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arXiv:2312.14547 · Physical Review A

Proposals for ruling out real quantum theories in an entanglement-swapping quantum network with causally independent sources

Jian Hua Yao, Hu Chen, Ya-Li Mao, Zheng-Da Li, Jingyun Fan

Published Jan 16, 2024• 10 citations
arXivHTMLPDF

Abstract

The question of whether complex numbers play a fundamental role in quantum theory has been debated since the inception of quantum mechanics. Recently, a feasible proposal to differentiate between real and complex quantum theories based on the technique of testing Bell nonlocalities has emerged [Renou et al., Nature (London) 600, 625 (2021)]. Based on this method, the real quantum theory has been falsified experimentally in both photonic and superconducting quantum systems [Li et al., Phys. Rev. Lett. 128, 040402 (2022); Chen, et al., Phys. Rev. Lett. 128, 040403 (2022)]. The quantum networks with multiple independent sources which are not causally connected have gained significant interest as they offer a new perspective on studying the nonlocalities. The independence of these sources imposes additional constraints on observable covariances and leads to new bounds for classical and quantum correlations. In this study, we examine the discrimination between the real and complex quantum theories with an entanglement swapping scenario under a stronger assumption that the two sources are causally independent, which was not made in previous works. Using a revised Navasc\'ues-Pironio-Ac\'{\i}n method and Bayesian optimization, we find a proposal with optimal coefficients of the correlation function which could give a larger discrimination between the real and quantum theories compared with the existing proposals. This work opens up avenues for further exploration of the discrimination between real and complex quantum theories within intricate quantum networks featuring causally independent parties.

