- Physical Review Journals
All JournalsPhysics Magazine

Physical Review D

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Geometric Rényi mutual information induced by localized particle excitations in quantum field theory

Willy A. Izquierdo1,*, David R. Junior1,2,3,†, and Gastão Krein1,‡

  • *Contact author: willy.izquierdo@unesp.br
  • †Contact author: david.rosa-junior@unesp.br
  • ‡Contact author: gastao.krein@unesp.br
PDF

Phys. Rev. D 113, 125001 – Published 2 June, 2026

DOI: https://doi.org/10.1103/fbz4-rr46

Abstract

Quantum field theory exhibits rich spatial correlation structures even in the vacuum, where entanglement entropy between regions scales with the area of their shared boundary. While this vacuum structure has been extensively studied, far less is understood about how localized wave packets influence correlations between field values in different spatial regions. In this work, we use the Schrödinger representation to study the Rényi mutual information between complementary spatial regions for a localized wave packet of a free massless scalar field in (d+1) dimensions. We find that the mutual information in this excited state includes both a vacuum term and an excitation-induced contribution. To obtain quantitative results, we specialize to 1+1 dimensions and evaluate the Rényi-2 mutual information between the negative and positive halves of the real line. We find that the excitation generates finite, positive correlations that are maximized when the wave packet sits at the boundary and decreases with its distance from it, at a rate determined by the wave packet’s width. Our findings constitute a contribution toward a more comprehensive understanding of quantum correlations in multiparticle states with nontrivial spatial localization, analyzed within a quantum field-theoretical framework.

View figure in article

Physics Subject Headings (PhySH)

See Also

Geometry of the Shannon mutual information in continuum QFT

David R. Junior and Luis E. Oxman
Phys. Rev. D 95, 125005 (2017)

Article Text

References (43)

