Testing Real Quantum Theory in an Optical Quantum Network
Zheng-Da Li, Ya-Li Mao, Mirjam Weilenmann, Armin Tavakoli, Hu Chen, Lixin Feng, Sheng-Jun Yang, Marc-Olivier Renou, David Trillo, Lê Phuc Thinh, Nicolas Gisin, Antonio Acín, Miguel Navascués, Zizhu Wang, Jingyun Fan
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.
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.
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.
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.
Complex numbers, i.e., numbers with a and an imaginary part, are essential for mathematical analysis, while their role in other subjects, such as electromagnetism or special relativity, is far less fundamental. Quantum is the only physical theory where these numbers seem to play an indispensible role, as the theory is explicitly formulated in terms of operators acting on complex Hilbert spaces. The occurrence of complex numbers within the quantum formalism has nonetheless puzzled countless physicists, including the fathers of the theory, for whom a version of quantum physics, where states and observables are represented by operators, seemed much more natural. In fact, previous works showed that such real quantum physics can reproduce the outcomes of any multipartite experiment, as long as the parts share arbitrary quantum states. Thus, are complex numbers really needed for a quantum description of nature? Here, we show this to be case by proving that and complex quantum make different predictions in network scenarios comprising independent quantum state sources. This allows us to devise a Bell-type quantum experiment whose input-output correlations cannot be approximated by any quantum model. The successful realization of such an experiment would disprove quantum physics, in the same way as standard Bell experiments disproved local physics.
Abstract Although complex numbers are essential in mathematics, they are not needed to describe physical experiments, as those are expressed in terms of probabilities, hence real numbers. Physics, however, aims to explain, rather than describe, experiments through theories. Although most theories of physics are based on real numbers, quantum theory was the first to be formulated in terms of operators acting on complex Hilbert spaces 1,2 . This has puzzled countless physicists, including the fathers of the theory, for whom a real version of quantum theory, in terms of real operators, seemed much more natural 3 . In fact, previous studies have shown that such a ‘real quantum theory’ can reproduce the outcomes of any multipartite experiment, as long as the parts share arbitrary real quantum states 4 . Here we investigate whether complex numbers are actually needed in the quantum formalism. We show this to be case by proving that real and complex Hilbert-space formulations of quantum theory make different predictions in network scenarios comprising independent states and measurements. This allows us to devise a Bell-like experiment, the successful realization of which would disprove real quantum theory, in the same way as standard Bell experiments disproved local physics.
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.
Standard quantum theory was formulated with complex-valued Schrödinger equations, wave functions, operators, and Hilbert spaces. Previous work attempted to simulate quantum systems using only real numbers by exploiting an enlarged Hilbert space. A fundamental question arises: are the complex numbers really necessary in the standard formalism of quantum theory? To answer this question, a quantum game has been developed to distinguish standard quantum theory from its real-number analog, by revealing a contradiction between a high-fidelity multiqubit quantum experiment and players using only real-number quantum theory. Here, using superconducting qubits, we faithfully realize the quantum game based on deterministic entanglement swapping with a state-of-the-art fidelity of 0.952. Our experimental results violate the real-number bound of 7.66 by 43 standard deviations. Our results disprove the real-number formulation and establish the indispensable role of complex numbers in the standard quantum theory.