DOI
10.1103/physreva.109.012211

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The question of whether complex numbers play a fundamental role in quantum theory has been debated since the inception of quantum mechanics. Recently, a feasible proposal to differentiate between real and complex quantum theories based on the technique of testing Bell nonlocalities has emerged [Renou et al., Nature (London) 600, 625 (2021)]. Based on this method, the real quantum theory has been falsified experimentally in both photonic and superconducting quantum systems [Li et al., Phys. Rev. Lett. 128, 040402 (2022); Chen, et al., Phys. Rev. Lett. 128, 040403 (2022)]. The quantum networks with multiple independent sources which are not causally connected have gained significant interest as they offer a new perspective on studying the nonlocalities. The independence of these sources imposes additional constraints on observable covariances and leads to new bounds for classical and quantum correlations. In this study, we examine the discrimination between the real and complex quantum theories with an entanglement swapping scenario under a stronger assumption that the two sources are causally independent, which was not made in previous works. Using a revised Navasc'ues-Pironio-Ac'{\i}n method and Bayesian optimization, we find a proposal with optimal coefficients of the correlation function which could give a larger discrimination between the real and quantum theories compared with the existing proposals. This work opens up avenues for further exploration of the discrimination between real and complex quantum theories within intricate quantum networks featuring causally independent parties.
The role of complex quantities in quantum theory has been puzzling physicists since the beginnings. It is, thus, natural to ask whether, in order to describe our experiments, the mathematical structure of the complex Hilbert spaces it is built on is really necessary. Recently, it was shown that this structure is inevitable in network scenarios with independent sources. More precisely, Real Quantum Theory cannot explain the predictions of (Complex) Quantum Theory [Renou et al., Nature (London) 600, 625 (2021).NATUAS0028-083610.1038/s41586-021-04160-4]. Here, we revisit the independence assumption underlying this work. We show that assuming partial independence is sufficient for showing the inadequacy of Real Quantum Theory. We derive a tradeoff between source independence and the Bell value achievable in Real Quantum Theory, which also lower bounds the source correlations required to explain previous experiments by means of real quantum systems. We further show that 1 bit of entanglement is necessary and sufficient for recovering the complex quantum correlations by means of Real Quantum Theory in the scenario from [Renou et al., Nature (London) 600, 625 (2021).NATUAS0028-083610.1038/s41586-021-04160-4]. Finally, building on [McKague et al., Phys. Rev. Lett. 102, 2009PRLTAO0031-900710.1103/PhysRevLett.102.020505], we provide a construction to simulate any complex quantum setup with m independent sources by means of Real Quantum Theory by allowing the sources to share an m real-qubit entangled state in the first round of the experiment.
Quantum theory is commonly formulated in complex Hilbert spaces. However, the question of whether complex numbers need to be given a fundamental role in the theory has been debated since its pioneering days. Recently it has been shown that tests in the spirit of a Bell inequality can reveal quantum predictions in entanglement swapping scenarios that cannot be modeled by the natural real-number analog of standard quantum theory. Here, we tailor such tests for implementation in state-of-the-art photonic systems. We experimentally demonstrate quantum correlations in a network of three parties and two independent EPR sources that violate the constraints of real quantum theory by over 4.5 standard deviations, hence disproving real quantum theory as a universal physical theory.
We present a method that allows the study of classical and quantum correlations in networks with causally independent parties, such as the scenario underlying entanglement swapping. By imposing relaxations of factorization constraints in a form compatible with semidefinite programing, it enables the use of the Navascués-Pironio-Acín hierarchy in complex quantum networks. We first show how the technique successfully identifies correlations not attainable in the entanglement-swapping scenario. Then we use it to show how the nonlocal power of measurements can be activated in a network: there exist measuring devices that, despite being unable to generate nonlocal correlations in the standard Bell scenario, provide a classical-quantum separation in an entanglement swapping configuration.
We find the minimal number of settings to test quantum theory based on real numbers, assuming separability of the sources, modifying the recent proposal [M.-O. Renou et al., Nature (London) 600, 625 (2021)]. The test needs only three settings for observers AAA and CCC, but the ratio of complex to real maximum is smaller than in the existing proposal. We also found that two settings and two outcomes for both observers are insufficient.
Whether the complex numbers of standard quantum theory are experimentally indispensable has remained open for decades. Real quantum theory (RQT), obtained by replacing complex amplitudes with real ones while retaining the usual Kronecker-product composition rule, reproduces all single-party and bipartite Bell correlations of quantum theory (QT), but its lack of local tomography suggested that the two theories might diverge in more general local experiments. This possibility appeared to be confirmed by Renou et al., who argued that a bilocal network experiment can falsify RQT without falsifying QT. Here we show that this conclusion relies on an experimentally untestable assumption. The key distinction is between product-state independence, which constrains the mathematical form of source states, and operational independence, which is defined entirely by the absence of observable cross-source correlations. We prove that, once source independence is imposed operationally, every finite network correlation achievable in QT is also achievable in RQT with the same locality structure of the measurements. We then extend this equivalence to arbitrary finite sequential multipartite protocols involving channels and measurements with prescribed locality structure. Thus, as long as no violation of QT is observed, RQT cannot be experimentally falsified. Our results restore the empirical indistinguishability of QT and RQT, while showing that they support markedly different pictures of the correlation structure underlying the same observed world.
Quantum mechanics is commonly formulated in a complex, rather than real, Hilbert space. However, whether quantum theory really needs the participation of complex numbers has been debated ever since its birth. Recently, a Bell-like test in an entanglement-swapping scenario has been proposed to distinguish standard quantum mechanics from its real-valued analog. Previous experiments have conceptually demonstrated, yet not satisfied, the central requirement of independent state preparation and measurements and leave several loopholes. Here, we implement such a Bell-like test with two separated independent sources delivering entangled photons to three separated parties under strict locality conditions that are enforced by spacelike separation of the relevant events, rapid random setting generation, and fast measurement. With the fair-sampling assumption and closed loopholes of independent source, locality, and measurement independence simultaneously, we violate the constraints of real-valued quantum mechanics by 5.30 standard deviations. Our results disprove the real-valued quantum theory to describe nature and ensure the indispensable role of complex numbers in quantum mechanics.