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Scientific Reports

Bell’s inequality tests via correlated diffraction of high-dimensional position-entangled two-photon states

Wei Li, Shengmei Zhao

Published Mar 13, 2018• 4 citations
Publisher

Abstract

Bell inequality testing, a well-established method to demonstrate quantum non-locality between remote two-partite entangled systems, is playing an important role in the field of quantum information. The extension to high-dimensional entangled systems, using the so-called Bell-CGLMP inequality, points the way in measuring joint probabilities, the kernel block to construct high dimensional Bell inequalities. Here we show that in theory the joint probability of a two-partite system entangled in a Hilbert space can be measured by choosing a set of basis vectors in its dual space that are related by a Fourier transformation. We next propose an experimental scheme to generate a high-dimensional position-entangled two-photon state aided by a combination of a multiple-slit and a 4 f system, and describe a method to test Bell's inequality using correlated diffraction. Finally, we discuss in detail consequences of such Bell-test violations and experimental requirements.

DOI
10.1038/s41598-018-23310-9

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Bell inequality testing, a well-established method to demonstrate quantum non-locality between remote two-partite entangled systems, is playing an important role in the field of quantum information. The extension to high-dimensional entangled systems, using the so-called Bell-CGLMP inequality, points the way in measuring joint probabilities, the kernel block to construct high dimensional Bell inequalities. Here we show that in theory the joint probability of a two-partite system entangled in a Hilbert space can be measured by choosing a set of basis vectors in its dual space that are related by a Fourier transformation. We next propose an experimental scheme to generate a high-dimensional position-entangled two-photon state aided by a combination of a multiple-slit and a 4 f system, and describe a method to test Bell's inequality using correlated diffraction. Finally, we discuss in detail consequences of such Bell-test violations and experimental requirements.
Violation of Bell inequalities is an essential requirement for many quantum information and communication protocols. In high-dimensional systems, Bell inequality tests face the challenge of implementing genuinely multi-outcome measurements, since the emulation of these with separate dichotomic projections opens a binarisation loophole that local hidden variable theories can exploit. Here we show that the joint spectral intensity of a two-photon entangled state contains access to the necessary multi-outcome measurements to overcome this obstacle and certify and violate a Bell inequality for high-dimensional states. This result is contrary to the belief that the joint spectral intensity is a phase-insensitive quantity and does not have sufficient information to certify entanglement or Bell-nonlocality. Using this approach, we violate the CGLMP Bell inequality up to dimension d = 8, all with negligible p-values, and for the first time close the binarisation loophole in high-dimensional Bell experiments. Guaranteeing Bell-nonlocal correlations using frequency-only measurements removes the technological hurdle of measurements in the temporal domain, thus greatly simplifying any practical implementation of future high-dimensional quantum information protocols.
Abstract Violation of Bell inequalities is an essential requirement for many quantum information and communication protocols. In high-dimensional systems, Bell inequality tests face the challenge of implementing genuinely multi-outcome measurements, since the emulation of these with separate dichotomic projections opens a binarisation loophole that local hidden variable theories can exploit. Here we show that the joint spectral intensity of a two-photon entangled state contains access to the necessary multi-outcome measurements to overcome this obstacle and certify nonlocality for high-dimensional states. This result is contrary to the belief that the joint spectral intensity is a phase-insensitive quantity and does not have sufficient information to certify entanglement or nonlocality. Using this approach, we violate the CGLMP Bell inequality up to dimension d = 8, all with negligible p-values, and for the first time close the binarisation loophole in high-dimensional Bell experiments. Guaranteeing nonlocal correlations using frequency-only measurements removes the technological hurdle of measurements in the temporal domain, thus greatly simplifying any practical implementation of future high-dimensional quantum information protocols.
