Quantum detailed balance via elementary transitions

Rocco Duvenhage; Kyle Oerder; Keagan van den Heuvel

doi:10.22331/q-2025-05-15-1743

Quantum detailed balance via elementary transitions

Rocco Duvenhage, Kyle Oerder, and Keagan van den Heuvel

Department of Physics, University of Pretoria, Pretoria 0002, South Africa

Published:	2025-05-15, volume 9, page 1743
Editor:	Philip Taranto
Eprint:	arXiv:2411.02339v3
Doi:	https://doi.org/10.22331/q-2025-05-15-1743
Citation:	Quantum 9, 1743 (2025).

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

Quantum detailed balance is formulated in terms of elementary transitions, in close analogy to detailed balance in a classical Markov chain on a finite set of points. An elementary transition is taken to be a pure state of two copies of the quantum system, as a quantum analogue of an ordered pair of classical points representing a classical transition from the first to the second point. This form of quantum detailed balance is shown to be equivalent to standard quantum detailed balance with respect to a reversing operation, thus providing a new conceptual foundation for the latter. Aspects of parity in quantum detailed balance are clarified in the process. The connection with the Accardi-Cecchini dual and the KMS dual (or Petz recovery map) is also elucidated.

Featured image: The diagram on the left represents two opposing classical transitions (the arrows) between two states represented by the dots. The probability for each transition is indicated. If they are equal for all pairs of states of the system, we have classical detailed balance. We ask: What is a natural quantum version of this?

Popular summary

A simple form of equilibrium for a process in a classical system is the situation where, at any time, the probability of a transition from any state of the system to another during this process, is the same as the probability for the opposite transition. This needs to be true for all pairs of states of the system. This equilibrium is called detailed balance. In such a classical setup, the available states are fixed. In a quantum system, however, the relevant states can conceivably depend on the observable being considered. Should one focus on a specific set of states, say energy states? Or be more general? Can one express the equality of probabilities in the two directions in a way directly analogous to the classical case? Or are we forced to approach the problem in some indirect way? Our strategy is simply to focus on the transitions, classically the pairs of states, rather than the states themselves, to open a clear path to the quantum case.

► BibTeX data

@article{Duvenhage2025quantumdetailed,
  doi = {10.22331/q-2025-05-15-1743},
  url = {https://doi.org/10.22331/q-2025-05-15-1743},
  title = {Quantum detailed balance via elementary transitions},
  author = {Duvenhage, Rocco and Oerder, Kyle and van den Heuvel, Keagan},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1743},
  month = may,
  year = {2025}
}

► References

[1] L. Accardi and C. Cecchini, Conditional expectations in von Neumann algebras and a theorem of Takesaki, J. Funct. Anal. 45 (1982), 245–273.
https://doi.org/10.1016/0022-1236(82)90022-2

[2] L. Accardi, F. Fagnola and R. Quezada, Weighted detailed balance and local KMS condition for non-equilibrium stationary states, in: Perspect. Nonequilibrium Statist. Phys., Busseikenkyu 97 (2011), 318–356.

[3] L. Accardi, F. Fagnola and R. Quezada, On three new principles in non-equilibrium statistical mechanics and Markov semigroups of weak coupling limit type, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 19 (2016), 1650009.
https://doi.org/10.1142/s0219025716500090

[4] L. Accardi and K. Imafuku, Dynamical detailed balance and local KMS condition for non-equilibrium states, Internat. J. Modern Phys. B 18 (2004), 435–467.
https://doi.org/10.1142/s0217979204024070

[5] G. S. Agarwal, Open quantum Markovian systems and the microreversibility, Z. Physik 258 (1973), 409–422.
https://doi.org/10.1007/bf01391504

[6] R. Alicki, On the detailed balance condition for non-Hamiltonian systems, Rep. Math. Phys. 10 (1976), 249–258.
https://doi.org/10.1016/0034-4877(76)90046-x

[7] H. Araki, Some properties of modular conjugation operator of von Neumann algebras and a non-commutative Radon-Nikodym theorem with a chain rule, Pacific J. Math. 50 (1974), 309–354.
https://doi.org/10.2140/pjm.1974.50.309

[8] P. Arrighi and C. Patricot, On quantum operations as quantum states, Ann. Physics 311 (2004), 26–52.
https://doi.org/10.1016/j.aop.2003.11.005

[9] H. Barnum and E. Knill, Reversing quantum dynamics with near-optimal quantum and classical fidelity, J. Math. Phys. 43 (2002), 2097–2106.
https://doi.org/10.1063/1.1459754

[10] J. R. Bolaños-Servin and R. Quezada, A cycle decomposition and entropy production for circulant quantum Markov semigroups, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 16 (2013), 1350016.
https://doi.org/10.1142/s0219025713500161

[11] J. R. Bolaños-Servin and R. Quezada, The $\Theta$-KMS adjoint and time reversed quantum Markov semigroups, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 18 (2015), 1550016.
https://doi.org/10.1142/s0219025715500162

