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Certifying steady-state properties of open quantum systems

Luke Mortimer1,*, Donato Farina2, Grazia Di Bello2, David Jansen1, Andreas Leitherer1, Pere Mujal1, and Antonio Acín1,3

  • *Contact author: lbmortimer1998@gmail.com
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Phys. Rev. Research 7, 033237 – Published 10 September, 2025

DOI: https://doi.org/10.1103/hbrt-cn8q

Abstract

Estimating the steady-state properties of open many-body quantum systems is a fundamental challenge in quantum science and technologies. In this work, we present a scalable approach based on semidefinite programming to derive certified bounds on the expectation value of an arbitrary observable in the steady state of Lindbladian dynamics. We illustrate our method on a series of many-body systems, including paradigmatic spin-1/2 chains and two-dimensional ladders, considering both equilibrium and nonequilibrium steady states. We benchmark our method with state-of-the-art tensor-network approaches that, unlike our method, are only able to provide estimates, with no guarantee, on steady-state quantities. For the tested models, only modest computational effort is needed to obtain certified nontrivial bounds for system sizes intractable by exact methods. Our method introduces the first general numerical tool for bounding steady-state properties of open quantum dynamics, opening an avenue in the understanding of stable configurations in many-body systems.

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References (48)

  1. H. Carmichael, An Open Systems Approach to Quantum Optics: Lectures Presented at the Université Libre de Bruxelles, October 28 to November 4, 1991 (Springer Science & Business Media, Berlin, Heidelberg, 2009), Vol. 18.
  2. A. Rivas and S. F. Huelga, Open Quantum Systems (Springer, Berlin, Heidelberg, 2012), Vol. 10.
  3. H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, USA, 2002).
  4. S. Lieu, R. Belyansky, J. T. Young, R. Lundgren, V. V. Albert, and A. V. Gorshkov, Symmetry breaking and error correction in open quantum systems, Phys. Rev. Lett. 125, 240405 (2020).
  5. G. T. Landi, A. L. Fonseca de Oliveira, and E. Buksman, Thermodynamic analysis of quantum error-correcting engines, Phys. Rev. A 101, 042106 (2020).
  6. R. Fazio, J. Keeling, L. Mazza, and M. Schirò, Many-body open quantum systems, SciPost Phys. Lect. Notes, 099 (2025).
  7. Y. Ilin and I. Arad, Learning a quantum channel from its steady-state, New J. Phys. 26, 073003 (2024).
  8. R. Rota, F. Storme, N. Bartolo, R. Fazio, and C. Ciuti, Critical behavior of dissipative two-dimensional spin lattices, Phys. Rev. B 95, 134431 (2017).
  9. J. W. Z. Lau, K. H. Lim, K. Bharti, L.-C. Kwek, and S. Vinjanampathy, Convex optimization for nonequilibrium steady states on a hybrid quantum processor, Phys. Rev. Lett. 130, 240601 (2023).
  10. D. Morrone, N. W. Talarico, M. Cattaneo, and M. A. C. Rossi, Estimating molecular thermal averages with the quantum equation of motion and informationally complete measurements, Entropy 26, 722 (2024).
  11. A. Nagy and V. Savona, Driven-dissipative quantum Monte Carlo method for open quantum systems, Phys. Rev. A 97, 052129 (2018).
  12. F. Vicentini, A. Biella, N. Regnault, and C. Ciuti, Variational neural-network ansatz for steady states in open quantum systems, Phys. Rev. Lett. 122, 250503 (2019).
  13. A. Nagy and V. Savona, Variational quantum Monte Carlo method with a neural-network ansatz for open quantum systems, Phys. Rev. Lett. 122, 250501 (2019).
  14. D. A. Hryniuk and M. H. Szymańska, Tensor-network-based variational Monte Carlo approach to the non-equilibrium steady state of open quantum systems, Quantum 8, 1475 (2024).
  15. J. Thingna and D. Manzano, Degenerated liouvillians and steady-state reduced density matrices, Chaos 31, 073114 (2021).
  16. D. Amato and P. Facchi, Number of steady states of quantum evolutions, Sci. Rep. 14, 14366 (2024).
  17. J. Guo, O. Hart, C.-F. Chen, A. J. Friedman, and A. Lucas, Designing open quantum systems with known steady states: Davies generators and beyond, Quantum 9, 1612 (2025).
  18. L. da Silva Souza and F. Iemini, Lindbladian reverse engineering for general non-equilibrium steady states: A scalable null-space approach, arXiv:2408.05302.
  19. X. Mi et al., Stable quantum-correlated many-body states through engineered dissipation, Science 383, 1332 (2024).
  20. G. Ferioli, A. Glicenstein, I. Ferrier-Barbut, and A. Browaeys, A non-equilibrium superradiant phase transition in free space, Nat. Phys. 19, 1345 (2023).
