- Physical Review Journals
All JournalsPhysics Magazine

Physical Review X

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

Clustering of Conditional Mutual Information and Quantum Markov Structure at Arbitrary Temperatures

Tomotaka Kuwahara

PDF

Phys. Rev. X 15, 041010 – Published 16 October, 2025

DOI: https://doi.org/10.1103/9hx7-pzxw

Abstract

Recent investigations have unveiled exotic quantum phases that elude characterization by simple bipartite correlation functions. In these phases, long-range entanglement arising from tripartite correlations plays a central role. Consequently, the study of multipartite correlations has become a focal point in modern physics. Here, conditional mutual information (CMI) is one of the most well-established information-theoretic measures, adept at encapsulating the essence of various exotic phases, including topologically ordered ones. Within the realm of quantum many-body physics, it has been a long-sought goal to establish a quantum analog to the Hammersley-Clifford theorem that bridges the two concepts of the Gibbs state and the Markov network. This theorem posits that the correlation length of CMI remains short-range across all thermal equilibrium quantum phases. In this work, we demonstrate that CMI exhibits exponential decay with respect to distance, with its correlation length increasing polynomially with respect to the inverse temperature. While this clustering theorem has previously been believed to hold for high temperatures devoid of thermal phase transitions, it has remained elusive at low temperatures, where genuine long-range entanglement is corroborated to exist by the quantum topological order. Our findings unveil that, even at low temperatures, a broad class of tripartite entanglement cannot manifest in the long-range regime. To achieve the proof, we establish a comprehensive formalism for analyzing the locality of effective Hamiltonians on subsystems, commonly known as the “entanglement Hamiltonian” or “Hamiltonian of mean force.” As one outcome of our analyses, we enhance the prior clustering theorem concerning bipartite entanglement. In essence, we investigate genuine bipartite entanglement that extends beyond the limitations of the positive-partial-transpose class.

View figure in article

Physics Subject Headings (PhySH)

Popular Summary

One of the biggest challenges in quantum physics is understanding how particles can remain connected, or entangled, across a system. While researchers know how to describe simple two-particle correlations, more exotic phases of matter depend on complex multipartite correlations that are harder to analyze. A central question is whether all quantum systems in thermal equilibrium obey the quantum Markov property, which says that two distant regions become independent once the region separating them is accounted for. In this study, we show that this property indeed holds by proving that conditional mutual information (CMI), a key way of measuring three-part correlations, always decays rapidly with distance at any temperature.

To reach this conclusion, we introduce a method that focuses on the entanglement Hamiltonian, an effective description of the energy associated with part of a quantum system. By analyzing this quantity, we demonstrate that CMI remains short-ranged, even in situations where standard correlations appear to stretch across the entire system. Using this approach, we also establish that, in certain systems, genuine bipartite entanglement, which includes all forms of two-particle quantum correlations, decreases quickly with distance regardless of temperature.

The main takeaway is that quantum systems at equilibrium are more localized than previously thought when we look through the lens of CMI. This discovery not only clarifies the structure of quantum correlations but also has practical implications for quantum computing, where efficient algorithms depend on such properties. In the future, we plan to extend this framework to larger subsystems in higher dimensions and to identify which forms of multipartite entanglement can survive at very low temperatures.

Article Text

Supplemental Material

References (181)

