arXiv subjects

Stefan Richter

Publications and source records attributed to Stefan Richter.

At least 19 recordsLinked to original sources

arXiv2026

Cyclicity via weak$^\ast$ sequential cyclicity in Radially weighted Besov spaces

A radially weighted Besov space $H$ is a space of holomorphic functions on the unit ball $\mathbb{B}_d \subseteq \mathbb{C}^d$ whose $N$-th radial derivative is square integrable with respect to a given admissible radial measure. We write $Mult(H)$ for its multiplier algebra. The cyclic vectors in $H$ are those functions $f$ whose multiplier multiples are dense in $H$. We call a multiplier has the complete Pick property. However, in more general radially weighted Besov spaces there may be multipliers that are cyclic, but not weak$^\ast$ sequentially cyclic. For bounded holomorphic functions $f$ with no zeros in $\mathbb{B}_d$, we obtain a condition on $\log f$ that implies the cyclicity of $f$ in $H$ and yields invertibility properties for $1/f$ within an associated Smirnov-type class. This condition is formulated in terms of weak$^\ast$ sequentially cyclic multipliers and can often be verified using a comparison principle: if $f, g \in Mult(H)$ satisfy $|f| \leq |g|$ and if $f$ is weak$^\ast$ sequentially cyclic, then $g$ is also weak$^\ast$ sequentially cyclic. These results provide new insights into cyclicity phenomena in radially weighted Besov spaces in settings, where $H$ fails to be a complete Pick space.

math.FA↗

arXiv2026

Bridging Reinforcement Learning and Optimal Control via Feasible Action Mapping

Operating constrained dynamical systems requires controllers to efficiently solve complex tasks while enforcing recursive feasibility and safety constraints. To address these competing requirements, we present Feasible Action for Optimal Control (FAOC), a novel control framework integrating Reinforcement Learning (RL) and Optimal Control (OC). The key contribution is a computationally efficient, optimization-based mapping algorithm that transforms the RL agent's action from a static abstract set into a state-dependent feasible parameter set of the Optimal Control Problem (OCP), guaranteeing strict satisfaction of the dynamical system's constraints. Thus, FAOC effectively combines the predictable safety of OC with the flexibility of RL. In contrast to prior work, the abstract action space of the RL agent does not require expert or heuristic design, and the OCP formulation is not compromised by the inability of RL to guarantee feasibility. We apply our approach to real-time motion planning for robot table tennis, which encapsulates these challenges. Via simulated experiments, we show that FAOC outperforms state-of-the-art baselines in both sample efficiency and closed-loop performance.

eess.SY↗

arXiv2025

Increased-Efficiency Multiple-Decoding-Attempts Error Correction for Continuous-Variable Quantum Key Distribution

In continuous-variable quantum key distribution (CV-QKD), the performance of the information reconciliation (IR) step is critical for the achievable secret key rate (SKR) and transmission distance. We show how to improve on the recently introduced implementation of an IR-protocol involving multiple decoding attempts (MDA) and validate the method on simulated data in different application scenarios. Throughout, we demonstrate meaningful SKR-gains compared to both the standard protocol of a single decoding attempt and to the original MDA-implementation, even at given decoding complexity.

quant-ph↗

arXiv2025

Inner factors of Dirichlet space functions

Every function in the Dirichlet space on the unit disc has an inner/outer factorization. We study which inner functions occur in this way. For Blaschke products, this is the well known question of which subsets of the disc are zero sets for the Dirichlet space. We also consider singular inner factors, and in particular prove an analogue of the Shapiro-Shields theorem in this setting. Our results on singular inner factors also yield new sufficient conditions for zero sets.

math.CV↗

arXiv2025

Spin-selective coherent light scattering from ion crystals

We study coherent light scattering from linear crystals with up to twelve $^{40}\text{Ca}^+$ ions, acting as single photon emitters. Light-scattering is induced by two-photon laser excitation, starting from the S$_{1/2}\rightarrow$ D$_{5/2}$ quadrupole transition at 729~nm followed by the D$_{5/2}\rightarrow$ P$_{3/2}$ dipole transition at 854~nm, from where the ions decay back to the S$_{1/2}$ ground state via emission of a photon near 393~nm. We realize spin-selective excitation from the Zeeman-split ground states S$_{1/2}$, m$= \pm 1/2$, of the $\text{Ca}^+$ ions and observe in the far field spin-dependent interference patterns displaying the spin textures of the ion crystals. We investigate their dynamics by measuring the temporal evolution of the spatial Fourier frequencies of the observed patterns.

