1,020 research outputs found
Operator-Schmidt decompositions and the Fourier transform, with applications to the operator-Schmidt numbers of unitaries
The operator-Schmidt decomposition is useful in quantum information theory
for quantifying the nonlocality of bipartite unitary operations. We construct a
family of unitary operators on C^n tensor C^n whose operator-Schmidt
decompositions are computed using the discrete Fourier transform. As a
corollary, we produce unitaries on C^3 tensor C^3 with operator-Schmidt number
S for every S in {1,...,9}. This corollary was unexpected, since it
contradicted reasonable conjectures of Nielsen et al [Phys. Rev. A 67 (2003)
052301] based on intuition from a striking result in the two-qubit case. By the
results of Dur, Vidal, and Cirac [Phys. Rev. Lett. 89 (2002) 057901
quant-ph/0112124], who also considered the two-qubit case, our result implies
that there are nine equivalence classes of unitaries on C^3 tensor C^3 which
are probabilistically interconvertible by (stochastic) local operations and
classical communication. As another corollary, a prescription is produced for
constructing maximally-entangled operators from biunimodular functions.
Reversing tact, we state a generalized operator-Schmidt decomposition of the
quantum Fourier transform considered as an operator C^M_1 tensor C^M_2 -->
C^N_1 tensor C^N_2, with M_1 x M_2 = N_1 x N_2. This decomposition shows (by
Nielsen's bound) that the communication cost of the QFT remains maximal when a
net transfer of qudits is permitted. In an appendix, a canonical procedure is
given for removing basis-dependence for results and proofs depending on the
"magic basis" introduced in [S. Hill and W. Wootters, "Entanglement of a pair
of quantum bits," Phys Rev. Lett 78 (1997) 5022-5025, quant-ph/9703041 (and
quant-ph/9709029)].Comment: More formal version of my talk at the Simons Conference on Quantum
and Reversible Computation at Stony Brook May 31, 2003. The talk slides and
audio are available at
http://www.physics.sunysb.edu/itp/conf/simons-qcomputation.html. Fixed typos
and minor cosmetic
A note on the error analysis of classical Gram-Schmidt
An error analysis result is given for classical Gram--Schmidt factorization
of a full rank matrix into where is left orthogonal (has
orthonormal columns) and is upper triangular. The work presented here shows
that the computed satisfies \normal{R}=\normal{A}+E where is an
appropriately small backward error, but only if the diagonals of are
computed in a manner similar to Cholesky factorization of the normal equations
matrix.
A similar result is stated in [Giraud at al, Numer. Math.
101(1):87--100,2005]. However, for that result to hold, the diagonals of
must be computed in the manner recommended in this work.Comment: 12 pages This v2. v1 (from 2006) has not the biliographical reference
set (at all). This is the only modification between v1 and v2. If you want to
quote this paper, please quote the version published in Numerische Mathemati
Unbiased bases (Hadamards) for 6-level systems: Four ways from Fourier
In quantum mechanics some properties are maximally incompatible, such as the
position and momentum of a particle or the vertical and horizontal projections
of a 2-level spin. Given any definite state of one property the other property
is completely random, or unbiased. For N-level systems, the 6-level ones are
the smallest for which a tomographically efficient set of N+1 mutually unbiased
bases (MUBs) has not been found. To facilitate the search, we numerically
extend the classification of unbiased bases, or Hadamards, by incrementally
adjusting relative phases in a standard basis. We consider the non-unitarity
caused by small adjustments with a second order Taylor expansion, and choose
incremental steps within the 4-dimensional nullspace of the curvature. In this
way we prescribe a numerical integration of a 4-parameter set of Hadamards of
order 6.Comment: 5 pages, 2 figure
Local RBF approximation for scattered data fitting with bivariate splines
In this paper we continue our earlier research [4] aimed at developing effcient methods of local approximation suitable for the first stage of a spline based two-stage scattered data fitting algorithm. As an improvement to the pure polynomial local approximation method used in [5], a hybrid polynomial/radial basis scheme was considered in [4], where the local knot locations for the RBF terms were selected using a greedy knot insertion algorithm. In this paper standard radial local approximations based on interpolation or least squares are considered and a faster procedure is used for knot selection, signicantly reducing the computational cost of the method. Error analysis of the method and numerical results illustrating its performance are given
Screening for Abdominal Aortic Aneurysm among Patients Referred to the Vascular Laboratory is Cost-effective
