Dynamical maps, quantum detailed balance and Petz recovery map

Abstract

Markovian master equations (formally known as quantum dynamical semigroups) can be used to describe the evolution of a quantum state $\rho$ when in contact with a memoryless thermal bath. This approach has had much success in describing the dynamics of real-life open quantum systems in the lab. Such dynamics increase the entropy of the state $\rho$ and the bath until both systems reach thermal equilibrium, at which point entropy production stops. Our main result is to show that the entropy production at time $t$ is bounded by the relative entropy between the original state and the state at time $2t$. The bound puts strong constraints on how quickly a state can thermalise, and we prove that the factor of $2$ is tight. The proof makes use of a key physically relevant property of these dynamical semigroups -- detailed balance, showing that this property is intimately connected with the field of recovery maps from quantum information theory. We envisage that the connections made here between the two fields will have further applications. We also use this connection to show that a similar relation can be derived when the fixed point is not thermal.

main text: 5 pages. Appendix: 11 pages. V3: The conjectures in V2 have been proven. This is close to the journal published version, but with improved citations

References in corpus (6)

• Assessing non-Markovian dynamics
• Quantum coherence, time-translation symmetry and thermodynamics
• On thermalization in Kitaev's 2D model
• Quantum Operation Time Reversal
• Multivariate Trace Inequalities
• Hypercontractivity of quasi-free quantum semigroups

Cited by in corpus (16)

• Finite-Time Quantum Landauer Principle and Quantum Coherence
• Quantum fluctuations hinder finite-time information erasure near the Landauer limit
• Entanglement Wedge Reconstruction using the Petz Map
• Fundamental limits on quantum dynamics based on entropy change
• Toward relaxation asymmetry: Heating is faster than cooling
• Reversing Lindblad Dynamics via Continuous Petz Recovery Map
• Approximate reversal of quantum Gaussian dynamics
• Duality for open fermion systems: energy-dependent weak coupling and quantum master equations
• Approximate quantum Markov chains
• Approximating Invertible Maps by Recovery Channels: Optimality and an Application to Non-Markovian Dynamics
• Joint fluctuation theorems for sequential heat exchange
• Geometrical bounds on irreversibility in squeezed thermal bath
• Jaynes-Cummings atoms coupled to a structured environment: Leakage elimination operators and the Petz recovery maps
• Fluctuation Theorems with Retrodiction rather than Reverse Processes
• Lower Bound on Irreversibility in Thermal Relaxation of Open Quantum Systems
• Thermal operations from informational equilibrium