[12] H. J. Carmichael and D. F. Walls, Detailed balance in open quantum Markoffian systems, Z. Physik B 23 (1976), 299–306.
https://doi.org/10.1007/bf01318974

[13] M.-D. Choi, Completely positive linear maps on complex matrices, Linear Algebra Appl. 10 (1975), 285–290.
https://doi.org/10.1016/0024-3795(75)90075-0

[14] A. Connes, Caractérisation des espaces vectoriels ordonnés sous-jacents aux algèbres de von Neumann, Ann. Inst. Fourier (Grenoble) 24 (1974), 121–155.
https://doi.org/10.5802/aif.534

[15] Y. N. Demkov, The principle of detailed balance in quantum mechanics and some identities for the amplitudes of scattering in the theory of collisions, (Russian), Dokl. Akad. Nauk SSSR (N.S.) 97 (1954), 1003–1006.

[16] J. de Pillis, Linear transformations which preserve hermitian and positive semidefinite operators, Pacific J. Math. 23 (1967), 129–137.
https://doi.org/10.2140/pjm.1967.23.129

[17] J. Derezynski and R. Fruboes, Fermi golden rule and open quantum systems, In: S. Attal et al. (eds.) Open Quantum Systems III, Lecture Notes in Mathematics 1882, Berlin-Heidelberg-New York: Springer, 2006, pp. 67–116.
https://doi.org/10.1007/3-540-33967-1_2

[18] P. A. M. Dirac, The conditions for statistical equilibrium between atoms, electrons and radiation Proc. Roy. Soc. London Ser. A 106 (1924), 581–596.
https://doi.org/10.1098/rspa.1924.0088

[19] R. Duvenhage, Optimal quantum channels, Phys. Rev. A 104 (2021), 032604.
https://doi.org/10.1103/physreva.104.032604

[20] R. Duvenhage, S. Skosana and M. Snyman, Extending quantum detailed balance through optimal transport, Rev. Math. Phys., 37 (2025), 2450040.
https://doi.org/10.1142/S0129055X24500405

[21] R. Duvenhage and M. Snyman, Detailed balance and entanglement, J. Phys. A 48 (2015), 155303.
https://doi.org/10.1088/1751-8113/48/15/155303

[22] R. Duvenhage and M. Snyman, Balance between quantum Markov semigroups, Ann. Henri Poincaré 19 (2018), 1747–1786.
https://doi.org/10.1007/s00023-018-0664-8

[23] F. Fagnola and R. Rebolledo, From classical to quantum entropy production, in: Proceedings of the 29th Conference on Quantum Probability and Related Topics, edited by H. Ouerdiane and A. Barhoumi, p 245–261, QP–PQ: Quantum Probability and White Noise Analysis, 25, World Scientific, Hackensack, NJ, 2010.
https://doi.org/10.1142/9789814295437_0017

[24] F. Fagnola and R. Rebolledo, Entropy production for quantum Markov semigroups, Comm. Math. Phys. 335 (2015), 547–570.
https://doi.org/10.1007/s00220-015-2320-1

[25] F. Fagnola and R. Rebolledo, Entropy production and detailed balance for a class of quantum Markov semigroups, Open Syst. Inf. Dyn. 22 (2015), 1550013.
https://doi.org/10.1142/s1230161215500134

[26] F. Fagnola and V. Umanità, Generators of KMS symmetric Markov semigroups on B(h) symmetry and quantum detailed balance, Comm. Math. Phys. 298 (2010), 523–547.
https://doi.org/10.1007/s00220-010-1011-1

[27] F. Fagnola and V. Umanità, Quantum detailed balance conditions with time reversal: three-level systems, Stochastics 84 (2012), 273–293.
https://doi.org/10.1080/17442508.2010.518707

[28] F. Fagnola and V. Umanità, Quantum detailed balance conditions with time reversal: the finite-dimensional case, Noncommutative harmonic analysis with applications to probability III, 159–174, Banach Center Publ., 96, Polish Academy of Sciences, Institute of Mathematics, Warsaw, 2012.
https://doi.org/10.4064/bc96-0-10

[29] C. W. Gardiner, Handbook of stochastic methods: for physics, chemistry and the natural sciences, Second edition, Springer Ser. Synergetics, 13, Springer-Verlag, Berlin, 1985.

