A universal constraint for relaxation rates for quantum Markov generators: complete positivity and beyond
arXiv:2505.24467 · doi:10.1088/1361-6633/ae075f
Abstract
Relaxation rates are key characteristics of quantum processes, as they determine how quickly a quantum system thermalizes, equilibrates, decoheres, and dissipates. While they play a crucial role in theoretical analyses, relaxation rates are also often directly accessible through experimental measurements. Recently, it was shown that for quantum processes governed by Markovian semigroups, the relaxation rates satisfy a universal constraint: the maximal rate is upper-bounded by the sum of all rates divided by the dimension of the Hilbert space. This bound, initially conjectured a few years ago, was only recently proven using classical Lyapunov theory. In this work, we present a new, purely algebraic proof of this constraint. Remarkably, our approach is not only more direct but also allows for a natural generalization beyond completely positive semigroups. We show that complete positivity can be relaxed to 2-positivity without affecting the validity of the constraint. This reveals that the bound is more subtle than previously understood: 2-positivity is necessary, but even when further relaxed to Schwarz maps, a slightly weaker -- yet still non-trivial -- universal constraint still holds. Finally, we explore the connection between these bounds and the number of steady states in quantum processes, uncovering a deeper structure underlying their behavior.
Contribution to special issue "Focus on Quantum Entanglement: State of the Art and Open Questions"
References in corpus (26)
- Quantum entanglement
- Separability of Mixed States: Necessary and Sufficient Conditions
- A bound on chaos
- Non-Markovian dynamics in open quantum systems
- Entanglement and non-Markovianity of quantum evolutions
- Quantum Non-Markovianity: Characterization, Quantification and Detection
- Assessing non-Markovian dynamics
- Canonical form of master equations and characterization of non-Markovianity
- On the degree of non-Markovianity of quantum evolution
- The Lyapunov Characteristic Exponents and their computation
- Dividing Quantum Channels
- Dynamics of initially entangled open quantum systems
- Open Quantum Dynamics: Complete Positivity and Entanglement
- Quantum logarithmic Sobolev inequalities and rapid mixing
- Physical accessibility of non-completely positive maps
- Lecture Notes on the Theory of Open Quantum Systems
- Measuring Non-Hermitian Operators via Weak Values
- Rapid mixing implies exponential decay of correlations
- The Generalized Lyapunov Theorem and its Application to Quantum Channels
- Restriction on relaxation times derived from the Lindblad-type master equations for 2-level systems
- On the universal constraints for relaxation rates for quantum dynamical semigroup
- Dissipative generators, divisible dynamical maps and Kadison-Schwarz inequality
- Exploring the Limits of Controlled Markovian Quantum Dynamics with Thermal Resources
- Quantum Lyapunov exponent in dissipative systems
- Number of steady states of quantum evolutions
- Understanding and Generalizing Unique Decompositions of Generators of Dynamical Semigroups