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Böttcher L, Porter MA. Complex networks with complex weights. Phys Rev E 2024; 109:024314. [PMID: 38491610 DOI: 10.1103/physreve.109.024314] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/15/2022] [Accepted: 12/20/2023] [Indexed: 03/18/2024]
Abstract
In many studies, it is common to use binary (i.e., unweighted) edges to examine networks of entities that are either adjacent or not adjacent. Researchers have generalized such binary networks to incorporate edge weights, which allow one to encode node-node interactions with heterogeneous intensities or frequencies (e.g., in transportation networks, supply chains, and social networks). Most such studies have considered real-valued weights, despite the fact that networks with complex weights arise in fields as diverse as quantum information, quantum chemistry, electrodynamics, rheology, and machine learning. Many of the standard network-science approaches in the study of classical systems rely on the real-valued nature of edge weights, so it is necessary to generalize them if one seeks to use them to analyze networks with complex edge weights. In this paper, we examine how standard network-analysis methods fail to capture structural features of networks with complex edge weights. We then generalize several network measures to the complex domain and show that random-walk centralities provide a useful approach to examine node importances in networks with complex weights.
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Affiliation(s)
- Lucas Böttcher
- Department of Computational Science and Philosophy, Frankfurt School of Finance and Management, 60322 Frankfurt am Main, Germany
- Department of Medicine, University of Florida, Gainesville, Florida, 32610, USA
| | - Mason A Porter
- Department of Mathematics, University of California, Los Angeles, California 90095, USA
- Department of Sociology, University of California, Los Angeles, California 90095, USA
- Santa Fe Institute, Santa Fe, New Mexico 87501, USA
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Makris N. Revisiting Schrödinger's fourth-order, real-valued wave equation and the implication from the resulting energy levels. ROYAL SOCIETY OPEN SCIENCE 2023; 10:230793. [PMID: 38126064 PMCID: PMC10731319 DOI: 10.1098/rsos.230793] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 06/12/2023] [Accepted: 11/22/2023] [Indexed: 12/23/2023]
Abstract
In his seminal part IV, Annalen der Physik vol. 81, 1926 paper, Schrödinger has developed a clear understanding about the wave equation that produces the correct quadratic dispersion relation for matter-waves and he first presents a real-valued wave equation that is fourth-order in space and second-order in time. In the view of the mathematical difficulties associated with the eigenvalue analysis of a fourth-order, differential equation in association with the structure of the Hamilton-Jacobi equation, Schrödinger splits the fourth-order real operator into the product of two, second-order, conjugate complex operators and retains only one of the two complex operators to construct his iconic second-order, complex-valued wave equation. In this paper, we show that Schrödinger's original fourth-order, real-valued wave equation is a stiffer equation that produces higher energy levels than his second-order, complex-valued wave equation that predicts with remarkable accuracy the energy levels observed in the atomic line spectra of the chemical elements. Accordingly, the fourth-order, real-valued wave equation is too stiff to predict the emitted energy levels from the electrons of the chemical elements; therefore, the paper concludes that quantum mechanics can only be described with the less stiff, second-order, complex-valued wave equation.
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Affiliation(s)
- Nicos Makris
- Department of Civil and Environmental Engineering, Southern Methodist University, Dallas, TX 75276, USA
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3
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Wootters WK. Optimal Information Transfer and the Uniform Measure over Probability Space. ENTROPY (BASEL, SWITZERLAND) 2023; 25:875. [PMID: 37372219 DOI: 10.3390/e25060875] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/01/2023] [Revised: 05/29/2023] [Accepted: 05/29/2023] [Indexed: 06/29/2023]
Abstract
For a quantum system with a d-dimensional Hilbert space, suppose a pure state |ψ⟩ is subjected to a complete orthogonal measurement. The measurement effectively maps |ψ⟩ to a point (p1,…,pd) in the appropriate probability simplex. It is a known fact-which depends crucially on the complex nature of the system's Hilbert space-that if |ψ⟩ is distributed uniformly over the unit sphere, then the resulting ordered set (p1,…,pd) is distributed uniformly over the probability simplex; that is, the resulting measure on the simplex is proportional to dp1⋯dpd-1. In this paper we ask whether there is some foundational significance to this uniform measure. In particular, we ask whether it is the optimal measure for the transmission of information from a preparation to a measurement in some suitably defined scenario. We identify a scenario in which this is indeed the case, but our results suggest that an underlying real-Hilbert-space structure would be needed to realize the optimization in a natural way.
