Related papers: Weaker quantization dimension results for self-sim…
We prove in this short report that for arbitrary weak converging sequence of sigma-finite Borelian measures in the separable Banach space there is a compact embedded separable subspace such that this measures not only are concentrated in…
Let \(\mu\) be a finite Borel measure on \((-\pi,\pi)\). Consider the one-dimensional Poisson equation \(-u''=\mu\), where equality holds in the sense of distributions, with Dirichlet boundary conditions \(u(\pm\pi)=0\). In this paper, we…
Using the recently developed soldering formalism we highlight certain features of quantum mechanical models. The complete correspondence between these models and self dual field theoretical models in odd dimensions is established. The…
Let $K\subset R^n$ be a compact basic semi-algebraic set. We provide a necessary and sufficient condition (with no a priori bounding parameter) for a real sequence $y=(y_\alpha)$, $\alpha\in N^n$, to have a finite representing Borel measure…
Fidelity is the standard measure for quantifying the similarity between two quantum states. It is equal to the square of the minimum Bhattacharyya coefficient between the probability distributions induced by quantum measurements on the two…
We quantify the pointwise doubling properties of self-similar measures using the notion of pointwise Assouad dimension. We show that all self-similar measures satisfying the open set condition are pointwise doubling in a set of full…
The quantization error in a fixed-size Self-Organizing Map (SOM) with unsupervised winner-take-all learning has previously been used successfully to detect, in minimal computation time, highly meaningful changes across images in medical…
Quantum physics constrains the accuracy of joint measurements of incompatible observables. Here we test tight measurement-uncertainty relations using single photons. We implement two independent, idealized uncertainty-estimation methods,…
The second part of the paper mainly deals with convergence of infinite determinantal measures, understood as the convergence of the approximating finite determinantal measures. In addition to the usual weak topology on the space of…
We identify shortcomings in two popular measures of localization of functions: the $L^p-L^q$ participation ratio and the mass concentration comparison. We then introduce a novel localization measure for functions on bounded subsets of…
For any pair of quantum states (the hypotheses), the task of binary quantum hypotheses testing is to derive the tradeoff relation between the probability $p_{01}$ of rejecting the null hypothesis and $p_{10}$ of accepting the alternative…
We show that whenever a separable subset $S$ of a complete metric space $X$ admits a $d$-dimensional weak tangent field, the set $S$ is close to being $d$-dimensional in the following sense. Whenever $\mu$ is a Borel finite measure on $X$…
We study quantum steering experiments without assuming that the trusted party can perfectly control their measurement device. Instead, we introduce a scenario in which these measurements are subject to small imprecision. We show that small…
We construct a planar homogeneous self-similar measure, with strong separation, dense rotations and dimension greater than $1$, such that there exist lines for which dimension conservation does not hold and the projection of the measure is…
We prove an analogue of the portmanteau theorem on weak convergence of probability measures allowing measures which are unbounded on an underlying metric space but finite on the complement of any Borel neighbourhood of a fixed element.
We examine a family of intrinsic performance measures in terms of probability distributions that generalize Hellinger distance and Fisher information. They are applied to quantum metrology to assess the uncertainty in the detection of…
Let $\mu$ be a self-similar measure generated by an IFS $\Phi=\{\phi_i\}_{i=1}^\ell$ of similarities on $\mathbb R^d$ ($d\ge 1$). When $\Phi$ is dimensional regular (see Definition~1.1), we give an explicit formula for the $L^q$-spectrum…
Mutually unbiased bases (MUBs) and symmetric informationally complete (SIC) positive operator-valued measurements (POVMs) are two related topics in quantum information theory. They are generalized to mutually unbiased measurements (MUMs)…
It is well known that when two or more quantum measurements suffer from imperfections they may lose their incompatibility. For a quantum system of finite dimension d we study the incompatibility of all projective measurements subjected to…
Self-testing is a promising approach to certifying quantum states or measurements. Originally, it relied solely on the outcome statistics of the measurements involved in a device-independent (DI) setup. Extra physical assumptions about the…