  1. H. Araki, Gibbs states of a one dimensional quantum lattice, Commun. Math. Phys. 14, 120 (1969).
  2. Y. M. Park and H. J. Yoo, Uniqueness and clustering properties of Gibbs states for classical and quantum unbounded spin systems, J. Stat. Phys. 80, 223 (1995).
  3. D. Ueltschi, Cluster expansions and correlation functions, Moscow Math. J. 4, 511 (2004).
  4. M. Kliesch, C. Gogolin, M. J. Kastoryano, A. Riera, and J. Eisert, Locality of temperature, Phys. Rev. X 4, 031019 (2014).
  5. J. Fröhlich and D. Ueltschi, Some properties of correlations of quantum lattice systems in thermal equilibrium, J. Math. Phys. (N.Y.) 56, 053302 (2015).
  6. M. B. Hastings, Lieb-Schultz-Mattis in higher dimensions, Phys. Rev. B 69, 104431 (2004).
  7. M. B. Hastings, Locality in quantum and Markov dynamics on lattices and networks, Phys. Rev. Lett. 93, 140402 (2004).
  8. M. Hastings and T. Koma, Spectral gap and exponential decay of correlations, Commun. Math. Phys. 265, 781 (2006).
  9. B. Nachtergaele and R. Sims, Lieb-Robinson bounds and the exponential clustering theorem, Commun. Math. Phys. 265, 119 (2006).
  10. L. Amico, R. Fazio, A. Osterloh, and V. Vedral, Entanglement in many-body systems, Rev. Mod. Phys. 80, 517 (2008).
  11. J. I. Latorre and A. Riera, A short review on entanglement in quantum spin systems, J. Phys. A 42, 504002 (2009).
  12. N. Laflorencie, Quantum entanglement in condensed matter systems, Phys. Rep. 646, 1 (2016).
  13. J. I. Cirac, D. Pérez-García, N. Schuch, and F. Verstraete, Matrix product states and projected entangled pair states: Concepts, symmetries, theorems, Rev. Mod. Phys. 93, 045003 (2021).
  14. J. Eisert, M. Cramer, and M. B. Plenio, Colloquium: Area laws for the entanglement entropy, Rev. Mod. Phys. 82, 277 (2010).
  15. M. B. Hastings, An area law for one-dimensional quantum systems, J. Stat. Mech. (2007) P08024.
  16. I. Arad, A. Kitaev, Z. Landau, and U. Vazirani, An area law and sub-exponential algorithm for 1D systems, arXiv:1301.1162.
  17. A. Anshu, I. Arad, and D. Gosset, An area law for 2d frustration-free spin systems, in Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2022 (Association for Computing Machinery, New York, NY, USA, 2022), pp. 12–18.
  18. 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).
  19. T. Kuwahara, A. M. Alhambra, and A. Anshu, Improved thermal area law and quasilinear time algorithm for quantum Gibbs states, Phys. Rev. X 11, 011047 (2021).
  20. H. Bernigau, M. J. Kastoryano, and J. Eisert, Mutual information area laws for thermal free fermions, J. Stat. Mech. (2015) P02008.
  21. M. Lemm and O. Siebert, Thermal area law for lattice bosons, Quantum 7, 1083 (2023).
  22. F. Verstraete, J. J. García-Ripoll, and J. I. Cirac, Matrix product density operators: Simulation of finite-temperature and dissipative systems, Phys. Rev. Lett. 93, 207204 (2004).
  23. U. Schollwöck, The density-matrix renormalization group, Rev. Mod. Phys. 77, 259 (2005).
  24. Z. Landau, U. Vazirani, and T. Vidick, A polynomial time algorithm for the ground state of one-dimensional gapped local Hamiltonians, Nat. Phys. 11, 566 (2015).
  25. I. Arad, Z. Landau, U. Vazirani, and T. Vidick, Rigorous RG algorithms and area laws for low energy eigenstates in 1D, Commun. Math. Phys. 356, 65 (2017).
  26. T. Kuwahara and K. Saito, Exponential clustering of bipartite quantum entanglement at arbitrary temperatures, Phys. Rev. X 12, 021022 (2022).
  27. X. Chen, Z.-C. Gu, and X.-G. Wen, Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order, Phys. Rev. B 82, 155138 (2010).
  28. X.-G. Wen, Colloquium: Zoo of quantum-topological phases of matter, Rev. Mod. Phys. 89, 041004 (2017).
  29. M. Christandl and A. Winter, “Squashed entanglement”: An additive entanglement measure, J. Math. Phys. (N.Y.) 45, 829 (2004).
  30. M. Berta, K. P. Seshadreesan, and M. M. Wilde, Rényi generalizations of the conditional quantum mutual information, J. Math. Phys. (N.Y.) 56, 022205 (2015).
  31. D. Sutter, Approximate Quantum Markov Chains (Springer, Cham, 2018).
