{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,1]],"date-time":"2026-09-01T17:19:57Z","timestamp":1788283197130,"version":"build-2803163510"},"reference-count":42,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2021,2,8]],"date-time":"2021-02-08T00:00:00Z","timestamp":1612742400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Probability is an important question in the ontological interpretation of quantum mechanics. It has been discussed in some trajectory interpretations such as Bohmian mechanics and stochastic mechanics. New questions arise when the probability domain extends to the complex space, including the generation of complex trajectory, the definition of the complex probability, and the relation of the complex probability to the quantum probability. The complex treatment proposed in this article applies the optimal quantum guidance law to derive the stochastic differential equation governing a particle\u2019s random motion in the complex plane. The probability distribution \u03c1c(t,x,y) of the particle\u2019s position over the complex plane z=x+iy is formed by an ensemble of the complex quantum random trajectories, which are solved from the complex stochastic differential equation. Meanwhile, the probability distribution \u03c1c(t,x,y) is verified by the solution of the complex Fokker\u2013Planck equation. It is shown that quantum probability |\u03a8|2 and classical probability can be integrated under the framework of complex probability \u03c1c(t,x,y), such that they can both be derived from \u03c1c(t,x,y) by different statistical ways of collecting spatial points.<\/jats:p>","DOI":"10.3390\/e23020210","type":"journal-article","created":{"date-parts":[[2021,2,9]],"date-time":"2021-02-09T23:43:16Z","timestamp":1612914196000},"page":"210","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Extending Quantum Probability from Real Axis to Complex Plane"],"prefix":"10.3390","volume":"23","author":[{"given":"Ciann-Dong","family":"Yang","sequence":"first","affiliation":[{"name":"Department of Aeronautics and Astronautics, National Cheng Kung University, Tainan 701, Taiwan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5357-4900","authenticated-orcid":false,"given":"Shiang-Yi","family":"Han","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, National University of Kaohsiung, Kaohsiung 811, Taiwan"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2021,2,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"166","DOI":"10.1103\/PhysRev.85.166","article-title":"A Suggested Interpretation of the Quantum Theory in Terms of Hidden Variables","volume":"85","author":"Bohm","year":"1952","journal-title":"Phys. Rev."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"208","DOI":"10.1103\/PhysRev.96.208","article-title":"Model of the Causal Interpretation of Quantum Theory in Terms of a Fluid with Irregular Fluctuations","volume":"96","author":"Bohm","year":"1954","journal-title":"Phys. Rev."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1079","DOI":"10.1103\/PhysRev.150.1079","article-title":"Derivation of the Schr\u00f6dinger Equation from Newtonian Mechanics","volume":"150","author":"Nelson","year":"1966","journal-title":"Phys. Rev."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"777","DOI":"10.1103\/PhysRev.47.777","article-title":"Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?","volume":"47","author":"Einstein","year":"1935","journal-title":"Phys. Rev."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1007\/BF02743566","article-title":"Quantum Interference and the Quantum Potential","volume":"52","author":"Philippidis","year":"1979","journal-title":"Nuovo Cim."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1016\/S0375-9601(01)00677-6","article-title":"Bohmian Trajectories for Photons","volume":"290","author":"Ghose","year":"2001","journal-title":"Phys. Lett. A"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"6109","DOI":"10.1088\/0953-8984\/14\/24\/312","article-title":"Particle Diffraction Studied Using Quantum Trajectories","volume":"14","author":"Sanz","year":"2002","journal-title":"J. Phys. Condens. Matter"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1351","DOI":"10.1103\/PhysRevLett.60.1351","article-title":"How the result of a measurement of a component of a spin-1\/2 particle can turn to be 100","volume":"60","author":"Aharonov","year":"1988","journal-title":"Phys. Rev. Lett."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"052111","DOI":"10.1103\/PhysRevA.72.052111","article-title":"A Quantum Averages of Weak Values","volume":"72","author":"Aharonov","year":"2005","journal-title":"Phys. Rev. A."