2025/10/22 by Muratore-Ginanneschi, Paolo, Kimura, Gen, Ende, Frederik vom +1
#Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Quantum Physics (quant-ph)
paper · doi:10.48550/arxiv.2510.19657
The spectral properties of positive maps provide pivotal information for understanding the dynamics of quantum systems interacting with their environment. Furthermore, central problems in quantum information such as the characterization of entanglement may be reformulated in terms of spectral properties of positive maps. The present work aims to contribute to a better understanding of the spectrum of positive maps. Specifically, our main result is a new proof of a universal bound on the d2-1 generically non vanishing decay rates Γi of time-autonomous quantum master equations on a d-dimensional Hilbert space: Γmax ≤ cd ∑i=1^d2-1Γi The prefactor cd , which we explicitly determine, depends only on the dimension d and varies depending on the sub-class of positive maps to which the semigroup solution of the master equation belongs. We provide a brief but self-consistent survey of these concepts. We obtain our main result by resorting to the theory of Lyapunov exponents, a central concept in the study of dynamical systems, control theory, and out-of-equilibrium statistical mechanics. We thus show that progress in understanding positive maps in quantum mechanics may require ideas at the crossroads between different disciplines. For this reason, we adopt a notation and presentation style aimed at reaching readers with diverse backgrounds.