Operator-Schmidt decomposition of the quantum Fourier transform on C^N1 tensor C^N2

Operator-Schmidt decompositions of the quantum Fourier transform on C^N1 tensor C^N2 are computed for all N1, N2 > 1. The decomposition is shown to be completely degenerate when N1 is a factor of N2 and when N1>N2. The first known special case, N1=N2=2^n, was computed by Nielsen in his study of the communication cost of computing the quantum Fourier transform of a collection of qubits equally distributed between two parties. [M. A. Nielsen, PhD Thesis, University of New Mexico (1998), Chapter 6, arXiv:quant-ph/0011036.] More generally, the special case N1=2^n1<2^n2=N2 was computed by Nielsen et. al. in their study of strength measures of quantum operations. [M.A. Nielsen et. al, (accepted for publication in Phys Rev A); arXiv:quant-ph/0208077.] Given the Schmidt decompositions presented here, it follows that in all cases the communication cost of exact computation of the quantum Fourier transform is maximal.Comment: 9 pages, LaTeX 2e; No changes in results. References and acknowledgments added. Changes in presentation added to satisfy referees: expanded introduction, inclusion of ommitted algebraic steps in the appendix, addition of clarifying footnote

Intensive care for extreme prematurity--moving beyond gestational age.

BACKGROUND: Decisions regarding whether to administer intensive care to extremely premature infants are often based on gestational age alone. However, other factors also affect the prognosis for these patients. METHODS: We prospectively studied a cohort of 4446 infants born at 22 to 25 weeks\u27 gestation (determined on the basis of the best obstetrical estimate) in the Neonatal Research Network of the National Institute of Child Health and Human Development to relate risk factors assessable at or before birth to the likelihood of survival, survival without profound neurodevelopmental impairment, and survival without neurodevelopmental impairment at a corrected age of 18 to 22 months. RESULTS: Among study infants, 3702 (83%) received intensive care in the form of mechanical ventilation. Among the 4192 study infants (94%) for whom outcomes were determined at 18 to 22 months, 49% died, 61% died or had profound impairment, and 73% died or had impairment. In multivariable analyses of infants who received intensive care, exposure to antenatal corticosteroids, female sex, singleton birth, and higher birth weight (per each 100-g increment) were each associated with reductions in the risk of death and the risk of death or profound or any neurodevelopmental impairment; these reductions were similar to those associated with a 1-week increase in gestational age. At the same estimated likelihood of a favorable outcome, girls were less likely than boys to receive intensive care. The outcomes for infants who underwent ventilation were better predicted with the use of the above factors than with use of gestational age alone. CONCLUSIONS: The likelihood of a favorable outcome with intensive care can be better estimated by consideration of four factors in addition to gestational age: sex, exposure or nonexposure to antenatal corticosteroids, whether single or multiple birth, and birth weight. (ClinicalTrials.gov numbers, NCT00063063 [ClinicalTrials.gov] and NCT00009633 [ClinicalTrials.gov].)

Advantages of Bayesian monitoring methods in deciding whether and when to stop a clinical trial: an example of a neonatal cooling trial.

BackgroundDecisions to stop randomized trials are often based on traditional P value thresholds and are often unconvincing to clinicians. To familiarize clinical investigators with the application and advantages of Bayesian monitoring methods, we illustrate the steps of Bayesian interim analysis using a recent major trial that was stopped based on frequentist analysis of safety and futility.MethodsWe conducted Bayesian reanalysis of a factorial trial in newborn infants with hypoxic-ischemic encephalopathy that was designed to investigate whether outcomes would be improved by deeper (32&nbsp;°C) or longer cooling (120&nbsp;h), as compared with those achieved by standard whole body cooling (33.5&nbsp;°C for 72&nbsp;h). Using prior trial data, we developed neutral and enthusiastic prior probabilities for the effect on predischarge mortality, defined stopping guidelines for a clinically meaningful effect, and derived posterior probabilities for predischarge mortality.ResultsBayesian relative risk estimates for predischarge mortality were closer to 1.0 than were frequentist estimates. Posterior probabilities suggested increased predischarge mortality (relative risk &gt; 1.0) for the three intervention groups; two crossed the Bayesian futility threshold.ConclusionsBayesian analysis incorporating previous trial results and different pre-existing opinions can help interpret accruing data and facilitate informed stopping decisions that are likely to be meaningful and convincing to clinicians, meta-analysts, and guideline developers.Trial registrationClinicalTrials.gov NCT01192776 . Registered on 31 August 2010