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Modular self-duality, symmetrized relative entropy, and Bogoliubov-Kubo-Mori susceptibility in quantum field theory

Rupak Chatterjee*

  • Department of Applied Physics, New York University, 2 MetroTech Center, Brooklyn, New York 11201, USA

  • *Contact author: Rupak.Chatterjee@nyu.edu
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Phys. Rev. D 114, 065016 – Published 14 September, 2026

DOI: https://doi.org/10.1103/vxkq-xb7l

Abstract

We develop an operator-algebraic framework for modular self-duality, symmetrized relative entropy, and Bogoliubov-Kubo-Mori (BKM) susceptibility of local states in quantum field theory. In finite dimensions, modular self-duality singles out fixed points at which a state coincides with its modularly reflected partner. At such points, the natural comparison functional is the symmetrized Umegaki relative entropy. It vanishes at coincidence, and its Hessian is governed by the BKM quantum Fisher information along the reflected tangent direction. We then extend this fixed-point construction to the local type III von Neumann algebras that arise in quantum field theory. Here, a local state is compared with the modular pullback of its commutant restriction, and the intrinsic comparison functional is the symmetrized Araki relative entropy. For sufficiently regular state deformations, the fixed-localization Hessian at the self-dual point defines a type III BKM susceptibility. This coefficient is obtained by evaluating the BKM bilinear form on the tangent selected by the modular pairing. Exact coherent-state realizations are obtained for the free scalar field on wedge algebras and for the chiral U(1) current on half-line algebras. In both examples, the comparison functional is exactly quadratic in the deformation parameter, and the susceptibility coefficients admit explicit boost-energy, stress-tensor, or half-line integral representations. We also apply the construction to a ball-shaped boundary region in the vacuum state of a conformal field theory with a semiclassical anti–de Sitter (AdS) gravitational dual. For linear bulk coherent excitations in an exact complementary-recovery model of this sector, bulk-boundary modular intertwining identifies the modular partner with the AdS-Rindler-reflected coherent state, and the symmetrized relative entropy is exactly quadratic. Its Hessian is twice the bulk canonical energy of the reflected-difference perturbation.

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References (44)

