Table of contents

Theory and experiment have not fully resolved the apparent dichotomy, which has agonized physics for the past eighty years: on the one hand, the description of microsystems by quantum mechanics and, on the other, the description of macrosystems by classical dynamics or statistical mechanics. Derivations of the time-irreversible Liouville equation for an open quantum system, based on projecting out its environment, have narrowed the gap between the quantum and classical descriptions. Yet our `classical' intuition continues to be confronted by quantum-mechanical results like the Einstein--Podolsky--Rosen paradox that challenges the classical notion of locality, or the quantum Zeno effect which suggests that the isolation of a system is not the only way to preserve its quantum state.

There are two key concepts in any discussion of such issues. The first, which is responsible for the most salient nonclassical properties, is entanglement, that is partial or complete correlation or, more generally, inseparability of the elements comprising a quantum ensemble. Even after their interaction has ceased, this inseparability, originating from their past interaction, can affect the state of one element when another element is subject to a nonunitary action, such as its measurement, tracing- out, or thermalization. The second key concept is decoherence of open quantum systems, which is the consequence of their entanglement with their environment, a `meter' or a thermal `reservoir', followed by the tracing-out of the latter. Despite new insights into entanglement and decoherence, there are still no complete, unequivocal answers to the fundamental questions of the transition from quantal to classical behaviour: how do irreversibility and classicality emerge from unitarity as systems and their environments become increasingly complex? At what stage does system--meter entanglement give rise to a classical readout of the meter? Is there an upper limit on the size or complexity of systems displaying entanglement?

Major developments have opened new vistas into controlled entanglement, which is the resource of quantum information processing. However, these developments have mainly focused on ensembles of simple two- and three-level systems that are thoroughly isolated from their environments. Treatments of coherence in quantum computing have mostly assumed that only a single or a few the elements of the quantum ensemble may simultaneously undergo an uncontrolled intervention---a quantum error. Decoherence-control protocols for more general types of errors are still incomplete.

In order to resolve the outstanding issues of the quantum--classical transition, and study the control of entanglement and decoherence without the foregoing restrictions, we must venture into the domain of Quantum Complex Systems (QUACS), either consisting of a large number of inseparable elements or having many coupled degrees of freedom.

Our conviction is that fundamental understanding and manipulation of entanglement within QUACS or their entanglement with the environment or a meter, call for the creation of a new conceptual framework or paradigm, that would encompass phenomena common to cold atoms in laser fields, large molecules, Josephson junctions, quantum gases and solids, with the view of employing these systems for quantum information processing and computing. Progress within this paradigm should allow us to answer the questions: does entanglement play an essential role in the evolution of large collections of complex systems? What are the size and complexity limits of systems and ensembles still controllable by an external intervention? What are the most appropriate decoherence protection schemes and control algorithms?

This special issue addresses the challenge of understanding in depth and manipulating the basic quantum properties of optical, atomic, molecular and condensed-matter QUACS, and large ensembles thereof.

In view of the interdisciplinary character of this issue, we find it expedient to look at the presented articles not only according to the physical objects they describe, but also from a unifying standpoint. Several unifying themes may be discerned:

• memory effects in dephasing and relaxation;

• protection of quantum information by its distribution throughout the ensemble;

• control of decoherence and entanglement by time-dependent interventions or measurements;

• manifestations of entanglement in scattering off statistical ensembles;

• measurment-induced dynamics.

The concerted discussion of the topics outlined above should help us advance the new paradigm that addresses our abilities to diagnose and manipulate the entangled states of complex quantum objects and their robustness against decoherence. These abilities are required for quantum information (QI) applications in matter--wave interferometry in molecular, semiconducting or superconducting systems. On the fundamental level, this book may help establish the notion of dynamical information exchange between quantum systems and chart in detail the route from unitarity to classicality.

Further developments within the outlined paradigm should yield novel, advantageous QI processing schemes and high-sensitivity interferometers, owing to decoherence suppression and effective control of many degrees of freedom. In the long run, these strategies may prove to be the first step towards the (hitherto unattempted) use of entanglement in nanotechnology, metrology and chemistry, with potentially remarkable novel applications.