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Wang NN, Pozas-Kerstjens A, Zhang C, Liu BH, Huang YF, Li CF, Guo GC, Gisin N, Tavakoli A. Certification of non-classicality in all links of a photonic star network without assuming quantum mechanics. Nat Commun 2023; 14:2153. [PMID: 37059704 PMCID: PMC10104853 DOI: 10.1038/s41467-023-37842-w] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/27/2022] [Accepted: 04/03/2023] [Indexed: 04/16/2023] Open
Abstract
Networks composed of independent sources of entangled particles that connect distant users are a rapidly developing quantum technology and an increasingly promising test-bed for fundamental physics. Here we address the certification of their post-classical properties through demonstrations of full network nonlocality. Full network nonlocality goes beyond standard nonlocality in networks by falsifying any model in which at least one source is classical, even if all the other sources are limited only by the no-signaling principle. We report on the observation of full network nonlocality in a star-shaped network featuring three independent sources of photonic qubits and joint three-qubit entanglement-swapping measurements. Our results demonstrate that experimental observation of full network nonlocality beyond the bilocal scenario is possible with current technology.
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Affiliation(s)
- Ning-Ning Wang
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, 230026, Hefei, China
- CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, 230026, Hefei, China
- Hefei National Laboratory, University of Science and Technology of China, 230088, Hefei, China
| | | | - Chao Zhang
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, 230026, Hefei, China.
- CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, 230026, Hefei, China.
- Hefei National Laboratory, University of Science and Technology of China, 230088, Hefei, China.
| | - Bi-Heng Liu
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, 230026, Hefei, China
- CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, 230026, Hefei, China
- Hefei National Laboratory, University of Science and Technology of China, 230088, Hefei, China
| | - Yun-Feng Huang
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, 230026, Hefei, China.
- CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, 230026, Hefei, China.
- Hefei National Laboratory, University of Science and Technology of China, 230088, Hefei, China.
| | - Chuan-Feng Li
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, 230026, Hefei, China.
- CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, 230026, Hefei, China.
- Hefei National Laboratory, University of Science and Technology of China, 230088, Hefei, China.
| | - Guang-Can Guo
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, 230026, Hefei, China
- CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, 230026, Hefei, China
- Hefei National Laboratory, University of Science and Technology of China, 230088, Hefei, China
| | - Nicolas Gisin
- Group of Applied Physics, University of Geneva, 1211, Geneva 4, Switzerland
- Constructor University, Geneva, Switzerland
| | - Armin Tavakoli
- Physics Department, Lund University, Box 118, 22100, Lund, Sweden.
- Institute for Quantum Optics and Quantum Information-IQOQI Vienna Austrian Academy of Sciences, Boltzmanngasse 3, 1090, Vienna, Austria.
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Wu D, Jiang YF, Gu XM, Huang L, Bai B, Sun QC, Zhang X, Gong SQ, Mao Y, Zhong HS, Chen MC, Zhang J, Zhang Q, Lu CY, Pan JW. Experimental Refutation of Real-Valued Quantum Mechanics under Strict Locality Conditions. PHYSICAL REVIEW LETTERS 2022; 129:140401. [PMID: 36240393 DOI: 10.1103/physrevlett.129.140401] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/11/2022] [Revised: 06/29/2022] [Accepted: 08/08/2022] [Indexed: 06/16/2023]
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.
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Affiliation(s)
- Dian Wu
- Hefei National Research Center for Physical Sciences at the Microscale and School of Physical Sciences, University of Science and Technology of China, Hefei 230026, China
- Shanghai 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
- Hefei National Laboratory, University of Science and Technology of China, Hefei 230088, China
| | - Yang-Fan Jiang
- Jinan Institute of Quantum Technology, Jinan 250101, China
| | - Xue-Mei Gu
- Hefei National Research Center for Physical Sciences at the Microscale and School of Physical Sciences, University of Science and Technology of China, Hefei 230026, China
- Shanghai 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
- Hefei National Laboratory, University of Science and Technology of China, Hefei 230088, China
| | - Liang Huang
- Hefei National Research Center for Physical Sciences at the Microscale and School of Physical Sciences, University of Science and Technology of China, Hefei 230026, China
- Shanghai 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
- Hefei National Laboratory, University of Science and Technology of China, Hefei 230088, China
| | - Bing Bai
- Hefei National Research Center for Physical Sciences at the Microscale and School of Physical Sciences, University of Science and Technology of China, Hefei 230026, China
- Shanghai 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
- Hefei National Laboratory, University of Science and Technology of China, Hefei 230088, China
| | - Qi-Chao Sun
- Hefei National Research Center for Physical Sciences at the Microscale and School of Physical Sciences, University of Science and Technology of China, Hefei 230026, China
- Shanghai 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
- Hefei National Laboratory, University of Science and Technology of China, Hefei 230088, China
| | - Xingjian Zhang
- Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing 100084, China
| | - Si-Qiu Gong
- Hefei National Research Center for Physical Sciences at the Microscale and School of Physical Sciences, University of Science and Technology of China, Hefei 230026, China
- Shanghai 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
- Hefei National Laboratory, University of Science and Technology of China, Hefei 230088, China
| | - Yingqiu Mao
