Paper · May 9, 2019

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

Sergiu I. Vacaru, Laurenţiu Bubuianu

physics.gen-pharXiv:1905.13015v3
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As on arXiv

Abstract

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.

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Submitted
May 9, 2019
Last revised
Jan 30, 2020 (v3)
Subjects
physics.gen-ph, math-ph, quant-ph
Journal
Quantum Inf. Process, QINP, 18 (2019) 376
DOI
10.1007/s11128-019-2487-z

Comments: latex2e 11pt, 25 pages with a table of content, v3 closed to published journal version, with up-dated co-affiliations and references

Perelman entropy

Its neighbourhood

Connected papers

The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2016, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Geometric Flows and Perelman's Thermodynamics for Black Ellipsoids in R² and Einstein Gravity Theories (Tamara Gheorghiu, Vyacheslav Ruchin, Olivia Vacaru et al., 2016)5Gheorghiu 2016 On Supersymmetric Geometric Flows and 𝓡² Inflation From Scale Invariant Supergravity (Subhash Rajpoot and Sergiu I. Vacaru, 2016)6Rajpoot 2016 Nonassociative geometric and quantum information flows and R-flux deformations of wormhole solutions in string gravity (Laurenţiu Bubuianu, Douglas Singleton, Sergiu I. Vacaru et al., 2024)1Bubuianu 2024 Innovative Dynamics: Utilizing Perelman's Entropy and Ricci Flow for Settler Position Models on Manifolds (Zeraoulia Rafik and Sobhan Sobhan Allah, 2024)7Rafik 2024 Nonassicative cosmological solitonic R-flux deformations in gauge gravity and G. Perelman geometric flow thermodynamics (L. Bubuianu, J. O. Seti, S. Vacaru et al., 2024)2Bubuianu 2024 Nonmetric geometric flows and quasicrystalline topological phases for dark energy and dark matter in f(Q) cosmology (L. Bubuianu, E. Nurlan, J. O. Seti et al., 2024)3Bubuianu 2024 The fundamental physical importance of generic off-diagonal solutions and Grigori Perelman entropy in the Einstein gravity theory (Sergiu I. Vacaru and Elşen V. Veliev, 2025)4Vacaru 2025 Stochastic Thermodynamics on Time-Evolving Curved Spaces (Rihito Nagase, Shoki Sugimoto, Asuka Takatsu et al., 2026)8Nagase 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Nonassociative geometric and quantum information flows and R-flux deformations of wormhole solutions in string gravity

    Laurenţiu Bubuianu, Douglas Singleton, Sergiu I. Vacaru et al.

    Same authors Sergiu I. Vacaru, Laurenţiu Bubuianu · Shared terms “information flows”, “quantum geometric”, “quantum information”

  2. Nonassicative cosmological solitonic R-flux deformations in gauge gravity and G. Perelman geometric flow thermodynamics

    L. Bubuianu, J. O. Seti, S. Vacaru et al.

    Topics Perelman entropy · Same authors Sergiu I. Vacaru, Laurenţiu Bubuianu · Shared terms “thermodynamic”, “relativistic”, “information theories”

  3. Nonmetric geometric flows and quasicrystalline topological phases for dark energy and dark matter in cosmology

    L. Bubuianu, E. Nurlan, J. O. Seti et al.

    Topics Perelman entropy · Same authors Sergiu I. Vacaru, Laurenţiu Bubuianu · Shared terms “geometric flows”, “statistical thermodynamic”, “elaborate”

  4. The fundamental physical importance of generic off-diagonal solutions and Grigori Perelman entropy in the Einstein gravity theory

    Sergiu I. Vacaru and Elşen V. Veliev

    Topics Perelman entropy · Same author Sergiu I. Vacaru · Shared terms “relativistic”, “thermodynamic”, “elaborate”

  5. Geometric Flows and Perelman's Thermodynamics for Black Ellipsoids in and Einstein Gravity Theories

    Tamara Gheorghiu, Vyacheslav Ruchin, Olivia Vacaru et al.

    Topics Perelman entropy · Same author Sergiu I. Vacaru · Shared terms “geometric flows”, “relativistic”, “statistical”

  6. On Supersymmetric Geometric Flows and Inflation From Scale Invariant Supergravity

    Subhash Rajpoot and Sergiu I. Vacaru

    Topics Perelman entropy · Same author Sergiu I. Vacaru · Shared terms “geometric flows”, “modelled”

  7. Innovative Dynamics: Utilizing Perelman's Entropy and Ricci Flow for Settler Position Models on Manifolds

    Zeraoulia Rafik and Sobhan Sobhan Allah

    Topics Perelman entropy · Shared terms “computation”

  8. Stochastic Thermodynamics on Time-Evolving Curved Spaces

    Rihito Nagase, Shoki Sugimoto, Asuka Takatsu et al.

    Topics Perelman entropy · Shared terms “thermodynamic”

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Perelman entropy Perelman found that Ricci flow is a gradient flow once a measure is fixed. His 𝓕-functional and 𝓦-entropy increase along the flow; a lower bound… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002)

In the guidebook

  1. 6A.12
    Entropy, Information and DiffusionBook 6A · The Heat Equation and Its Relatives

    Entropy along the heat flow, the log-Sobolev inequality and Li–Yau: the ancestors of Perelman’s entropy.

  2. 6A.13
    Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives

    Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.

  3. 6B.16
    Eigenvalue Problems and Perelman’s λBook 6B · Linear PDE: Characteristics, Waves and Regularity

    Rayleigh quotients and min–max, Faber–Krahn and Cheeger’s inequality, ground states of Schrödinger operators, spectral clustering, and the first eigenvalue of −4Δ + R that begins Perelman’s first preprint.