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Jaynes’ principle for quantum Markov processes: generalized Gibbs–von Neumann states rule

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Abstract

We prove that any asymptotics of a finite-dimensional quantum Markov processes can be formulated in the form of a generalized Jaynes’ principle in the discrete as well as in the continuous case. Surprisingly, we find that the open-system dynamics does not require maximization of von Neumann entropy. In fact, the natural functional to be extremized is the quantum relative entropy and the resulting asymptotic states or trajectories are always of the exponential Gibbs-like form. Three versions of the principle are presented for different settings, each treating different prior knowledge: for asymptotic trajectories of fully known initial states, for asymptotic trajectories incompletely determined by known expectation values of some constants of motion and for stationary states incompletely determined by expectation values of some integrals of motion. All versions are based on the knowledge of the underlying dynamics. Hence, our principle is primarily rooted in the inherent physics and it is not solely an information construct. The found principle coincides with the MaxEnt principle in the special case of unital quantum Markov processes. We discuss how the generalized principle modifies fundamental relations of statistical physics.

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Acknowledgements

J. N., J.M. and I. J. were supported by the Grant Agency of the Czech Technical University in Prague, Grant No. SGS22/181/OHK4/3T/14. This publication was funded by the project “Centre for Advanced Applied Sciences,” Registry No. CZ.02.1.01/0.0/0.0/16 019/0000778, supported by the Operational Programme Research, Development and Education, co-financed by the European Structural and Investment Funds and the state budget of the Czech Republic. The support for J. N. and I. J. by GAČR of the Czech Republic under Grant No. 23-07169 S is gratefully acknowledged.

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Novotný, J., Maryška, J. & Jex, I. Jaynes’ principle for quantum Markov processes: generalized Gibbs–von Neumann states rule. Eur. Phys. J. Plus 138, 657 (2023). https://doi.org/10.1140/epjp/s13360-023-04272-y

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