  32. A. Kitaev and J. Preskill, Topological entanglement entropy, Phys. Rev. Lett. 96, 110404 (2006).
  33. M. Levin and X.-G. Wen, Detecting topological order in a ground state wave function, Phys. Rev. Lett. 96, 110405 (2006).
  34. K. Kato, F. Furrer, and M. Murao, Information-theoretical analysis of topological entanglement entropy and multipartite correlations, Phys. Rev. A 93, 022317 (2016).
  35. D. Ding, P. Hayden, and M. Walter, Conditional mutual information of bipartite unitaries and scrambling, J. High Energy Phys. 12 (2016) 145.
  36. E. Iyoda and T. Sagawa, Scrambling of quantum information in quantum many-body systems, Phys. Rev. A 97, 042330 (2018).
  37. B. Bertini and L. Piroli, Scrambling in random unitary circuits: Exact results, Phys. Rev. B 102, 064305 (2020).
  38. G. L. Monaco, L. Innocenti, D. Cilluffo, D. A. Chisholm, S. Lorenzo, and G. M. Palma, Quantum scrambling via accessible tripartite information, Quantum Sci. Technol. 8, 035006 (2023).
  39. M. J. Gullans and D. A. Huse, Dynamical purification phase transition induced by quantum measurements, Phys. Rev. X 10, 041020 (2020).
  40. M. Ippoliti, M. J. Gullans, S. Gopalakrishnan, D. A. Huse, and V. Khemani, Entanglement phase transitions in measurement-only dynamics, Phys. Rev. X 11, 011030 (2021).
  41. J. M. Koh, S.-N. Sun, M. Motta, and A. J. Minnich, Measurement-induced entanglement phase transition on a superconducting quantum processor with mid-circuit readout, Nat. Phys. 19, 1314 (2023).
  42. P. Hayden, M. Headrick, and A. Maloney, Holographic mutual information is monogamous, Phys. Rev. D 87, 046003 (2013).
  43. M. Rota, Tripartite information of highly entangled states, J. High Energy Phys. 04 (2016) 075.
  44. V. Marić and M. Fagotti, Universality in the tripartite information after global quenches, Phys. Rev. B 108, L161116 (2023).
  45. V. Marić and M. Fagotti, Universality in the tripartite information after global quenches: (Generalised) quantum XY models, J. High Energy Phys. 06 (2023) 140.
  46. D. Petz, Sufficient subalgebras and the relative entropy of states of a von Neumann algebra, Commun. Math. Phys. 105, 123 (1986).
  47. O. Fawzi and R. Renner, Quantum conditional mutual information and approximate Markov chains, Commun. Math. Phys. 340, 575 (2015).
  48. F. G. S. L. Brandão, A. W. Harrow, J. Oppenheim, and S. Strelchuk, Quantum conditional mutual information, reconstructed states, and state redistribution, Phys. Rev. Lett. 115, 050501 (2015).
  49. M. M. Wilde, Recoverability in quantum information theory, Proc. R. Soc. A 471, 20150338 (2015).
  50. M. Junge, R. Renner, D. Sutter, M. M. Wilde, and A. Winter, Universal recovery maps and approximate sufficiency of quantum relative entropy, Ann. Henri Poincaré 19, 2955 (2018).
  51. D. Sutter and R. Renner, Necessary criterion for approximate recoverability, Ann. Henri Poincaré 19, 3007 (2018).
  52. J. M. Hammersley and P. Clifford, Markov fields on finite graphs and lattices (unpublished).
  53. M. Leifer and D. Poulin, Quantum graphical models and belief propagation, Ann. Phys. (Amsterdam) 323, 1899 (2008).
  54. W. Brown and D. Poulin, Quantum Markov networks and commuting Hamiltonians, arXiv:1206.0755.
  55. F. G. Jouneghani, M. Babazadeh, R. Bayramzadeh, and H. Movla, Investigation of commuting Hamiltonian in quantum Markov network, Int. J. Theor. Phys. 53, 2521 (2014).
  56. K. Kato and F. G. S. L. Brandão, Quantum approximate Markov chains are thermal, Commun. Math. Phys. 370, 117 (2019).
  57. T. Kuwahara, K. Kato, and F. G. S. L. Brandão, Clustering of conditional mutual information for quantum Gibbs states above a threshold temperature, Phys. Rev. Lett. 124, 220601 (2020).
  58. T. Kuwahara, K. Kato, and F. G. S. L. Brandão, Clustering of conditional mutual information for quantum Gibbs states above a threshold temperature, Phys. Rev. Lett. 124, 220601 (2020); 134, 199901(E) (2025).
  59. C. Castelnovo and C. Chamon, Topological order in a three-dimensional toric code at finite temperature, Phys. Rev. B 78, 155120 (2008).
