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

Commitment versus persuasion in the three-party constrained voter model

In the framework of the three-party constrained voter model, where voters of two radical parties (A and B) interact with "centrists" (C and Cz), we study the competition between a persuasive majority and a committed minority. In this model, A's and B's are incompatible voters that can convince centrists or be swayed by them. Here, radical voters are more persuasive than centrists, whose sub-population consists of susceptible agents C and a fraction zeta of centrist zealots Cz. Whereas C's may adopt the opinions A and B with respective rates 1+delta_A and 1+delta_B (with delta_A>=delta_B>0), Cz's are committed individuals that always remain centrists. Furthermore, A and B voters can become (susceptible) centrists C with a rate 1. The resulting competition between commitment and persuasion is studied in the mean field limit and for a finite population on a complete graph. At mean field level, there is a continuous transition from a coexistence phase when zeta= Delta_c. In a finite population of size N, demographic fluctuations lead to centrism consensus and the dynamics is characterized by the mean consensus time tau. Because of the competition between commitment and persuasion, here consensus is reached much slower (zeta=Delta_c) than in the absence of zealots (when tau\simN). In fact, when zeta<Delta_c and there is an initial minority of centrists, the mean consensus time asymptotically grows as tau\simN^{-1/2} e^{N gamma}, where gamma is determined. The dynamics is thus characterized by a metastable state where the most persuasive voters and centrists coexist when delta_A>delta_B, whereas all species coexist when delta_A=delta_B. When zeta>=Delta_c and the initial density of centrists is low, one finds tau\simln N (when N>>1). Our analytical findings are corroborated by stochastic simulations.Comment: 25 pages, 6 figures. Final version for the Journal of Statistical Physics (special issue on the "applications of statistical mechanics to social phenomena"

Large-scale discovery of novel genetic causes of developmental disorders

Despite three decades of successful, predominantly phenotype-driven discovery of the genetic causes of monogenic disorders1, up to half of children with severe developmental disorders of probable genetic origin remain without a genetic diagnosis. Particularly challenging are those disorders rare enough to have eluded recognition as a discrete clinical entity, those with highly variable clinical manifestations, and those that are difficult to distinguish from other, very similar, disorders. Here we demonstrate the power of using an unbiased genotype-driven approach2 to identify subsets of patients with similar disorders. By studying 1,133 children with severe, undiagnosed developmental disorders, and their parents, using a combination of exome sequencing3,4,5,6,7,8,9,10,11 and array-based detection of chromosomal rearrangements, we discovered 12 novel genes associated with developmental disorders. These newly implicated genes increase by 10% (from 28% to 31%) the proportion of children that could be diagnosed. Clustering of missense mutations in six of these newly implicated genes suggests that normal development is being perturbed by an activating or dominant-negative mechanism. Our findings demonstrate the value of adopting a comprehensive strategy, both genome-wide and nationwide, to elucidate the underlying causes of rare genetic disorders