  21. G. M. Andolina, P. A. Erdman, F. Noé, J. Pekola, and M. Schirò, Dicke superradiant enhancement of the heat current in circuit QED quantum electrodynamics, Phys. Rev. Res. 6, 043128 (2024).
  22. V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, Completely positive dynamical semigroups of N-level systems, J. Math. Phys. 17, 821 (1976).
  23. G. Lindblad, On the generators of quantum dynamical semigroups, Commun. Math. Phys. 48, 119 (1976).
  24. D. Manzano, A short introduction to the Lindblad master equation, AIP Adv. 10, 025106 (2020).
  25. In our work, we set ℏ=1 and kB=1 for simplicity.
  26. See Supplemental Material at http://link.aps.org/supplemental/10.1103/hbrt-cn8q for descriptions on the density matrix formulation of the problem, clarification on the difference between ansatz and certification methods, a perspective on how one might add heat-current constraints, and a set of auxiliary data tables for different systems, which also contains Refs.  [39, 40].
  27. D. Jaschke, S. Montangero, and L. D. Carr, One-dimensional many-body entangled open quantum systems with tensor network methods, Quantum Sci. Technol. 4, 013001 (2018).
  28. H. Fawzi, O. Fawzi, and S. O. Scalet, Certified algorithms for equilibrium states of local quantum Hamiltonians, Nat. Commun. 15, 7394 (2024).
  29. M. Navascués, S. Pironio, and A. Acín, Bounding the set of quantum correlations, Phys. Rev. Lett. 98, 010401 (2007).
  30. M. Navascués, S. Pironio, and A. Acín, A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations, New J. Phys. 10, 073013 (2008).
  31. S. Pironio, M. Navascués, and A. Acín, Convergent relaxations of polynomial optimization problems with noncommuting variables, SIAM J. Optim. 20, 2157 (2010).
  32. D. Wu et al., Variational benchmarks for quantum many-body problems, Science 386, 296 (2024).
  33. G. Robichon and A. Tilloy, Bootstrapping the stationary state of bosonic open quantum systems, arXiv:2410.07384.
  34. S. P. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University Press, Cambridge, 2004).
  35. I. Kogias, P. Skrzypczyk, D. Cavalcanti, A. Acín, and G. Adesso, Hierarchy of steering criteria based on moments for all bipartite quantum systems, Phys. Rev. Lett. 115, 210401 (2015).
  36. A. Pozas-Kerstjens, R. Rabelo, Ł. Rudnicki, R. Chaves, D. Cavalcanti, M. Navascués, and A. Acín, Bounding the sets of classical and quantum correlations in networks, Phys. Rev. Lett. 123, 140503 (2019).
  37. J. Wang, J. Surace, I. Frérot, B. Legat, M.-O. Renou, V. Magron, and A. Acín, Certifying ground-state properties of many-body systems, Phys. Rev. X 14, 031006 (2024).
  38. L. Mortimer, Lumorti/thermo NPA: For paper submission (Paper), Zenodo (2025), https://doi.org/10.5281/zenodo.16095113.
  39. R. Alicki and R. Kosloff, Introduction to quantum thermodynamics: History and prospects, in Thermodynamics in the Quantum Regime: Fundamental Aspects and New Directions (Springer, Cham, 2018), pp. 1–33.
  40. A. Levy and R. Kosloff, The local approach to quantum transport may violate the second law of thermodynamics, Europhys. Lett. 107, 20004 (2014).
  41. P. P. Hofer, M. Perarnau-Llobet, L. D. M. Miranda, G. Haack, R. Silva, J. B. Brask, and N. Brunner, Markovian master equations for quantum thermal machines: Local versus global approach, New J. Phys. 19, 123037 (2017).
  42. S. Khandelwal, N. Palazzo, N. Brunner, and G. Haack, Critical heat current for operating an entanglement engine, New J. Phys. 22, 073039 (2020).
  43. D. Farina, B. Benazout, F. Centrone, and A. Acín, Thermodynamic precision in the nonequilibrium exchange scenario, Phys. Rev. E 109, 034112 (2024).
  44. This is consistent with the analytical solution given in Eqs. (6)–(8) in Ref.  [42], in which there are also six nonzero real variables and corresponding linear equations (see Supplemental Material for more details [26]).
  45. T. Prosen, Third quantization: A general method to solve master equations for quadratic open fermi systems, New J. Phys. 10, 043026 (2008).
  46. U. Marzolino and T. Prosen, Computational complexity of nonequilibrium steady states of quantum spin chains, Phys. Rev. A 93, 032306 (2016).
  47. F. Verstraete, M. M. Wolf, and J. Ignacio Cirac, Quantum computation and quantum-state engineering driven by dissipation, Nat. Phys. 5, 633 (2009).
  48. B. Requena, G. Muñoz Gil, M. Lewenstein, V. Dunjko, and J. Tura, Certificates of quantum many-body properties assisted by machine learning, Phys. Rev. Res. 5, 013097 (2023).

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Phys. Rev. Research 7, 033237– Published 10 September, 2025
  • Received 29 October 2024
  • Revised 12 February 2025
  • Accepted 16 August 2025
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