Classical and quantum geometric information flows and entanglement of relativistic mechanical systems

  • Sergiu I. VacaruEmail author
  • Laurenţiu Bubuianu


This article elaborates on entanglement entropy and quantum information theory of geometric flows of (relativistic) Lagrange–Hamilton mechanical systems. A set of basic geometric and quantum mechanics and probability concepts together with methods of computation are developed in general covariant form for curved phase spaces modelled as cotangent Lorentz bundles. The constructions are based on ideas relating the Grigori Perelman’s entropy for geometric flows and associated statistical thermodynamic systems to the quantum von Neumann entropy, classical and quantum relative and conditional entropy, mutual information, etc. We formulate the concept of the entanglement entropy of quantum geometric information flows and study properties and inequalities for quantum, thermodynamic and geometric entropies characterizing such systems.


Perelman W-entropy Quantum geometric information flows Relativistic Lagrange-Hamilton mechanics Entanglement entropy of quantum geometric information flows 

Mathematics Subject Classification

53C44 53C50 53C80 81P45 82D99 83C15 83C55 83C99 83D99 35Q75 37J60 37D35 



This research develops the former programs partially supported by IDEI, PN-II-ID-PCE-2011-3-0256, CERN 2012-2014, DAAD-2015 and QGR 2016-2017 and contains certain results for new grant proposals. The UAIC affiliation for S. V. refers to a Project IDEI hosted by that University during 2012–2015, when the bulk of geometric ideas and methods of this and partner works were elaborated (to put a relevant co-/affiliation for further related results was the condition of that grant). Performing rigorous mathematical proofs and respective manuscripts request many years of technical work and further collaborations. S. V. is grateful to D. Singleton, S. Rajpoot and P. Stavrinos for collaboration and supporting his research on geometric methods in physics.


