## Fokker-Planck equations for eigenstate distributions

Letters in Mathematical Physics 25, 161-174 (1992) A simple model of relaxation phenomena is defined with a variable strength of interaction and where the interaction term is given by a Gaussian unitary ensemble of random matrices. A set of Fokker-Planck equations are derived which describe the gradual delocalization with increasing strength of interaction of the eigenstates with respect to the unperturbed energy eigenbasis. The effect of localization on the time evolution is a nonergodic property: the system has a memory of the initial state.Gzipped postscript file

## Determinism and randomness in quantum dynamics

Journal of Physics A 26, 7193-7211 (1993) A class of models is considered where a finite or infinite quantum dynamical system in a stationary state is probed by sequences of observations acting on a specified finite subsystem. The whole set of such experiments is described by a time-ordered (causal) and stationary quantum correlation kernel. It is shown that any such kernel can be decomposed in a unique way into a convex combination of two kernels, here called the regular and singular components. A singular system has a strong deterministic property, the predictability of the future from the knowledge of the past is limited only by the inevitable indeterminism of quantum measurements. Furthermore, in this case the full set of correlation functions of arbitrary time order, and hence the dynamical system itself, is determined by the causal kernel. Finite systems and infinite systems satisfying the KMS condition at finite temperature are of this type. In the regular case the dynamics contains a shift, there is a genuine asymptotic randomness and the dynamical system cannot be reconstructed in a unique way from the causal kernel. Nontrivial quantum Markov processes are shown to belong to this class.Gzipped postscript file

## Irreversibility and randomness in linear response theory

Journal of Statistical Physics 72, 539-554 (1993) Recall that the fluctuation-dissipation theorem connects the response function of a passive linear system and the spectral density of the stationary stochastic process which describes the thermal fluctuations in the system. It is shown that the classical limit \hbar = 0 of the fluctuation dissipation theorem implies a correspondence between systems which are reversible in the sense that the energy used to drive it away from equilibrium is completely recoverable as work and processes which are deterministic in the sense of Wiener's prediction theory, while irreversible systems correspond to non-deterministic processes. This correspondence is expressed by a simple transformation between the operator kernel which determines the optimal choice of the time-dependent force and the linear predictor for the stochastic process. For quantum systems this correspondence does not hold, the fluctuations are always of the deterministic type for any finite temperature, but the system is not necessarily reversible. For irreversible systems a formula is derived for the instantaneous entropy production which is a generalization of the standard one for Markovian dynamics.Gzipped postscript file

## Decoherence properties of finite quantum systems

1993, unpublished In this paper some of the properties of the correlation functions for finite quantum systems (with unitary dynamics and discrete spectra) are investigated using methods from operator algebras and quantum statistical mechanics. Their information content and decoherence properties are measured by entropy functionals in a formalism which is capable of dealing with approximate measurements as well as the ideal ones defined by orthogonal projectors. Necessary and sufficient decoherence conditions are found for the observables to form a classical commutative system. These conditions involve the correlation functions of all orders, and the systems satisfying them exactly are exceptional. On the other hand it is shown that for large but finite quantum systems most observables will approximately satisfy a chaotic form of decoherence condition for correlation functions of not too high order. It is also possible to choose observables which define nearly deterministic (and hence decoherent) histories over a finite time.Gzipped postscript file

## On the existence of quantum subdynamics

Journal of Physics A 29, 4197-4207 (1996) It is shown, using only elementary operator algebra, that an open quantum system coupled to its environment will have a subdynamics (reduced dynamics) as an exact consequence of the reversible dynamics of the composite system only when the states of system and environment are uncorrelated. Furthermore, it is proved that for a finite temperature the KMS condition for the lowest order correlation function cannot be reproduced by any type of linear subdynamics except the reversible Hamiltonian one of a closed system. The first statement can be seen as a particular case of a much more general theorem of Takesaki on the properties of conditional expectations in von Neumann algebras. The concept of subdynamics used here allows for memory effects, no assumption is made of a Markov property. For dynamical systems based on commutative algebras of observables the subdynamics always exists as a stochastic process in the random variable defining the open subsystem.Gzipped postscript file

## Brownian motion of quantum harmonic oscillators: existence of a subdynamics

Journal of Mathematical Physics 39(5), 2763-80 (1998) The effects of system-environment correlations on the dynamics of an open quantum system are investigated for the standard model of a set of quantum harmonic oscillators interacting with a heat bath of oscillators. The observable properties of the open system are completely defined by a matrix-valued covariance function, this fact allows a simple but detailed analysis. There is a subdynamics when this function comes from transformations of the open system observables alone. We show that this happens only when the states of system and environment are uncorrelated, while for classical systems there is always a subdynamics. A quantum subdynamics cannot have the properties we associate with thermal fluctuations, the KMS relation for the covariance function at a finite temperature implies that the system must be closed. The conditions for having a subdynamics as a good approximation to the exact closed dynamics are investigated, and so are the similar but stronger conditions for a Markovian dynamics. It is also shown how a subdynamics defines the response of the open system to some types of time dependent external forces.Gzipped postscript file

## A general no-cloning theorem

Letters in Mathematical Physics 47(2), 189-196 (1999) A simple theorem is proved which has most of the known no-cloning and no-broadcasting results as corollaries. It also implies the standard restrictions on measuring non-commuting observables.Gzipped postscript file