physics.atom-ph↗

arXiv2025

Cyclicity and iterated logarithms in the Dirichlet space

Let $D(μ)$ denote a harmonically weighted Dirichlet space on the unit disc $\mathbb D$. We show that outer functions $f\in D(μ)$ are cyclic in $D(μ)$, whenever $\log f$ belongs to the Pick-Smirnov class $N^+(D(μ))$. If $f$ has $H^\infty$-norm less than or equal to 1, then cyclicity can also be checked via iterated logarithms. For example, we show that such outer functions $f$ are cyclic, whenever $\log(1+ \log(1/f))\in N^+(D(μ))$. This condition can be checked by verifying that $\log(1+ \log(1/f))\in D(μ)$. If $f$ satisfies a mild extra condition, then the conditions also become necessary for cyclicity.

math.FA↗

arXiv2024

Ad-hoc hybrid-heterogeneous metropolitan-range quantum key distribution network

This paper presents the development and implementation of a versatile ad-hoc metropolitan-range Quantum Key Distribution (QKD) network. The approach presented integrates various types of physical channels and QKD protocols, and a mix of trusted and untrusted nodes. Unlike conventional QKD networks that predominantly depend on either fiber-based or free-space optical (FSO) links, the testbed presented amalgamates FSO and fiber-based links, thereby overcoming some inherent limitations. Various network deployment strategies have been considered, including permanent infrastructure and provisional ad-hoc links to eradicate coverage gaps. Furthermore, the ability to rapidly establish a network using portable FSO terminals and to investigate diverse link topologies is demonstrated. The study also showcases the successful establishment of a quantum-secured link to a cloud server.

quant-ph↗

arXiv2024

Composable free-space continuous-variable quantum key distribution using discrete modulation

Continuous-variable (CV) quantum key distribution (QKD) allows for quantum secure communication with the benefit of being close to existing classical coherent communication. In recent years, CV QKD protocols using a discrete number of displaced coherent states have been studied intensively, as the modulation can be directly implemented with real devices with a finite digital resolution. However, the experimental demonstrations until now only calculated key rates in the asymptotic regime. To be used in cryptographic applications, a QKD system has to generate keys with composable security in the finite-size regime. In this paper, we present a CV QKD system using discrete modulation that is especially designed for urban atmospheric channels. For this, we use polarization encoding to cope with the turbulent but non-birefringent atmosphere. This will allow to expand CV QKD networks beyond the existing fiber backbone. In a first laboratory demonstration, we implemented a novel type of security proof allowing to calculate composable finite-size key rates against i.i.d. collective attacks without any Gaussian assumptions. We applied the full QKD protocol including a QRNG, error correction and privacy amplification to extract secret keys. In particular, we studied the impact of frame errors on the actual key generation.

quant-ph↗

arXiv2024

Financial market geometry: The tube oscillator

Based on geometrical considerations, we propose a new oscillator for technical market analysis, the tube oscillator. This oscillator measures the trending behavior of a fixed market instrument based on its past history. It is shown in an empirical analysis of the German DAX and the Forex EUR/USD exchange rate that a simple trading strategy based on this oscillator and fixed threshold leads to consistent positive monthly returns of average magnitude of 2% or more. The oscillator is derived from a broader understanding of the geometric behavior of prices throughout a fixed period, which we term financial market geometry. The remarkable profit results of the presented technique show that 1) prices of financial market instruments have a strong underlying deterministic component which can be detected and quantified with a matching approach and 2) financial market geometry is capable of providing such detectors.

q-fin.TR↗

arXiv2023

Photon counting intensity interferometry in the blue at a 0.5 m telescope

Intensity interferometry is a re-emerging interferometry tool that alleviates some of the challenges of amplitude interferometry at the cost of reduced sensitivity. We demonstrate the feasibility of intensity interferometry with fast single photon counting detectors at small telescopes by utilising a telescope of diameter of merely $0.5$\,m. The entire measurement setup, including collimation, optical filtering, and two single photon detectors, is attached directly to the telescope without the use of optical fibres, facilitated by the large area of our single photon detectors. For digitisation and timing, we utilise a Time-To-Amplitude-Converter. Observing $α$ Lyrae (Vega) for a total exposure time of $32.4$\,h over the course of six nights, an auto-correlation signal with a contrast of $(9.5 \pm 2.7) \times 10^-3$ and a coherence time of $(0.34 \pm 0.12)$ ps at a SNR of 2.8 is measured. The result fits well to preceding laboratory tests as well as expectations calculated from the optical and electronic characteristics of our measurement setup. This measurement, to our knowledge, constitutes the first time that a bunching signal with starlight was measured in the B band with single photon counting detectors. Simultaneously, this is to date the stellar intensity interferometry measurement utilising the smallest telescope. Our successful measurement shows that intensity interferometry can be adopted not only at large scale facilities, but also at readily available and inexpensive smaller telescopes.