  60. M. B. Hastings, Topological order at nonzero temperature, Phys. Rev. Lett. 107, 210501 (2011).
  61. L. Eldar and A. W. Harrow, Local Hamiltonians whose ground states are hard to approximate, in 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS) (IEEE, Berkeley, CA, USA, 2017), pp. 427–438.
  62. A. Anshu, N. P. Breuckmann, and C. Nirkhe, NLTS Hamiltonians from good quantum codes, in Proceedings of the 55th Annual ACM Symposium on Theory of Computing, STOC 2023 (Association for Computing Machinery, New York, NY, USA, 2023), pp. 1090–1096.
  63. D. H. Ackley, G. E. Hinton, and T. J. Sejnowski, A learning algorithm for Boltzmann machines, Cogn. Sci. 9, 147 (1985).
  64. R. Salakhutdinov and G. Hinton, Deep Boltzmann machines, in Proceedings of the 12th International Conference on Artificial Intelligence and Statistics, Proceedings of Machine Learning Research, Vol. 5, edited by D. van Dyk and M. Welling (PMLR, Hilton Clearwater Beach Resort, Clearwater Beach, Florida USA, 2009), pp. 448–455.
  65. A. Anshu, S. Arunachalam, T. Kuwahara, and M. Soleimanifar, Sample-efficient learning of interacting quantum systems, Nat. Phys. 17, 931 (2021).
  66. A. Anshu, S. Arunachalam, T. Kuwahara, and M. Soleimanifar, Efficient learning of commuting Hamiltonians on lattices, Electron. Notes 25 (2021), https://anuraganshu.seas.harvard.edu/sites/g/files/omnuum6156/files/anshu/files/learning_commuting_hamiltonian.pdf.
  67. D. Poulin and P. Wocjan, Sampling from the thermal quantum Gibbs state and evaluating partition functions with a quantum computer, Phys. Rev. Lett. 103, 220502 (2009).
  68. F. G. S. L. Brandão and M. J. Kastoryano, Finite correlation length implies efficient preparation of quantum thermal states, Commun. Math. Phys. 365, 1 (2019).
  69. I. H. Kim, Markovian marignals, arXiv:1609.08579.
  70. I. H. Kim, Entropy scaling law and the quantum marginal problem, Phys. Rev. X 11, 021039 (2021).
  71. S. L. Lauritzen, Graphical Models (Clarendon Press, New York, 1996), Vol. 17.
  72. D. Koller and N. Friedman, Probabilistic Graphical Models: Principles and Techniques (MIT Press, Cambridge, England, 2009).
  73. See Supplemental Material at http://link.aps.org/supplemental/10.1103/9hx7-pzxw for the technical details, which includes Refs. [74–109].
  74. M. J. Kastoryano and F. G. S. L. Brandão, Quantum Gibbs samplers: The commuting case, Commun. Math. Phys. 344, 915 (2016).
  75. I. H. Kim, Markovian matrix product density operators: Efficient computation of global entropy, arXiv:1709.07828.
  76. L. Lami, Non-classical correlations in quantum mechanics and beyond, arXiv:1803.02902.
  77. M. Berta and M. Tomamichel, Entanglement monogamy via multivariate trace inequalities, Commun. Math. Phys. 405, 29 (2024).
  78. B. Nachtergaele, Y. Ogata, and R. Sims, Propagation of correlations in quantum lattice systems, J. Stat. Phys. 124, 1 (2006).
  79. M. Foss-Feig, Z.-X. Gong, C. W. Clark, and A. V. Gorshkov, Nearly linear light cones in long-range interacting quantum systems, Phys. Rev. Lett. 114, 157201 (2015).
  80. C.-F. Chen and A. Lucas, Finite speed of quantum scrambling with long range interactions, Phys. Rev. Lett. 123, 250605 (2019).
  81. T. Kuwahara and K. Saito, Strictly linear light cones in long-range interacting systems of arbitrary dimensions, Phys. Rev. X 10, 031010 (2020).
  82. M. C. Tran, A. Y. Guo, C. L. Baldwin, A. Ehrenberg, A. V. Gorshkov, and A. Lucas, Lieb-Robinson light cone for power-law interactions, Phys. Rev. Lett. 127, 160401 (2021).
  83. R. Scharf, The Campbell-Baker-Hausdorff expansion for classical and quantum kicked dynamics, J. Phys. A 21, 2007 (1988).
  84. J. Haah, A. W. Harrow, Z. Ji, X. Wu, and N. Yu, Sample-optimal tomography of quantum states, in Proceedings of the 48th Annual ACM Symposium on Theory of Computing (Association for Computing Machinery, New York, NY, USA, 2016), pp. 913–925, 10.1145/2897518.2897585.
  85. T. Kuwahara, Fundamental inequalities in quantum many-body systems, Ph.D. thesis, University of Tokyo, 2015.
