Related papers: Quantization coefficients in infinite systems

Qualitative and infinitesimal probability schemes are consistent with the axioms of probability theory, but avoid the need for precise numerical probabilities. Using qualitative probabilities could substantially reduce the effort for…

We investigated the asymptotics of high-rate constrained quantization errors for a compactly supported probability measure P on Euclidean spaces whose quantizers are confined to a closed set S. The key tool is the metric projection of K…

In this paper, for a given family of constraints and the classical Cantor distribution we determine the constrained optimal sets of $n$-points, $n$th constrained quantization errors for all positive integers $n$. We also calculate the…

New measures for the quantization of systems with constraints are discussed and applied to several examples, in particular, examples of alternative but equivalent formulations of given first-class constraints, as well as a comparison of…

For a measurable map $T$ and a sequence of $T$-invariant probability measures $\mu_n$ that converges in some sense to a $T$-invariant probability measure $\mu$, an estimate from below for the Kolmogorov--Sinai entropy of $T$ with respect to…

Let $\mathcal{M}(X,\mathcal{A},\mu)$ be the ring of all real-valued measurable functions constructed over a measure space $(X,\mathcal{A},\mu)$. A topology on $\mathcal{M}(X,\mathcal{A},\mu)$, called the {$F_\mu$-topology} weaker than the {…

The existence of incompatible measurements, i.e. measurements which cannot be performed simultaneously on a single copy of a quantum state, constitutes an important distinction between quantum mechanics and classical theories. While…

The representation of a given quantity with less information is often referred to as `quantization' and it is an important subject in information theory. In this paper, we have considered absolutely continuous probability measures on unit…

We discuss two ways to construct standard probability measures, called push-down measures, from internal probability measures. We show that the Wasserstein distance between an internal probability measure and its push-down measure is…

Let $ T:[0,1]\to[0,1] $ be an expanding Markov map with a finite partition. Let $ \mu_\phi $ be the invariant Gibbs measure associated with a H\"older continuous potential $ \phi $. In this paper, we investigate the size of the uniform…

We show that the order dimension of the partial order of all finite subsets of $\kappa$ under set inclusion is ${\log}_{2}({\log}_{2}(\kappa))$ whenever $\kappa$ is an infinite cardinal. We also show that the order dimension of any locally…

We propose a new measure of relative incompatibility for a quantum system with respect to two non-commuting observables, and call it quantumness of relative incompatibility. In case of a classical state, order of observation is…

For a partially ordered set $(S, \mathord\preceq)$, the order (monotone) dimension is the minimum cardinality of total orders (respectively, real-valued order monotone functions) on $S$ that characterize the order $\preceq$. In this paper…

In this paper we study the quantization problem for probability measures on Riemannian manifolds. Under a suitable assumption on the growth at infinity of the measure we find asymptotic estimates for the quantization error, generalizing the…

In this paper, we introduce and develop the concept of conditional quantization for Borel probability measures on $\mathbb{R}^k,$ considering both constrained and unconstrained frameworks. For each setting, we define the associated…

The basic goal of quantization for probability distribution is to reduce the number of values, which is typically uncountable, describing a probability distribution to some finite set and thus to make an approximation of a continuous…

Let $E$ be a Moran set on $\mathbb{R}^1$ associated with a closed interval $J$ and two sequences $(n_k)_{k=1}^\infty$ and $(\mathcal{C}_k=(c_{k,j})_{j=1}^{n_k})_{k\geq1}$. Let $\mu$ be the infinite product measure (Moran measure) on $E$…

Indeterminacy associated with probing of a quantum state is commonly expressed through spectral distances (metric) featured in the outcomes of repeated experiments. Here we express it as an effective amount (measure) of distinct outcomes…

In this article, we study the approximation of a probability measure $\mu$ on $\mathbb{R}^{d}$ by its empirical measure $\hat{\mu}_{N}$ interpreted as a random quantization. As error criterion we consider an averaged $p$-th moment…

We generalize recent work of Massar and Popescu dealing with the amount of classical data that is produced by a quantum measurement on a quantum state ensemble. In the previous work it was shown how spurious randomness generally contained…