Optimal estimation of qubit states with continuous time measurements

We propose an adaptive, two steps strategy, for the estimation of mixed qubit states. We show that the strategy is optimal in a local minimax sense for the trace norm distance as well as other locally quadratic figures of merit. Local minimax optimality means that given $n$ identical qubits, there exists no estimator which can perform better than the proposed estimator on a neighborhood of size $n^{-1/2}$ of an arbitrary state. In particular, it is asymptotically Bayesian optimal for a large class of prior distributions. We present a physical implementation of the optimal estimation strategy based on continuous time measurements in a field that couples with the qubits. The crucial ingredient of the result is the concept of local asymptotic normality (or LAN) for qubits. This means that, for large $n$, the statistical model described by $n$ identically prepared qubits is locally equivalent to a model with only a classical Gaussian distribution and a Gaussian state of a quantum harmonic oscillator. The term `local' refers to a shrinking neighborhood around a fixed state $\rho_{0}$. An essential result is that the neighborhood radius can be chosen arbitrarily close to $n^{-1/4}$. This allows us to use a two steps procedure by which we first localize the state within a smaller neighborhood of radius $n^{-1/2+\epsilon}$, and then use LAN to perform optimal estimation.Comment: 32 pages, 3 figures, to appear in Commun. Math. Phy

Towards Machine Wald

The past century has seen a steady increase in the need of estimating and predicting complex systems and making (possibly critical) decisions with limited information. Although computers have made possible the numerical evaluation of sophisticated statistical models, these models are still designed \emph{by humans} because there is currently no known recipe or algorithm for dividing the design of a statistical model into a sequence of arithmetic operations. Indeed enabling computers to \emph{think} as \emph{humans} have the ability to do when faced with uncertainty is challenging in several major ways: (1) Finding optimal statistical models remains to be formulated as a well posed problem when information on the system of interest is incomplete and comes in the form of a complex combination of sample data, partial knowledge of constitutive relations and a limited description of the distribution of input random variables. (2) The space of admissible scenarios along with the space of relevant information, assumptions, and/or beliefs, tend to be infinite dimensional, whereas calculus on a computer is necessarily discrete and finite. With this purpose, this paper explores the foundations of a rigorous framework for the scientific computation of optimal statistical estimators/models and reviews their connections with Decision Theory, Machine Learning, Bayesian Inference, Stochastic Optimization, Robust Optimization, Optimal Uncertainty Quantification and Information Based Complexity.Comment: 37 page

Measurement of the Bottom-Strange Meson Mixing Phase in the Full CDF Data Set

We report a measurement of the bottom-strange meson mixing phase \beta_s using the time evolution of B0_s -> J/\psi (->\mu+\mu-) \phi (-> K+ K-) decays in which the quark-flavor content of the bottom-strange meson is identified at production. This measurement uses the full data set of proton-antiproton collisions at sqrt(s)= 1.96 TeV collected by the Collider Detector experiment at the Fermilab Tevatron, corresponding to 9.6 fb-1 of integrated luminosity. We report confidence regions in the two-dimensional space of \beta_s and the B0_s decay-width difference \Delta\Gamma_s, and measure \beta_s in [-\pi/2, -1.51] U [-0.06, 0.30] U [1.26, \pi/2] at the 68% confidence level, in agreement with the standard model expectation. Assuming the standard model value of \beta_s, we also determine \Delta\Gamma_s = 0.068 +- 0.026 (stat) +- 0.009 (syst) ps-1 and the mean B0_s lifetime, \tau_s = 1.528 +- 0.019 (stat) +- 0.009 (syst) ps, which are consistent and competitive with determinations by other experiments.Comment: 8 pages, 2 figures, Phys. Rev. Lett 109, 171802 (2012

Some approximations to power functions of ϕ-divergence tests in parametric models

Asymptotic distributions, ϕ-divergences, Pitman type alternatives, power, testing statistical hypotheses, 62B10, 62E20,

Does avian species richness in natural patch mosaics follow the forest fragmentation paradigm?

As one approaches the north-eastern limit of pinyon (Pinus spp.) juniper (Juniperus spp.) vegetation on the Colorado Plateau, USA, woodland patches become increasingly disjunct, grading into sagebrush (Artemisia spp.)-dominated landscapes. Patterns of avian species richness in naturally heterogeneous forests may or may not respond to patch discontinuity in the same manner as bird assemblages in fragmented agricultural systems. We used observational data from naturally patchy woodlands and predictions derived from studies of human-modified agricultural forests to estimate the effects of patch area, shape, isolation and distance to contiguous woodland on avian species richness. We predicted that patterns of species richness in naturally patchy juniper woodlands would differ from those observed in fragmented agricultural systems. Our objectives were to (1) estimate the effect of naturally occurring patch structure on avian species richness with respect to habitat affinity and migratory strategy and (2) assess the concordance of the effects to predictions from agricultural forest systems. We used the analogy between populations and communities to estimate species richness, where species are treated as individuals in the application of traditional capture-recapture theory. Information-theoretic model selection showed that overall species richness was explained primarily by the species area relationship. There was some support for a model with greater complexity than the equilibrium theory of island biogeography where the isolation of large patches resulted in greater species richness. Species richness of woodland-dwelling birds was best explained by the equilibrium hypothesis with partial landscape complementation by open-country species in isolated patches. Species richness within specific migratory strategies showed concomitant increases and no shifts in species composition along the patch area gradient. Our results indicate that many patterns of species richness considered to be fragmentation effects may be general consequences of patch discontinuity and are ubiquitous in naturally heterogeneous systems. There was no evidence for the effects of patch shape and distance to contiguous woodland in juniper woodland, suggesting edge effects and dependence upon regional species pools are characteristics of fragmented agricultural systems. Natural patch mosaics may provide benchmarks for evaluating fragmentation effects and managing forests by mimicking natural landscape patterns