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APL Quantum

Automated Bell inequality violation searches for estimating entanglement quality in fiber

Evan Dowling, Gerald Baumgartner, R. A. Brewster, Mark B. Morris, Anne Marie Richards, Rajarshi Roy, Thomas E. Murphy

Published Feb 18, 2025• 0 citations
Publisher

Abstract

In a future quantum network, two spatially separated individuals sharing a polarization entanglement source may need to assess the entanglement quality of the source without the presence of classical auxiliary signals. When the two are separated by a single-mode fiber, automated methods are necessary for the validation and estimation of polarization entanglement. We experimentally examine real-time iterative methods to search for maximal Bell violations, specifically the Clauser–Horne–Shimony–Holt (CHSH) inequality, between two observers sharing polarization entangled photons to evaluate the entanglement quality of the source. Our source allows us to tune the degree of entanglement of their shared photon pair by changing the temporal overlap of the two photons at an entangling 50/50 beam splitter so that we can compare the CHSH parameter these methods find to the theoretical values our source produces. The iterative methods used within our experiment are the Nelder–Mead optimization method, stochastic gradient descent, and Bayesian optimization. This is the first feedback experiment to study automated Bell violations in fiber and the first to compare all three of these iterative methods to one another in a quantum polarization control experiment. In our experiment, all methods are able to find Bell violations, but the Nelder–Mead method performed the best in terms of the speed and accuracy in finding the maximal violation.

DOI
10.1063/5.0239623

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In a future quantum network, two spatially separated individuals sharing a polarization entanglement source may need to assess the entanglement quality of the source without the presence of classical auxiliary signals. When the two are separated by a single-mode fiber, automated methods are necessary for the validation and estimation of polarization entanglement. We experimentally examine real-time iterative methods to search for maximal Bell violations, specifically the Clauser–Horne–Shimony–Holt (CHSH) inequality, between two observers sharing polarization entangled photons to evaluate the entanglement quality of the source. Our source allows us to tune the degree of entanglement of their shared photon pair by changing the temporal overlap of the two photons at an entangling 50/50 beam splitter so that we can compare the CHSH parameter these methods find to the theoretical values our source produces. The iterative methods used within our experiment are the Nelder–Mead optimization method, stochastic gradient descent, and Bayesian optimization. This is the first feedback experiment to study automated Bell violations in fiber and the first to compare all three of these iterative methods to one another in a quantum polarization control experiment. In our experiment, all methods are able to find Bell violations, but the Nelder–Mead method performed the best in terms of the speed and accuracy in finding the maximal violation.
Many experiments have been made to test the Clauser–Horne–Shimony–Holt (CHSH)–Bell inequality (CBI). However, the Cirel’son’s limit has not been reached yet, due to various practically existing imperfections. In this paper, we experimentally investigate how to select the measurement settings for obtaining the optimal violation of CBI for a mixed entangled state. With the commercialized entangled source, we first determine the entanglement degree of the generated photon pairs by using the tomographic technique, and then numerically optimize the measurement settings. Finally, we experimentally obtain the optimal violation of the CBI with the present commercialized entangled source without the maximal entanglement.
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.
The violation of Bell-like inequalities unambiguously verifies the entanglement of a multipartite quantum state. However, experimental impairments and nonidealities often compromise the entanglement quality, and this degradation translates into a nonmaximal violation of these inequalities. We present a detailed analysis of the influence of experimental impairments on the quality of nonlocal polarization entanglement for a two-qubit state as measured by the Clauser–Horne–Shimony–Holt parameter, with a particular emphasis on beamsplitters, polarization controllers, filters, and photon sources. This analysis allows the identification of the impairments that are critical and that need to be specifically monitored at the experimental level.
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.
Linearly polarized projections are the tacit means for performing Clauser-Horne-Shimony-Holt (CHSH) Bell-inequality tests using polarization-entangled photon pairs. The inequality is valid for all states on the Poincaré-Bloch sphere, but few laboratory studies have investigated violations with the full sphere. In this article, we explore the experimental verifications of the predicted violations of the CHSH inequality with Bell and non-Bell states with same and different linear and elliptically polarized basis states for each photon. We find that Bell states violate CHSH when using the same basis for both photons, regardless of their ellipticity, whereas they show no violations for photon projections in different bases. We found non-Bell maximally-entangled states for which the converse is true.