545 research outputs found
The fundamental limit on the rate of quantum dynamics: the unified bound is tight
The question of how fast a quantum state can evolve has attracted a
considerable attention in connection with quantum measurement, metrology, and
information processing. Since only orthogonal states can be unambiguously
distinguished, a transition from a state to an orthogonal one can be taken as
the elementary step of a computational process. Therefore, such a transition
can be interpreted as the operation of "flipping a qubit", and the number of
orthogonal states visited by the system per unit time can be viewed as the
maximum rate of operation.
A lower bound on the orthogonalization time, based on the energy spread
DeltaE, was found by Mandelstam and Tamm. Another bound, based on the average
energy E, was established by Margolus and Levitin. The bounds coincide, and can
be exactly attained by certain initial states if DeltaE=E; however, the problem
remained open of what the situation is otherwise.
Here we consider the unified bound that takes into account both DeltaE and E.
We prove that there exist no initial states that saturate the bound if DeltaE
is not equal to E. However, the bound remains tight: for any given values of
DeltaE and E, there exists a one-parameter family of initial states that can
approach the bound arbitrarily close when the parameter approaches its limit
value. The relation between the largest energy level, the average energy, and
the orthogonalization time is also discussed. These results establish the
fundamental quantum limit on the rate of operation of any
information-processing system.Comment: 4 pages 1 PS figure Late
Thermodynamic cost of reversible computing
Since reversible computing requires preservation of all information
throughout the entire computational process, this implies that all errors that
appear as a result of the interaction of the information-carrying system with
uncontrolled degrees of freedom must be corrected. But this can only be done at
the expense of an increase in the entropy of the environment corresponding to
the dissipation, in the form of heat, of the ``noisy'' part of the system's
energy.
This paper gives an expression of that energy in terms of the effective noise
temperature, and analyzes the relationship between the energy dissipation rate
and the rate of computation. Finally, a generalized Clausius principle based on
the concept of effective temperature is presented.Comment: 5 pages; added two paragraphs and fixed a number of typo
Why 'scaffolding' is the wrong metaphor : the cognitive usefulness of mathematical representations.
The metaphor of scaffolding has become current in discussions of the cognitive help we get from artefacts, environmental affordances and each other. Consideration of mathematical tools and representations indicates that in these cases at least (and plausibly for others), scaffolding is the wrong picture, because scaffolding in good order is immobile, temporary and crude. Mathematical representations can be manipulated, are not temporary structures to aid development, and are refined. Reflection on examples from elementary algebra indicates that Menary is on the right track with his ‘enculturation’ view of mathematical cognition. Moreover, these examples allow us to elaborate his remarks on the uniqueness of mathematical representations and their role in the emergence of new thoughts.Peer reviewe
Quantum lattice gases and their invariants
The one particle sector of the simplest one dimensional quantum lattice gas
automaton has been observed to simulate both the (relativistic) Dirac and
(nonrelativistic) Schroedinger equations, in different continuum limits. By
analyzing the discrete analogues of plane waves in this sector we find
conserved quantities corresponding to energy and momentum. We show that the
Klein paradox obtains so that in some regimes the model must be considered to
be relativistic and the negative energy modes interpreted as positive energy
modes of antiparticles. With a formally similar approach--the Bethe ansatz--we
find the evolution eigenfunctions in the two particle sector of the quantum
lattice gas automaton and conclude by discussing consequences of these
calculations and their extension to more particles, additional velocities, and
higher dimensions.Comment: 19 pages, plain TeX, 11 PostScript figures included with epsf.tex
(ignore the under/overfull \vbox error messages
Triggering rogue waves in opposing currents
We show that rogue waves can be triggered naturally when a stable wave train
enters a region of an opposing current flow. We demonstrate that the maximum
amplitude of the rogue wave depends on the ratio between the current velocity,
, and the wave group velocity, . We also reveal that an opposing
current can force the development of rogue waves in random wave fields,
resulting in a substantial change of the statistical properties of the surface
elevation. The present results can be directly adopted in any field of physics
in which the focusing Nonlinear Schrodinger equation with non constant
coefficient is applicable. In particular, nonlinear optics laboratory
experiments are natural candidates for verifying experimentally our results.Comment: 5 pages, 5 figure
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