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"043A01","DOI":"10.1093\/ptep\/ptv032","article-title":"Quantum Trajectories Based on the Weak Value","volume":"2015","author":"Mori","year":"2015","journal-title":"Prog. Theor. Exp. Phys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1170","DOI":"10.1126\/science.1202218","article-title":"Observing the Average Trajectories of Single Photons in a Two-Slit Interferometer","volume":"332","author":"Kocsis","year":"2011","journal-title":"Science"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"e1501466","DOI":"10.1126\/sciadv.1501466","article-title":"Experimental Nonlocal and Surreal Bohmian Trajectories","volume":"2","author":"Mahler","year":"2016","journal-title":"Sci. Adv."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1038\/nature12539","article-title":"Observing Single Quantum Trajectories of a Superconducting Quantum Bit","volume":"502","author":"Murch","year":"2013","journal-title":"Nature"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"163601","DOI":"10.1103\/PhysRevLett.123.163601","article-title":"Observing and Verifying the Quantum Trajectory of a Mechanical Resonator","volume":"123","author":"Rossi","year":"2019","journal-title":"Phys. Rev. Lett."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"329","DOI":"10.1023\/A:1021212410819","article-title":"Modified de Broglie-Bohm approach to quantum mechanics","volume":"15","author":"John","year":"2002","journal-title":"Found. Phys. Lett."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"231103","DOI":"10.1063\/1.2400851","article-title":"Bohmian Mechanics with Complex Action: A New Trajectory-Based Formulation of Quantum Mechanics","volume":"125","author":"Goldfarb","year":"2006","journal-title":"J. Chem. Phys."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1103\/PhysRevLett.50.3","article-title":"Hamitlon-Jacobi Theory and the Quantum Action Variable","volume":"50","author":"Leacock","year":"1983","journal-title":"Phys. Rev. Lett."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2876","DOI":"10.1016\/j.aop.2006.07.008","article-title":"Quantum Hamilton Mechanics: Hamilton Equations of Quantum Motion, Origin of Quantum Operators, and Proof of Quantization Axiom","volume":"321","author":"Yang","year":"2006","journal-title":"Ann. Phys."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1016\/j.cplett.2008.04.103","article-title":"Interplay of Causticity and Verticality within the Complex Quantum Hamilton-Jacobi Formalism","volume":"458","author":"Sanz","year":"2008","journal-title":"Chem. Phys. Lett."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"316","DOI":"10.1016\/j.chaos.2008.01.019","article-title":"On the Existence of Complex Spacetime in Relativistic Quantum Mechanics","volume":"38","author":"Yang","year":"2008","journal-title":"Chaos Solitons Fractals"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"851","DOI":"10.1007\/BF02058638","article-title":"Hypercomplex Quantum Mechanics","volume":"26","author":"Horwitz","year":"1996","journal-title":"Found. Phys."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1016\/S0034-4877(99)80024-X","article-title":"De Donder-Weyl Theory and a Hypercomplex Extension of Quantum Mechanics to Field Theory","volume":"43","author":"Kanatchikov","year":"1999","journal-title":"Rep. Math. Phys."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"15044","DOI":"10.1038\/ncomms15044","article-title":"Single-Photon Test of Hyper-Complex Quantum Theories Using a Metamaterial","volume":"8","author":"Procopio","year":"2017","journal-title":"Nat. Commun."},{"key":"ref_24","first-page":"417","article-title":"A Stochastic Variational Treatment of Quantum Mechanics","volume":"450","author":"Rosenbrock","year":"1995","journal-title":"Proc. R. Soc. Lond. Ser. A Math. Phys. Sci."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"3319","DOI":"10.1103\/PhysRevLett.79.3319","article-title":"Stochastic Interpretation of Quantum Mechanics in Complex Space","volume":"79","author":"Wang","year":"1997","journal-title":"Phys. Rev. Lett."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1016\/j.aop.2013.08.001","article-title":"Optimal Guidance Law in Quantum Mechanics","volume":"338","author":"Yang","year":"2013","journal-title":"Ann. Phys."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"19984","DOI":"10.1038\/s41598-019-56357-3","article-title":"Quantum Mechanics can be Understood through Stochastic Optimization on Spacetimes","volume":"9","author":"Lindgren","year":"2019","journal-title":"Sci. Rep."