Bipartite quantum entangled systems can exhibit measurement correlations that violate Bell inequalities, revealing the profoundly counter-intuitive nature of the physical universe. These correlations reflect the impossibility of constructing a joint probability distribution for all values of all the different properties observed in Bell inequality tests. Physically, the impossibility of measuring such a distribution experimentally, as a set of relative frequencies, is due to the quantum back-action of projective measurements. Weakly coupling to a quantum probe, however, produces minimal back-action, and so enables a weak measurement of the projector of one observable, followed by a projective measurement of a noncommuting observable. By this technique it is possible to empirically measure weak-valued probabilities for all of the values of the observables relevant to a Bell test. The marginals of this joint distribution, which we experimentally determine, reproduces all of the observable quantum statistics including a violation of the Bell inequality, which we independently measure. This is possible because our distribution, like the weak values for projectors on which it is built, is not constrained to the interval [0,1][0,1][0,1]. It was first pointed out by Feynman that, for explaining singlet-state correlations within ``a [local] hidden variable view of nature \dots{} everything works fine if we permit negative probabilities.'' However, there are infinitely many such theories. Our method, involving ``weak-valued probabilities,'' singles out a unique set of probabilities, and moreover does so empirically.
Entangled photons play a crucial role in quantum applications, and determining and characterising their entanglement is vital to using them effectively. High-dimensional entangled states offer richer possibilities, but their additional measurement degrees of freedom make them increasingly demanding to characterise. However, adaptive Bell-test methods based on complex simultaneous perturbation stochastic approximation (CSPSA) have so far focused mainly on qubits. Here we numerically investigate a Bell-inequality-violation-based method for detecting entanglement in unknown quantum states. We extend CSPSA to high-dimensional Bell testing by using the Collins-Gisin-Linden-Massar-Popescu (CGLMP) inequality for bipartite qudits. The resulting protocol can detect Bell-nonlocal correlations in unknown entangled states, whether pure or mixed, without first reconstructing their density matrix. Using 100 optimisation iterations in each of 100 independent finite-shot runs per number of dimensions d, we demonstrate certified CGLMP violations throughout d=2-8. For isotropic mixed states tested at a visibility of just 0.05 above the standard-Fourier violation threshold, we likewise observe confidence-certified CGLMP violations throughout the range of d studied. We compare this direct stochastic approach with quantum state tomography, the standard method for characterising an unknown state. In the matched benchmark, CSPSA uses fewer measurement configurations per attempt from d=6, whereas tomography requires fewer detected pairs per certified result through d=8. We also derive the phase dependence of the CGLMP parameter and clarify the features of its landscape that govern the adaptive search. Because the measurement-setting cost of each CSPSA iteration is independent of dimension, the method offers a particularly attractive route to the certification of high-dimensional entanglement.
Quantum correlations between spatially separated parts of a ddd-dimensional bipartite system (d≥2d\geq 2d≥2) have no classical analog. Such correlations, also called entanglements, are not only conceptually important, but also have a profound impact on information science. In theory the violation of Bell inequalities based on local realistic theories for ddd-dimensional systems provides evidence of quantum nonlocality. Experimental verification is required to confirm whether a quantum system of extremely large dimension can possess this feature, however it has never been performed for large dimension (e.g., d≥1000d\geq 1000d≥1000). Here, we report that Bell inequalities are experimentally violated for bipartite quantum systems of extremely high dimensionality with the usual ensembles of polarization-entangled photon pairs. Our entanglement source violates Bell inequalities for extremely high dimensionality of d>4000d>4000d>4000. The designed scenario offers a possible new method to investigate the entanglement of multipartite systems of large dimensionality and their application in quantum information processing.
Quantum correlations between spatially separated parts of a d-dimensional bipartite system (d ≥ 2) have no classical analog. Such correlations, also called entanglements, are not only conceptually important, but also have a profound impact on information science. In theory the violation of Bell inequalities based on local realistic theories for d-dimensional systems provides evidence of quantum nonlocality. Experimental verification is required to confirm whether a quantum system of extremely large dimension can possess this feature, however it has never been performed for large dimension. Here, we report that Bell inequalities are experimentally violated for bipartite quantum systems of dimensionality d = 16 with the usual ensembles of polarization-entangled photon pairs. We also estimate that our entanglement source violates Bell inequalities for extremely high dimensionality of d > 4000. The designed scenario offers a possible new method to investigate the entanglement of multipartite systems of large dimensionality and their application in quantum information processing.