AbstractScreening for abdominal aortic aneurysm (AAA) in high-risk groups has been recommended based on a high prevalence of disease, while being questioned due to a high frequency of co-morbidities and inferior life-expectancy. We evaluated the long-term outcome and the cost-effectiveness of selective AAA screening among patients referred to the vascular laboratory for arterial examination.MethodsA total of 5924 patients, referred to the vascular laboratory of a university hospital, were screened for AAA with ultrasound (definition: ∅≥30mm), 1993–2005. Outcome data were gathered through hospital records and the national population registry. A Markov model was used for health–economic evaluation.ResultsAn AAA was detected in 181 patients (mean age 72.8 years), of whom 21.5% underwent elective repair (perioperative mortality 5.1%) after 7.5 years of follow-up. Four of six patients diagnosed with AAA rupture were operated upon. Relative 5-year survival compared with the general Swedish population, controlled for age and sex, was 80.4% (95% confidence interval (CI): 70.8–88.8). The cost-effectiveness was robust in base-case (11 084 Euro/life year gained) and in sensitivity analyses of prevalence, cost and survival.ConclusionsPatients in whom AAA was detected at selective screening had inferior long-term survival and were operated on less frequently, compared with AAA patients described in previous studies. Yet, selective screening at the vascular laboratory was cost-effective
Constructing Mutually Unbiased Bases in Dimension Six
The density matrix of a qudit may be reconstructed with optimal efficiency if
the expectation values of a specific set of observables are known. In dimension
six, the required observables only exist if it is possible to identify six
mutually unbiased complex 6x6 Hadamard matrices. Prescribing a first Hadamard
matrix, we construct all others mutually unbiased to it, using algebraic
computations performed by a computer program. We repeat this calculation many
times, sampling all known complex Hadamard matrices, and we never find more
than two that are mutually unbiased. This result adds considerable support to
the conjecture that no seven mutually unbiased bases exist in dimension six.Comment: As published version. Added discussion of the impact of numerical
approximations and corrected the number of triples existing for non-affine
families (cf Table 3
Evolution of the interfacial structure of LaAlO3 on SrTiO3
The evolution of the atomic structure of LaAlO3 grown on SrTiO3 was
investigated using surface x-ray diffraction in conjunction with
model-independent, phase-retrieval algorithms between two and five monolayers
film thickness. A depolarizing buckling is observed between cation and oxygen
positions in response to the electric field of polar LaAlO3, which decreases
with increasing film thickness. We explain this in terms of competition between
elastic strain energy, electrostatic energy, and electronic reconstructions.
The findings are qualitatively reproduced by density-functional theory
calculations. Significant cationic intermixing across the interface extends
approximately three monolayers for all film thicknesses. The interfaces of
films thinner than four monolayers therefore extend to the surface, which might
affect conductivity
Unit cell of graphene on Ru(0001): a 25 x 25 supercell with 1250 carbon atoms
The structure of a single layer of graphene on Ru(0001) has been studied
using surface x-ray diffraction. A surprising superstructure has been
determined, whereby 25 x 25 graphene unit cells lie on 23 x 23 unit cells of
Ru. Each supercell contains 2 x 2 crystallographically inequivalent subcells
caused by corrugation. Strong intensity oscillations in the superstructure rods
demonstrate that the Ru substrate is also significantly corrugated down to
several monolayers, and that the bonding between graphene and Ru is strong and
cannot be caused by van der Waals bonds. Charge transfer from the Ru substrate
to the graphene expands and weakens the C-C bonds, which helps accommodate the
in-plane tensile stress. The elucidation of this superstructure provides
important information in the potential application of graphene as a template
for nanocluster arrays.Comment: 9 pages, 3 figures, paper submitted to peer reviewed journa
Constructive updating/downdating of oblique projectors: a generalization of the Gram-Schmidt process
A generalization of the Gram-Schmidt procedure is achieved by providing
equations for updating and downdating oblique projectors. The work is motivated
by the problem of adaptive signal representation outside the orthogonal basis
setting. The proposed techniques are shown to be relevant to the problem of
discriminating signals produced by different phenomena when the order of the
signal model needs to be adjusted.Comment: As it will appear in Journal of Physics A: Mathematical and
Theoretical (2007
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