  86. A. Bluhm, A. Capel, P. Gondolf, and A. Pérez-Hernández, Continuity of quantum entropic quantities via almost convexity, IEEE Trans. Inf. Theory 69, 5869 (2023).
  87. G. Lindblad, Expectations and entropy inequalities for finite quantum systems, Commun. Math. Phys. 39, 111 (1974).
  88. A. Uhlmann, Relative entropy and the Wigner-Yanase-Dyson-Lieb concavity in an interpolation theory, Commun. Math. Phys. 54, 21 (1977).
  89. M. Berta, M. Lemm, and M. M. Wilde, Monotonicity of quantum relative entropy and recoverability, Quantum Inf. Comput. 15, 1333 (2015).
  90. K. M. R. Audenaert and J. Eisert, Continuity bounds on the quantum relative entropy, J. Math. Phys. (N.Y.) 46, 102104 (2005).
  91. I. Schur, Uber eine Klasse von Mittelbildungen mit Anwendungen auf die Determinantentheorie, Sitzungsber. Berliner Math. Ges. 22, 51 (1923).
  92. A. Horn, Doubly stochastic matrices and the diagonal of a rotation matrix, Am. J. Math. 76, 620 (1954).
  93. M. A. Nielsen and G. Vidal, Majorization and the interconversion of bipartite states, Quantum Inf. Comput. 1, 76 (2001).
  94. T. Kuwahara, T. V. Vu, and K. Saito, Effective light cone and digital quantum simulation of interacting bosons, Nat. Commun. 15, 2520 (2024).
  95. I. Pinelis, Exact lower and upper bounds on the incomplete gamma function, Math. Inequalities Appl. 23, 1261 (2020).
  96. S. O. Skrøvseth and S. D. Bartlett, Phase transitions and localizable entanglement in cluster-state spin chains with Ising couplings and local fields, Phys. Rev. A 80, 022316 (2009).
  97. A. C. Doherty and S. D. Bartlett, Identifying phases of quantum many-body systems that are universal for quantum computation, Phys. Rev. Lett. 103, 020506 (2009).
  98. S. Hernández-Santana, A. Riera, K. V. Hovhannisyan, M. Perarnau-Llobet, L. Tagliacozzo, and A. Acín, Locality of temperature in spin chains, New J. Phys. 17, 085007 (2015).
  99. A. Winter, Tight uniform continuity bounds for quantum entropies: Conditional entropy, relative entropy distance and energy constraints, Commun. Math. Phys. 347, 291 (2016).
  100. D. Sutter, O. Fawzi, and R. Renner, Universal recovery map for approximate Markov chains, Proc. R. Soc. A 472, 20150623 (2016).
  101. A. Bluhm, Á. Capel, and A. Pérez-Hernández, Exponential decay of mutual information for Gibbs states of local Hamiltonians, Quantum 6, 650 (2022).
  102. M. E. Shirokov, Tight uniform continuity bounds for the quantum conditional mutual information, for the Holevo quantity, and for capacities of quantum channels, J. Math. Phys. (N.Y.) 58, 102202 (2017).
  103. G. Bouch, Complex-time singularity and locality estimates for quantum lattice systems, J. Math. Phys. (N.Y.) 56, 123303 (2015).
  104. X. Bonet-Monroig, R. Babbush, and T. E. O’Brien, Nearly optimal measurement scheduling for partial tomography of quantum states, Phys. Rev. X 10, 031064 (2020).
  105. J. Cotler and F. Wilczek, Quantum overlapping tomography, Phys. Rev. Lett. 124, 100401 (2020).
  106. H.-Y. Huang, R. Kueng, and J. Preskill, Predicting many properties of a quantum system from very few measurements, Nat. Phys. 16, 1050 (2020).
  107. T. Kuwahara and K. Saito, Polynomial-time classical simulation for one-dimensional quantum Gibbs states, arXiv:1807.08424.
  108. S. L. Adler, Taylor expansion and derivative formulas for matrix logarithms.
  109. H. E. Haber, Notes on the matrix exponential and logarithm, Santa Cruz Institute for Particle Physics, University of California, Santa Cruz, CA, USA, 2018.
  110. P. Svetlichnyy and T. A. B. Kennedy, Decay of quantum conditional mutual information for purely generated finitely correlated states, J. Math. Phys. (N.Y.) 63, 072201 (2022).
  111. P. Svetlichnyy, S. Mittal, and T. A. B. Kennedy, Matrix product states and the decay of quantum conditional mutual information, J. Math. Phys. (N.Y.) 65, 022201 (2024).
  112. C.-F. Chen, K. Kato, and F. G. S. L. Brandão, Matrix product density operators: When do they have a local parent Hamiltonian?, arXiv:2010.14682.
