Related papers: Quantization Dimension of $1$-variable Random Self…
Analyzing the dimension of an unknown quantum system in a device-independent manner, i.e., using only the measurement statistics, is a fundamental task in quantum physics and quantum information theory. In this paper, we consider this…
The scale of quantum mechanical effects in matter is set by Planck's constant, $\hbar$. This represents the quantisation scale for material objects. In this article, we present a simple argument why the quantisation scale for space, and…
Let $(g_i)_{i=1}^M$ be a family of contractive similitudes satisfying the open set condition. Let $\nu$ be a self-similar measure associated with $(g_i)_{i=1}^M$. We study the quantization problem for the in-homogeneous self-similar measure…
We show the existence of the local dimension of an invariant probability measure on an infinitely generated self-affine set, for almost all translations. This implies that an ergodic probability measure is exactly dimensional. Furthermore…
We prove that for random affine code tree fractals the affinity dimension is almost surely equal to the unique zero of the pressure function. As a consequence, we show that the Hausdorff, packing and box counting dimensions of such systems…
Mode separation, namely how sharply a distribution fragments into barrier-separated clusters, is a fundamental geometric property of densities, difficult to quantify in high dimensions. It is structurally distinct from dispersion, yet…
The spectra of random feature matrices provide essential information on the conditioning of the linear system used in random feature regression problems and are thus connected to the consistency and generalization of random feature models.…
Quantum physics, despite its observables being intrinsically of a probabilistic nature, does not have a quantum entropy assigned to them. We propose a quantum entropy that quantify the randomness of a pure quantum state via a conjugate pair…
Metric mean dimension is a dynamical counterpart of the box dimension in fractal geometry to characterize the topological complexity of infinite entropy systems. The classical variational principle states that topological entropy equals the…
In this work, we investigate the possibility of compressing a quantum system to one of smaller dimension in a way that preserves the measurement statistics of a given set of observables. In this process, we allow for an arbitrary amount of…
Notions of (pointwise) tangential dimension are considered, for measures of R^n. Under regularity conditions (volume doubling), the upper resp. lower dimension at a point x of a measure can be defined as the supremum, resp. infimum, of…
Randomized measurements are useful for analyzing quantum systems especially when quantum control is not fully perfect. However, their practical realization typically requires multiple rotations in the complex space due to the adoption of…
We show for a free action of a countable group $\Gamma$ on a finite-dimensional, compact metric space by homeomorphisms that the dynamic asymptotic dimension is either infinite or coincides with the asymptotic dimension of $\Gamma$.
We investigate the problem of metric fluctuations in the presence of the vacuum fluctuations of matter fields and critically assess the usual assertion that vacuum energy implies a Planckian cosmological constant. A new stochastic classical…
In this paper, we discuss quantization effects in rigid formation control systems when target formations are described by inter-agent distances. Because of practical sensing and measurement constraints, we consider in this paper distance…
We consider scenarios where the dynamics of a quantum system are partially determined by prior local measurements of some interacting environmental degrees of freedom. The resulting effective system dynamics are described by a disordered…
The effect of quantum steering arises from the judicious combination of an entangled state with a set of incompatible measurements. Recently, it was shown that this form of quantum correlations can be quantified in terms of a dimension,…
The distortion-rate performance of certain randomly-designed scalar quantizers is determined. The central results are the mean-squared error distortion and output entropy for quantizing a uniform random variable with thresholds drawn…
The quantization scheme in probability theory deals with finding a best approximation of a given probability distribution by a probability distribution that is supported on finitely many points. For a given $k\geq 2$, let $\{S_j : 1\leq…
The paper studies homogenization problem for a non-autonomous parabolic equation with a large random rapidly oscillating potential in the case of one dimensional spatial variable. We show that if the potential is a statistically homogeneous…