  1. 1.
  2. 2.
    Solodukhin, S.N.: Entanglement entropy of black holes. Living Rev. Relativ. 14, 8 (2011). arXiv:1104.3712 ADSzbMATHCrossRefGoogle Scholar
  3. 3.
    Aolita, L., de Melo, F., Davidovich, L.: Opens-system dynamics of entanglement. Rep. Progr. Phys. 78, 042001 (2015). arXiv:1402.3713 ADSCrossRefGoogle Scholar
  4. 4.
    Ionicioiu, R.: Schrödinger’s cat: where does the entanglement come from? Quanta 6, 57–60 (2017). arXiv:1603.07986 MathSciNetCrossRefGoogle Scholar
  5. 5.
    Stoica, O.C.: Revisiting the black hole entropy and information paradox. Adv. High. Energy Phys., art. ID 4130417 (2018). arXiv:1807.05864
  6. 6.
    Nishioka, T.: Entanglement entropy: holography and renormalization group. Rev. Mod. Phys. 90, 03500 (2018). arXiv:1801.10352 MathSciNetCrossRefGoogle Scholar
  7. 7.
    Witten, E.: Notes on some entanglement properties of quantum field theory. Rev. Mod. Phys. 90, 45003 (2018). arXiv:1803.04993
  8. 8.
    Witten, E.: A mini-introduction to information theory. arXiv:1805.11965
  9. 9.
    Ecker, C.: Entanglement Entropy from Numerical Holography, Ph.D. thesis. arXiv:1809.05529
  10. 10.
    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information, 10th Anniversary edn. Cambridge University Press, Cambridge (2010)Google Scholar
  11. 11.
    Ryu, S., Takayanagi, T.: Holographic derivation of entanglement entropy from AdS/CFT. Phys. Rev. Lett. 96, 181602 (2006). arXiv:hep-th/0603001 ADSMathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Vidal, G., Latorre, J.I., Rico, E., Kitaev, A.: Entanglement in quantum critical phenomena. Phys. Rev. Lett. 90, 227902 (2003). arXiv:quant-ph/0211074 ADSCrossRefGoogle Scholar
  13. 13.
    Kitaev, A., Preskill, J.: Topological entanglement entropy. Phys. Rev. Lett. 96, 110404 (2006). arXiv:hep-th/0510092 ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    Fendley, P., Fisher, M.P.A., Nayak, C.: Topological entanglement entropy from the holographic partition function. J. Stat. Phys. 126, 1111 (2007). arXiv:cond-mat/0609072 ADSMathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Van Raamsdonk, M.: Building up spacetime with quantum entanglement. Gen. Relat. Gravity 42, 2323 (2010) [Int. J. Mod. Phys. D 19, 2429 (2010) ]; arXiv:1005.3035
  16. 16.
    Maldacena, J., Susskind, L.: Cool horizons for entangled black holes. Fortschr. Phys. 61, 781–811 (2013). arXiv:1306.0533 MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Vacaru, S.: Entropy functionals for nonholonomic geometric flows, quasiperiodic Ricci solitons, and emergent gravity. arXiv:1903.04920
  18. 18.
    Vacaru, S., Bubuianu, L.: Exact solutions for E. Verlinde emergent gravity and generalized G. Perelman entropy for geometric flows. arXiv:1904.05149
  19. 19.
    Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159
  20. 20.
    Perelman, G.: Ricci flow with surgery on three-manifolds. arXiv:math.DG/0303109
  21. 21.
    Perelman, G.: Finite extinction time for the solutions to the Ricci flow on certain three-manifolds. arXiv:math.DG/0307245
  22. 22.
    Hamilton, R.S.: Three-manifolds with positive Ricci curvature. J. Differ. Geom. 17, 255–306 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Hamilton, R.S.: The Ricci flow on surfaces. Math Gen Relativ Contemp Math 71, 237–262 (1988) MathSciNetCrossRefGoogle Scholar
  24. 24.
    Hamilton, R.S.: In: Surveys in Differential Geometry, vol. 2, pp. 7–136. International Press, Vienna (1995)Google Scholar
  25. 25.
    Friedan, D.: Nonlinear Models in \(2+\varepsilon \) Dimensions, Ph.D. Thesis (Berkely) LBL-11517, UMI-81-13038 (1980)Google Scholar
  26. 26.
    Friedan, D.: Nonlinear models in \(2+\varepsilon \) dimensions. Phys. Rev. Lett. 45, 1057–1060 (1980)ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    Friedan, D.: Nonlinear models in \(2+\varepsilon \) dimensions. Ann. Phys. 163, 318–419 (1985)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Bubuianu, L., Vacaru, S.: Black holes with MDRs and Bekenstein-Hawking and Perelman entropies for Finsler-Lagrange-Hamilton-spaces. Ann. Phys. N. Y. 404, 10–38 (2019). arXiv:1812.02590 ADSMathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Vacaru, S.: Geometric information flows and G. Perelman entropy for relativistic classical and quantum mechanical systems [under elaboration]Google Scholar
  30. 30.
    Cao, H.-D., Zhu, H.-P.: A complete proof of the Poincaré and geometrization conjectures–application of the Hamilton-Perelman theory of the Ricci flow. Asian J. Math. 10, 165–495 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Morgan, J.W., Tian, G.: Ricci Flow and the Poincaré Conjecture. Clay Mathematics Monographs, vol. 3. AMS, Providence (2007)Google Scholar
  32. 32.
    Kleiner, B., Lott, J.: Notes on Perelman’s papers. Geom. Topol. 12, 2587–2855 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Vacaru, S.: Locally anisotropic kinetic processes and thermodynamics in curved spaces. Ann. Phys. (N.Y.) 290, 83–123 (2001). arXiv:gr-qc/0001060 ADSMathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Vacaru, S.: Spectral functionals, nonholonomic Dirac operators, and noncommutative Ricci flows. J. Math. Phys. 50, 073503 (2009). arXiv:0806.3814 ADSMathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Rajpoot, S., Vacaru, S.: On supersymmetric geometric flows and R2 inflation from scale invariant supergravity. Ann. Phys. N. Y. 384, 20–60 (2017). arXiv:1606.06884 ADSzbMATHCrossRefGoogle Scholar
  36. 36.
    Ruchin, V., Vacaru, O., Vacaru, S.: Perelman’s W-entropy and statistical and relativistic thermodynamic description of gravitational fields. Eur. Phys. J. C 77, 184 (2017). arXiv:1312.2580 ADSCrossRefGoogle Scholar
  37. 37.
    Gheorghiu, T., Ruchin, V., Vacaru, O., Vacaru, S.: Geometric flows and Perelman’s thermodynamics for black ellipsoids in R2 and Einstein gravity theories. Ann. Phys. N. Y. 369, 1–35 (2016). arXiv:1602.08512 ADSzbMATHCrossRefGoogle Scholar