  113. Y. Kimura and T. Kuwahara, Clustering theorem in 1D long-range interacting systems at arbitrary temperatures, Commun. Math. Phys. 406, 65 (2025).
  114. Donghoon Kim (private communication).
  115. P. Gondolf, S. O. Scalet, A. R. de Alarcon, A. M. Alhambra, and A. Capel, Conditional independence of 1D Gibbs states with applications to efficient learning, arXiv:2402.18500.
  116. V. P. Belavkin and P. Staszewski, C-algebraic generalization of relative entropy and entropy, Ann. l’I. H. P. Phys. Théor. 37, 51 (1982), https://www.numdam.org/item/AIHPA_1982__37_1_51_0.pdf.
  117. M. Horodecki, P. Horodecki, and R. Horodecki, Mixed-state entanglement and distillation: Is there a “bound” entanglement in nature?, Phys. Rev. Lett. 80, 5239 (1998).
  118. P. Horodecki, M. Horodecki, and R. Horodecki, Bound entanglement can be activated, Phys. Rev. Lett. 82, 1056 (1999).
  119. K. Audenaert, J. Eisert, E. Jané, M. B. Plenio, S. Virmani, and B. De Moor, Asymptotic relative entropy of entanglement, Phys. Rev. Lett. 87, 217902 (2001).
  120. K. Audenaert, B. De Moor, K. G. H. Vollbrecht, and R. F. Werner, Asymptotic relative entropy of entanglement for orthogonally invariant states, Phys. Rev. A 66, 032310 (2002).
  121. A. Miranowicz and S. Ishizaka, Closed formula for the relative entropy of entanglement, Phys. Rev. A 78, 032310 (2008).
  122. M. W. Girard, G. Gour, and S. Friedland, On convex optimization problems in quantum information theory, J. Phys. A 47, 505302 (2014).
  123. G. Vidal and R. F. Werner, Computable measure of entanglement, Phys. Rev. A 65, 032314 (2002).
  124. F. G. S. L. Brandão, M. Christandl, and J. Yard, Faithful squashed entanglement, Commun. Math. Phys. 306, 805 (2011).
  125. K. Li and A. Winter, Relative entropy and squashed entanglement, Commun. Math. Phys. 326, 63 (2014).
  126. H. Jeng, S. Tserkis, J. Y. Haw, H. M. Chrzanowski, J. Janousek, T. C. Ralph, P. K. Lam, and S. M. Assad, Entanglement properties of a measurement-based entanglement distillation experiment, Phys. Rev. A 99, 042304 (2019).
  127. V. Vedral, M. B. Plenio, M. A. Rippin, and P. L. Knight, Quantifying entanglement, Phys. Rev. Lett. 78, 2275 (1997).
  128. C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, Mixed-state entanglement and quantum error correction, Phys. Rev. A 54, 3824 (1996).
  129. H. Li and F. D. M. Haldane, Entanglement spectrum as a generalization of entanglement entropy: Identification of topological order in non-Abelian fractional quantum Hall effect states, Phys. Rev. Lett. 101, 010504 (2008).
  130. I. Peschel and V. Eisler, Reduced density matrices and entanglement entropy in free lattice models, J. Phys. A 42, 504003 (2009).
  131. E. Bilgin and D. Poulin, Coarse-grained belief propagation for simulation of interacting quantum systems at all temperatures, Phys. Rev. B 81, 054106 (2010).
  132. F. Parisen Toldin and F. F. Assaad, Entanglement Hamiltonian of interacting fermionic models, Phys. Rev. Lett. 121, 200602 (2018).
  133. V. Eisler, G. D. Giulio, E. Tonni, and I. Peschel, Entanglement Hamiltonians for non-critical quantum chains, J. Stat. Mech. (2020) 103102.
  134. C. Kokail, R. van Bijnen, A. Elben, B. Vermersch, and P. Zoller, Entanglement Hamiltonian tomography in quantum simulation, Nat. Phys. 17, 936 (2021).
  135. T. V. Zache, C. Kokail, B. Sundar, and P. Zoller, Entanglement spectroscopy and probing the Li-Haldane conjecture in topological quantum matter, Quantum 6, 702 (2022).
  136. P. Talkner and P. Hänggi, Colloquium: Statistical mechanics and thermodynamics at strong coupling: Quantum and classical, Rev. Mod. Phys. 92, 041002 (2020).
  137. J. D. Cresser and J. Anders, Weak and ultrastrong coupling limits of the quantum mean force Gibbs state, Phys. Rev. Lett. 127, 250601 (2021).
  138. A. S. Trushechkin, M. Merkli, J. D. Cresser, and J. Anders, Open quantum system dynamics and the mean force Gibbs state, AVS Quantum Sci. 4, 012301 (2022).
  139. P. C. Burke, G. Nakerst, and M. Haque, Structure of the Hamiltonian of mean force, Phys. Rev. E 110, 014111 (2024).

