Finding stationary solutions of the Lindblad equation by analyzing the entropy production functional
Informatics and Automation, Complex analysis, mathematical physics, and applications, Tome 301 (2018), pp. 276-286.

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A necessary and sufficient condition is derived for a density operator to be a stationary solution for a certain class of Lindblad equations in the theory of open quantum systems. This condition is based on the properties of a functional that in some cases corresponds to entropy production. Examples are given where this condition is used to find stationary solutions.
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A. S. Trushechkin. Finding stationary solutions of the Lindblad equation by analyzing the entropy production functional. Informatics and Automation, Complex analysis, mathematical physics, and applications, Tome 301 (2018), pp. 276-286. http://geodesic.mathdoc.fr/item/TRSPY_2018_301_a19/

[1] Accardi L., Kozyrev S., “Lectures on quantum interacting particle systems”, Quantum interacting particle systems, Lecture notes of the Volterra–CIRM Int. Sch. (Trento, 2000), QP–PQ: Quantum Probab. White Noise Anal., 14, World Sci., Singapore, 2002, 1–195 | DOI | MR

[2] Accardi L., Kozyrev S. V., “Coherent population trapping and partial decoherence in the stochastic limit”, Int. J. Theor. Phys., 45:4 (2006), 661–678 | DOI | MR

[3] Accardi L., Lu Y. G., Volovich I., Quantum theory and its stochastic limit, Springer, Berlin, 2002 | MR | Zbl

[4] Aref'eva I., “Multiplicity and thermalization time in heavy-ions collisions”, EPJ Web Conf., 125 (2016), 01007 | DOI

[5] Aref'eva I. Ya., Khramtsov M. A., “AdS/CFT prescription for angle-deficit space and winding geodesics”, J. High Energy Phys., 2016:4 (2016), 121 | MR

[6] Aref'eva I. Ya., Khramtsov M. A., Tikhanovskaya M. D., “Thermalization after holographic bilocal quench”, J. High Energy Phys., 2017:9 (2017), 115 | DOI | MR | Zbl

[7] I. Ya. Aref'eva, I. V. Volovich, S. V. Kozyrev, “Stochastic limit method and interference in quantum many-particle systems”, Theor. Math. Phys., 183:3 (2015), 782–799 | DOI | DOI | MR | Zbl

[8] Barra F., “The thermodynamic cost of driving quantum systems by their boundaries”, Sci. Rep., 5 (2015), 14873 | DOI

[9] H.-P. Breuer, F. Petruccione, The Theory of Open Quantum Systems, Oxford Univ. Press, Oxford, 2002 | MR | Zbl

[10] Davies E. B., “Markovian master equations”, Commun. Math. Phys., 39 (1974), 91–110 | DOI | MR | Zbl

[11] Yu. N. Drozhzhinov, “Multidimensional Tauberian theorems for generalized functions”, Russ. Math. Surv., 71:6 (2016), 1081–1134 | DOI | DOI | MR | Zbl

[12] Gorini V., Kossakowski A., Sudarshan E. C. G., “Completely positive dynamical semigroups of $N$-level systems”, J. Math. Phys., 17:5 (1976), 821–825 | DOI | MR

[13] A. K. Gushchin, “Solvability of the Dirichlet problem for an inhomogeneous second-order elliptic equation”, Sb. Math., 206:10 (2015), 1410–1439 | DOI | DOI | MR | Zbl

[14] A. K. Gushchin, “$L_p$-estimates for the nontangential maximal function of the solution to a second-order elliptic equation”, Sb. Math., 207:10 (2016), 1384–1409 | DOI | DOI | MR | Zbl

[15] M. O. Katanaev, “Lorentz invariant vacuum solutions in general relativity”, Proc. Steklov Inst. Math., 290 (2015), 138–142 | DOI | DOI | MR | Zbl

[16] M. O. Katanaev, “Killing vector fields and a homogeneous isotropic universe”, Phys. Usp., 59:7 (2016), 689–700 | DOI | DOI

