Purification of quantum trajectories

A key concept in the modern theory of open quantum systems is the notion of indirect measurement as introduced by Kraus [Kra]. An indirect measurement on a quantum system is a (direct) measurement of some quantity in its environment, made after some interaction with the system has taken place. When we make such a measurement, our description of the quantum system changes in two ways: we account for the flow of time by a unitary transformation (following Schrödinger), and we update our knowledge of the system by conditioning on the measurement outcome (following von Neumann). If we then repeat the indirect measurement indefinitely, we obtain a chain of random outcomes. In the course of time we may keep record of the updated density matrix Θt, which at time t reflects our best estimate of all observable quantities of the quantum system, given the observations made up to that time. This information can in its turn be used to predict later measurements outcomes. The stochastic process Θt of updated states, is the quantum trajectory associated to the repeated measurement process. By taking the limit of continuous time, we arrive at the modern models of continuous observation: quantum trajectories in continuous time satisfying stochastic Schrödinger equations [Dav], [Gis], [Car], [BGM]. These models are employed with great success for calculations and computer simulations of laboratory experiments such as photon counting and homodyne field detection. In this paper we consider the question, what happens to the quantum trajectory at large times. We do so only for the case of discrete time, not a serious restriction indeed, since asymptotic behaviour remains basically unaltered in the continuous time limit. We focus on the case of perfect measurement, i.e. the situation where no information flows into the system, and all information which leaks out is indeed observed. In classical probability such repeated perfect measurement would lead to a further and further narrowing of the distribution of the system, until it either becomes pure, i.e. an atomic measure, or it remains spread out over some area, thus leaving a certain amount of information ‘in the dark’ forever. Using a fundamental inequality of Nielsen [Nie] we prove that in quantum mechanics the situation is quite comparable: the density matrix tends to purify, until it hits upon some family of ‘dark’ subspaces, if such exist, i.e. spaces from which no information can leak out. A crucial difference with the classical case is, however, that even after all available information has been extracted by observation, the state continues to move about in a random fashion between the ‘dark’ subspaces, thus continuing to produce ‘quantum noise’. The structure of this paper is as follows. In Section 2 we introduce quantum measurement on a finite system, in particular Kraus measurement. In Section 3 repeated