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

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 Grigory 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 characterising such systems.

[1]  L. Aolita,et al.  Open-system dynamics of entanglement:a key issues review , 2014, Reports on progress in physics. Physical Society.

[2]  S. Vacaru,et al.  Axiomatic formulations of modified gravity theories with nonlinear dispersion relations and Finsler–Lagrange–Hamilton geometry , 2018, The European Physical Journal C.

[3]  S. Vacaru,et al.  Black holes with MDRs and Bekenstein–Hawking and Perelman entropies for Finsler–Lagrange–Hamilton Spaces , 2018, Annals of Physics.

[4]  A. Lichnerowicz.  Proof of the Strong Subadditivity of Quantum-Mechanical Entropy , 2018 .

[5]  Ovidiu Cristinel Stoica,et al.  Revisiting the Black Hole Entropy and the Information Paradox , 2018, Advances in High Energy Physics.

[6]  H. Umegaki.  CONDITIONAL EXPECTATION IN AN OPERATOR ALGEBRA, II , 1954 .

[7]  Salman Beigi,et al.  Sandwiched Rényi divergence satisfies data processing inequality , 2013, 1306.5920.

[8]  C. Vafa,et al.  Microscopic origin of the Bekenstein-Hawking entropy , 1996, hep-th/9601029.

[9]  D. Friedan,et al.  NONLINEAR MODELS IN 2 +<-DIMENSIONS , 1980 .

[10]  G. Vidal,et al.  Entanglement in quantum critical phenomena. , 2002, Physical review letters.

[11]  J. Morgan,et al.  Ricci Flow and the Poincare Conjecture , 2006, math/0607607.

[12]  E. Witten.  APS Medal for Exceptional Achievement in Research: Invited article on entanglement properties of quantum field theory , 2018, Reviews of Modern Physics.

[13]  Isaac L. Chuang,et al.  Quantum Computation and Quantum Information (10th Anniversary edition) , 2011 .

[14]  A. Rényi.  On Measures of Entropy and Information , 1961 .

[15]  R. Ionicioiu.  Schrödinger's Cat: Where Does The Entanglement Come From? , 2016, 1603.07986.

[16]  L. Susskind,et al.  Cool horizons for entangled black holes , 2013, 1306.0533.

[17]  Stephen W. Hawking,et al.  Particle Creation by Black Holes , 1993, Resonance.

[18]  V. Vedral.  The role of relative entropy in quantum information theory , 2001, quant-ph/0102094.

[19]  S. Vacaru,et al.  Perelman's W--entropy and Statistical and Relativistic Thermodynamic Description of Gravitational Fields , 2013 .

[20]  J. Cirac,et al.  Three qubits can be entangled in two inequivalent ways , 2000, quant-ph/0005115.

[21]  H. Umegaki.  Conditional expectation in an operator algebra. IV. Entropy and information , 1962 .

[22]  Elliott H. Lieb,et al.  Entropy inequalities , 1970 .

[23]  T. Takayanagi,et al.  Holographic Derivation of Entanglement Entropy from AdS/CFT , 2006, hep-th/0603001.

[24]  Nonholonomic Ricci flows. II. Evolution equations and dynamics , 2007, math/0702598.

[25]  The entropy of Lagrange-Finsler spaces and Ricci flows , 2007, math/0701621.

[26]  Matthew B Hastings,et al.  Area laws in quantum systems: mutual information and correlations. , 2007, Physical review letters.

[27]  S. Vacaru,et al.  On supersymmetric geometric flows and R2 inflation from scale invariant supergravity , 2016, 1606.06884.

[28]  C. Ecker.  Entanglement Entropy from Numerical Holography , 2018, 1809.05529.

[29]  S. Vacaru.  Spectral Functionals, Nonholonomic Dirac Operators, and Noncommutative Ricci Flows , 2008, 0806.3814.

[30]  S. Yau,et al.  Surveys in Differential Geometry , 1999 .

[31]  T. Takayanagi,et al.  Holographic derivation of entanglement entropy from the anti-de Sitter space/conformal field theory correspondence. , 2006, Physical review letters.

[32]  W. Thirring,et al.  FROM RELATIVE ENTROPY TO ENTROPY , 1985 .

[33]  Sergey N. Solodukhin,et al.  Entanglement Entropy of Black Holes , 2011, Living reviews in relativity.