  1. M. Srednicki, Entropy and area, Phys. Rev. Lett. 71, 666 (1993).
  2. C. Holzhey, F. Larsen, and F. Wilczek, Geometric and renormalized entropy in conformal field theory, Nucl. Phys. B424, 443 (1994).
  3. P. Calabrese and J. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. (2004) P06002.
  4. H. Casini and M. Huerta, Entanglement entropy in free quantum field theory, J. Phys. A 42, 504007 (2009).
  5. T. Nishioka, Entanglement entropy: Holography and renormalization group, Rev. Mod. Phys. 90, 035007 (2018).
  6. J. Lee, L. McGough, and B. R. Safdi, Rényi entropy and geometry, Phys. Rev. D 89, 125016 (2014).
  7. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, England, 2000).
  8. B. Groisman, S. Popescu, and A. Winter, Quantum, classical, and total amount of correlations in a quantum state, Phys. Rev. A 72, 032317 (2005).
  9. M. M. Wolf, F. Verstraete, M. B. Hastings, and J. I. Cirac, Area laws in quantum systems: Mutual information and correlations, Phys. Rev. Lett. 100, 070502 (2008).
  10. S. O. Scalet, Á. M. Alhambra, G. Styliaris, and J. I. Cirac, Computable Rényi mutual information: Area laws and correlations, Quantum 5, 541 (2021).
  11. V. Balasubramanian, M. B. McDermott, and M. Van Raamsdonk, Momentum-space entanglement and renormalization in quantum field theory, Phys. Rev. D 86, 045014 (2012).
  12. T.-C. L. Hsu, M. B. McDermott, and M. Van Raamsdonk, Momentum-space entanglement for interacting fermions at finite density, J. High Energy Phys. 11 (2013) 121.
  13. M. H. M. Costa, J. v. d. Brink, F. S. Nogueira, and G. I. Krein, Momentum space entanglement from the Wilsonian effective action, Phys. Rev. D 106, 065024 (2022).
  14. M. Gluza, T. Schweigler, B. Rauer, C. Krumnow, J. Schmiedmayer, and J. Eisert, Quantum read-out for cold atomic quantum simulators, Commun. Phys. 3, 12 (2020).
  15. M. Tajik, I. Kukuljan, S. Sotiriadis, B. Rauer, T. Schweigler, F. Cataldini, J. Sabino, F. Møller, P. Schüttelkopf, S.-C. Ji, D. Sels, E. Demler, and J. Schmiedmayer, Verification of the area law of mutual information in a quantum field simulator, Nat. Phys. 19, 1022 (2023).
  16. D. R. Junior and L. E. Oxman, Geometry of the shannon mutual information in continuum QFT, Phys. Rev. D 95, 125005 (2017).
  17. J.-M. Stéphan, S. Furukawa, G. Misguich, and V. Pasquier, Shannon and entanglement entropies of one- and two-dimensional critical wave functions, Phys. Rev. B 80, 184421 (2009).
  18. J.-M. Stéphan, G. Misguich, and V. Pasquier, Phase transition in the Rényi-Shannon entropy of Luttinger liquids, Phys. Rev. B 84, 195128 (2011).
  19. M. P. Zaletel, J. H. Bardarson, and J. E. Moore, Logarithmic terms in entanglement entropies of 2D quantum critical points and Shannon entropies of spin chains, Phys. Rev. Lett. 107, 020402 (2011).
  20. M. G. Nezhadhaghighi and M. A. Rajabpour, Quantum entanglement entropy and classical mutual information in long-range harmonic oscillators, Phys. Rev. B 88, 045426 (2013).
  21. F. C. Alcaraz and M. A. Rajabpour, Universal behavior of the Shannon mutual information of critical quantum chains, Phys. Rev. Lett. 111, 017201 (2013).
  22. F. C. Alcaraz and M. A. Rajabpour, Generalized mutual information of quantum critical chains, Phys. Rev. B 91, 155122 (2015).
  23. F. C. Alcaraz, M. I. n. Berganza, and G. Sierra, Entanglement of low-energy excitations in conformal field theory, Phys. Rev. Lett. 106, 201601 (2011).
  24. T. Pálmai, Excited state entanglement in one-dimensional quantum critical systems: Extensivity and the role of microscopic details, Phys. Rev. B 90, 161404 (2014).
  25. O. A. Castro-Alvaredo, C. D. Fazio, B. Doyon, and I. M. Szécsényi, Entanglement content of quantum particle excitations. Part I. Free field theory, J. High Energy Phys. 10 (2018) 039.
  26. O. A. Castro-Alvaredo, C. D. Fazio, B. Doyon, and I. M. Szécsényi, Entanglement content of quantum particle excitations. part II. Disconnected regions and logarithmic negativity, J. High Energy Phys. 11 (2019) 058.
  27. O. A. Castro-Alvaredo, C. De Fazio, B. Doyon, and I. M. Szécsényi, Entanglement content of quantum particle excitations. III. Graph partition functions, J. Math. Phys. (N.Y.) 60, 082301 (2019).
  28. J. Zhang and M. A. Rajabpour, Excited state Rényi entropy and subsystem distance in two-dimensional non-compact bosonic theory. Part I. Single-particle states, J. High Energy Phys. 12 (2020) 160.
  29. J. Zhang and M. A. Rajabpour, Excited state Rényi entropy and subsystem distance in two-dimensional non-compact bosonic theory. Part II. Multi-particle states, J. High Energy Phys. 08 (2021) 106.
  30. D. D. Blanco, H. Casini, L.-Y. Hung, and R. C. Myers, Relative entropy and holography, J. High Energy Phys. 08 (2013) 060.
  31. G. Wong, I. Klich, L. A. Pando Zayas, and D. Vaman, Entanglement temperature and entanglement entropy of excited states, J. High Energy Phys. 12 (2013) 020.
  32. L. Caffarelli and L. Silvestre, An extension problem related to the fractional laplacian, Commun. Partial Differ. Eq. 32, 1245 (2007).
  33. N. S. Landkof, Foundations of Modern Potential Theory (Springer-Verlag, Berlin, 1972).
  34. B. Hatfield, Quantum Field Theory of Point Particles and Strings, Frontiers in Physics (Westview Press, Boca Raton, 1998).
  35. Q.-Y. GUAN and Z.-M. MA, Boundary problems for fractional Laplacians, Stochastics Dyn. 05, 385 (2005).
  36. L. A. Caffarelli and P. R. Sting a, Fractional elliptic equations caccioppoli estimates and regularity, Ann. l’Inst. Henri Poincaré C 33, 767 (2016).
  37. C. Bucur, Some observations on the Green function for the ball in the fractional Laplace framework, Commun. Pure Appl. Anal. 15, 657 (2016).
  38. L. A. Rubel, Necessary and sufficient conditions for Carlson’s theorem on entire functions, Proc. Natl. Acad. Sci. U.S.A. 41, 601 (1955).
  39. J. L. Cardy, O. A. Castro-Alvaredo, and B. Doyon, Form factors of branch-point twist fields in quantum integrable models and entanglement entropy, J. Stat. Phys. 130, 129 (2008).
  40. O. A. Castro-Alvaredo and B. Doyon, Bi-partite entanglement entropy in integrable models with backscattering, J. Phys. A 41, 275203 (2008).
  41. W. A. I. Inga, Shannon mutual information of localized one-particle states in scalar quantum field theory, Dissertação (mestrado), IFT–D.005/25, Instituto de Física Teórica, Universidade Estadual Paulista (UNESP), São Paulo, Brazil, 2025.
  42. M. Pavšič, Localized states in quantum field theory, Adv. Appl. Clifford Algebr. 28, 89 (2018).
  43. M. Al-Hashimi and U.-J. Wiese, Minimal position-velocity uncertainty wave packets in relativistic and non-relativistic quantum mechanics, Ann. Phys. (Amsterdam) 324, 2599 (2009).

Outline

Information

Phys. Rev. D 113, 125001– Published 2 June, 2026
  • Received 2 December 2025
  • Accepted 13 April 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation