## INTRODUCTION

The quantum entanglement makes a significant difference between the classical and the quantum world. The debate over the interpretation of this entanglement remained central for much of the 20th century. That an interpretation based on hidden variables can be excluded on the basis of experimental observation is the essence of Bell's inequality and follows the groundbreaking work of Freedman and Clauser (* 1 *) and Aspect * et al. * (* 2 * – * 4 *) many groups worldwide have used nonlocal correlations between photon pairs to show the violation of this kind of inequality. In particular, recent improvements to the schema design and performance of the components have made it possible to simultaneously close the various gaps that were present in previous demonstrations (* 5 * ̵

*7*).

The violation of a Bell's inequality is a fundamental manifestation of a quantum system. It not only confirms the quantum laxity of the behavior of a system, but also evaluates the performance of these systems when they are involved in certain quantum protocols. For example, certain quantum protocols require the performance of Bell-type non-local behaviors, such as device-independent protocols (* 8 * – * 10 *). One quantum technology that is currently of interest is quantum imaging, which attempts to exploit the quantum behavior of light in order to perform new types of imaging that can exceed the limits of classical methods. Capturing images of one of the most basic quantum effects is therefore evidence that images can be used to access the full range of possibilities allowed in the quantum world.

The Violation of Bell's Inequalities Against Hidden Variable Interpretations of Quantum Mechanics has usually relied on the sequential measurement of correlation rates as a function of analyzer settings (eg, the relative angles of the linear polarizers) that rely on the act on two separate photons in space. A violation of Bell's inequality causes the correlation rate to depend not only on the angle of a polarizer, but on the combination of the two. Due to the non-locality of this correlation, the term "spooky distance effect" is used. Measurement of polarization is convenient, but interlacing tests were also performed on other variables (* 11 * – * 14 [19459006)]) and although the original Bell inequality was applied to variables in a two-dimensional Hilbert space , a similar logic can be used for design tests in higher-dimensional state spaces ( 15 ). *

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To understand our present work, it is important to consider two of the other high-dimensional realms in which entanglements can be explored. The first high-dimensional domain is that instead of analyzing the polarization and thus the spin angular momentum of the photons, alternatively the orbital angular momentum (OAM) of the photons is measured. The OAM of ℓℏ per photon results from the helical phase structure of the beam described as exp (* i * ℓφ) (* 16 ** 17 *.) Although early experiments with this OAM are based on observation of his mechanical manifestations (* 18 *), later works examined the correlations of the OAM between photons generated by parametric downconversion and entanglement (* 19 *) and subsequently a violation of a Bell's inequality in two-dimensional (* 20 *) and higher-dimensional (* 21 *) OAM subspaces.) The second high-dimensional domain concerns Einstein et al. (* 22 *), who express their concern about the Completeness of quantum mechanics was expressed by the EPR paradox (Einstein-Podolsky-Rosen) This paradox concerns the correlations between position and imp uls correlations that could occur between the two entangled particles. For photon pairs generated by parametric downconversion, both spatial correlations and impulse anticorrelations can be observed in the image plane and in the far field of the source, respectively. These correlations form the basis for quantum ghost imaging (* 23 ** 24 *), where one of the two down-converted beams is directed to the object with a single pixel (not spatially resolving). Detector, which collects the interacting light, and the other beam directed to an imaging detector. The data from neither detector yields an image of the object, but a summation of the correlations between the non-image detector and the image detector. When the object and the spatial resolution detector are in the image planes of the source, the image is upright with respect to the object; If the object and the detector are in the far field of the source, the image is inverted with respect to the object. The upright or inverted nature of the image results from the position correlation or the impulse anticorrelation and can be regarded as an image-based manifestation of the EPR paradox (* 25 *). In another context, the demonstration of an EPR paradox in imaging has been performed in a series of experiments (* 26 ** 27 ** 28 ** 29 *) in which cameras were used to perform the Position and impulse of entangled photon pairs. This eventually led to the demonstration of an EPR paradox within individual frames of a detector array (* 30 *). The question to be presented in this work is what kind of imaging could reveal a Bell's inequality.

The key to measurements in the polarization experiments is to recognize that the orientation of the linear polarizer analyzer is actually a measurement of the phase difference between a superposition of the right and left circular polarization states (ie the angular momenta states). A superposition of right and left OAMs (ℓ = ± 1) has a π-phase step across the diameter of the beam, with the phase difference between the two OAM states determining the orientation of the step. Note that a circle contains all possible edge orientations and therefore as an object a circle corresponds to a full rotation of the polarizer in the polarization case. This was exploited in an earlier demonstration in which the non-local detection of phase steps showed an edge enhancement depending on the edge orientation, indicating a violation of Bell's inequality (* 31 *). However, this finding was not based on imaging because the data was sequentially acquired by scanning an object in a single-mode spatial optical setup. The data from this set of measurements was finally recombined to be displayed in the form of an image. In contrast, in this work, we use a full field imaging configuration in which a phase edge filter has not been placed locally with respect to a circular phase object to obtain a ghost image whose intensity characteristics reveal the expected violation of a bell inequality for OAM. No sampling is required for this experiment, since a quantum state must be used which can exhibit correlations in OAM space or in direct Cartesian space. It shows that bell-type non-local behavior can be demonstrated within a full field quantum mapping protocol. Since we do not fill all the gaps, our demonstration can not be interpreted as another absolute demonstration that the world does not behave locally. However, these gaps are not fundamentally associated with the experimental paradigm presented here and could, in principle, be concluded with more technically advanced detectors and phase-image displays. In addition, as will be discussed below, our results may be interpreted as the first experimental demonstration that an imaging protocol can be used to uncover the Bell-type violation by making only some physically reasonable assumptions about the source involved in the demonstration become behavior of a quantum system. Conversely, our results indicate that Bell-type non-local behavior can be exploited to perform special types of imaging that could not be performed with a conventional classical source.

