The quantum formalism is a "measurement" formalism--a phenomenological formalism describing certain macroscopic regularities. We argue that it can be regarded, and best be understood, as arising from Bohmian mechanics, which is what emerges from Schr6dinger's equation for a system of particles when we merely insist that "particles " means particles. While distinctly non-Newtonian, Bohmian mechanics is a fully deterministic theory of particles in motion, a motion choreographed by the wave function. We find that a Bohmian universe, though deterministic, evolves in such a manner that an appearance of randomness emerges, precisely as described by the quantum formalism and given, for example, by "p = IV [ 2.,, A crucial ingredient in our analysis of the origin of this randomness is the notion of the effective wave function of a subsystem, a notion of interest in its own right and of relevance to any discussion of quantum theory. When the quantum formalism is regarded as arising in this way, the paradoxes and perplexities so often associated with (nonrelativistic) quantum theory simply evaporate.

In order to arrive at Bohmian mechanics from standard nonrelativistic quantum mechanics one need do almost nothing! One need only complete the usual quantum description in what is really the most obvious way: by simply including the positions of the particles of a quantum system as part of the state description of that system, allowing these positions to evolve in the most natural way. The entire quantum formalism, including the uncertainty principle and quantum randomness, emerges from an analysis of this evolution. This can be expressed succinctly—though in fact not succinctly enough—by declaring that the essential innovation of Bohmian mechanics is the insight that particles move! 1 Bohmian Mechanics is Minimal Is it not clear from the smallness of the scintillation on the screen that we have to do with a particle? And is it not clear, from the diffraction and interference 1 patterns, that the motion of the particle is directed by a wave? De Broglie showed in detail how the motion of a particle, passing through just one of two holes in screen, could be influenced by waves propagating through both holes.

We explore in the framework of Quantum Computation the notion of computability, which holds a central position in Mathematics and Theoretical Computer Science. A quantum algorithm that exploits the quantum adiabatic which is equivalent to the Turing halting problem and known to be mathematically noncomputable. Generalised quantum algorithms are also considered for some other mathematical noncomputables in the same and of different noncomputability classes. The key element of all these algorithms is the measurability of both the values of physical observables and of the quantum-mechanical probability distributions for these values. It is argued that computability, and thus the limits of Mathematics, ought to be determined not

Bohmian mechanics is the most naively obvious embedding imaginable of Schrödinger’s equation into a completely coherent physical theory. It describes a world in which particles move in a highly non-Newtonian sort of way, one which may at first appear to have little to do with the spectrum of predictions of quantum mechanics. It turns out, however, that as a consequence of the defining dynamical equations of Bohmian mechanics, when a system has wave function ψ its configuration is typically random, with probability density ρ given by |ψ | 2, the quantum equilibrium distribution. It also turns out that the entire quantum formalism, operators as observables and all the rest, naturally emerges in Bohmian mechanics from the analysis of “measurements. ” This analysis reveals the status of operators as observables in the description of quantum phenomena, and facilitates a clear view of the range of applicability of the usual quantum mechanical formulas. Dedicated to Elliott Lieb on the occasion of his 70th birthday. Elliott will be (we fear unpleasantly) surprised to learn that he bears a greater responsibility for this paper than he could possibly imagine. We would of course like to think that our work addresses in some way the concern suggested by the title of his recent talks, The

Abstract. The purpose of these lecture notes is to provide readers, who have some mathematical background but little or no exposure to quantum mechanics and quantum computation, with enough material to begin reading

This article reviews the limitations of the standard computing paradigm and sketches the concept of quantum neural computing. Implications of this idea for the understanding of biological information processing and design of new kinds of computing machines are described. Arguments are presented in support of the thesis that brains are to be viewed as quantum systems with their neural structures representing the classical measurement hardware. From a performance point of view, a quantum neural computer may be viewed as a collection of many conventional computers that are designed to solve different problems. A quantum neural computer is a single machine that reorganizes itself, in response to a stimulus, to perform a useful computation. Selectivity offered by such a reorganization appears to be at the basis of the gestalt style of biological information processing. Clearly, a quantum neural computer is more versatile than the conventional computing machine.

Quantum theory has found a new field of applications in the realm of information and computation during the recent years. This paper reviews how quantum physics allows information coding in classically unexpected and subtle nonlocal ways, as well as information processing with an efficiency largely surpassing that of the present and foreseeable classical computers. Some outstanding aspects of classical and quantum information theory will be addressed here. Quantum teleportation, dense coding, and quantum cryptography are discussed as a few samples of the impact of quanta in the transmission of information. Quantum logic gates and quantum algorithms are also discussed as instances of the improvement in information processing by a quantum computer. We provide finally some examples of current experimental

This article presents a general discussion of several aspects of our present understanding of quantum mechanics. The emphasis is put on the very special correlations that this theory makes possible: they are forbidden by very general arguments based on realism and local causality. In fact, these correlations are completely impossible in any circumstance, except the very special situations designed by physicists especially to observe these purely quantum effects. Another general point that is emphasized is the necessity for the theory to predict the emergence of a single result in a single realization of an experiment. For this purpose, orthodox quantum mechanics introduces a special postulate: the reduction of the state vector, which comes in addition to the Schrödinger evolution postulate. Nevertheless, the presence in parallel of two evolution processes of the same object (the state vector) may be a potential source for conflicts; various attitudes that are possible

Abstract. We show that the particle motion in Bohmian mechanics, given by the solution of an ordinary differential equation, exists globally: For a large class of potentials the singularities of the velocity field and infinity will not be reached in finite time for typical initial values. A substantial part of the analysis is based on the probabilistic significance of the quantum flux. We elucidate the connection between the conditions necessary for global existence and the self-adjointness of the Schrödinger Hamiltonian.

Carrying out a research program outlined by John S. Bell in 1987, we arrive at a relativistic version of the Ghirardi–Rimini–Weber (GRW) model of spontaneous wavefunction collapse. As suggested by Bell, we take the primitive ontology, or local beables, of our model to be a discrete set of space-time points, at which the collapses are centered. This set is random with distribution determined by the initial wavefunction. The model is nonlocal and violates Bell’s inequality though it does not make use of a preferred slicing of space-time or any other sort of synchronization of spacelike separated points. Like the GRW model, it reproduces the quantum probabilities in all cases presently testable, though it entails deviations from the quantum formalism that are in principle testable. Our model works in Minkowski space-time as well as in (well-behaved) curved background space-times. PACS numbers: 03.65.Ta; 03.65.Ud; 03.30.+p. Key words: spontaneous wavefunction collapse; relativity; quantum theory without observers. 1