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

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

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

Incidence of Acute Thrombo-Embolic Occlusion of the Superior Mesenteric Artery—A Population-based Study

AbstractObjective. To determine the incidence of acute thrombo-embolic occlusion of the superior mesenteric artery (AOSMA) in a population-based study.Material. All clinical (n=23,446) and forensic (n=7569) autopsies performed in the city of Malmö between 1970 and 1982 (population 264,000–230,000 inhabitants). The autopsy rate was 87%.Methods. Calculation of the incidence of AOSMA with intestinal gangrene in those autopsies coded for bowel ischaemia (997/23,446 clinical and 9/7569 forensic autopsies). The operative procedures performed in 1970, 1976 and 1982 were also analysed.Results. Two forensic and 211 clinical autopsies demonstrated AOSMA with intestinal gangrene. Previous suspicion of intestinal ischaemia was noted in only 33%. Sixteen patients were operated. The cause-specific mortality was 6.0/1000 deaths. The incidence was 8.6/100,000 person years, increasing exponentially with age (p<0.001). Mortality was 93%.Conclusions. The incidence and mortality of AOSMA is higher than previously reported from clinical series. There is seldom any suspicion of the diagnosis prior to death

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

Delayed maximum northern European summer temperatures during the Last Interglacial as a result of Greenland Ice Sheet melt

This is the author accepted manuscript. The final version is available from the Geological Society of America via the DOI in this record.Here we report a new quantitative mean July temperature reconstruction using non-biting midges (chironomids) from the Danish Last Interglacial (LIG) site Hollerup (spanning 127–116 ka). We find that peak mean July temperatures of 17.5 °C, similar to those of the present day (1961–1990 CE), were reached shortly before the onset of the regional Carpinus pollen zone. Through comparison to terrestrial and marine sequences we demonstrate that peak summer warmth took place some three millennia after the onset of LIG warming in Europe, a marked delay in line with records from the North Atlantic. Crucially, the warmest northern European summer temperatures appear to follow maximum Greenland Ice Sheet mass loss, implying that meltwater substantially reduced Atlantic Meridional Overturning Circulation and depressed European temperatures during the early part of the interglacial.Turney and Fogwill thank the Australian Research Council (grants FL100100195, FT120100004, LP120200724). Thanks to Bjørn Buchardt for providing the C:N data, Angela Self for help with statistical analysis, David Campbell and Alan Bedford for laboratory work, and three reviewers for their constructive comments

Better Nonlinear Models from Noisy Data: Attractors with Maximum Likelihood

A new approach to nonlinear modelling is presented which, by incorporating the global behaviour of the model, lifts shortcomings of both least squares and total least squares parameter estimates. Although ubiquitous in practice, a least squares approach is fundamentally flawed in that it assumes independent, normally distributed (IND) forecast errors: nonlinear models will not yield IND errors even if the noise is IND. A new cost function is obtained via the maximum likelihood principle; superior results are illustrated both for small data sets and infinitely long data streams.Comment: RevTex, 11 pages, 4 figure

Spectrum of mutations in the CFTR gene of patients with classical and atypical forms of cystic fibrosis from southwestern Sweden: Identification of 12 novel mutations

Cystic fibrosis (CF) is caused by mutations in the CFTR gene. The spectrum of CFTR mutations varies between populations and depends on different factors, such as ethnic background and geographical location. The extensive CFTR mutation screening of 129 patients with classical or atypical CF from the south-western region of Sweden revealed the presence of 37 CFTR mutations, including 12 novel alleles. The overall mutation detection rate in this study population was 92%, the highest among all tested regions in Sweden. Eight mutations with a frequency above 1% (ΔF508, 394delTT, R117C, 3659delC, E60X, 1112delT, R764X, and 621 + 1G → T) accounted for 78% of CF chromosomes and have been recommended for inclusion in the CFTR mutation screening panel for molecular diagnosis of CF in this region. The multiple occurrence of specific CFTR alleles less common than the predominant ΔF508 mutation (394delTT, R117C, 3659delC) allowed for genotype-phenotype comparisons and revealed consistent relationships between these mutations and disease severity.published_or_final_versio

When the optimal is not the best: parameter estimation in complex biological models

Background: The vast computational resources that became available during the past decade enabled the development and simulation of increasingly complex mathematical models of cancer growth. These models typically involve many free parameters whose determination is a substantial obstacle to model development. Direct measurement of biochemical parameters in vivo is often difficult and sometimes impracticable, while fitting them under data-poor conditions may result in biologically implausible values. Results: We discuss different methodological approaches to estimate parameters in complex biological models. We make use of the high computational power of the Blue Gene technology to perform an extensive study of the parameter space in a model of avascular tumor growth. We explicitly show that the landscape of the cost function used to optimize the model to the data has a very rugged surface in parameter space. This cost function has many local minima with unrealistic solutions, including the global minimum corresponding to the best fit. Conclusions: The case studied in this paper shows one example in which model parameters that optimally fit the data are not necessarily the best ones from a biological point of view. To avoid force-fitting a model to a dataset, we propose that the best model parameters should be found by choosing, among suboptimal parameters, those that match criteria other than the ones used to fit the model. We also conclude that the model, data and optimization approach form a new complex system, and point to the need of a theory that addresses this problem more generally