This paper provides an introduction to quantum filtering theory. An probability theory is given, focusing on the spectral theorem and conditional expectation as a least squares estimate, culminating in construction of Wiener Poisson processes Fock space. We describe Itô calculus its use modeling physical systems. both reference innovations methods obtain equations for system-probe models from optics.

We obtain nonasymptotic bounds on the spectral norm of random matrices with independent entries that improve significantly earlier results. If $X$ is $n\times n$ symmetric matrix $X_{ij}\sim N(0,b_{ij}^{2})$, we show \[\mathbf{E}\Vert X\Vert \lesssim\max_{i}\sqrt{\sum_{j}b_{ij}^{2}}+\max_{ij}\vert b_{ij}\vert \sqrt{\log n}.\] This bound optimal in sense a matching lower holds under mild assumptions, and constants are sufficiently sharp can often capture precise edge spectrum. Analogous...

Feedback control of quantum mechanical systems must take into account the probabilistic nature measurement. We formulate feedback as a problem stochastic nonlinear by considering separately filtering and state for filter. explore use Lyapunov techniques design controllers spin demonstrate possibility stabilizing one outcome measurement with unit probability.

No quantum measurement can give full information on the state of a system; hence any feedback control problem is necessarily one with partial observations and generally be converted into completely observed for an appropriate filter as in classical stochastic theory. Here we study properties controlled filtering equations differential equations. We then develop methods, using combination geometric probabilistic techniques, global stabilization class filters around particular eigenstate operator.

The discovery of particle filtering methods has enabled the use nonlinear in a wide array applications. Unfortunately, approximation error filters typically grows exponentially dimension underlying model. This phenomenon rendered limited complex data assimilation problems. In this paper, we argue that it is often possible, at least principle, to develop local algorithms whose dimension-free. key such developments decay correlations property, which spatial counterpart much better understood...

We characterize the long-time projective behavior of stochastic master equation describing a continuous, collective spin measurement an atomic ensemble both analytically and numerically. By adding state-based feedback, we show that it is possible to prepare highly entangled Dicke states deterministically.

The engineering and control of devices at the quantum mechanical level—such as those consisting small numbers atoms photons—is a delicate business. fundamental uncertainty that is inherently present this scale manifests itself in unavoidable presence noise, making novel field application for stochastic estimation theory. In expository paper we demonstrate feedback systems what essentially noncommutative version binomial model popular mathematical finance. extremely rich allows full...

extremals of the alexandrov-fenchel inequality 95 settings considered by Stanley, where N i is number linear extensions a partially ordered set for which distinguished element has rank i.Such extremal problems appear to be inaccessible currently known methods enumerative or algebraic combinatorics.This example highlights significance questions in this paper other areas mathematics, and hints at possibility that structures developed here might have analogues outside convexity; brief...

The goal of this article is to provide a largely self-contained introduction the modelling controlled quantum systems under continuous observation, and design feedback controls that prepare particular states.We describe bottom-up approach, where field-theoretic model subjected statistical inference ultimately controlled.As an example, formalism applied highly idealized interaction atomic ensemble with optical field.Our aim unified outline for modelling, from first principles, realistic...

We consider a discrete time hidden Markov model where the signal is stationary chain. When conditioned on observations, chain in random environment under conditional measure. It shown that this weakly ergodic when and observations are nondegenerate. This permits delicate exchange of intersection supremum $σ$-fields, which key for stability nonlinear filter partially resolves long-standing gap proof result Kunita [J. Multivariate Anal. 1 (1971) 365--393]. A similar obtained also continuous...

Consider a parametrized family of general hidden Markov models, where both the observed and unobserved components take values in complete separable metric space. We prove that maximum likelihood estimator (MLE) parameter is strongly consistent under rather minimal set assumptions. As special cases our main result, we obtain consistency large class nonlinear state space as well results on linear Gaussian models finite models. A novel aspect approach an information-theoretic technique for...

This paper proposes a general method for establishing non-asymptotic converses in information theory via reverse hypercontractivity of Markov semigroups. In contrast to the blowing-up approach strong converses, proposed is applicable non-discrete settings, and yields optimal order second-order term rate expansion (square root blocklength) regime non-vanishing error probability.

Let $X$ be a $d\times d$ symmetric random matrix with independent but non-identically distributed Gaussian entries. It has been conjectured by Lata\l{a} that the spectral norm of is always same order as largest Euclidean its rows. A positive resolution this conjecture would provide sharp understanding probabilistic mechanisms control inhomogeneous matrices. This paper establishes up to dimensional factor $\sqrt{\log\log d}$. Moreover, dimension-free bounds are developed optimal leading and...

At the heart of convex geometry lies observation that volume bodies behaves as a polynomial. Many geometric inequalities may be expressed in terms coefficients this polynomial, called mixed volumes. Among deepest results theory is Alexandrov-Fenchel inequality, which subsumes many known special cases. The aim note to give new proofs inequality and its matrix counterpart, Alexandrov’s for discriminants, appear conceptually technically simpler than earlier clarify underlying structure. Our...

In a seminal paper "Volumen und Oberfläche" (1903), Minkowski introduced the basic notion of mixed volumes and corresponding inequalities that lie at heart convex geometry. The fundamental importance characterizing extremals these was already emphasized by himself, but has to date only been resolved in special cases. this paper, we completely settle Minkowski's quadratic inequality, confirming conjecture R. Schneider. Our proof is based on representation arbitrary bodies as Dirichlet forms...

A hidden Markov model is called observable if distinct initial laws give rise to of the observation process. Observability implies stability nonlinear filter when signal process tight, but this need not be case unstable. This paper introduces a stronger notion uniform observability which guarantees in absence assumptions on signal. By developing certain approximation properties convolution operators, we subsequently demonstrate that condition satisfied for various classes filtering models...