  1. R. Haag, Local Quantum Physics: Fields, Particles, Algebras, 2nd ed. (Springer-Verlag, Berlin, 1996).
  2. E. Witten, APS Medal for Exceptional Achievement in Research: Invited article on entanglement properties of quantum field theory, Rev. Mod. Phys. 90, 045003 (2018).
  3. R. Chatterjee, Tomita–Takesaki theory and quantum concurrence, Phys. Rev. D 110, 065012 (2024).
  4. R. Chatterjee, Entanglement harvesting and curvature of entanglement: A modular operator approach, Phys. Rev. D 112, 085025 (2025).
  5. D. Petz, Quasi-entropies for states of a von Neumann algebra, Publ. Res. Inst. Math. Sci. Kyoto Univ. 21, 787 (1985).
  6. A. Lesniewski and M. B. Ruskai, Monotone Riemannian metrics and relative entropy on noncommutative probability spaces, J. Math. Phys. (N.Y.) 40, 5702 (1999).
  7. A. Jenčová, Quantum information geometry and non-commutative Lp spaces, Infinite Dimensional Analysis, Quantum Probab. Relat. Top. 8, 215 (2005).
  8. Z. H. Saleem, A. Shaji, A. M. Babu, D.-W. Luo, Q. Langfitt, T. Yu, and S. K. Gray, Quantum Fisher information and the curvature of entanglement, arXiv:2504.13729.
  9. H. Araki, Relative entropy of states of von Neumann algebras, Publ. RIMS 11, 809 (1975).
  10. H. Araki, Relative entropy for states of von Neumann algebras II, Publ. RIMS 13, 173 (1977).
  11. H. Casini, S. Grillo, and D. Pontello, Relative entropy for coherent states from Araki formula, Phys. Rev. D 99, 125020 (2019).
  12. R. Longo, Entropy distribution of localised states, Commun. Math. Phys. 373, 473 (2020).
  13. R. Longo, Entropy of coherent excitations, Lett. Math. Phys. 109, 2587 (2019).
  14. H. Bostelmann, D. Cadamuro, and S. Del Vecchio, Relative entropy of coherent states on general CCR algebras, Commun. Math. Phys. 389, 661 (2022).
  15. S. Hollands, Relative entropy for coherent states in chiral CFT, Lett. Math. Phys. 110, 713 (2020).
  16. A. Garbarz and G. Palau, Relative entropy of an interval for a massless boson at finite temperature, Phys. Rev. D 107, 125016 (2023).
  17. J. J. Bisognano and E. H. Wichmann, On the duality condition for a Hermitian scalar field, J. Math. Phys. (N.Y.) 16, 985 (1975).
  18. J. J. Bisognano and E. H. Wichmann, On the duality condition for quantum fields, J. Math. Phys. (N.Y.) 17, 303 (1976).
  19. J. M. Maldacena, The large-N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2, 231 (1998).
  20. S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, Gauge theory correlators from non-critical string theory, Phys. Lett. B 428, 105 (1998).
  21. E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2, 253 (1998).
  22. S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96, 181602 (2006).
  23. X. Dong, D. Harlow, and A. C. Wall, Reconstruction of bulk operators within the entanglement wedge in Gauge-gravity duality, Phys. Rev. Lett. 117, 021601 (2016).
  24. H. Casini, M. Huerta, and R. C. Myers, Towards a derivation of holographic entanglement entropy, J. High Energy Phys. 05 (2011) 036.
  25. 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.
  26. N. Lashkari and M. Van Raamsdonk, Canonical energy is quantum Fisher information, J. High Energy Phys. 04 (2016) 153.
  27. D. L. Jafferis, A. Lewkowycz, J. Maldacena, and S. J. Suh, Relative entropy equals bulk relative entropy, J. High Energy Phys. 06 (2016) 004.
  28. S. Hollands and R. M. Wald, Stability of black holes and black branes, Commun. Math. Phys. 321, 629 (2013).
  29. F. Benini, V. Godet, and A. H. Singh, Entanglement asymmetry in conformal field theory and holography, Prog. Theor. Exp. Phys. 2025, 063B05 (2025).
  30. D. Petz and G. Tóth, The Bogoliubov inner product in quantum statistics, Lett. Math. Phys. 27, 205 (1993).
  31. O. Bratteli and D. Robinson, Operator Algebras and Quantum Statistical Mechanics II, 2nd ed. (Springer Verlag, Berlin, 1997)
  32. T. Faulkner, The holographic map as a conditional expectation, arXiv:2008.04810.
  33. H. Umegaki, Conditional expectation in an operator algebra. IV. Entropy and information, Kodai Math. Sem. Rep. 14, 59 (1962).
  34. D. Petz, Monotone metrics on matrix spaces, Linear Algebra Appl. 244, 81 (1996).
  35. P. W. Michor, D. Petz, and A. Andai, On the curvature of a certain Riemannian space of matrices, Inf. Dim. Anal. Quant. Prob. Rel. Topics 03, 199 (2000).
  36. D. Guido, Modular theory for the von Neumann Algebras of Local Quantum Physics, in Aspects of Operator Algebras and Applications, edited by P. Ara, F. Lledo, and F. Perera (American Mathematical Society, Providence, 2011).
  37. H. Hasegawa, α-Divergence of the non-commutative information geometry, Rep. Math. Phys. 33, 87 (1993).
  38. J. Dereziński and C. Gérard, Mathematics of Quantization and Quantum Fields (Cambridge University Press, Cambridge, England, 2013).
  39. H. J. Borchers and J. Yngvason, Modular groups of quantum fields in thermal states, J. Math. Phys. (N.Y.) 40, 601 (1999).
  40. P. D. Hislop and R. Longo, Modular structure of the local algebras associated with the free massless scalar field theory, Commun. Math. Phys. 84, 71 (1982).
  41. R. Brunetti, D. Guido, and R. Longo, Modular structure and duality in conformal quantum field theory, Commun. Math. Phys. 156, 201 (1993).
  42. V. Iyer and R. M. Wald, Some properties of Noether charge and a proposal for dynamical black hole entropy, Phys. Rev. D 50, 846 (1994).
  43. W. Donnelly and L. Freidel, Local subsystems in gauge theory and gravity, J. High Energy Phys. 09 (2016) 102.
  44. D. Harlow and J.-q. Wu, Covariant phase space with boundaries, J. High Energy Phys. 10 (2020) 146.

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Phys. Rev. D 114, 065016– Published 14 September, 2026
  • Received 22 May 2026
  • Accepted 27 August 2026
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