{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,1]],"date-time":"2026-03-01T02:36:52Z","timestamp":1772332612679,"version":"3.50.1"},"reference-count":45,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2018,6,18]],"date-time":"2018-06-18T00:00:00Z","timestamp":1529280000000},"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>We revisit the concept of entanglement within the Bohmian approach to quantum mechanics. Inspired by Bohmian dynamics, we introduce two partial measures for the amount of entanglement corresponding to a pure state of a pair of quantum particles. One of these measures is associated with the statistical correlations exhibited by the joint probability density of the two Bohmian particles in configuration space. The other partial measure corresponds to the correlations associated with the phase of the joint wave function, and describes the non-separability of the Bohmian velocity field. The sum of these two components is equal to the total entanglement of the joint quantum state, as measured by the linear entropy of the single-particle reduced density matrix.<\/jats:p>","DOI":"10.3390\/e20060473","type":"journal-article","created":{"date-parts":[[2018,6,18]],"date-time":"2018-06-18T10:57:11Z","timestamp":1529319431000},"page":"473","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":16,"title":["Revisiting Entanglement within the Bohmian Approach to Quantum Mechanics"],"prefix":"10.3390","volume":"20","author":[{"given":"Claudia","family":"Zander","sequence":"first","affiliation":[{"name":"Physics Department, University of Pretoria, Pretoria 0002, South Africa"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Angel Ricardo","family":"Plastino","sequence":"additional","affiliation":[{"name":"CeBio y Secretaria de Investigaciones, Universidad Nacional del Noroeste de la Prov. de Buenos Aires-UNNOBA y CONICET, Roque Saenz Pe\u00f1a 456,6000 Jun\u00edn, Argentina"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,6,18]]},"reference":[{"key":"ref_1","unstructured":"Holland, P.R. (1995). The Quantum Theory of Motion: An Account of the de Broglie-Bohm Causal Interpretation of Quantum Mechanics, Cambridge University Press."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Sanz, A.S., and Miret-Art\u00e9s, S. (2012). A Trajectory Description of Quantum Processes. I. Fundamentals, Springer.","DOI":"10.1007\/978-3-642-18092-7"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Sanz, A.S., and Miret-Art\u00e9s, S. (2014). A Trajectory Description of Quantum Processes. II. Applications, Springer.","DOI":"10.1007\/978-3-642-17974-7"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Nassar, A.B., and Miret-Art\u00e9s, S. (2017). Bohmian Mechanics, Open Quantum Systems and Continuous Measurements, Springer International Publishing.","DOI":"10.1007\/978-3-319-53653-8"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"317","DOI":"10.1016\/j.aop.2013.01.009","article-title":"Quantum trajectories and the Bohm time constant","volume":"331","author":"Nassar","year":"2013","journal-title":"Ann. Phys."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"150401","DOI":"10.1103\/PhysRevLett.111.150401","article-title":"Dividing line between quantum and classical trajectories in a measurement problem: Bohmian time constant","volume":"111","author":"Nassar","year":"2013","journal-title":"Phys. Rev. Lett."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2522","DOI":"10.1016\/j.aop.2012.07.005","article-title":"Position measurements in the de Broglie-Bohm interpretation of quantum mechanics","volume":"327","author":"Erez","year":"2012","journal-title":"Ann. Phys."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"036203","DOI":"10.1103\/PhysRevE.79.036203","article-title":"Origin of chaos near critical points of quantum flow","volume":"79","author":"Efthymiopoulos","year":"2009","journal-title":"Phys. Rev. E"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"40004","DOI":"10.1209\/0295-5075\/96\/40004","article-title":"Geometrical view of quantum entanglement","volume":"96","author":"Ramsak","year":"2011","journal-title":"EPL"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"195304","DOI":"10.1088\/1751-8113\/43\/19\/195304","article-title":"Relative dispersion and quantum thermal equilibrium in de Broglie-Bohm mechanics","volume":"43","author":"Bennett","year":"2010","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"012131","DOI":"10.1103\/PhysRevA.97.012131","article-title":"Quantum