, again highlighting the nonlocal nature of quantum mechanics. We demonstrate violations of Leggett-type inequalities, thereby discounting nonlocal hidden variables for describing correlations. Lastly, we have looked into the entan- glement of topological vortex structures formed from a special superposition of OAM modes and show violations of Bell-type inequalities confined to a finite, isolated volume.
Recently, an alternative form of Hardy's paradox was introduced for two-setting high-dimensional systems [Phys. Rev. A 88, 062116 (2013)], for which the maximum probability of the nonlocal event was shown to increase as the dimension goes larger. Here, we generalize the result to a general scenario with multisetting high-dimensional systems. The general Hardy's paradox (i) reduces to the one by Chen et al. [Phys. Rev. A 88, 062116 (2013)] for two settings, (ii) is equivalent to the ladder proof of nonlocality without inequalities given by Boschi et al. [Phys. Rev. Lett. 79, 2755 (1997)] for two-dimensional systems, and (iii) increases the maximum probability of the nonlocal event for, e.g., three-dimensional systems. In particular, the maximum probability for the five-setting three-dimensional system is increased to 0.401840.401\phantom{\rule{0.16em}{0ex}}840.40184, which is larger than 0.171 originally by Cabello [Phys. Rev. A 58, 1687 (1998)].