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.
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.
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.
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, d-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.
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.
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.
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,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 k and dimension d. 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.
Orbital angular momentum (OAM) is one of the most recently discovered properties of light, and it is only in the past decade its quantum properties have been the subject of experimental investigations and have found applications. Unlike polarization, which is only bidimensional, orbital angular momentum provides, with relative ease, unprecedented access to a theoretically unbounded discrete state space. The process of spontaneous parametric down-conversion has long been used as a source of two-photon states that can be entangled in several degrees of freedom, including OAM. In this thesis, the properties of the natural OAM spectrum associated with the entangled states produced by parametric down-conversion were investigated. Chapters 2 and 3 describe the production and detection of tunable high-dimensional OAM entanglement in a down-conversion system. By tuning the phase-matching conditions and improving the detection stage, a substantial increase in the half-width of the OAM correlation spectrum was observed. The conjugate variable of OAM, angular position, was also considered when examining high-dimensional states entangled in OAM. In order to efficiently determine their dimension, high-dimensional entangled states were probed by implementing a technique based on phase masks composed of multiple angular sectors, as opposed to narrow single-sector analysers. Presented in chapter 4, this technique allows the measurements of tight angular correlations while maintaining high optical throughput. The states so produced were then used for a number of applications centred around the concept of mutually unbiased bases. One can define sets of mutually unbiased bases for arbitrary subspaces of the OAM state space. Two bases are mutually unbiased if the measurement of a state in one basis provides no information about the state as described in the other basis. Complete measurements in mutually unbiased bases of high-dimensional OAM spaces are presented in chapter 5. Measurements in sets of mutually unbiased bases are integral to quantum science and can be used in a variety of protocols that fully exploit the large size of the OAM state space; we describe their use in efficient quantum state tomography, quantum key distribution and entanglement detection. Caution is however necessary when dealing with state spaces embedded in higher-dimensional spaces, such as that provided by OAM. Experimental tests of Bell-type inequalities allow us to rule out local hidden variable theories in the description of quantum correlations. Correlations inconsistent with the states observed, or even with quantum mechanics, known as super-quantum correlations, have however been recorded previously in experiments that fail to comply with the fair-sampling conditions. Chapter 6 describes an experiment that uses a particular choice of transverse spatial modes for which super-quantum correlations persist even if the detection is made perfectly efficient. The sets of modes carrying OAM allow a complete description of the transverse field. The ability to control and combine additional degrees of freedom provides the possibility for richer varieties of entanglement and can make quantum protocols more powerful and versatile. One such property of light, associated with transverse modes possessing radial nodes in the field distribution, can be accessed within the same type of experimental apparatus used for OAM. In chapter 7, the radial degree of freedom is explored, together with OAM, in the context of Hong-Ou-Mandel interference.