work in the Bohmian framework","volume":"97","author":"Sampaio","year":"2018","journal-title":"Phys. Rev. A"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"023516","DOI":"10.1103\/PhysRevD.75.023516","article-title":"Noninflationary model with scale invariant cosmological perturbations","volume":"75","author":"Peter","year":"2007","journal-title":"Phys. Rev. D"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"063504","DOI":"10.1103\/PhysRevD.86.063504","article-title":"Wheeler-DeWitt quantization can solve the singularity problem","volume":"86","author":"Falciano","year":"2012","journal-title":"Phys. Rev. D"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"083506","DOI":"10.1103\/PhysRevD.85.083506","article-title":"Quantum-to-classical transition of primordial cosmological perturbations in de Broglie-Bohm quantum theory","volume":"85","author":"Santos","year":"2012","journal-title":"Phys. Rev. D"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"104046","DOI":"10.1103\/PhysRevD.82.104046","article-title":"n-dimensional FLRW quantum cosmology","volume":"82","author":"Letelier","year":"2010","journal-title":"Phys. Rev. D"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"3602","DOI":"10.3390\/e15093602","article-title":"Unification of quantum and gravity by non classical information entropy space","volume":"15","author":"Resconi","year":"2013","journal-title":"Entropy"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"62","DOI":"10.3390\/e16010062","article-title":"Nonadiabatic molecular dynamics based on trajectories","volume":"16","author":"Bouduban","year":"2013","journal-title":"Entropy"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"286","DOI":"10.1140\/epjd\/e2014-50222-4","article-title":"Applied Bohmian mechanics","volume":"68","author":"Benseny","year":"2014","journal-title":"Eur. Phys. J. D"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1007\/s10698-017-9277-4","article-title":"The relationship between chemistry and physics from the perspective of Bohmian mechanics","volume":"19","author":"Fortin","year":"2017","journal-title":"Found. Chem."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"3285","DOI":"10.1088\/1751-8113\/40\/12\/S24","article-title":"Astrophysical and cosmological tests of quantum theory","volume":"40","author":"Valentini","year":"2007","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"063513","DOI":"10.1103\/PhysRevD.82.063513","article-title":"Inflationary cosmology as a probe of primordial quantum mechanics","volume":"82","author":"Valentini","year":"2010","journal-title":"Phys. Rev. D"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"055102","DOI":"10.1103\/PhysRevE.70.055102","article-title":"Derivation of the upper limit of temperature from the field theory of thermodynamics","volume":"70","year":"2004","journal-title":"Phys. Rev. E"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Valentini, A., and Westman, H. (arXiv, 2005). Dynamical origin of quantum probabilities, arXiv.","DOI":"10.1098\/rspa.2004.1394"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1016\/j.physa.2014.02.021","article-title":"Pilot wave approach to the NRT nonlinear Schr\u00f6dinger equation","volume":"403","author":"Pennini","year":"2014","journal-title":"Physica A"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Plastino, A.R., and Wedemann, R.S. (2017). Nonlinear wave equations related to nonextensive thermostatistics. Entropy, 19.","DOI":"10.3390\/e19020060"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1016\/j.aop.2014.05.021","article-title":"Bohmian trajectories of Airy packets","volume":"348","author":"Nassar","year":"2014","journal-title":"Ann. Phys."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"505","DOI":"10.1016\/j.aop.2004.09.008","article-title":"Computing the wavefunction from trajectories: Particle and wave pictures in quantum mechanics and their relation","volume":"315","author":"Holland","year":"2005","journal-title":"Ann. Phys."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1016\/S0375-9601(01)00143-8","article-title":"Bohmian quantum theory of motion for particles with position-dependent effective mass","volume":"281","author":"Plastino","year":"2001","journal-title":"Phys. Lett. A"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Gisin, N. (2018). Why Bohmian mechanics? One- and two-time position measurements, Bell inequalities, philosophy, and physics. Entropy, 20.","DOI":"10.3390\/e20020105"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"68","DOI":"10.1038\/scientificamerican0201-68","article-title":"100 Years of quantum mysteries","volume":"284","author":"Tegmark","year":"2001","journal-title":"Sci. Am."