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arXiv:2410.23689

Impact of High-Brightness Entangled Photon Pairs on CHSH Inequality Experiment

Kim, Jin-Woo, Suseong Lim, Heonoh Kim, June‐Koo Kevin Rhee

Published Oct 31, 2024• 0 citations
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Abstract

Verifying the violation of Bell's inequality is one of the most representative methods to demonstrate that entangled photon pairs prepared in a quantum optics-based system exhibit quantum properties. While experiments on Bell inequality violations have been theoretically well-established and extensively conducted to implement various quantum information technologies in laboratory settings, mathematical modeling for accurately predicting the distribution of high-intensity entangled photon pairs in high-loss environments remains an issue that requires further research. As the brightness of the entangled photon pairs increases, the influence of multi-photon effects becomes more significant, leading to a decrease in the CHSH value $S$ and also a reduction in the standard deviation of the CHSH value $ΔS$. Therefore, a new analysis of the $(S-2)/ΔS$ value is required to more precisely confirm the degree of CHSH inequality violation including the reliability of $S$. In this paper, we propose a mathematical model to predict the $(S-2)/ΔS$ value as a function of the brightness of the entangled photon pair source, and we also suggest the need to optimize the brightness of this source. Additionally, we provide experimental evidence supporting this model. The experiment confirms that when the mean photon number is $μ=0.026$ in an entanglement distribution setup with a total loss of $-19.03$ dB, the CHSH value drops to 2.69, while the $(S-2)/ΔS$ value increases to 60.95.

