2,365 research outputs found
Quantum complexities of ordered searching, sorting, and element distinctness
We consider the quantum complexities of the following three problems:
searching an ordered list, sorting an un-ordered list, and deciding whether the
numbers in a list are all distinct. Letting N be the number of elements in the
input list, we prove a lower bound of \frac{1}{\pi}(\ln(N)-1) accesses to the
list elements for ordered searching, a lower bound of \Omega(N\log{N}) binary
comparisons for sorting, and a lower bound of \Omega(\sqrt{N}\log{N}) binary
comparisons for element distinctness. The previously best known lower bounds
are {1/12}\log_2(N) - O(1) due to Ambainis, \Omega(N), and \Omega(\sqrt{N}),
respectively. Our proofs are based on a weighted all-pairs inner product
argument.
In addition to our lower bound results, we give a quantum algorithm for
ordered searching using roughly 0.631 \log_2(N) oracle accesses. Our algorithm
uses a quantum routine for traversing through a binary search tree faster than
classically, and it is of a nature very different from a faster algorithm due
to Farhi, Goldstone, Gutmann, and Sipser.Comment: This new version contains new results. To appear at ICALP '01. Some
of the results have previously been presented at QIP '01. This paper subsumes
the papers quant-ph/0009091 and quant-ph/000903
Modeling the Daily Variations of the Coronal X-ray Spectral Irradiance with Two Temperatures and Two Emission Measures
The Miniature X-ray Solar Spectrometer (MinXSS-1) CubeSat observed solar
X-rays between 0.5 and 10 keV. A two-temperature, two-emission measure model is
fit to each daily averaged spectrum. These daily average temperatures and
emission measures are plotted against the corresponding daily solar 10.7 cm
radio flux (F10.7) value and a linear correlation is found between each that we
call the Schwab Woods Mason (SWM) model. The linear trends show that one can
estimate the solar spectrum between 0.5 keV and 10 keV based on the F10.7
measurement alone. The cooler temperature component of this model represents
the quiescent sun contribution to the spectra and is essentially independent of
solar activity, meaning the daily average quiescent sun is accurately described
by a single temperature (1.70 MK) regardless of solar intensity and only the
emission measure corresponding to this temperature needs to be adjusted for
higher or lower solar intensity. The warmer temperature component is shown to
represent active region contributions to the spectra and varies between 5 MK to
6 MK. GOES XRS-B data between 1-8 Angstroms is used to validate this model and
it is found that the ratio between the SWM model irradiance and the GOES XRS-B
irradiance is close to unity on average. MinXSS-1 spectra during quiescent
solar conditions have very low counts beyond around 3 keV. The SWM model can
generate MinXSS-1 or DAXSS spectra at very high spectral resolution and with
extended energy ranges to fill in gaps between measurements and extend
predictions back to 1947
Quantum correlations from local amplitudes and the resolution of the Einstein-Podolsky-Rosen nonlocality puzzle
The Einstein-Podolsky-Rosen nonlocality puzzle has been recognized as one of
the most important unresolved issues in the foundational aspects of quantum
mechanics. We show that the problem is resolved if the quantum correlations are
calculated directly from local quantities which preserve the phase information
in the quantum system. We assume strict locality for the probability amplitudes
instead of local realism for the outcomes, and calculate an amplitude
correlation function.Then the experimentally observed correlation of outcomes
is calculated from the square of the amplitude correlation function. Locality
of amplitudes implies that the measurement on one particle does not collapse
the companion particle to a definite state. Apart from resolving the EPR
puzzle, this approach shows that the physical interpretation of apparently
`nonlocal' effects like quantum teleportation and entanglement swapping are
different from what is usually assumed. Bell type measurements do not change
distant states. Yet the correlations are correctly reproduced, when measured,
if complex probability amplitudes are treated as the basic local quantities. As
examples we discuss the quantum correlations of two-particle maximally
entangled states and the three-particle GHZ entangled state.Comment: Std. Latex, 11 pages, 1 table. Prepared for presentation at the
International Conference on Quantum Optics, ICQO'2000, Minsk, Belaru
Interpolation of Hilbert and Sobolev Spaces: Quantitative Estimates and Counterexamples
This paper provides an overview of interpolation of Banach and Hilbert
spaces, with a focus on establishing when equivalence of norms is in fact
equality of norms in the key results of the theory. (In brief, our conclusion
for the Hilbert space case is that, with the right normalisations, all the key