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"012101","DOI":"10.1088\/1742-6596\/1348\/1\/012101","article-title":"Probability Representation of the Quantum Mechanics and Star-product Quantization","volume":"1348","author":"Chernega","year":"2019","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"074031","DOI":"10.1088\/0031-8949\/90\/7\/074031","article-title":"Quantum Tomography Tewnty Years Later","volume":"90","author":"Asorey","year":"2015","journal-title":"Phys. Scr."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1007\/s10946-019-09778-4","article-title":"Probability Representation of Quantum States as Renaissance of Hidden Variables-God Plays coins","volume":"40","author":"Chernega","year":"2019","journal-title":"J. Russ. Laser Res."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"033041","DOI":"10.1088\/1367-2630\/13\/3\/033041","article-title":"Weak Measurement of Photon Polarization by Back-Action-Induced Path Interference","volume":"13","author":"Iinuma","year":"2011","journal-title":"New J. Phys."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"042121","DOI":"10.1103\/PhysRevA.95.042121","article-title":"Experimental Observation of Anomalous Trajectories of Single Photons","volume":"95","author":"Zhou","year":"2017","journal-title":"Phys. Rev. A"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1630","DOI":"10.1038\/s41467-019-08951-2","article-title":"Interfering Trajectories in Experimental Quantum-Enhanced Stochastic Simulation","volume":"10","author":"Ghafari","year":"2019","journal-title":"Nat. Commun."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"240501","DOI":"10.1103\/PhysRevLett.125.240501","article-title":"Experimental Parity-Time Symmetric Quantum Walks for Centrality Ranking on Directed Graphs","volume":"125","author":"Wu","year":"2020","journal-title":"Phys. Rev. Lett."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"899","DOI":"10.1007\/s10559-005-0029-4","article-title":"Probability Distribution of an Integral Quadratic Functional on the Trajectories of a Complex-Valued Ornstein-Uhlenbeck Process","volume":"40","author":"Virchenko","year":"2004","journal-title":"Cybern. Syst. Anal."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"044702","DOI":"10.1103\/PhysRevA.64.044702","article-title":"Complex Density Probability in non-Hermitian Quantum Mechanics: Interpretation and a Formula for Resonant Tunneling Probability Amplitude","volume":"64","author":"Barkay","year":"2001","journal-title":"Phys. Rev. A"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"044101","DOI":"10.1103\/PhysRevA.78.044101","article-title":"Considerations on the Probability Density in Complex Space","volume":"78","author":"Chou","year":"2008","journal-title":"Phys. Rev. A"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"220","DOI":"10.1016\/j.aop.2008.09.007","article-title":"Probability and Complex Quantum Trajectories","volume":"324","author":"John","year":"2009","journal-title":"Ann. Phys."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"2332","DOI":"10.1016\/j.aop.2010.02.011","article-title":"Probability Density in the Complex Plane","volume":"325","author":"Bender","year":"2010","journal-title":"Ann. Phys."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"428","DOI":"10.1002\/qua.25059","article-title":"Synthesizing Quantum Probability by a Single Chaotic Complex-Valued Trajectory","volume":"116","author":"Yang","year":"2016","journal-title":"Int. J. Quantum Chem."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"478","DOI":"10.1080\/21642583.2015.1108885","article-title":"The Paradigm of Complex Probability and the Brownian Motion","volume":"3","author":"Jaoude","year":"2015","journal-title":"Syst. Sci. Control Eng."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"960","DOI":"10.1007\/s10701-020-00363-3","article-title":"Trajectory Interpretation of Correspondence Principle: Solution of Nodal Issue","volume":"50","author":"Yang","year":"2020","journal-title":"Found. Phys."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/2\/210\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:21:32Z","timestamp":1760160092000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/2\/210"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,2,8]]},"references-count":42,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2021,2]]}},"alternative-id":["e23020210"],"URL":"https:\/\/doi.org\/10.3390\/e23020210","relation":{"has-preprint":[{"id-type":"doi","id":"10.20944\/preprints202012.0733.v1","asserted-by":"object"}]},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,2,8]]}}}