On existence of quantum trajectories for the linear deterministic processes. (English) Zbl 07858411
Summary: The method of “quantum trajectories”, i.e. transitions from a pure to a pure quantum state, is a useful tool in the open quantum systems theory and applications. This method relies on the nonlinear stochastic differential equations as a dynamical model. In contrast to this, we pose the question of existence of quantum trajectories for the pure states, each of which would be a solution to a linear, deterministic master equation. It turns out that this task is rather delicate. In its full generality, the task is practically intractable. On the other hand, we do not obtain a general answer even for certain well-established and widely used Markovian processes. Only for a few models for the case of the environment on the absolute zero temperature, we obtain existence of the desired quantum trajectories. In all other cases, there is not even a single such quantum trajectory. In conjunction with the standard method of quantum trajectories, our findings pose some nontrivial challenges for the foundations of the open systems theory and some interpretational and cosmological contexts.
MSC:
| 81Vxx | Applications of quantum theory to specific physical systems |
| 81-XX | Quantum theory |
| 81Pxx | Foundations, quantum information and its processing, quantum axioms, and philosophy |
Keywords:
quantum trajectories; open quantum systems; master equations; Markov processes; quantum interpretations; quantum cosmologyReferences:
| [1] | Breuer, H-P; Petruccione, F., The Theory of Open Quantum Systems, 2002, Oxford: Clarendon Press, Oxford · Zbl 1053.81001 |
| [2] | Joos, E.; Zeh, H-D; Kiefer, C.; Giulini, DJW; Kupsch, J.; Stamatescu, I-O, Decoherence and the Appearance of a Classical World in Quantum Theory, 2003, Berlin: Springer, Berlin · doi:10.1007/978-3-662-05328-7 |
| [3] | Schlosshauer, M., Decoherence and the Quantum-to-Classical Transition, 2009, Heidelberg/Berlin: Springer, Heidelberg/Berlin |
| [4] | Davies, EB, Quantum stochastic processes, Commun. Math. Phys., 15, 277-304, 1969 · Zbl 0184.54102 · doi:10.1007/BF01645529 |
| [5] | Dalibard, J.; Castin, Y.; Molmer, K., Wave-function approach to dissipative processes in quantum optics, Phys. Rev. Lett., 68, 580-583, 1992 · doi:10.1103/PhysRevLett.68.580 |
| [6] | Carmichael H.: An Open Systems Approach to Quantum Optics, Volume 18 in Lecture Notes in Physics. Springer-Verlag, Berlin (1993) · Zbl 1151.81300 |
| [7] | Gardiner C.W., Zoller P.: Quantum Noise. Springer-Verlag, Berlin/Heidelberg (2004) Springer · Zbl 1072.81002 |
| [8] | Scully, MO; Zubairy, S., Quantum Optics, 2001, Cambridge: Cambridge University Press, Cambridge |
| [9] | Wiseman, H.; Milburn, GJ, Quantum Measurement and Control, 2010, Cambridge: Cambridge University Press, Cambridge · Zbl 1350.81004 |
| [10] | Gisin, N., Quantum measurements and stochastic processes, Phys. Rev. Lett., 52, 1665, 1984 · doi:10.1103/PhysRevLett.52.1657 |
| [11] | Ghiradi, GC; Rimini, A.; Weber, T., Unified dynamics for microscopic and macroscopic systems, Phys. Rev. D, 34, 470-491, 1986 · Zbl 1222.82047 · doi:10.1103/PhysRevD.34.470 |
| [12] | Bassi, A.; Lochan, K.; Satin, S.; Singh, TP; Ulbricht, H., Models of wave-function collapse, underlying theories, and experimental tests, Rev. Mod. Phys., 85, 471-527, 2013 · doi:10.1103/RevModPhys.85.471 |
| [13] | Griffiths, RB, Consistent Quantum Theory, 2003, Cambridge: Cambridge University Press, Cambridge |
| [14] | Wallace, D.; Saunders, S.; Barrett, J.; Kent, A.; Wallace, D., Decoherence and Ontology (or: How I learned to stop worrying and love FAPP), Many Worlds? Everett, Quantum Theory, and Reality, 53-72, 2010, Oxford: Oxford University Press, Oxford · Zbl 1197.81035 · doi:10.1093/acprof:oso/9780199560561.003.0002 |