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## RESULTS

### Experimental Design and Principle of Demonstration

Our experimental system shown in Figure 1 consists of a β-barium borate (BBO) crystal pumped at 355 nm using a quasicontinuous laser to give spatially interlaced Photon pairs at 710 nm were generated by the process of spontaneous parametric downconversion (SPDC). The two photons are separated on a beam splitter and spread out into two different optical systems (arms). The first photon is reflected by a spatial light modulator (SLM) located in an image plane of the crystal and displaying a phase object before being collected in a single-mode fiber (SMF) followed by a single-photon avalanche diode (SPAD). The second photon moving through the other arm is reflected by an SLM located in a Fourier plane of the crystal (corresponding to the object's Fourier plane) and displays a spatial π-phase step filter. The photon then propagates through approximately 20 meters of image-sparing delay line before being finally detected by an ICCD (Intensified Charge Coupled Device) camera. The ICCD camera is triggered on the condition that the SPAD in the first arm detects a photon. This delay line ensures that the images obtained from the ICCD camera are match images with respect to the SPAD recognition. The presence of the delay line in the second arm compensates for the camera's shutter lag and ensures that the second photon falls onto the camera during the 4 ns gate of the image intensifier.

Using such a triggering mechanism, we have implemented a quantum illumination protocol and captured images with less than one photon per pixel (* 32 *) and in the context of phase unis d Amplitude images ng (* 33 *). We also used a similar setup to test the experimental limits of ghost imaging and ghost diffraction (* 34 ** 35 *). In the present work, our scheme uses phase mapping to improve the edges by spatial filtering. Here, the object (a circular phase step) and the filter (a rectilinear phase step) are not located locally in separate optical arms and examined by two spatially separated but entangled photons. The resulting edge-enhanced image of the circle is a result of the non-local interference between the object and the spatial filter being examined by the two-photon wave function. However, simply obtaining an edge enhanced image under these circumstances is not evidence of the nonlocal nature of the behavior of the two photons, as it may possibly be reproduced by classical means such as in the context of ghost imaging (* 24 *). By proving the violation of a Bell inequality, one can still produce images that are not reproducible by classical means.

From the realization that the implementation shown in Fig. 1 can violate Bell's inequality, it is possible to derive the understanding. A π-phase step has both ℓ = 1 and ℓ = -1 contributions if it is in an OAM Basis is expressed. Indeed, a π-phase step can be represented as a linear superposition of ℓ = -1 and ℓ = 1, and thus on a Bloch-Poincaré sphere (* 36 *), which describes a two-dimensional OAM basis. The phase difference θ between the two modes ℓ = -1 and ℓ = 1 determines the orientation angle θ of the π-phase stage in the two-dimensional transverse plane. One can therefore use these phase steps as a filter to perform measurements in this particular two-dimensional OAM space (* 20 *). Projected purely into such a space, the two-photon wavefunctions can be written as follows

(1) . This is the result of preserving the entire OAM from the pump photons (ℓ = 0) to those from the SPDC process. Such a state violates a Bell's inequality of the form (* 20 *)

(2) with [19659055] S

(3) 19659046] and

$$E({\mathrm{\theta}}_{1}{\mathrm{\theta}}_{\mathrm{]\; 2}})=\frac{\mathrm{C\; [19659050]{\mathrm{\&\; thgr;}}_{1}[19659059]?2)+({\mathrm{\theta}}_{1}+\frac{\mathrm{\pi}}{2}[19659059]2+\frac{\mathrm{}}{2})\u2013C({\mathrm{\theta}}_{1}+\frac{\mathrm{\pi}}{}{\mathrm{\theta ]\; 2}}_{})\u2013C({\mathrm{\theta}}_{1}{\mathrm{\theta}}_{2}+\frac{\mathrm{\pi}}{2})C({\mathrm{\theta}}_{1}{\mathrm{\theta}}_{2})+C\theta 1+\frac{\mathrm{\pi}}{2}{\mathrm{\theta}}_{2}+\frac{\mathrm{\pi}}{2}+C({\mathrm{\theta}}_{1}+\; [19659113]\; \pi 2{\mathrm{\theta}}_{\mathrm{2\; [1\; 9659129]\; +}}]\; C({\mathrm{\theta}}_{1}{\mathrm{\theta}}_{2}+\frac{\mathrm{\pi}}{\mathrm{2(4)whereC(\theta 1\theta 2)\; is\; the\; recorded\; coincidence\; rate\; when\; the\; first\; photon\; is\; detected\; thereafter\; a\; phase\; step\; with\; the\; orientation\; \theta 1and\; when\; the\; second\; photon\; after\; a\; phase\; step\; with\; the\; orientation\; \theta 2is\; measured.\; The\; inequality\; (Eq.\; 2)\; is\; a\; Clauser-Horne-Shimony-Holt\; (CHSH)\; Bell\; inequality\; (37).\; As\; in\; a\; demonstration\; with\; the\; degree\; of\; polarization\; freedom,\; the\; condition\; (Equation\; 1)\; shows\; a\; maximum\; violation\; of\; the\; inequality\; (Equation\; 2)\; when\; the\; settings\; are\; chosen\; as\; follows:\; \theta 1=\; 22.5\; \xb0,$ {\mathrm{\&\; thgr;\; ;}}_{1}^{[}=67.5\xb0$\&\; thgr;2=\; 0\; \xb0\; andsrc="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js">(adsbygoogle\; =\; window.adsbygoogle\; ||\; []).push(\{\});$ {\mathrm{thgr;}}_{2}^{[}=45\xb0$}}}}{}$$