  38. 38.
    Vacaru, S.: On axiomatic formulation of gravity and matter field theories with MDRs and Finsler–Lagrange–Hamilton geometry on (co) tangent Lorentz bundles. arXiv:1801.06444
  39. 39.
    Bubuianu, L., Vacaru, S.: Axiomatic formulations of modified gravity theories with nonlinear dispersion relations and Finsler–Lagrange–Hamilton geometry. Eur. Phys. J. C 78, 969 (2018)ADSCrossRefGoogle Scholar
  40. 40.
    Vacaru, S.: Nonholonomic Ricci flows: II. Evolution equations and dynamics. J. Math. Phys. 49, 043504 (2008). arXiv:math.DG/0702598 ADSMathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Vacaru, S.: The entropy of Lagrange–Finsler spaces and Ricci flows. Rep. Math. Phys. 63, 95–110 (2009). arXiv:math.DG/0701621 ADSMathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Ruppeiner, G.: Riemannian geometry in thermodynamic fluctuation theory, Rev. Mod. Phys. 67, 605–659 (1995), Erratum: 68 (1996) 313Google Scholar
  43. 43.
    Quevedo, H.: Geometrothermodynamics. J. Math. Phys. 48, 013506 (2007)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Vacaru, S.: Nonholonomic relativistic diffusion and exact solutions for stochastic Einstein spaces. Eur. Phys. J. P. 127, 32 (2012)CrossRefGoogle Scholar
  45. 45.
    Castro Perelman, C.: Thermal relativity, corrections of black hole entropy, Born’s reciprocal relativity theory and quantum gravity. Can. J. Phys. (2019). CrossRefGoogle Scholar
  46. 46.
    Greenberger, D.M., Horne, M.A., Zeilinger, A.: Going beyond Bells theorem. Kafatos, M. (ed.) Bells Theorem, Quantum Theory and Conceptions of the Universe, pp. 69–72. Springer, Berlin (1989) Google Scholar
  47. 47.
    Greenberger, D.M., Horne, M.A., Shimony, A., Zeilinger, A.: Bells theorem without inequalities. Am. J. Phys. 58, 1131–1141 (1990)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Dur, W., Vidal, G., Cirac, J.I.: Three qubits can be entangled in two inequivalent ways. Phys. Rev. A 62, 062314 (2000)ADSMathSciNetCrossRefGoogle Scholar
  49. 49.
    Akraki, H., Lieb, E.H.: Entropy inequalities. Commun. Math. Phys. 18, 160–170 (1970)ADSMathSciNetCrossRefGoogle Scholar
  50. 50.
    Lieb, E.H., Urskai, M.B.: Proof of the strong subadditivity of quantum-mechanical entropy. J. Math. Phys. 14, 1938–1941 (1973)ADSMathSciNetCrossRefGoogle Scholar
  51. 51.
    Narnhofer, H., Thirring, W.E.: From relative entropy to entropy. Fizika 17, 257–265 (1985)Google Scholar
  52. 52.
    Umegaki, H.: Conditional expectation in an operator algebra. IV. Entropy and information. Kodai Math. Semin. Rep. 14, 59–85 (1962)MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    Vedral, V.: The role of relative entropy in quantum information theory. Rev. Mod. Phys. 74, 197–234 (2002)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  54. 54.
    Ohya, M., Pertz, D.: Quantum Entropy and Its Use [corrected second printing]. Springer, Berlin (2004)Google Scholar
  55. 55.
    Wolf, M.M., Verstraete, F., Hasings, M.B., Cirac, J.I.: Area laws in quantum systems: mutual information and correlations. Phys. Rev. Lett. 100, 070502 (2008)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  56. 56.
    Rényi, A.: On measures of entropy in information. In: Fourth Berkeley Symposium on Mathematical Statistics and Probability, pp. 547–561 (1961)Google Scholar
  57. 57.
    Zyczkowski, K.: Rényi extrapolation of Shannon entropy. Open Syst. Inf. Dyn. 10, 297–310 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  58. 58.
    Müller-Lennert, M., Dupius, F., Szehr, O., Fehr, S., Tomamichel, M.: On quantum Rényi entropies: a new generalization and some properties. J. Math. Phys. 54, 122203 (2013)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  59. 59.
    Wilde, M.M., Winter, A., Yang, D.: Strong converse for the classical capacity of entanglement-breaking channels. arXiv:1306.1586
  60. 60.
    Adesso, G., Girolami, D., Serafini, A.: Measuring Gaussian quantum information and correlation using the Rényi entropy of order 2. Phys. Rev. Lett. 109, 190502 (2012)ADSCrossRefGoogle Scholar
  61. 61.
    Beingi, S.: Sandwiched Rényi divergence satisfied data processing inequality. J. Math. Phys. 54, 122202 (2013)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  62. 62.
    Bekenstein, J.D.: Black holes and the second law. Nuovo Cim. Lett. 4, 737–740 (1972)ADSCrossRefGoogle Scholar
  63. 63.
    Bekenstein, J.D.: Black holes and entropy. Phys. Rev. D 7, 2333–2346 (1973)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  64. 64.
    Bardeen, J.M., Carter, B., Hawking, S.W.: The four laws of black hole mechanics. Commun. Math. Phys. 31, 161 (1973) ADSMathSciNetzbMATHCrossRefGoogle Scholar
  65. 65.
    Hawking, S.W.: Particle creation by black holes. Commun. Math. Phys. 43, 199–220 (1975)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  66. 66.
    Strominger, A., Vafa, C.: Microscopic origin of the Bekenstein–Hawking entropy. Phys. Lett. B 379, 99 (1996). arXiv:hep-th/9601029 ADSMathSciNetzbMATHCrossRefGoogle Scholar
  67. 67.
    Faulkner, T., Guica, M., Harman, T., Myers, R.C., Van Raamsdonk, M.: Gravitation from entanglement and holographic CFTs. J. High Energy Phys. 1403, 051 (2015). arXiv:1312.7856 ADSMathSciNetzbMATHGoogle Scholar
  68. 68.
    Swingle, B.: Entanglement renormalization and holography. Phys. Rev. D 86, 065007 (2012). arXiv:0905.1317 ADSCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Physics DepartmentCalifornia State University at FresnoFresnoUSA
  2. 2.Department of Theoretical Physics and Computer Modelling, Institute of Applied-Physics and Computer SciencesYuriy Fedkovych Chernivtsi National UniversityChernivtsiUkraine
  3. 3.Project IDEI - 2011University “Al. I. Cuza”IaşiRomania
  4. 4.SRTV - Studioul TVR IaşiIaşiRomania
  5. 5.University ApolloniaIaşiRomania

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