Fabio Benatti

OPEN QUANTUM DYNAMICS: COMPLETE POSITIVITY AND ENTANGLEMENT

We review the standard treatment of open quantum systems in relation to quantum entanglement, analyzing, in particular, the behavior of bipartite systems immersed in the same environment. We first focus upon the notion of complete positivity, a physically motivated algebraic constraint on the quantum dynamics, in relation to quantum entanglement, i.e. the existence of statistical correlations which can not be accounted for by classical probability. We then study the entanglement power of heat baths versus their decohering properties, a topic of increasing importance in the framework of the fast developing fields of quantum information, communication and computation. The presentation is self contained and, through several examples, it offers a detailed survey of the physics and of the most relevant and used techniques relative to both quantum open system dynamics and quantum entanglement.

Open system approach to neutrino oscillations

Neutrino oscillations are studied in the general framework of open quantum systems by means of extended dynamics that take into account possible dissipative effects. These new phenomena induce modifications in the neutrino oscillation pattern that in general can be parametrized by means of six phenomenological constants. Although very small, stringent bounds on these parameters are likely to be given by future planned neutrino experiments.

ASYMPTOTIC ENTANGLEMENT OF TWO INDEPENDENT SYSTEMS IN A COMMON BATH

Two non-interacting systems immersed in a common bath and evolving with a Markovian, completely positive dynamics can become initially entangled via a purely noisy mechanism. Remarkably, for certain phenomenologically relevant environments, the quantum correlations can persist even in the asymptotic long-time regime.

Complete positivity and the neutral kaon system

New experiments on neutral K-mesons might turn out to be promising tests of the hypothesis of Complete Positivity in the physics of open quantum systems. In particular, a consistent dynamical description of correlated neutral kaons seems to ask for Complete Positivity.

OPEN QUANTUM SYSTEMS AND COMPLETE POSITIVITY

The dynamics of an open quantum system is usually assumed to be generated by time-evolution equations of Lindblad form. It follows that the time evolutors are completely positive. Far from being just a useful technicality, complete positivity is one of the many aspects of quantum entanglement and is necessary to avoid physical inconsistencies. This issue is addressed by examining the open dynamics of two level systems in situations typical of chemical physics and quantum optics.The dynamics of an open quantum system is usually assumed to be generated by time-evolution equations of Lindblad form. It follows that the time evolutors are completely positive. Far from being just a useful technicality, complete positivity is one of the many aspects of quantum entanglement and is necessary to avoid physical inconsistencies. This issue is addressed by examining the open dynamics of two level systems in situations typical of chemical physics and quantum optics.

Entanglement and algebraic independence in fermion systems

In the case of systems composed of identical particles, a typical instance in quantum statistical mechanics, the standard approach to separability and entanglement ought to be reformulated and rephrased in terms of correlations between operators from subalgebras localized in spatially disjoint regions. While this algebraic approach is straightforward for bosons, in the case of fermions it is subtler since one has to distinguish between micro-causality, that is the anti-commutativity of the basic creation and annihilation operators, and algebraic independence that is the commutativity of local observables. We argue that a consistent algebraic formulation of separability and entanglement should be compatible with micro-causality rather than with algebraic independence.

Discrete dynamical systems embedded in Cantor sets

While the notion of chaos is well established for dynamical systems on manifolds, it is not so for dynamical systems over discrete spaces with N variables, as binary neural networks and cellular automata. The main difficulty is the choice of a suitable topology to study the limit N→∞. By embedding the discrete phase space into a Cantor set we provided a natural setting to define topological entropy and Lyapunov exponents through the concept of error profile. We made explicit calculations both numerical and analytic for well-known discrete dynamical models.

Semi-Classical Localisation Properties of Quantum Oscillators on a Noncommutative Configuration Space

When dealing with the classical limit of two quantum mechanical oscillators on a noncommutative configuration space, the limits corresponding to the removal of configuration-space noncommutativity and position-momentum noncommutativity do not commute. We address this behaviour from the point of view of the phase-space localisation properties of the Wigner functions of coherent states under the two limits.

A non-Markovian Dissipative Maryland Model

The so-called Maryland model is a linear version of the quantum kicked rotor; it exhibits Anderson localization in momentum space. By turning the kicks into a Markovian stochastic process, the dynamics becomes a dissipative quantum process described by a discrete family of completely positive maps that allows to explicitly study the relation between divisibility of the maps and the degree of memory of the process.

On Deciding Whether a Boolean Function is Constant or Not

We study the probability of making an error if, by querying an oracle a fixed number of times, we declare constant a randomly chosen n-bit Boolean function. We compare the classical and the quantum case, and we determine for how many oracle-queries k and for how many bits n one querying procedure is more efficient than the other.