Experimental Refutation of Real-Valued Quantum Mechanics under Strict Locality Conditions

Dian Wu, Yang-Fan Jiang, Xue-Mei Gu, Liang Huang, Bing Bai, Qi-Chao Sun, Xingjian Zhang, Si-Qiu Gong, Yingqiu Mao, Han-Sen Zhong, Ming-Cheng Chen, Jun Zhang, Qiang Zhang, Chao-Yang Lu, and Jian-Wei Pan

Phys. Rev. Lett. 129, 140401 – Published 26 September 2022

Abstract

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.

Received 11 January 2022
Revised 29 June 2022
Accepted 8 August 2022

DOI:https://doi.org/10.1103/PhysRevLett.129.140401

Physics Subject Headings (PhySH)

Nonlocality Optical tests of quantum theory Quantum correlations, foundations & formalism Quantum foundations

General PhysicsAtomic, Molecular & Optical

Authors & Affiliations

Dian Wu ^1,2,3,*, Yang-Fan Jiang ^4,*, Xue-Mei Gu ^1,2,3,*, Liang Huang^1,2,3, Bing Bai^1,2,3, Qi-Chao Sun ^1,2,3, Xingjian Zhang⁵, Si-Qiu Gong ^1,2,3, Yingqiu Mao^1,2,3, Han-Sen Zhong^1,2,3, Ming-Cheng Chen^1,2,3, Jun Zhang^1,2,3, Qiang Zhang^1,2,3, Chao-Yang Lu ^1,2,3, and Jian-Wei Pan^1,2,3

¹Hefei National Research Center for Physical Sciences at the Microscale and School of Physical Sciences, University of Science and Technology of China, Hefei 230026, China
²Shanghai Research Center for Quantum Science and CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China
³Hefei National Laboratory, University of Science and Technology of China, Hefei 230088, China
⁴Jinan Institute of Quantum Technology, Jinan 250101, China
⁵Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing 100084, China

^*These authors contributed equally to this work.

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Vol. 129, Iss. 14 — 30 September 2022

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Figure 1
The scheme to disprove RQM. Two independent sources $S_{1}$ and $S_{2}$ distribute EPR pairs between Alice and Bob, and Bob and Claire, respectively. Alice and Claire perform measurements ( $A_{x}$ and $C_{z}$ ) on their received photons with random inputs (labeled as $x$ and $z$ ) and give measurement outcomes (labeled as $a$ and $c$ ). Bob performs a full Bell-state measurement (BSM) on his received two photons, which outputs one of the four measurement results $b$ . The spatial distances between Alice- $S_{1}$ , $S_{1}$ -Bob, Bob- $S_{2}$ , and $S_{2}$ -Claire are 104, 106, 89, and 110 m, with fiber links of length 112.63, 123.84, 108.75, and 125.48 m, respectively.
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Figure 2
Experimental setup. (a) In each source, a 1558 nm distributed feedback (DFB) laser is frequency doubled in a periodically poled MgO-doped lithium niobate (PPMgLN) crystal after passing through an erbium-doped fiber amplifier (EDFA), for producing 779 nm pump pulses. The PPMgLN crystal in a polarization-based Sagnac loop is then pumped to create polarization-entangled photon pairs. The pulse pattern generator (PPG) in $S_{2}$ offers 250 MHz synchronization signals (CLK, black dashed lines) to all parties and 12.5 GHz signals to trigger the PPG at $S_{1}$ . (b) An electro-optic polarization modulation (EOPM) consisting of a polarization beam splitter (PBS), two Faraday rotators (FR), and a fiber-coupled electro-optic phase modulator (PM) is implemented for modulating the photon’s polarization [29]. (c) Alice performs measurements $A_{x}$ , which are determined by her quantum random number generator ( ${QRNG}_{A}$ ) and a fast polarization modulator (constructed by two EOPMs and fixed wave plates), and records measurement outcomes with a time-to-digital converter (TDC). (d) Similar to (c), Claire performs measurements $C_{z}$ and records the outcomes with a TDC. (e) A partial BSM analyzer is constructed by three PBSs, two HWPs, four $50 ∶ 50$ beam splitters (BSs), and eight superconducting nanowire single photon detectors (SNSPDs). In front of the first PBS, a polarization modulator is situated there. Bob performs the full BSM of four possible outcomes which are randomly decided by outcomes of ${QRNG}_{B}$ and random photon clicks at SNSPDs. The choice of ${QRNG}_{B}$ is recorded by a TDC, and the photon detection results are analyzed in real time and recorded by a field-programmable gate array (FPGA). PC, polarization controller; DWDM, dense wavelength-division multiplexer; DM, dichroic mirrors; OPM, off-axis parabolic mirrors; FBG, fiber Bragg grating; QWP, quarter-wave plate; $λ / 8$ , eighth wave plate.
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Figure 3
Space-time configuration of relevant events in our experiment. Origins of the axes are displaced to reflect the relative space and time difference between them. (a) Relative spatial configuration of the two independent sources ( $S_{1}$ and $S_{2}$ ) and three players (Alice, Bob, and Claire). (b) Spacelike separation between state emission events from sources $S_{1}$ and $S_{2}$ . (c) Spacelike separation between setting choice events ${QRNG}_{A}$ and ${QRNG}_{C}$ and between setting choice event ${QRNG}_{A}$ ( ${QRNG}_{C}$ ) and measurement event $M_{C}$ ( $M_{A}$ ). (d) Spacelike separation between setting choice event ${QRNG}_{B}$ and state emission events from sources $S_{1}$ and $S_{2}$ . (e) Spacelike separation between setting choice events ${QRNG}_{A}$ and ${QRNG}_{B}$ and between setting choice event ${QRNG}_{A}$ ( ${QRNG}_{B}$ ) and measurement event $M_{B}$ ( $M_{A}$ ) shown on the left side of the vertical axis, with the state emission event $S_{1}$ on the right side. (f) Similar to (e), ${QRNG}_{C}$ is the setting choice event of Claire and the state emission event is $S_{2}$ . Blue vertical bars indicate the time elapsing for events, with the start and end marked by circles and a horizontal line, respectively. All time-space relations are drawn to scale. Hence, further relations can be inferred, e.g., the spacelike separation between ${QRNG}_{A} − S_{2}$ and ${QRNG}_{C} − S_{1}$ can be implied by (e) and (f) (for details, see Ref. [29]).
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Figure 4
Experimental results. The upper bounds of $F$ for classical mechanics, RQM, and CQM are 6, 7.66, and 8.49, respectively. The results for Bob’s four measurement outcomes are 7.80(6), 7.89(6), 7.78(6), and 7.84(6). The average result is 7.83(3), which violates the RQM bound by 5.30 standard deviations. Error bars indicate one standard deviation based on Poisson statistics.
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