J. J. Halliwell

Information-theoretic measure of uncertainty due to quantum and thermal fluctuations

We study an information-theoretic measure of uncertainty for quantum systems. It is the Shannon information I of the phase-space probability distribution , where |z> are coherent states and ρ is the density matrix. As shown by Lieb I ≥ 1, and this bound represents a strengthened version of the uncertainty principle. For a harmonic oscillator in a thermal state, I coincides with von Neumann entropy, −Tr(ρlnρ), in the high-temperature regime, but unlike entropy, it is nonzero (and equal to the Lieb bound) at zero temperature. It therefore supplies a nontrivial measure of uncertainty due to both quantum and thermal fluctuations

Somewhere in the universe: Where is the information stored when histories decohere?

In the context of the decoherent histories approach to quantum theory, we investigate the idea that decoherence is connected with the storage of information about the decohering system somewhere in the universe. The known connection between decoherence and the existence of records is extended from the case of pure initial states to mixed states, where it is shown that records may still exist but are necessarily imperfect. We formulate an information-theoretic conjecture about decoherence due to an environment; the number of bits required to describe a set of decoherent histories is approximately equal to the number of bits of information thrown away to the environment in the coarse-graining process. This idea is verified in a simple model consisting of a particle coupled to an environment that can store only one bit of information. We explore the decoherence and information storage in the quantum Brownian motion model, in which a particle trajectory is decohered as a result of coupling to an environment of harmonic oscillators in a thermal state. It is shown that the variables that the environment naturally measures and stores information about are nonlocal functions of time, which are essentially the Fourier components of the function x(t) ~describing the particle trajectory!. In particular, the records storing the information about the Fourier modes are the positions and momenta of the environmental oscillators at the final time. We show that it is possible to achieve decoherence even if there is only one oscillator in the environment. The information count of the histories and records in the environment adds up according to our conjecture. These results give quantitative content to the idea that decoherence is related to ‘‘information lost.’’ Some implications of these ideas for quantum cosmology are discussed. @S0556-2821~99!07320-8#

Alternative derivation of the Hu-Paz-Zhang master equation of quantum Brownian motion

Hu, Paz and Zhang [ B.L. Hu, J.P. Paz and Y. Zhang, Phys. Rev. D {\bf 45} (1992) 2843] have derived an exact master equation for quantum Brownian motion in a general environment via path integral techniques. Their master equation provides a very useful tool to study the decoherence of a quantum system due to the interaction with its environment. In this paper, we give an alternative and elementary derivation of the Hu-Paz-Zhang master equation, which involves tracing the evolution equation for the Wigner function. We also discuss the master equation in some special cases.

DECOHERENT HISTORIES AND QUANTUM STATE DIFFUSION

We demonstrate a close connection between the decoherent histories (DH) approach to quantum mechanics and the quantum state diffusion (QSD) picture, for open quantum systems described by a master equation of Lindblad form. The (physically unique) set of variables that localize in the QSD picture also define an approximately decoherent set of histories in the DH approach. The degree of localization is related to the degree of decoherence, and the probabilities for histories prescribed by each approach are essentially the same.

Probabilities in Quantum Cosmological Models: A Decoherent Histories Analysis Using a Complex Potential

In the quantization of simple cosmological models (minisuperspace models) described by the Wheeler-DeWitt equation, an important step is the construction, from the wave function, of a probability distribution answering various questions of physical interest, such as the probability of the system entering a given region of configuration space at any stage in its entire history. A standard but heuristic procedure is to use the flux of (components of) the wave function in a WKB approximation. This gives sensible semiclassical results but lacks an underlying operator formalism. In this paper, we address the issue of constructing probability distributions linked to the Wheeler-DeWitt equation using the decoherent histories approach to quantum theory. The key step is the construction of class operators characterizing questions of physical interest. Taking advantage of a recent decoherent histories analysis of the arrival time problem in non-relativistic quantum mechanics, we show that the appropriate class operators in quantum cosmology are readily constructed using a complex potential. The class operator for not entering a region of configuration space is given by the S-matrix for scattering off a complex potential localized in that region. We thus derive the class operators for entering one or more regions in configuration space. The class operators commute with the Hamiltonian, have a sensible classical limit and are closely related to an intersection number operator. The definitions of class operators given here handles the key case in which the underlying classical system has multiple crossings of the boundaries of the regions of interest. We show that oscillatory WKB solutions to the Wheeler-DeWitt equation give approximate decoherence of histories, as do superpositions of WKB solutions, as long as the regions of configuration space are sufficiently large. The corresponding probabilities coincide, in a semiclassical approximation, with standard heuristic procedures. In brief, we exhibit the welldefined operator formalism underlying the usual heuristic interpretational methods in quantum cosmology. PACS numbers: 04.60.-m, 04.60.Kz, 04.60.Ds, 03.65.Yz

Generalized uncertainty relations and long time limits for quantum Brownian motion models

We study the time evolution of the reduced Wigner function for a class of quantum Brownian motion models. We derive two generalized uncertainty relations. The first consists of a sharp lower bound on the uncertainty function,

Quantum state diffusion, density matrix diagonalization, and decoherent histories: A model.

U = (\Delta p)^2 (\Delta q)^2

Decoherent histories and hydrodynamic equations

, after evolution for time

Arrival times, complex potentials, and decoherent histories

t

Decoherent histories analysis of minisuperspace quantum cosmology

in the presence of an environment. The second, a stronger and simpler result, consists of a lower bound at time