Weighted graph states naturally arise when spin systems interact via an Ising-type interaction. First, we abstractly define the class of weighted graph states and demonstrate its computational accessibility. We show how reduced density matrices of a small number of spins (≈10) can be computed from arbitrarily large systems using weighted graph techniques and projected entangled pair techniques, and we discuss various entanglement measures accessible from these reduced density matrices. Second, we apply these findings to spin chains and lattices with long-range interactions and analytically derive area laws for the scaling of block-wise entanglement. Then, we turn to disordered spin systems, spin gases, which are connected to random weighted graph states and which share their entanglement properties. Finally, we use a spin gas as a bath that introduces decoherence in single as well as multipartite spin systems. The microscopic, exact decoherence model we obtain can operate in different regimes and exhibit non-Markovian features as well as spatially correlated noise effects.

We examine entanglement dynamics via concurrence among four two-state systems labelled A, a, B, b. The four systems are arranged on an addressable 'lattice' in such a way that A and a at one location labelled Aa can interact with each other via excitation exchange, and the same for B and b at location Bb. The Aa location is prepared entangled with the Bb location, but their mutual complete isolation prevents interaction in the interval between actions of an external addressing agent. There are six pairwise concurrences on the lattice, and we follow their evolution in the interval between external actions. We show how entanglement evolves and may exhibit the non-analytic effect termed entanglement sudden death (ESD), with periodic recovery. These loss and gain processes may be interpreted as entanglement transfers between the subsystems.

We present and compare stochastic open-loop techniques aimed at controlling quantum coherence in dissipative environments. One approach describes the evolution time as a random non-Gaussian variable. The other implements dynamical control on non-Markovian time-scales via stochastic modulation of the system–bath coupling.

In this paper, we develop, step by step, the framework for universal dynamical control of two-level systems (TLS) or qubits experiencing amplitude or phase noise (AN or PN) due to coupling to a thermal bath. A comprehensive arsenal of modulation schemes is introduced and applied to either AN or PN, resulting in completely analogous formulae for the decoherence rates, thus underscoring the unified nature of this universal formalism. We then address the extension of this formalism to multipartite decoherence control, where symmetries are exploited to overcome decoherence.

We have identified a class of many-body problems with analytic solution beyond the mean-field approximation. This is the case where each body can be considered as an element of an assembly of interacting particles that are translationally frozen multi-level quantum systems and that do not change significantly their initial quantum states during the evolution. In contrast, the entangled collective state of the assembly experiences an appreciable change. We apply this approach to interacting three-level systems.

Methods of optimal control are applied to a model system of interacting two-level particles (e.g., spin-half atomic nuclei or electrons or two-level atoms) to produce high-fidelity quantum gates while simultaneously negating the detrimental effect of decoherence. One set of particles functions as the quantum information processor, whose evolution is controlled by a time-dependent external field. The other particles are not directly controlled and serve as an effective environment, coupling to which is the source of decoherence. The control objective is to generate target one- and two-qubit unitary gates in the presence of strong environmentally-induced decoherence and under physically motivated restrictions on the control field. The quantum-gate fidelity, expressed in terms of a novel state-independent distance measure, is maximized with respect to the control field using combined genetic and gradient algorithms. The resulting high-fidelity gates demonstrate the feasibility of precisely guiding the quantum evolution via optimal control, even when the system complexity is exacerbated by environmental coupling. It is found that the gate duration has an important effect on the control mechanism and resulting fidelity. An analysis of the sensitivity of the gate performance to random variations in the system parameters reveals a significant degree of robustness attained by the optimal control solutions.

We consider the STIRAP process in a three-level atom. Viewed as a closed system, no geometric phase is acquired. But in the presence of spontaneous emission and/or collisional relaxation we numerically show that a non-vanishing, purely real, geometric phase is acquired during STIRAP, whose magnitude grows with the decay rates. We speculate that rather than viewing this decoherence-induced geometric phase as a nuisance, it can be considered an example of 'beneficial decoherence': the environment provides a mechanism for the generation of geometric phases which would otherwise require an extra experimental control knob.