  140. This can happen when the temperature is sufficiently low. For example, consider a Hamiltonian whose ground state is a product state (e. g., the XX model with magnetic fields). At low temperatures, numerical simulations indicate that nonlocal interactions can be induced by tracing out the spin at the right end (or the left end). In this example, the CMI is small at low temperatures because the quantum Gibbs state is close to a product state.

  141. H. Hakoshima, S. Endo, K. Yamamoto, Y. Matsuzaki, and N. Yoshioka, Localized virtual purification, Phys. Rev. Lett. 133, 080601 (2024).
  142. M. B. Hastings, Quantum belief propagation: An algorithm for thermal quantum systems, Phys. Rev. B 76, 201102(R) (2007).
  143. I. H. Kim, Perturbative analysis of topological entanglement entropy from conditional independence, Phys. Rev. B 86, 245116 (2012).
  144. D. Pérez-García and A. Pérez-Hernández, Locality estimates for complex time evolution in 1D, Commun. Math. Phys. 399, 929 (2023).
  145. A. Bakshi, A. Liu, A. Moitra, and E. Tang, Learning quantum Hamiltonians at any temperature in polynomial time, in Proceedings of the 56th Annual ACM Symposium on Theory of Computing, STOC 2024 (Association for Computing Machinery, New York, NY, USA, 2024), pp. 1470–1477.
  146. J. Haah, R. Kothari, and E. Tang, Learning quantum Hamiltonians from high-temperature Gibbs states and real-time evolutions, Nat. Phys. 20, 1027 (2024).
  147. S. Narayanan, Improved algorithms for learning quantum Hamiltonians, via flat polynomials, arXiv:2407.04540.
  148. E. Lieb and D. Robinson, The finite group velocity of quantum spin systems, Commun. Math. Phys. 28, 251 (1972).
  149. S. Bravyi, M. B. Hastings, and F. Verstraete, Lieb-Robinson bounds and the generation of correlations and topological quantum order, Phys. Rev. Lett. 97, 050401 (2006).
  150. S. Blanes, F. Casas, J. Oteo, and J. Ros, The Magnus expansion and some of its applications, Phys. Rep. 470, 151 (2009).
  151. M. Bukov, L. D’Alessio, and A. Polkovnikov, Universal high-frequency behavior of periodically driven systems: From dynamical stabilization to Floquet engineering, Adv. Phys. 64, 139 (2015).
  152. T. Kuwahara, T. Mori, and K. Saito, Floquet-Magnus theory and generic transient dynamics in periodically driven many-body quantum systems, Ann. Phys. (Amsterdam) 367, 96 (2016).
  153. T. Mori, T. Kuwahara, and K. Saito, Rigorous bound on energy absorption and generic relaxation in periodically driven quantum systems, Phys. Rev. Lett. 116, 120401 (2016).
  154. M. B. Hastings, On Lieb-Robinson bounds for the double bracket flow, arXiv:2201.07141.
  155. R. Alicki and M. Fannes, Continuity of quantum conditional information, J. Phys. A 37, L55 (2004).
  156. I. Arad, T. Kuwahara, and Z. Landau, Connecting global and local energy distributions in quantum spin models on a lattice, J. Stat. Mech. (2016) 033301.
  157. C. Castelnovo and C. Chamon, Entanglement and topological entropy of the toric code at finite temperature, Phys. Rev. B 76, 184442 (2007).
  158. D. Mazáč and A. Hamma, Topological order, entanglement, and quantum memory at finite temperature, Ann. Phys. (Amsterdam) 327, 2096 (2012).
  159. Y. A. Lee and G. Vidal, Entanglement negativity and topological order, Phys. Rev. A 88, 042318 (2013).