[30] S. Goldstein and J.M. Lindsay, KMS-symmetric Markov semigroups, Math. Z. 219 (1995), 591–608.
https://doi.org/10.1007/bf02572383

[31] U. Haagerup, The standard form of von Neumann algebras, Math. Scand. 37 (1975), 271–283.
https://doi.org/10.7146/math.scand.a-11606

[32] A. Jamiołkowski, Linear transformations which preserve trace and positive semidefiniteness of operators, Rep. Mathematical Phys. 3 (1972), 275–278.
https://doi.org/10.1016/0034-4877(72)90011-0

[33] A. JenÄovÃ¡, Reversibility conditions for quantum operations, Rev. Math. Phys. 24 (2012), 1250016.
https://doi.org/10.1142/s0129055x1250016x

[34] M. Jiang, S. Luo and S. Fu, Channel-state duality, Phys. Rev. A, 87 (2013) 022310.
https://doi.org/10.1103/physreva.87.022310

[35] A. Kossakowski, A. Frigerio, V. Gorini and M. Verri, Quantum detailed balance and KMS condition, Commun. Math. Phys. 57 (1977), 97–110.
https://doi.org/10.1007/bf01625769

[36] W. A. Majewski, The detailed balance condition in quantum statistical mechanics, J. Math. Phys. 25 (1984), 614–616.
https://doi.org/10.1063/1.526164

[37] W. A. Majewski and R. F. Streater, Detailed balance and quantum dynamical maps, J. Phys. A 31 (1998), 7981–7995.
https://doi.org/10.1088/0305-4470/31/39/013

[38] D. Petz, A dual in von Neumann algebras with weights, Quart. J. Math. Oxford Ser. (2) 35 (1984), 475–483.
https://doi.org/10.1093/qmath/35.4.475

[39] D. Petz, Sufficient subalgebras and the relative entropy of states of a von Neumann algebra, Comm. Math. Phys. 105 (1986), 123–131.
https://doi.org/10.1007/bf01212345

[40] D. Petz, Sufficiency of channels over von Neumann algebras, Quart. J. Math. Oxford Ser. (2) 39 (1988), 97–108.
https://doi.org/10.1093/qmath/39.1.97

[41] D. Roberts, A. Lingenfelter and A.A. Clerk, Hidden Time-Reversal Symmetry, Quantum Detailed Balance and Exact Solutions of Driven-Dissipative Quantum Systems, PRX Quantum 2 (2021), 020336.
https://doi.org/10.1103/prxquantum.2.020336

[42] J. Schwinger, Unitary operator bases, Proc. Nat. Acad. Sci. U.S.A. 46 (1960), 570–579.
https://doi.org/10.1073/pnas.46.4.570

[43] M. E. Shirokov, Reversibility of a quantum channel: general conditions and their applications to bosonic linear channels, J. Math. Phys. 54 (2013), 112201.
https://doi.org/10.1063/1.4827436

[44] L. M. Sieberer, A. Chiocchetta, A. Gambassi, U. C. Täuber and S. Diehl, Thermodynamic equilibrium as a symmetry of the Schwinger-Keldysh action, Phys. Rev. B 92 (2015), 134307.
https://doi.org/10.1103/physrevb.92.134307

[45] G. Stragier, J. Quaegebeur and A. Verbeure, Quantum detailed balance, Ann. Inst. H. Poincaré Phys. Théor. 41 (1984), 25–36.

[46] M. Takesaki, Tomita's theory of modular Hilbert algebras and its applications, Lecture Notes in Math., Vol. 128, Springer-Verlag, Berlin-New York, 1970.
https://doi.org/10.1007/bfb0065832

[47] M. Takesaki, Theory of operator algebras. II, Encyclopaedia of Mathematical Sciences, 125. Operator Algebras and Non-commutative Geometry, 6. Springer-Verlag, Berlin, 2003.
https://doi.org/10.1007/978-3-662-10451-4

[48] K. Temme, M. J. Kastoryano, M. B. Ruskai, M. M. Wolf and F. Verstraete, The $\chi^2$-divergence and mixing times of quantum Markov processes, J. Math. Phys. 51 (2010), 122201.
https://doi.org/10.1063/1.3511335

[49] R. C. Tolman, The principles of statistical mechanics, Oxford at the Clarendon Press, 1938.

[50] M. Tsang, Quantum reversal: a general theory of coherent quantum absorbers, Quantum 9 (2025), 1650.
https://doi.org/10.22331/q-2025-02-26-1650

[51] M. Tsang, Quantum Onsager relations, Quantum Sci. Technol. 10 (2025), 015015.
https://doi.org/10.1088/2058-9565/ad8513

[52] A. Verbeure, Detailed balance and equilibrium, in: Quantum Probability and Applications II, edited by L. Accardi and W. von Waldenfels, p 504, Lecture Notes in Mathematics 1136, Springer-Verlag, Berlin, 1985.
https://doi.org/10.1007/bfb0074497

[53] A. Vourdas, Quantum systems with finite Hilbert space, Rep. Progr. Phys. 67 (2004), no. 3, 267–320.
https://doi.org/10.1088/0034-4885/67/3/r03

[54] H. Weyl, The theory of groups and quantum mechanics, reprint of the 1931 English translation, Dover Publications, Inc., New York, 1950.

[55] M. M. Wilde, Quantum information theory, second edition, Cambridge University Press, Cambridge, 2017.
https://doi.org/10.1017/9781316809976

Cited by

On Crossref's cited-by service no data on citing works was found (last attempt 2026-08-08 20:29:02). On SAO/NASA ADS no data on citing works was found (last attempt 2026-08-08 20:29:02).

This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.