| [15] | Gell-Mann, M., Hartle, J.B.: Quantum mechanics in the light of quantum cosmology. In: Zurek, W. (ed.) Complexity, Entropy, and the Physics of Information, SFI Studies in the Sciences of Complexity, vol. VIII. Addison Wesley, Reading, MA (1990) |
| [16] | Hartle J.B.: The quantum mechanics of cosmology. In: S. Coleman, J.B. Hartle, T. Piran, and S. Weinberg (eds) Quantum Cosmology and Baby Universes, Proceedings of the 1989 Jerusalem Winter School for Theoretical Physics, pp.65-157. World Scientific, Singapore (1991) |
| [17] | Kraus, K., States, Effects and Operations: Fundamental Notions of Quantum Theory, 1983, Berlin: Springer Verlag, Berlin · Zbl 0545.46049 · doi:10.1007/3-540-12732-1 |
| [18] | Nielsen, MA; Chuang, IA, Quantum comuptation and quantum information, 2000, Cambridge: Cambridge University Press, Cambridge · Zbl 1049.81015 |
| [19] | Ghirardi, G-C; Pearle, P.; Rimini, A., Markov processes in Hilbert space and continuous spontaneous localization of systems of identical particles, Phys. Rev. A, 42, 78, 1990 · doi:10.1103/PhysRevA.42.78 |
| [20] | Diosi, L., Models for universal reduction of macroscopic quantum fluctuations, Phys. Rev. A, 40, 1165, 1989 · doi:10.1103/PhysRevA.40.1165 |
| [21] | Donvil, B.; Muratore-Gianneschi, P., Quantum trajectory framework for general time-local master equations, Nat. Communicat., 13, 4140, 2022 · doi:10.1038/s41467-022-31533-8 |
| [22] | Vovk, T.; Pichler, H., Entanglement-optimal trajectories of many-body quantum Markov processes, Phys. Rev. Lett., 128, 243601, 2022 · doi:10.1103/PhysRevLett.128.243601 |
| [23] | Kolodrubetz, M., Optimality of Lindblad unfolding in measurement phase transitions, Phys. Rev. B, 107, L140301, 2023 · doi:10.1103/PhysRevB.107.L140301 |
| [24] | Zhang, Z.; Karniadakis, Z., Numerical methods for stochastic partial differential equations with white noise, 2017, Berlin: Springer, Berlin · Zbl 1380.65021 · doi:10.1007/978-3-319-57511-7 |
| [25] | Gisin, N., Stochastic quantum dynamics and relativity, Helv. Phys. Acta, 62, 363-371, 1989 · Zbl 0632.46068 |
| [26] | Bahrami, M.; Grossardt, A.; Donadi, S.; Bassi, A., The Schrödinger-Newton equation and its foundations, New J. Phys., 16, 115007, 2014 · Zbl 1451.81381 · doi:10.1088/1367-2630/16/11/115007 |
| [27] | Tilloy, A.; Wiseman, HW, Non-Markovian wave-function collapse models are Bohmian-like theories in disguise, Quantum, 5, 594, 2021 · doi:10.22331/q-2021-11-29-594 |
| [28] | Rivas, Á.; Huelga, SF, Open Quantum Systems, 2011, Berlin: An Introduction. Springer-Briefs in Physics, Berlin |
| [29] | Jeknić-Dugić, J.; Arsenijević, M.; Dugić, M., Invertibility as a witness of Markovianity of the quantum dynamical maps, Braz. J. Phys., 53, 58, 2023 · Zbl 1371.81329 · doi:10.1007/s13538-023-01274-0 |
| [30] | Bassi, A.; Dürr, D.; Hinrichs, G., Uniqueness of the equation for quantum state vector collapse, Phys. Rev. Lett., 111, 210401, 2013 · doi:10.1103/PhysRevLett.111.210401 |
| [31] | Sandulescu, A.; Scutary, H., Open quantum systems and the damping of collective modes in deep inelastic collisions, Ann. Phys., 173, 277-317, 1987 · doi:10.1016/0003-4916(87)90162-X |
| [32] | Isar, A.; Sandulescu, A., Scheid W: Purity and decoherence in the theory of a damped harmonic oscillator, Phys. Rev. E, 60, 6371-6381, 1999 · Zbl 1062.82505 · doi:10.1103/PhysRevE.60.6371 |
| [33] | Hornberger, K., Master equation for a quantum particle in a gas, Phys. Rev. Lett., 97, 060601, 2006 · doi:10.1103/PhysRevLett.97.060601 |
| [34] | Arsenijević, M.; Jeknić-Dugić, J.; Dugić, M., Generalized kraus operators for the one-qubit depolarizing quantum channel, Braz. J. Phys., 47, 339, 2017 · Zbl 1359.81032 · doi:10.1007/s13538-017-0502-3 |
| [35] | Dann, R.; Levy, A.; Kosloff, R., Time-dependent Markovian quantum master equation, Phys. Rev. A, 98, 052129, 2018 · doi:10.1103/PhysRevA.98.052129 |
| [36] | Walls, DF; Collet, MJ; Milburn, GJ, Analysis of a quantum measurement, Phys. Rev. D, 32, 3208-3215, 1985 · doi:10.1103/PhysRevD.32.3208 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