DOI
10.48550/arxiv.2410.23689

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Verifying the violation of Bell's inequality is one of the most representative methods to demonstrate that entangled photon pairs prepared in a quantum optics-based system exhibit quantum properties. While experiments on Bell inequality violations have been theoretically well-established and extensively conducted to implement various quantum information technologies in laboratory settings, mathematical modeling for accurately predicting the distribution of high-intensity entangled photon pairs in high-loss environments remains an issue that requires further research. As the brightness of the entangled photon pairs increases, the influence of multi-photon effects becomes more significant, leading to a decrease in the CHSH value SSS and also a reduction in the standard deviation of the CHSH value ΔSΔSΔS. Therefore, a new analysis of the (S−2)/ΔS(S-2)/ΔS(S−2)/ΔS value is required to more precisely confirm the degree of CHSH inequality violation including the reliability of SSS. In this paper, we propose a mathematical model to predict the (S−2)/ΔS(S-2)/ΔS(S−2)/ΔS value as a function of the brightness of the entangled photon pair source, and we also suggest the need to optimize the brightness of this source. Additionally, we provide experimental evidence supporting this model. The experiment confirms that when the mean photon number is μ=0.026μ=0.026μ=0.026 in an entanglement distribution setup with a total loss of −19.03-19.03−19.03 dB, the CHSH value drops to 2.69, while the (S−2)/ΔS(S-2)/ΔS(S−2)/ΔS value increases to 60.95.
Verifying the violation of Bell's inequality is one of the most representative methods to demonstrate that entangled photon pairs prepared in a quantum optics-based system exhibit quantum properties. While experiments on Bell inequality violations have been theoretically well-established and extensively conducted to implement various quantum information technologies in laboratory settings, mathematical modeling for systematically describing the distribution of high-brightness entangled photon pairs in high-loss environments remains an issue that requires further research. As the brightness of the entangled photon pairs increases, the influence of multi-photon effects becomes more significant, leading to a decrease in the CHSH valueSSSand also a reduction in the standard deviation of the CHSH valueΔS\Delta SΔS. Therefore, a new analysis of the(S−2)/ΔS(S-2)/\Delta S(S−2)/ΔSvalue is required to more precisely confirm the degree of CHSH inequality violation including the reliability ofSSS. In this paper, we propose a mathematical model to predict the(S−2)/ΔS(S-2)/\Delta S(S−2)/ΔSvalue as a function of the brightness of the entangled photon pair source, and we also suggest the need to optimize the brightness of this source. Additionally, we provide experimental evidence supporting this model. The experiment confirms that when the mean photon number isμ=0.026\mu =0.026μ=0.026in an entanglement distribution setup with a total loss of−19.03-19.03−19.03dB, the CHSH value drops to 2.69, while the(S−2)/ΔS(S-2)/\Delta S(S−2)/ΔSvalue increases to 60.95.
Optimization of a Pulsed Source of Entangled Photons for the Violation of the CHSH Inequality This thesis focuses on my work carried out in the Quantum Communications laboratory with the Quantum Future research group of the Department of Information Engeneering, in which I worked on the creation and optimization of a pulsed source of polarization entangled photons. The main focus in the laboratory was to optimize the experimental setup to prove the entangled nature of the photons produced. This was done by violating the Bell inequality in the form presented by Clauser, Horne, Shimony and Holt (CHSH) in 1969. The present work's objective is to provide an overview of all the crucial physical phenomena exploited in the experiment to obtain the wanted results. In section 1 a presentation of the phenomenon of entanglement and of Bell's argument proving the incapability to describe its properties with a both real and local theory is given, along with the derivation of the ``CHSH inequality and the possibility to violate it with polarization entangled photons. Sections 2 and 3 focus on the properties of the Gaussian beam used in the experiment and its source: the mode-locked Mira laser. A brief discussion about mode-locked lasers and their properties is provided. In section 4 the nonlinear optical phenomena of second harmonic generation and spontaneous parametric down conversion are discussed, the former from a classical standpoint and the latter with a few quantum mechanical considerations. Section 5 focuses on polarization optics and the both classical and quantum mechanical formalism of the Jones vectors. A discussion of the effects of the birefringent devices used in the experiment on the photons' state is provided. Finally, sections 6 and 7 display a complete overview of the experimental setup and the results obtained for the violation of the inequality. A systematic explanation of the experimental optimization procedure is provided in section 6 as it may be useful for the future experimenters, since it proved to be tricky. Throughout all the work, various characteristics of the devices used in the experimental setup are given, hopefully making the most important and peculiar aspects of the experiment clear.
It is well known that the violation of Bell's inequality in the form given by Clauser, Horne, Shimony, and Holt (CHSH) in two-qubit systems requires entanglement, but not vice versa, i.e., there are entangled states which do not violate the CHSH inequality. Here we compare some standard entanglement measures with violations of the CHSH inequality (as given by the Horodecki measure) for two-qubit states generated by Monte Carlo simulations. We describe states that have extremal entanglement according to the negativity, concurrence, and relative entropy of entanglement for a given value of the CHSH violation. We explicitly find these extremal states by applying the generalized method of Lagrange multipliers based on the Karush-Kuhn-Tucker conditions. The found minimal and maximal states define the range of entanglement accessible for any two-qubit states that violate the CHSH inequality by the same amount. We also find extremal states for the concurrence versus negativity by considering only such states which do not violate the CHSH inequality. Furthermore, we describe an experimentally efficient linear-optical method to determine the highest Horodecki degree of the CHSH violation for arbitrary mixed states of two polarization qubits. By assuming to have access simultaneously to two copies of the states, our method requires only six discrete measurement settings instead of nine settings, which are usually considered.
The aim of this paper is to attract the attention of experimenters to the original Bell (OB) inequality that was shadowed by the common consideration of the Clauser-Horne-Shimony-Holt (CHSH) inequality. There are two reasons to test the OB inequality and not the CHSH inequality. First of all, the OB inequality is a straightforward consequence to the Einstein-Podolsky-Rosen (EPR) argumentation. In addition, only this inequality is directly related to the EPR-Bohr debate. The second distinguishing feature of the OB inequality was emphasized by Itamar Pitowsky. He pointed out that the OB inequality provides a higher degree of violations of classicality than the CHSH inequality. For the CHSH inequality, the fraction of the quantum (Tsirelson) bound Q CHSH = 2 2 to the classical bound C CHSH = 2 , i.e., F CHSH = Q CHSH C CHSH = 2 is less than the fraction of the quantum bound for the OB inequality Q OB = 3 2 to the classical bound C OB = 1 , i.e., F OB = Q OB C OB = 3 2 . Thus, by violating the OB inequality, it is possible to approach a higher degree of deviation from classicality. The main problem is that the OB inequality is derived under the assumption of perfect (anti-) correlations. However, the last few years have been characterized by the amazing development of quantum technologies. Nowadays, there exist sources producing, with very high probability, the pairs of photons in the singlet state. Moreover, the efficiency of photon detectors was improved tremendously. In any event, one can start by proceeding with the fair sampling assumption. Another possibility is to use the scheme of the Hensen et al. experiment for entangled electrons. Here, the detection efficiency is very high.
We theoretically and experimentally investigate the optimal conditions for the Bell experiment using spontaneous parametric down-conversion (SPDC) sources. In theory, we show that a relatively large average photon number (typically $\ensuremath{\sim}0.5$) is desirable to observe the maximum violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality. Moreover, we show that the violation converges to 0.27 in the limit of large average photon number. In experiment, we perform the Bell experiment without postselection using polarization-entangled photon pairs at 1550-nm telecommunication wavelength generated from SPDC sources. While the violation of the CHSH inequality is not directly observed due to the overall detection efficiencies of our system, the experimental values agree well with those obtained by the theory with experimental imperfections. Furthermore, in the range of small average photon numbers ($\ensuremath{\le}0.1$), we propose and demonstrate a method to estimate the ideal CHSH value intrinsically contained in the tested state from the lossy experimental data without assuming the input quantum state.
Nonlocality is a fascinating and counterintuitive aspect of nature, revealed by the violation of a Bell inequality. The standard and easiest configuration in which Bell inequalities can be measured has been proposed by Clauser-Horne-Shimony-Holt (CHSH). However, alternative nonlocality tests can also be carried out. In particular, Bell inequalities requiring multiple measurement settings can provide deeper fundamental insights about quantum nonlocality, as well as offering advantages in the presence of noise and detection inefficiency. In this paper we show how these nonlocality tests can be performed using a commercially available source of entangled photon pairs. We report the violation of a series of these nonlocality tests (I3322{I}_{3322}I3322​, I4422{I}_{4422}I4422​, and chained inequalities). With the violation of the chained inequality with 4 settings per side we put an upper limit at 0.49 on the local content of the states prepared by the source (instead of 0.63 attainable with CHSH). We also quantify the amount of true randomness that has been created during our experiment (assuming fair sampling of the detected events).