[17] M. O. Katanaev, “Cosmological models with homogeneous and isotropic spatial sections”, Theor. Math. Phys., 191:2 (2017), 661–668 | DOI | DOI | MR | Zbl

[18] Lindblad G., “On the generators of quantum dynamical semigroups”, Commun. Math. Phys., 48 (1976), 119–130 | DOI | MR | Zbl

[19] Luchnikov I. A., Filippov S. N., “Quantum evolution in the stroboscopic limit of repeated measurements”, Phys. Rev. A, 95:2 (2017), 022113 | DOI

[20] N. G. Marchuk, “Demonstration representation and tensor products of Clifford algebras”, Proc. Steklov Inst. Math., 290 (2015), 143–154 | DOI | DOI | MR | Zbl

[21] Marchuk N. G., Shirokov D. S., “General solutions of one class of field equations”, Rep. Math. Phys., 78:3 (2016), 305–326 | DOI | MR | Zbl

[22] A. N. Pechen, N. B. Il'in, “On critical points of the objective functional for maximization of qubit observables”, Russ. Math. Surv., 70:4 (2015), 782–784 | DOI | DOI | MR | Zbl

[23] A. N. Pechen, N. B. Il'in, “Existence of traps in the problem of maximizing quantum observable averages for a qubit at short times”, Proc. Steklov Inst. Math., 289 (2015), 213–220 | DOI | MR | Zbl

[24] A. N. Pechen, N. B. Il'in, “On the problem of maximizing the transition probability in an $n$-level quantum system using nonselective measurements”, Proc. Steklov Inst. Math., 294 (2016), 233–240 | DOI | DOI | MR | Zbl

[25] Spohn H., “Entropy production for quantum dynamical semigroups”, J. Math. Phys., 19:5 (1978), 1227–1230 | DOI | MR | Zbl

[26] Spohn H., Lebowitz J. L., “Irreversible thermodynamics for quantum systems weakly coupled to thermal reservoirs”, Adv. Chem. Phys., 38 (1978), 109–142

[27] Trushechkin A., “Semiclassical evolution of quantum wave packets on the torus beyond the Ehrenfest time in terms of Husimi distributions”, J. Math. Phys., 58:6 (2017), 062102 | DOI | MR | Zbl

[28] A. S. Trushechkin, “On general definition of entropy production in Markovian open quantum systems”, Quantum Computations, Itogi Nauki Tekh., Ser.: Sovrem. Mat. Prilozh. Temat. Obz., 138, VINITI, Moscow, 2017, 82–98

[29] Trushechkin A. S., Volovich I. V., “Perturbative treatment of inter-site couplings in the local description of open quantum networks”, Europhys. Lett., 113:3 (2016), 30005 | DOI

[30] B. O. Volkov, “Lévy Laplacians and instantons”, Proc. Steklov Inst. Math., 290 (2015), 210–222 | DOI | DOI | MR | Zbl

[31] Volkov B. O., “Stochastic Lévy differential operators and Yang–Mills equations”, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 20:2 (2017), 1750008 | DOI | MR | Zbl

[32] Volovich I. V., “Cauchy–Schwarz inequality-based criteria for the non-classicality of sub-Poisson and antibunched light”, Phys. Lett. A, 380:1–2 (2016), 56–58 | DOI | MR | Zbl

[33] I. V. Volovich, S. V. Kozyrev, “Manipulation of states of a degenerate quantum system”, Proc. Steklov Inst. Math., 294 (2016), 241–251 | DOI | MR | Zbl

[34] V. V. Zharinov, “Conservation laws, differential identities, and constraints of partial differential equations”, Theor. Math. Phys., 185:2 (2015), 1557–1581 | DOI | DOI | MR | Zbl

[35] V. V. Zharinov, “Bäcklund transformations”, Theor. Math. Phys., 189:3 (2016), 1681–1692 | DOI | DOI | MR | Zbl