[34]  B. Chow,et al.  The Ricci flow on surfaces , 2004 .

[35]  Edward Witten,et al.  A mini-introduction to information theory , 2018, La Rivista del Nuovo Cimento.

[36]  S. Vacaru.  Geometric information flows and G. Perelman entropy for relativistic classical and quantum mechanical systems , 2019, The European Physical Journal C.

[37]  Dong Yang,et al.  Strong converse for the classical capacity of entanglement-breaking channels , 2013, ArXiv.

[38]  A. Serafini,et al.  Measuring Gaussian quantum information and correlations using the Rényi entropy of order 2. , 2012, Physical review letters.

[39]  Thierry Paul,et al.  Quantum computation and quantum information , 2007, Mathematical Structures in Computer Science.

[40]  E. Witten.  Notes on Some Entanglement Properties of Quantum Field Theory , 2018, 1803.04993.

[41]  R. Hamilton.  Three-manifolds with positive Ricci curvature , 1982 .

[42]  A. Zeilinger,et al.  Going Beyond Bell’s Theorem , 2007, 0712.0921.

[43]  J. Bekenstein.  Black Holes and Entropy , 1973, Jacob Bekenstein.

[44]  Locally Anisotropic Kinetic Processes and Thermodynamics in Curved Spaces , 2000, gr-qc/0001060.

[45]  George Ruppeiner,et al.  Riemannian geometry in thermodynamic fluctuation theory , 1995 .

[46]  R. Hamilton,et al.  The formations of singularities in the Ricci Flow , 1993 .

[47]  J. Bekenstein,et al.  Black holes and the second law , 2019, Jacob Bekenstein.

[48]  A. Shimony,et al.  Bell’s theorem without inequalities , 1990 .

[49]  H. Quevedo.  Black hole geometrothermodynamics , 2017 .

[50]  Karol Zyczkowski,et al.  Rényi Extrapolation of Shannon Entropy , 2003, Open Syst. Inf. Dyn..

[51]  G. Perelman.  Ricci flow with surgery on three-manifolds , 2003, math/0303109.

[52]  Topological Entanglement Entropy from the Holographic Partition Function , 2006, cond-mat/0609072.

[53]  G. Perelman.  Finite extinction time for the solutions to the Ricci flow on certain three-manifolds , 2003, math/0307245.

[54]  Brandon Carter,et al.  The four laws of black hole mechanics , 1973 .

[55]  G. Perelman.  The entropy formula for the Ricci flow and its geometric applications , 2002, math/0211159.

[56]  B. Swingle,et al.  Entanglement Renormalization and Holography , 2009, 0905.1317.

[57]  S. Vacaru,et al.  Exact solutions for E. Verlinde emergent gravity and generalized G. Perelman entropy for geometric flows , 2019, 1904.05149.

[58]  Huai-Dong Cao,et al.  A Complete Proof of the Poincaré and Geometrization Conjectures - application of the Hamilton-Perelman theory of the Ricci flow , 2006 .

[59]  D. Petz,et al.  Quantum Entropy and Its Use , 1993 .

[60]  S. Vacaru,et al.  Geometric Flows and Perelman's Thermodynamics for Black Ellipsoids in $R^2$ and Einstein Gravity Theories , 2016, 1602.08512.

[61]  T. Nishioka.  Entanglement entropy: Holography and renormalization group , 2018, Reviews of Modern Physics.

[62]  John Preskill,et al.  Topological entanglement entropy. , 2005, Physical Review Letters.

[63]  S. Vacaru.  Entropy functionals for nonholonomic geometric flows, quasiperiodic Ricci solitons, and emergent gravity , 2019, 1903.04920.

[64]  S. Vacaru.  Nonholonomic relativistic diffusion and exact solutions for stochastic Einstein spaces , 2010, 1010.0647.

[65]  Richard S. Hamilton,et al.  The Ricci flow on surfaces , 1986 .

[66]  Bruce Kleiner,et al.  Notes on Perelman's papers , 2006 .

[67]  Mark Van Raamsdonk.  Building up spacetime with quantum entanglement , 2010 .

[68]  Serge Fehr,et al.  On quantum Rényi entropies: A new generalization and some properties , 2013, 1306.3142.