Related papers: Quantization for uniform distributions on equilate…
Four common optimality criteria for measurements are formulated using relations in the set of observables, and their connections are clarified. As case studies, 1-0 observables, localization observables, and photon counting observables are…
We enumerate all dissections of an equilateral triangle into smaller equilateral triangles up to size 20, where each triangle has integer side lengths. A perfect dissection has no two triangles of the same side, counting up- and…
Accurate approximation of probability measures is essential in numerical applications. This paper explores the quantization of probability measures using the maximum mean discrepancy (MMD) distance as a guiding metric. We first investigate…
The number of triangulations of a planar n point set is known to be $c^n$, where the base $c$ lies between $2.43$ and $30.$ The fastest known algorithm for counting triangulations of a planar n point set runs in $O^*(2^n)$ time. The fastest…
We study the convergence rate of the optimal quantization for a probability measure sequence $(\mu_{n})_{n\in\mathbb{N}^{*}}$ on $\mathbb{R}^{d}$ converging in the Wasserstein distance in two aspects: the first one is the convergence rate…
The maximum entropy principle is a powerful tool for solving underdetermined inverse problems. This paper considers the problem of discretizing a continuous distribution, which arises in various applied fields. We obtain the approximating…
We discuss the problem of finding the best measurement strategy for estimating the value of a quantum system parameter. In general the optimum quantum measurement, in the sense that it maximizes the quantum Fisher information and hence…
With the advance of quantum information technology, the question of how to most efficiently test quantum circuits is becoming of increasing relevance. Here we introduce the statistics of lengths of measurement sequences that allows one to…
In this paper, we study an asymptotic approximation of the Fisher information for the estimation of a scalar parameter using quantized measurements. We show that, as the number of quantization intervals tends to infinity, the loss of Fisher…
The study of finite approximations of probability measures has a long history. In (Xu and Berger, 2017), the authors focus on constrained finite approximations and, in particular, uniform ones in dimension $d=1$. The present paper gives an…
Efficient methods for characterizing the performance of quantum measurements are important in the experimental quantum sciences. Ideally, one requires both a physically relevant distinguishability measure between measurement operations and…
We establish a general transference principle for the irrationality measure of points with $\mathbb{Q}$-linearly independent coordinates in $\mathbb{R}^{n+1}$, for any given integer $n\geq 1$. On this basis, we recover an important…
A sphere is a fundamental geometric object widely used in (computer aided) geometric design. It possesses rational parameterizations but no parametric polynomial parameterization exists. The present study provides an approach to the optimal…
We present some new and explicit error bounds for the approximation of distributions. The approximation error is quantified by the maximal density ratio of the distribution $Q$ to be approximated and its proxy $P$. This non-symmetric…
We introduce a new Partition of Unity Method for the numerical homogenization of elliptic partial differential equations with arbitrarily rough coefficients. We do not restrict to a particular ansatz space or the existence of a finite…
We study the problem of quantization of discrete probability distributions, arising in universal coding, as well as other applications. We show, that in many situations this problem can be reduced to the covering problem for the unit…
In the last decade, it was understood that quantum networks involving several independent sources of entanglement which are distributed and measured by several parties allowed for completely novel forms of nonclassical quantum correlations,…
Let $\{S_j : 1\leq j\leq 3\}$ be a set of three contractive similarity mappings such that $S_j(x)=rx+\frac {j-1}{2}(1-r)$ for all $x\in \mathbb R$, and $1\leq j\leq 3$, where $0<r<\frac 1 3$. Let $P=\sum_{j=1}^3 \frac 13 P\circ S_j^{-1}$.…
Entanglement does not correspond to any observable and its evaluation always corresponds to an estimation procedure where the amount of entanglement is inferred from the measurements of one or more proper observables. Here we address…
We consider the problem of estimating the ensemble average of an observable on an ensemble of equally prepared identical quantum systems. We show that, among all kinds of measurements performed jointly on the copies, the optimal unbiased…