results hold with equality of norms.) In the final section we apply the Hilbert
space results to the Sobolev spaces and
, for and an open . We exhibit examples in one and two dimensions of sets
for which these scales of Sobolev spaces are not interpolation scales. In the
cases when they are interpolation scales (in particular, if is
Lipschitz) we exhibit examples that show that, in general, the interpolation
norm does not coincide with the intrinsic Sobolev norm and, in fact, the ratio
of these two norms can be arbitrarily large
Evolution of a global string network in a matter dominated universe
We evolve the network of global strings in the matter-dominated universe by
means of numerical simulations. The existence of the scaling solution is
confirmed as in the radiation-dominated universe but the scaling parameter
takes a slightly smaller value, , which is
defined as with the energy density of
global strings and the string tension per unit length. The change of
from the radiation to the matter-dominated universe is consistent with
that obtained by Albrecht and Turok by use of the one-scale model. We also
study the loop distribution function and find that it can be well fitted with
that predicted by the one-scale model, where the number density of
the loop with the length is given by with and . Thus, the evolution of the
global string network in the matter-dominated universe can be well described by
the one-scale model as in the radiation-dominated universe.Comment: 10 pages, 5 figure
Quantum state transfer and entanglement distribution among distant nodes in a quantum network
We propose a scheme to utilize photons for ideal quantum transmission between
atoms located at spatially-separated nodes of a quantum network. The
transmission protocol employs special laser pulses which excite an atom inside
an optical cavity at the sending node so that its state is mapped into a
time-symmetric photon wavepacket that will enter a cavity at the receiving node
and be absorbed by an atom there with unit probability. Implementation of our
scheme would enable reliable transfer or sharing of entanglement among
spatially distant atoms.Comment: 4 pages, 3 postscript figure
Scaling Property of the global string in the radiation dominated universe
We investigate the evolution of the global string network in the radiation
dominated universe by use of numerical simulations in 3+1 dimensions. We find
that the global string network settles down to the scaling regime where the
energy density of global strings, , is given by with the string tension per unit length and the scaling parameter,
, irrespective of the cosmic time. We also find that the
loop distribution function can be fitted with that predicted by the so-called
one scale model. Concretely, the number density, , of the loop with
the length, , is given by
where and is related with the Nambu-Goldstone(NG)
boson radiation power from global strings, , as with
. Therefore, the loop production function also scales and
the typical scale of produced loops is nearly the horizon distance. Thus, the
evolution of the global string network in the radiation dominated universe can
be well described by the one scale model in contrast with that of the local
string network.Comment: 18 pages, 9 figures, to appear in Phys. Rev.
The Satellite Luminosity Function of M101 into the Ultra-Faint Dwarf Galaxy Regime
We have obtained deep Hubble Space Telescope (HST) imaging of four faint and
ultra-faint dwarf galaxy candidates in the vicinity of M101 - Dw21, Dw22, Dw23
and Dw35, originally discovered by Bennet et al. (2017). Previous distance
estimates using the surface brightness fluctuation technique have suggested
that these four dwarf candidates are the only remaining viable M101 satellites
identified in ground based imaging out to the virial radius of M101 (D~250
kpc). Advanced Camera for Surveys imaging of all four dwarf candidates shows no
associated resolved stellar populations, indicating that they are thus
background galaxies. We confirm this by generating simulated HST color
magnitude diagrams of similar brightness dwarfs at the distance of M101. Our
targets would have displayed clear, resolved red giant branches with dozens of
stars if they had been associated with M101. With this information, we
construct a satellite luminosity function for M101, which is 90% complete to
M_V=-7.7 mag and 50% complete to M_V=-7.4 mag, that extends into the
ultra-faint dwarf galaxy regime. The M101 system is remarkably poor in
satellites in comparison to the Milky Way and M31, with only eight satellites
down to an absolute magnitude of M_V=-7.7 mag, compared to the 14 and 26 seen
in the Milky Way and M31, respectively. Further observations of Milky Way
analogs are needed to understand the halo-to-halo scatter in their faint
satellite systems, and connect them with expectations from cosmological
simulations.Comment: 9 Pages, 3 Figures, 1 Table, Accepted by ApJ
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