astro-ph.IM↗

arXiv2023

Cyclicity and iterated logarithms in the Drury-Arveson space

Let $H^2_d$ be the Drury-Arveson space, and let $f\in H^2_d$ have bounded argument and no zeros in $\mathbb{B}_d$. We show that $f$ is cyclic in $H^2_d$ if and only if $\log f$ belongs to the Pick-Smirnov class $N^+(H^2_d)$. Furthermore, for non-vanishing functions $f\in H^2_d$ with bounded argument and $H^\infty$-norm less than 1, cyclicity can also be tested via iterated logarithms. For example, we show that $f$ is cyclic if and only if $\log(1+\log (1/f))\in N^+(H^2_d)$. Thus, a sufficient condition for cyclicity is that $\log(1+\log (1/f))\in H^2_d$. More generally, our results hold for all radially weighted Besov spaces that also are complete Pick spaces.

math.FA↗

arXiv2023

Cyclicity in the Drury-Arveson space and other weighted Besov spaces

Let $\mathcal{H}$ be a space of analytic functions on the unit ball $\mathbb B_d$ in $\mathbb C^d$ with multiplier algebra $\mathrm{Mult}(\mathcal{H})$. A function $f\in \mathcal{H}$ is called cyclic if the set $[f]$, the closure of $\{φf:φ\in \mathrm{Mult}(\mathcal{H})\}$, equals $\mathcal{H}$. For multipliers we also consider a weakened form of the cyclicity concept. Namely for $n\in \mathbb N_0$ we consider the classes $$\mathcal{C}_n(\mathcal{H})=\{φ\in \mathrm{Mult}(\mathcal H):φ\ne 0, [φ^n]=[φ^{n+1}]\}.$$ Many of our results hold for $N$:th order radially weighted Besov spaces on $\mathbb B_d$, but we describe our results only for the Drury-Arveson space $H^2_d$ here. Letting $\mathbb C_{stable}[z]$ denote the stable polynomials for $\mathbb B_d$, i.e. the $d$-variable complex polynomials without zeros in $\mathbb B_d $, we show that \begin{align*} &\text{ if }d \text{ is odd, then } \mathbb C_{stable}[z]\subseteq \mathcal C_{\frac{d-1}{2}}(H^2_d), \text{ and }\\ &\text{ if }d \text{ is even, then } \mathbb C_{stable}[z]\subseteq \mathcal C_{\frac{d}{2}-1}(H^2_d).\end{align*} For $d=2$ and $d=4$ these inclusions are the best possible, but in general we can only show that if $0\le n\le \frac{d}{4}-1$, then $\mathbb C_{stable}[z]\nsubseteq \mathcal C_n(H^2_d)$. For functions other than polynomials we show that if $f,g\in H^2_d$ such that $f/g\in H^\infty$ and $f$ is cyclic, then $g$ is cyclic. We use this to prove that if $f,g\in H^2_d$ extend to be analytic in a neighborhood of $\overline{\mathbb B_d }$, have no zeros in $\mathbb B_d $, and their zero sets coincide on the boundary, then $f$ is cyclic if and only if $g$ is cyclic. Furthermore, if for $f\in H^2_d\cap C(\overline{\mathbb B_d })$ the set $Z(f)\cap \partial \mathbb B_d$ embeds a cube of real dimension $\ge 3$, then $f$ is not cyclic in the Drury-Arveson space.

math.FA↗

arXiv2022

Multivariable versions of a lemma of Kaluza's

Let $d\in \mathbb{N}$ and $f(z)= \sum_{α\in \mathbb{N}_0^d} c_αz^α$ be a convergent multivariable power series in $z=(z_1,\dots,z_d)$. In this paper we present two conditions on the positive coefficients $c_α$ which imply that $f(z)=\frac{1}{1-\sum_{α\in \mathbb{N}_0^d} q_αz^α}$ for non-negative coefficients $q_α$. If $d=1$, then both of our results reduce to a lemma of Kaluza's. For $d>1$ we present examples to show that our two conditions are independent of one another. It turns out that functions of the type $$f(z)= \int_{[0,1]^d} \frac{1}{1-\sum_{j=1}^d t_j z_j} dμ(t)$$ satisfy one of our conditions, whenever $dμ(t) = dμ_1(t_1) \times \dots \times dμ_d(t_d)$ is a product of probability measures $μ_j$ on $[0,1]$. Our results have applications to the theory of Nevanlinna-Pick kernels.

