13 research outputs found
Noiseless Quantum Circuits for the Peres Separability Criterion
In this Letter we give a method for constructing sets of simple circuits that
can determine the spectrum of a partially transposed density matrix, without
requiring either a tomographically complete POVM or the addition of noise to
make the spectrum non-negative. These circuits depend only on the dimension of
the Hilbert space and are otherwise independent of the state.Comment: 4 pages RevTeX, 7 figures encapsulated postscript. v5: title changed
slightly, more-or-less equivalent to the published versio
The quantum to classical transition for random walks
We look at two possible routes to classical behavior for the discrete quantum
random walk on the line: decoherence in the quantum ``coin'' which drives the
walk, or the use of higher-dimensional coins to dilute the effects of
interference. We use the position variance as an indicator of classical
behavior, and find analytical expressions for this in the long-time limit; we
see that the multicoin walk retains the ``quantum'' quadratic growth of the
variance except in the limit of a new coin for every step, while the walk with
decoherence exhibits ``classical'' linear growth of the variance even for weak
decoherence.Comment: 4 pages RevTeX 4.0 + 2 figures (encapsulated Postscript). Trimmed for
length. Minor corrections + one new referenc
Quantum Walks driven by many coins
Quantum random walks have been much studied recently, largely due to their
highly nonclassical behavior. In this paper, we study one possible route to
classical behavior for the discrete quantum random walk on the line: the use of
multiple quantum ``coins'' in order to diminish the effects of interference
between paths. We find solutions to this system in terms of the single coin
random walk, and compare the asymptotic limit of these solutions to numerical
simulations. We find exact analytical expressions for the time-dependence of
the first two moments, and show that in the long time limit the ``quantum
mechanical'' behavior of the one-coin walk persists. We further show that this
is generic for a very broad class of possible walks, and that this behavior
disappears only in the limit of a new coin for every step of the walk.Comment: 36 pages RevTeX 4.0 + 5 figures (encapsulated Postscript). Submitted
to Physical Review
On the logical structure of Bell theorems without inequalities
Bell theorems show how to experimentally falsify local realism. Conclusive
falsification is highly desirable as it would provide support for the most
profoundly counterintuitive feature of quantum theory - nonlocality. Despite
the preponderance of evidence for quantum mechanics, practical limits on
detector efficiency and the difficulty of coordinating space-like separated
measurements have provided loopholes for a classical worldview; these loopholes
have never been simultaneously closed. A number of new experiments have
recently been proposed to close both loopholes at once. We show some of these
novel designs fail in the most basic way, by not ruling out local hidden
variable models, and we provide an explicit classical model to demonstrate
this. They share a common flaw, which reveals a basic misunderstanding of how
nonlocality proofs work. Given the time and resources now being devoted to such
experiments, theoretical clarity is essential. Our explanation is presented in
terms of simple logic and should serve to correct misconceptions and avoid
future mistakes. We also show a nonlocality proof involving four participants
which has interesting theoretical properties.Comment: 8 pages, text clarified, explicit LHV model provided for flawed
nonlocality tes
Maximum Power Efficiency and Criticality in Random Boolean Networks
Random Boolean networks are models of disordered causal systems that can
occur in cells and the biosphere. These are open thermodynamic systems
exhibiting a flow of energy that is dissipated at a finite rate. Life does work
to acquire more energy, then uses the available energy it has gained to perform
more work. It is plausible that natural selection has optimized many biological
systems for power efficiency: useful power generated per unit fuel. In this
letter we begin to investigate these questions for random Boolean networks
using Landauer's erasure principle, which defines a minimum entropy cost for
bit erasure. We show that critical Boolean networks maximize available power
efficiency, which requires that the system have a finite displacement from
equilibrium. Our initial results may extend to more realistic models for cells
and ecosystems.Comment: 4 pages RevTeX, 1 figure in .eps format. Comments welcome, v2: minor
clarifications added, conclusions unchanged. v3: paper rewritten to clarify
it; conclusions unchange
Evanescence in Coined Quantum Walks
In this paper we complete the analysis begun by two of the authors in a
previous work on the discrete quantum walk on the line [J. Phys. A 36:8775-8795
(2003) quant-ph/0303105 ]. We obtain uniformly convergent asymptotics for the
"exponential decay'' regions at the leading edges of the main peaks in the
Schr{\"o}dinger (or wave-mechanics) picture. This calculation required us to
generalise the method of stationary phase and we describe this extension in
some detail, including self-contained proofs of all the technical lemmas
required. We also rigorously establish the exact Feynman equivalence between
the path-integral and wave-mechanics representations for this system using some
techniques from the theory of special functions. Taken together with the
previous work, we can now prove every theorem by both routes.Comment: 32 pages AMS LaTeX, 5 figures in .eps format. Rewritten in response
to referee comments, including some additional references. v3: typos fixed in
equations (131), (133) and (134). v5: published versio
Three routes to the exact asymptotics for the one-dimensional quantum walk
We demonstrate an alternative method for calculating the asymptotic behaviour
of the discrete one-coin quantum walk on the infinite line, via the Jacobi
polynomials that arise in the path integral representation. This is
significantly easier to use than the Darboux method. It also provides a single
integral representation for the wavefunction that works over the full range of
positions, including throughout the transitional range where the behaviour
changes from oscillatory to exponential. Previous analyses of this system have
run into difficulties in the transitional range, because the approximations on
which they were based break down here. The fact that there are two different
kinds of approach to this problem (Path Integral vs. Schr\"{o}dinger wave
mechanics) is ultimately a manifestation of the equivalence between the
path-integral formulation of quantum mechanics and the original formulation
developed in the 1920s. We discuss how and why our approach is related to the
two methods that have already been used to analyse these systems.Comment: 25 pages, AMS preprint format, 4 figures as encapsulated postscript.
Replaced because there were sign errors in equations (80) & (85) and Lemma 2
of the journal version (v3