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From Quantum Relative Entropy to the Semiclassical Einstein Equations

Philipp Dorau* and Albert Much†

  • *Contact author: philipp.dorau@uni-leipzig.de
  • †Contact author: much@itp.uni-leipzig.de
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Phys. Rev. Lett. 136, 091602 – Published 2 March, 2026

DOI: https://doi.org/10.1103/lmq8-nsty

Abstract

We provide arguments indicating that the semiclassical Einstein equations follow from quantum relative entropy and its proportionality to an area variation. Using modular theory, we establish that the relative entropy between the vacuum state and coherent excitations of a scalar quantum field on a bifurcate Killing horizon is given by the energy flux across the horizon. Under the assumption of the Bekenstein-Hawking entropy-area formula, this energy flux is proportional to a variation in the surface area of the horizon cross section. The semiclassical Einstein equations follow automatically from this identification. Our approach provides a quantum field theoretic generalization of Jacobson’s thermodynamic derivation of the Einstein equations, replacing classical thermodynamic entropy with the well-defined quantum relative (Araki-Uhlmann) entropy. This suggests that quantum information plays a central role in what is often seen as a zeroth order approximation of a theory of quantum gravity, namely quantum field theory in curved spacetimes.

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References (59)

  1. T. Jacobson, Thermodynamics of spacetime: The Einstein equation of state, Phys. Rev. Lett. 75, 1260 (1995).
  2. J. D. Bekenstein, Black holes and entropy, Phys. Rev. D 7, 2333 (1973).
  3. S. W. Hawking, Particle creation by black holes, Commun. Math. Phys. 43, 199 (1975).
  4. S. W. Hawking, Black holes and thermodynamics, Phys. Rev. D 13, 191 (1976).
  5. W. G. Unruh, Notes on black-hole evaporation, Phys. Rev. D 14, 870 (1976).
  6. T. Jacobson, Entanglement equilibrium and the Einstein equation, Phys. Rev. Lett. 116, 201101 (2016).
  7. T. Faulkner, M. Guica, T. Hartman, R. C. Myers, and M. Van Raamsdonk, Gravitation from entanglement in holographic CFTs, J. High Energy Phys. 03 (2014) 051.
  8. M. Rangamani and T. Takayanagi, Holographic Entanglement Entropy (Springer, New York, 2017).
  9. E. Witten, APS medal for exceptional achievement in research: Invited article on entanglement properties of quantum field theory, Rev. Mod. Phys. 90, 045003 (2018).
  10. H. Casini and M. Huerta, Lectures on entanglement in quantum field theory, Proc. Sci. TASI2021 (2023) 002.
  11. A. Alonso-Serrano, M. Liška, and M. Piotrak, Semi-classical spacetime thermodynamics, arXiv:2509.05052.
  12. A. Alonso-Serrano, L. J. Garay, M. Liška, and C. López Pineros, Gravity from equilibrium thermodynamics of stretched light cones, Phys. Rev. D 112, 124026 (2025).
  13. E. Curiel, F. Finster, and J. M. Isidro, Two-dimensional area and matter flux in the theory of causal fermion systems, Int. J. Mod. Phys. D 29, 2050098 (2020).
  14. R. Longo, Entropy of coherent excitations, Lett. Math. Phys. 109, 2587 (2019).
  15. S. Hollands, Relative entropy for coherent states in chiral CFT, Lett. Math. Phys. 110, 713 (2020).
  16. S. Hollands and A. Ishibashi, News versus information, Classical Quantum Gravity 36, 195001 (2019).
  17. H. Casini, S. Grillo, and D. Pontello, Relative entropy for coherent states from Araki formula, Phys. Rev. D 99, 125020 (2019).
  18. H. Casini, Relative entropy and the Bekenstein bound, Classical Quantum Gravity 25, 205021 (2008).

  19. More precisely, the von Neumann entropy is ill defined for local QFT algebras because sharp localization yields type III von Neumann algebras with infinitely many UV-entangled modes across any boundary, so that neither a trace nor a density matrix exists to make the entropy finite.

  20. For example, in the simpler case of quantum mechanics (type I algebras), the relative entropy is defined in terms of density operators ρ, ϱ as Srel(ρ∥ϱ)=Tr(ρlogρ)−Tr(ρlogϱ).

  21. G. Bianconi, Gravity from entropy, Phys. Rev. D 111, 066001 (2025).
  22. G. Bianconi, The quantum relative entropy of the Schwarzschild black hole and the area law, Entropy 27, 266 (2025).
  23. H. Araki, Relative entropy of states of von Neumann algebras, Publ. RIMS, Kyoto Univ. 11, 809 (1976).
  24. A. Uhlmann, Relative entropy and the Wigner-Yanase-Dyson-Lieb concavity in an interpolation theory, Commun. Math. Phys. 54, 21 (1977).
  25. M. Takesaki, Tomita’s Theory of Modular Hilbert Algebras and its Applications, Lecture Notes in Mathematics (Springer, New York, 1970).
  26. F. Kurpicz, N. Pinamonti, and R. Verch, Temperature and entropy-area relation of quantum matter near spherically symmetric outer trapping horizons, Lett. Math. Phys. 111, 110 (2021).
  27. B. S. Kay and R. M. Wald, Theorems on the uniqueness and thermal properties of stationary, nonsingular, quasifree states on spacetimes with a bifurcate Killing horizon, Phys. Rep. 207, 49 (1991).
  28. R. M. Wald, General Relativity (The University of Chicago Press, Chicago, 1984).
  29. S. J. Summers and R. Verch, Modular inclusion, the Hawking temperature, and quantum field theory in curved spacetime, Lett. Math. Phys. 37, 145 (1996).
  30. C. J. Fewster and K. Rejzner, Algebraic quantum field theory—an introduction, in Progress and Visions in Quantum Theory in View of Gravity: Bridging Foundations of Physics and Mathematics (Birkhäuser, Basel, Switzerland, 2020), pp. 1–61.
  31. R. Haag, Local Quantum Physics—Fields, Particles, Algebras, 2nd ed. (Springer, New York, 1996).
  32. R. Haag, H. Narnhofer, and U. Stein, On quantum field theory in gravitational background, Commun. Math. Phys. 94, 219 (1984).