We consider two two-level atoms, which are located in two independent dissipative cavities. The two atoms are initially prepared in the singlet state. We investigate the influence of dissipation on the entanglement between the two atoms. In the case of resonance the degree of the entanglement can fall abruptly to zero, which is the so-called sudden death of entanglement. It is noted that when two atoms are in a common environment, the singlet state is a decoherence-free state. When the dispersive limit is fulfilled, the degree of entanglement between the two atoms is oscillating at first, then it arrives at a steady value, which increases with the increase of the damping constant κ; when both fields are initially in the vacuum state, the entanglement between the two atoms will not decay.

In this paper, we propose a neutral atom implementation of the two-qubit conditional phase gate, based on the large Rydberg–Rydberg interaction induced energy shift. Contrary to previous proposals, this gate does not substantially populate the Rydberg levels. After describing the model in detail, we provide numerical simulations and discuss the performance of our gate. In particular we address the influence on the fidelity of spontaneous emission and atomic motion.

We apply the general formalism of nilpotent polynomials (Mandilara et al 2006 Phys. Rev. A 74 022331) to the problem of pure-state multipartite entanglement classification in four qubits. In addition to establishing contact with the existing results, we explicitly show how the nilpotent formalism naturally suggests constructions of entanglement measures invariant under the required unitary or invertible class of local operations. A candidate measure of four-partite entanglement is also suggested, and its behaviour numerically tested on random pure states.

We calculate the radiative properties for a linear dipole-coupled chain of qubits. Using the explicit energy eigenstates of the system, we find the radiation patterns for spontaneous transitions from the one-photon eigenstates to the ground state of the system. We show that depending on the excitation of a specific atom, the radiation tends to be focused either along or perpendicular to the chain. We conclude with a derivation of the total decay rate of the one-photon eigenstates, and find the interesting result that for systems where the photon wavenumber is not much larger than the interatomic spacing, up to 94% of the eigenstates are subradiant, that is, they decay significantly slower than a single atom in isolation.

With a model of a one-dimensional two-electron atom we study the entanglement properties between two particles with non-overlapping wavefunctions that interact via the Coulombic, long-range potential. We calculate the two-electron eigenstates and monitor the bipartite entanglement using the von Neumann entropy. We observe non-trivial entangled states as we approach the double-excited state and Bell-like states at the single-ionization potential.

The utility of classical mechanics to compute the dynamics of decoherence in continuous variable systems, as defined by the time dependence of the system purity, is examined. Its accuracy, in the semiclassical domain, is shown to depend on the nature of the coupling potentials and, less so, on the initial conditions. For a large range of cases, agreement between quantum and classical computations in small systems is shown to be excellent.

We propose a scheme for the generation of maximally entangled states involving internal electronic degrees of freedom of two distant trapped ions, each of them located in a cavity. This is achieved by using a single flying atom to distribute entanglement. For certain specific interaction times, the proposed scheme leads to the non-probabilistic generation of a perfect Bell-type state. At the end of the protocol, the flying atom completely disentangles from the rest of the system, leaving both ions in a Bell-type state. Moreover, the scheme is insensitive to the cavity field state and cavity losses. We also address the situation in which dephasing and dissipation must be taken into account for the flying atom on its way from one cavity to the other, and discuss the applicability of the resulting noisy channel for performing quantum teleportation.

It is shown that the stochastic dephasing of a qubit caused by the random frequency modulation in which the stochastic variable obeys the stationary Gauss–Markov process can be suppressed by applying a sequence of π pulses to the qubit. It is found that the application of around ten π pulses within the correlation time of the stochastic variable is enough to suppress the decoherence of the qubit. The recovery of the entanglement of the Werner state by means of π pulses is also discussed.

By controlling externally applied pulses, we find suppression of decoherence on the basis of an exactly solvable quantum mechanical model. This is intuitively recognized in terms of an effective potential function produced by pulse application. Time evolution of spin variables is explicitly determined.