Abstract
What can one infer about the dynamical evolution of quantum systems just by symmetry considerations? For Markovian dynamics in finite dimensions, we present a simple construction that assigns to each symmetry of the generator a family of scalar functions over quantum states that are monotonic under the time evolution. The aforementioned monotones can be utilized to identify states that are non-reachable from an initial state by the time evolution and include all constraints imposed by conserved quantities, providing a generalization of Noether's theorem for this class of dynamics. As a special case, the generator itself can be considered a symmetry, resulting in non-trivial constraints over the time evolution, even if all conserved quantities trivialize. The construction utilizes tools from quantum information-geometry, mainly the theory of monotone Riemannian metrics. We analyze the prototypical cases of dephasing and Davies generators.
Highlights
One of the main tasks in the study of nonrelativistic quantum dynamical systems is predicting how quantum states and observables evolve over time given some dynamical law, for instance, a Hamiltonian operator and the associated equations of motion
Since Markovian dynamics is fully characterized by its generator, we invoke an infinitesimal version of distinguishability measures, connecting with monotone Riemannian metrics in the space of quantum states
The present manuscript constitutes an attempt to provide a simple correspondence between symmetries in the generators of quantum Markovian dynamics and monotones of the corresponding evolution
Summary
One of the main tasks in the study of nonrelativistic quantum dynamical systems is predicting how quantum states and observables evolve over time given some dynamical law, for instance, a Hamiltonian operator and the associated equations of motion. Markovian dynamics with a unique steady state does not admit any non-trivial conserved quantities [4] For this reason, we approach the problem by instead seeking to utilize symmetries to obtain monotones, i.e., functions of the quantum state that are monotonic (in our case, non-increasing) under the time evolution. The generator itself can be considered a symmetry of the dynamics, resulting in nontrivial constraints over the time evolution, even if all conserved quantities trivialize. Since Markovian dynamics is fully characterized by its generator, we invoke an infinitesimal version of distinguishability measures, connecting with monotone Riemannian metrics in the space of quantum states.
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