M. Hossein Partovi

Irreversibility, reduction, and entropy increase in quantum measurements

Abstract Using the strong subadditivity property of entropy, it is shown that the interaction of the measuring device with the environment brings about the reduction process characteristic of a quantum measurement. This entails an entropy increase which has an inviolable lower limit. This limit is found and explicitly calculated for illustrative examples.

Majorization formulation of uncertainty in quantum mechanics

Heisenbergs uncertainty principle is formulated for a set of generalized measurements within the framework of majorization theory, resulting in a partial uncertainty order on probability vectors that is stronger than those based on quasientropic measures. The theorem that emerges from this formulation guarantees that the uncertainty of the results of a set of generalized measurements without a common eigenstate has an inviolable lower bound which depends on the measurement set but not the state. A corollary to this theorem yields a parallel formulation of the uncertainty principle for generalized measurements corresponding to the entire class of quasientropic measures. Optimal majorization bounds for two and three mutually unbiased bases in two dimensions are calculated. Similarly, the leading term of the majorization bound for position and momentum measurements is calculated which provides a strong statement of Heisenbergs uncertainty principle in direct operational terms. Another theorem provides a majorization condition for the least-uncertain generalized measurement of a given state with interesting physical implications.

Entanglement detection using majorization uncertainty bounds

Entanglement detection criteria are developed within the framework of the majorization formulation of uncertainty. The primary results are two theorems asserting linear and nonlinear separability criteria based on majorization relations, the violation of which would imply entanglement. Corollaries to these theorems yield infinite sets of scalar entanglement detection criteria based on quasi-entropic measures of disorder. Examples are analyzed to probe the efficacy of the derived criteria in detecting the entanglement of bipartite Werner states. Characteristics of the majorization relation as a comparator of disorder uniquely suited to information-theoretical applications are emphasized throughout.

Entropic formulation of chaos for quantum dynamics

Abstract A general formulation of quantum chaos based on the asymptotic growth rate of measurement entropy is presented and shown to be closely related to the classical formulation in terms of Lyapunov characteristics exponents and the Kolmogorov-Sinai invariant. Examples are used to illustrate the formalism.

Algorithmic complexity and randomness

Algorithmic information content is an important fundamental idea for dealing with systems that contain or in the course of time produce large amounts of information. This notion provides a rigorous meaning to the intuitive concept of randomness, and thereby plays an important role in formal mathematics as well as in theoretical computer science. More importantly for applications, however, this idea has succeeded in providing a quantitatively precise way of characterizing the output of a deterministic dynamical system as random, or equivalently, algorithmically complex. This presentation will provide an introduction to the basic ideas of algorithmic complexity and randomness, as well to the applications of these ideas to the characterization of dynamical systems as examples of complex behavior.

Entropy, Chaos, and Quantum Mechanics

A characteristic feature of quantum theory is the nontrivial manner in which information about a quantum system is inferred from measurements. These measurements are in general incomplete and do not provide an exhaustive determination of the state of the system. Incomplete information is thus an inherent feature of microphysics and naturally calls for the application of entropie methods. Such methods have in recent years been successfully applied to a number of important problems in quantum dynamics. Here, we will concentrate on a recent formulation of chaos using entropic methods. This formalism, which is equally valid for classical and quantum dynamics, provides a new way of extracting Lyapunov exponents from measured or computed data. The basic quantity is the measurement entropy associated with a dynamical variable of the system, the characteristic exponents being related to the asymptotic growth rate of this entropy. Examples of the use of this method in quantum and classical dynamics, as well as a recent demonstration of the absence of sensitivity to initial conditions in quantum mechanics are discussed. Other applications of entropie methods such as the quantum maximum entropy principle, quantum thermodynamics, and decay of correlations in an open system are briefly recalled.