Symmetries and monotones in Markovian quantum dynamics

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.

arXiv commentAccepted for publication in Quantum. Comments welcome! v3: new appendices

References in corpus (21)

• Quantum Resource Theories
• A Theory of Quantum Error-Correcting Codes
• Open Quantum Systems. An Introduction
• Reference frames, superselection rules, and quantum information
• Symmetries and conserved quantities in Lindblad master equations
• A note on symmetry reductions of the Lindblad equation: transport in constrained open spin chains
• Extending Noether's theorem by quantifying the asymmetry of quantum states
• Stabilizing Quantum Information
• The theory of manipulations of pure state asymmetry: basic tools and equivalence classes of states under symmetric operations
• Monotone Riemannian Metrics and Relative Entropy on Non-Commutative Probability Spaces
• Metric adjusted skew information
• Analysis of quantum semigroups with GKS--Lindblad generators II. General
• Quantum logarithmic Sobolev inequalities and rapid mixing
• The ${\chi }^2$-divergence and Mixing times of quantum Markov processes
• A Mini-Introduction To Information Theory
• Davies maps for qubits and qutrits
• Markovian evolution of quantum coherence under symmetric dynamics
• Dynamical Response Theory for Driven-Dissipative Quantum Systems
• Asymptotics of quantum channels: conserved quantities, an adiabatic limit, and matrix product states
• A Cutoff Phenomenon for Quantum Markov Chains
• Asymptotics of quantum Markov processes: From algebraic structure to characterization of asymptotic states

Cited by in corpus (6)

• Continuous thermomajorization and a complete set of laws for Markovian thermal processes
• KAM-Stability for Conserved Quantities in Finite-Dimensional Quantum Systems
• A resource theoretical unification of Mpemba effects: classical and quantum
• Strong symmetries in collision models and physical dilations of covariant quantum maps
• Universal Bound on the Eigenvalues of 2-Positive Trace-Preserving Maps
• Non-Markovian Noise in Symmetry-Preserving Quantum Dynamics