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"052109","DOI":"10.1103\/PhysRevA.66.052109","article-title":"Bohm\u2019s interpretation and maximally entangled states","volume":"66","author":"Durt","year":"2002","journal-title":"Phys. Rev. A"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"060406","DOI":"10.1103\/PhysRevLett.110.060406","article-title":"Proposal to observe the nonlocality of Bohmian trajectories with entangled photons","volume":"110","author":"Braverman","year":"2013","journal-title":"Phys. Rev. Lett."},{"key":"ref_33","unstructured":"Jammer, M. (2009). Concepts of Mass in Contemporary Physics and Philosophy, Princeton University Press."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"225303","DOI":"10.1088\/1751-8113\/44\/22\/225303","article-title":"Entropic dynamics, time and quantum theory","volume":"44","author":"Caticha","year":"2011","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Bell, J.S. (2004). Speakable and Unspeakable in Quantum Mechanics, Cambridge University Press. [2nd ed.].","DOI":"10.1017\/CBO9780511815676"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"2861","DOI":"10.1088\/1751-8113\/40\/11\/020","article-title":"Entanglement and the speed of evolution of multi-partite quantum systems","volume":"40","author":"Zander","year":"2007","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Mensky, M.B. (2017). Continuous Quantum Measurement and Path Integrals, Routledge.","DOI":"10.1201\/9781315140292"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"3589","DOI":"10.1063\/1.1678812","article-title":"On the Schr\u00f6dinger-langevin equation","volume":"57","author":"Kostin","year":"1972","journal-title":"J. Chem. Phys."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1652","DOI":"10.1063\/1.525861","article-title":"Nonlinear Schr\u00f6dinger-type field equation for the description of dissipative systems. I. Derivation of the nonlinear field equation and one-dimensional example","volume":"24","author":"Schuch","year":"1983","journal-title":"J. Math. Phys."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"391","DOI":"10.1002\/qua.560250210","article-title":"Nonlinear Schr\u00f6dinger-type field equation for the description of dissipative systems. II. Frictionally damped motion in a magnetic field","volume":"25","author":"Schuch","year":"1984","journal-title":"Int. J. Quantum Chem."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"3086","DOI":"10.1063\/1.526024","article-title":"Nonlinear Schr\u00f6dinger-type field equation for the description of dissipative systems. III. Frictionally damped free motion as an example for an aperiodic motion","volume":"25","author":"Schuch","year":"1984","journal-title":"J. Math. Phys."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"012042","DOI":"10.1088\/1742-6596\/174\/1\/012042","article-title":"Complex nonlinear relations in classical and quantum physics","volume":"174","author":"Schuch","year":"2009","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"1561","DOI":"10.1002\/qua.560290546","article-title":"From macroscopic irreversibility to microscopic reversibility via a nonlinear Schr\u00f6dinger-type field equation","volume":"29","author":"Schuch","year":"1986","journal-title":"Int. J. Quantum Chem."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"012019","DOI":"10.1088\/1742-6596\/538\/1\/012019","article-title":"Relations between nonlinear Riccati equations and other equations in fundamental physics","volume":"538","author":"Schuch","year":"2014","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_45","unstructured":"Pask, C. (2013). Magnificent Principia: Exploring Isaac Newton\u2019s Masterpiece, Prometheus Books."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/6\/473\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:09:08Z","timestamp":1760195348000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/6\/473"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,6,18]]},"references-count":45,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2018,6]]}},"alternative-id":["e20060473"],"URL":"https:\/\/doi.org\/10.3390\/e20060473","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,6,18]]}}}