  160. M. Gabbrielli, A. Smerzi, and L. Pezzè, Multipartite entanglement at finite temperature, Sci. Rep. 8, 15663 (2018).
  161. H. Shapourian and S. Ryu, Finite-temperature entanglement negativity of free fermions, J. Stat. Mech. (2019) 043106.
  162. T.-C. Lu, T. H. Hsieh, and T. Grover, Detecting topological order at finite temperature using entanglement negativity, Phys. Rev. Lett. 125, 116801 (2020).
  163. K.-H. Wu, T.-C. Lu, C.-M. Chung, Y.-J. Kao, and T. Grover, Entanglement Renyi negativity across a finite temperature transition: A Monte Carlo study, Phys. Rev. Lett. 125, 140603 (2020).
  164. T.-C. Lu and T. Grover, Structure of quantum entanglement at a finite temperature critical point, Phys. Rev. Res. 2, 043345 (2020).
  165. D. Kim, T. Kuwahara, and K. Saito, Thermal area law in long-range interacting systems, Phys. Rev. Lett. 134, 020402 (2025).
  166. R. Achutha, D. Kim, Y. Kimura, and T. Kuwahara, Provably efficient simulation of 1D long-range interacting systems at any temperature, Phys. Rev. Lett. 134, 190404 (2025).
  167. A. Miyake, Classification of multipartite entangled states by multidimensional determinants, Phys. Rev. A 67, 012108 (2003).
  168. R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Quantum entanglement, Rev. Mod. Phys. 81, 865 (2009).
  169. S. Szalay, Multipartite entanglement measures, Phys. Rev. A 92, 042329 (2015).
  170. S. Xie and J. H. Eberly, Triangle measure of tripartite entanglement, Phys. Rev. Lett. 127, 040403 (2021).
  171. J. L. Beckey, N. Gigena, P. J. Coles, and M. Cerezo, Computable and operationally meaningful multipartite entanglement measures, Phys. Rev. Lett. 127, 140501 (2021).
  172. M. Ma, Y. Li, and J. Shang, Multipartite entanglement measures: A review, Fundamental Research 10.1016/j. fmre.2024.03.031 (2024).
  173. D. Poilblanc, Entanglement spectra of quantum Heisenberg ladders, Phys. Rev. Lett. 105, 077202 (2010).
  174. C. Li, R.-Z. Huang, Y.-M. Ding, Z. Y. Meng, Y.-C. Wang, and Z. Yan, Relevant long-range interaction of the entanglement Hamiltonian emerges from a short-range gapped system, Phys. Rev. B 109, 195169 (2024).
  175. G. Giudici, T. Mendes-Santos, P. Calabrese, and M. Dalmonte, Entanglement Hamiltonians of lattice models via the Bisognano-Wichmann theorem, Phys. Rev. B 98, 134403 (2018).
  176. C.-F. Chen, A. Anshu, and Q. T. Nguyen, Learning quantum Gibbs states locally and efficiently, arXiv:2504.02706.
  177. S. U. Lee, C. Oh, Y. Wong, S. Chen, and L. Jiang, Universal spreading of conditional mutual information in noisy random circuits, Phys. Rev. Lett. 133, 200402 (2024).
  178. Y. Zhang and S. Gopalakrishnan, Nonlocal growth of quantum conditional mutual information under decoherence, Phys. Rev. A 110, 032426 (2024).
  179. S. Sang and T. H. Hsieh, Stability of mixed-state quantum phases via finite Markov length, Phys. Rev. Lett. 134, 070403 (2025).
  180. K. Kato and T. Kuwahara, Clustering of conditional mutual information via quantum belief-propagation channels, arXiv:2504.02235.
  181. C.-F. Chen and C. Rouzé, Quantum Gibbs states are locally Markovian, arXiv:2504.02208.

Outline

Information

Phys. Rev. X 15, 041010– Published 16 October, 2025
  • Received 23 July 2024
  • Revised 7 May 2025
  • Accepted 19 September 2025
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.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation