GENERIC q-MARKOV SEMIGROUPS AND SPEED OF CONVERGENCE OF q-ALGORITHMS

Abstract

We study a special class of generic quantum Markov semigroups, on the algebra of all bounded operators on a Hilbert space [Formula: see text], arising in the stochastic limit of a generic system interacting with a boson–Fock reservoir. This class depends on an orthonormal basis of [Formula: see text]. We obtain a new estimate for the trace distance of a state from a pure state and use this estimate to prove that, under the action of a semigroup of this class, states with finite support with respect to the given basis converge to equilibrium with a speed which is exponential, but with a polynomial correction which makes the convergence increasingly worse as the dimension of the support increases (Theorem 5.1). We interpret the semigroup as an algorithm, its initial state as input and, following Belavkin and Ohya,10 the dimension of the support of a state as a measure of complexity of the input. With this interpretation, the above results mean that the complexity of the input "slows down" the convergence of the algorithm. Even if the convergence is exponential and the slow down the polynomial, the constants involved may be such that the convergence times become unacceptable from a computational standpoint. This suggests that, in the absence of estimates of the constants involved, distinctions such as "exponentially fast" and "polynomially slow" may become meaningless from a constructive point of view. We also show that, for arbitray states, the speed of convergence to equilibrium is controlled by the rate of decoherence and the rate of purification (i.e. of concentration of the probability on a single pure state). We construct examples showing that the order of magnitude of these two decays can be quite different.