- Hefei National Research Center for Physical Sciences at the Microscale and School of Physical Sciences, University of Science and Technology of China, Hefei 230026, China
- Shanghai 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
- Hefei National Laboratory, University of Science and Technology of China, Hefei 230088, China
| | - Han-Sen Zhong
- Hefei National Research Center for Physical Sciences at the Microscale and School of Physical Sciences, University of Science and Technology of China, Hefei 230026, China
- Shanghai 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
- Hefei National Laboratory, University of Science and Technology of China, Hefei 230088, China
| | - Ming-Cheng Chen
- Hefei National Research Center for Physical Sciences at the Microscale and School of Physical Sciences, University of Science and Technology of China, Hefei 230026, China
- Shanghai 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
- Hefei National Laboratory, University of Science and Technology of China, Hefei 230088, China
| | - Jun Zhang
- Hefei National Research Center for Physical Sciences at the Microscale and School of Physical Sciences, University of Science and Technology of China, Hefei 230026, China
- Shanghai 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
- Hefei National Laboratory, University of Science and Technology of China, Hefei 230088, China
| | - Qiang Zhang
- Hefei National Research Center for Physical Sciences at the Microscale and School of Physical Sciences, University of Science and Technology of China, Hefei 230026, China
- Shanghai 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
- Hefei National Laboratory, University of Science and Technology of China, Hefei 230088, China
| | - Chao-Yang Lu
- Hefei National Research Center for Physical Sciences at the Microscale and School of Physical Sciences, University of Science and Technology of China, Hefei 230026, China
- Shanghai 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
- Hefei National Laboratory, University of Science and Technology of China, Hefei 230088, China
| | - Jian-Wei Pan
- Hefei National Research Center for Physical Sciences at the Microscale and School of Physical Sciences, University of Science and Technology of China, Hefei 230026, China
- Shanghai 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
- Hefei National Laboratory, University of Science and Technology of China, Hefei 230088, China
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Tavakoli A, Pozas-Kerstjens A, Luo MX, Renou MO. Bell nonlocality in networks. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2022; 85:056001. [PMID: 34883470 DOI: 10.1088/1361-6633/ac41bb] [Citation(s) in RCA: 17] [Impact Index Per Article: 8.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/15/2021] [Accepted: 12/09/2021] [Indexed: 06/13/2023]
Abstract
Bell's theorem proves that quantum theory is inconsistent with local physical models. It has propelled research in the foundations of quantum theory and quantum information science. As a fundamental feature of quantum theory, it impacts predictions far beyond the traditional scenario of the Einstein-Podolsky-Rosen paradox. In the last decade, the investigation of nonlocality has moved beyond Bell's theorem to consider more sophisticated experiments that involve several independent sources which distribute shares of physical systems among many parties in a network. Network scenarios, and the nonlocal correlations that they give rise to, lead to phenomena that have no counterpart in traditional Bell experiments, thus presenting a formidable conceptual and practical challenge. This review discusses the main concepts, methods, results and future challenges in the emerging topic of Bell nonlocality in networks.
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Affiliation(s)
- Armin Tavakoli
- Institute for Quantum Optics and Quantum Information-IQOQI Vienna, Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria
- Institute for Atomic and Subatomic Physics, Vienna University of Technology, 1020 Vienna, Austria
| | - Alejandro Pozas-Kerstjens
- Departamento de Análisis Matemático, Universidad Complutense de Madrid, 28040 Madrid, Spain
- Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), Madrid, Spain
| | - Ming-Xing Luo
- Information Coding & Transmission Key Laboratory of Sichuan Province, School of Information Science & Technology, Southwest Jiaotong University, Chengdu 610031, People's Republic of China
- Shenzhen Institute for Quantum Science and Engineering, Southern University of Science and Technology, Shenzhen 518055, China
| | - Marc-Olivier Renou
- ICFO-Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain
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Linear Superposition as a Core Theorem of Quantum Empiricism. UNIVERSE 2022. [DOI: 10.3390/universe8040217] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
Abstract
Clarifying the nature of the quantum state |Ψ⟩ is at the root of the problems with insight into counter-intuitive quantum postulates. We provide a direct—and math-axiom free—empirical derivation of this object as an element of a vector space. Establishing the linearity of this structure—quantum superposition—is based on a set-theoretic creation of ensemble formations and invokes the following three principia: (I) quantum statics, (II) doctrine of the number in the physical theory, and (III) mathematization of matching the two observations with each other (quantum covariance). All of the constructs rest upon a formalization of the minimal experimental entity—the registered micro-event, detector click. This is sufficient for producing the C-numbers, axioms of linear vector space (superposition principle), statistical mixtures of states, eigenstates and their spectra, and non-commutativity of observables. No use is required of the spatio-temporal concepts. As a result, the foundations of theory are liberated to a significant extent from the issues associated with physical interpretations, philosophical exegeses, and mathematical reconstruction of the entire quantum edifice.
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Avella A. Quantum Mechanics Must Be Complex. PHYSICS 2022. [DOI: 10.1103/physics.15.7] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 02/04/2023]
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