Spectral delineation of Markov Generators: Classical vs Quantum

Abstract

The celebrated theorem of Perron and Frobenius implies that spectra of classical Markov operators, represented by stochastic matrices, are restricted to the unit disk. This property holds also for spectra of quantum stochastic maps (quantum channels), which describe quantum Markovian evolution in discrete time. Moreover, the spectra of stochastic $N \times N$ matrices are additionally restricted to a subset of the unit disk, called Karpelevi\u{c} region, the shape of which depends on $N$. We address the question of whether the spectra of generators, which induce Markovian evolution in continuous time, can be bound in a similar way. We propose a rescaling that allows us to answer this question affirmatively. The eigenvalues of the rescaled classical generators are confined to the modified Karpelevi\u{c} regions, whereas the eigenvalues of the rescaled quantum generators fill the entire unit disk.