  33. States satisfying the KMS condition generalize the notion of Gibbs ensembles beyond finite systems, see Refs. [34, 35, 36].

  34. R. Kubo, Statistical-mechanical theory of irreversible processes. I. General theory and simple applications to magnetic and conduction problems, J. Phys. Soc. Jpn. 12, 570 (1957).
  35. P. C. Martin and J. Schwinger, Theory of many-particle systems. I, Phys. Rev. 115, 1342 (1959).
  36. R. Haag, N. M. Hugenholtz, and M. Winnink, On the equilibrium states in quantum statistical mechanics, Commun. Math. Phys. 5, 215 (1967).
  37. H. J. Borchers, On revolutionizing quantum field theory with Tomita’s modular theory, J. Math. Phys. (N.Y.) 41, 3604 (2000).
  38. I. M. Gelfand and M. A. Naimark, On the imbedding of normed rings into the ring of operators in Hilbert space, Mat. Sb. 54, 197 (1943).
  39. I. E. Segal, Irreducible representations of operator algebras, Bull. Am. Math. Soc. 53, 73 (1947).
  40. J. J. Bisognano and E. H. Wichmann, On the duality condition for a Hermitian scalar field, J. Math. Phys. (N.Y.) 16, 985 (1975).

  41. For improved readability, we adopt the mild abuse of notation that W and Φ are implicitly understood in their appropriate representation, see Ref. [30] for details.

  42. Note that for noncoherent states, Formula (10) does not hold, so that the expression for the relative entropy generally becomes significantly more involved.

  43. In fact, the relative entropy can be interpreted as a Noether charge. This was explicitly proven for de Sitter spacetime in [44] and the argument extends to the case at hand. We also refer to [45] for a broader discussion.

  44. M. B. Fröb, A. Much, and K. Papadopoulos, Relative entropy in de Sitter spacetime is a Noether charge, Phys. Rev. D 108, 105004 (2023).
  45. V. Iyer and R. M. Wald, Some properties of the Noether charge and a proposal for dynamical black hole entropy, Phys. Rev. D 50, 846 (1994).

  46. Here, normal ordering is understood in the sense of Hadamard regularization.

  47. We use the fact that since W implements linear canonical transformations on the field, one thus has W:O:W†=:WOW†:.

  48. M. Combescure and D. Robert, Coherent States and Applications in Mathematical Physics, Theoretical and Mathematical Physics (Springer, New York, 2012).
  49. A. Degner, Properties of states of low energy on cosmological spacetimes, Dissertation, University of Hamburg, Hamburg, 2013.
  50. E. D’Angelo, Entropy for spherically symmetric, dynamical black holes from the relative entropy between coherent states of a scalar quantum field, Classical Quantum Gravity 38, 175001 (2021).
  51. E. D’Angelo, M. B. Fröb, S. Galanda, P. Meda, A. Much, and K. Papadopoulos, Entropy-area law and temperature of de Sitter horizons from modular theory, Prog. Theor. Exp. Phys. 2024, 021A01 (2024).

  52. Although the Rindler horizon cross section has infinite surface area, one can still meaningfully define its variation δA, as in Jacobson’s original argument [1]. We also refer to [53] for a related discussion yielding an area variation closely aligned with Eq. (20).

  53. E. Bianchi and A. Satz, Mechanical laws of the Rindler horizon, Phys. Rev. D 87, 124031 (2013).
  54. E. Poisson, A Relativist’s Toolkit. The Mathematics of Black-Hole Mechanics (Cambridge University Press, Cambridge, England, 2004).
  55. R. M. Wald, The back reaction effect in particle creation in curved spacetime, Commun. Math. Phys. 54, 1 (1977).
  56. S. L. Adler and J. Lieberman, Trace anomaly of the stress-energy tensor for massless vector particles propagating in a general background metric, Ann. Phys. (N.Y.) 113, 294 (1978).
  57. N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved Space, Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, England, 1982).
  58. S. Hollands and R. Longo, A new proof of the QNEC, Commun. Math. Phys. 406, 269 (2025).
  59. V. Morinelli, Y. Tanimoto, and B. Wegener, Modular operator for null plane algebras in free fields, Commun. Math. Phys. 395, 331 (2022).

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Phys. Rev. Lett. 136, 091602– Published 2 March, 2026
  • Received 12 November 2025
  • Accepted 9 February 2026
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