math.FA↗

arXiv2022

von Neumann's inequality for row contractive matrix tuples

We prove that for all $n\in \mathbb{N}$, there exists a constant $C_{n}$ such that for all $d \in \mathbb{N}$, for every row contraction $T$ consisting of $d$ commuting $n \times n$ matrices and every polynomial $p$, the following inequality holds: \[ \|p(T)\| \le C_{n} \sup_{z \in \mathbb{B}_d} |p(z)| . \] We apply this result and the considerations involved in the proof to several open problems from the pertinent literature. First, we show that Gleason's problem cannot be solved contractively in $H^\infty(\mathbb{B}_d)$ for $d \ge 2$. Second, we prove that the multiplier algebra $\operatorname{Mult}(\mathcal{D}_a(\mathbb{B}_d))$ of the weighted Dirichlet space $\mathcal{D}_a(\mathbb{B}_d)$ on the ball is not topologically subhomogeneous when $d \ge 2$ and $a \in (0,d)$. In fact, we determine all the bounded finite dimensional representations of the norm closed subalgebra $A(\mathcal{D}_a(\mathbb{B}_d))$ of $\operatorname{Mult}(\mathcal{D}_a(\mathbb{B}_d))$ generated by polynomials. Lastly, we also show that there exists a uniformly bounded nc holomorphic function on the free commutative ball $\mathfrak{C}\mathfrak{B}_d$ that is levelwise uniformly continuous but not globally uniformly continuous.

math.FA↗

arXiv2022

Multiplier tests and subhomogeneity of multiplier algebras

Multipliers of reproducing kernel Hilbert spaces can be characterized in terms of positivity of $n \times n$ matrices analogous to the classical Pick matrix. We study for which reproducing kernel Hilbert spaces it suffices to consider matrices of bounded size $n$. We connect this problem to the notion of subhomogeneity of non-selfadjoint operator algebras. Our main results show that multiplier algebras of many Hilbert spaces of analytic functions, such as the Dirichlet space and the Drury-Arveson space, are not subhomogeneous, and hence one has to test Pick matrices of arbitrarily large matrix size $n$. To treat the Drury-Arveson space, we show that multiplier algebras of certain weighted Dirichlet spaces on the disc embed completely isometrically into the multiplier algebra of the Drury-Arveson space.

math.FA↗

arXiv2022

Free outer functions in complete Pick spaces

Jury and Martin establish an analogue of the classical inner-outer factorization of Hardy space functions. They show that every function $f$ in a Hilbert function space with a normalized complete Pick reproducing kernel has a factorization of the type $f=φg$, where $g$ is cyclic, $φ$ is a contractive multiplier, and $\|f\|=\|g\|$. In this paper we show that if the cyclic factor is assumed to be what we call free outer, then the factors are essentially unique, and we give a characterization of the factors that is intrinsic to the space. That lets us compute examples. We also provide several applications of this factorization.

math.FA↗

arXiv2022

Collective photon emission patterns from two atoms in free space

Modification of spontaneous decay in space and time is a central topic of quantum physics. It has been predominantly investigated in the context of cavity quantum electrodynamics (QED), gaining new interest recently in the domain of nano-optics. Beyond cavity-QED, spontaneous emission may be modified also in free space due to correlations among the photon emitters, a phenomenon known as super- and sub-radiance. Correlations may stem either from direct interactions between the particles, from long-range exchange of photons, or by measuring single photons in a common mode. Yet, the genuine spatial spontaneous emission pattern of an atomic ensemble in an entangled quantum state has not been observed so far, due to the lack of ultra-fast cameras with high spatial resolution suited for recording single photons from single atoms. Preparing two trapped ions in free space in entangled Dicke states via photon detection, we study the resulting collective spontaneous emission patterns. Depending on the symmetry of the Dicke states, associated with the direction of detection of the first state-determining photon, we observe fundamentally different emission patterns for the subsequently scattered photon, including super- and sub-radiance. Our results demonstrate that the detection of a single photon can profoundly modify the collective emission of an atomic array, here represented by its most elementary building block of two atoms in free space.

quant-ph↗

arXiv2022

Comparing Different Approaches for Stellar Intensity Interferometry

Stellar intensity interferometers correlate photons within their coherence time and could overcome the baseline limitations of existing amplitude interferometers. Intensity interferometers do not rely on phase coherence of the optical elements and thus function without high grade optics and light combining delay lines. However, the coherence time of starlight observed with realistic optical filter bandwidths (> 0.1 nm) is usually much smaller than the time resolution of the detection system (> 10 ps), resulting in a greatly reduced correlation signal. Reaching high signal to noise in a reasonably short measurement time can be achieved in different ways: either by increasing the time resolution, which increases the correlation signal height, or by increasing the photon rate, which decreases statistical uncertainties of the measurement. We present laboratory measurements employing both approaches and directly compare them in terms of signal to noise ratio. A high time-resolution interferometry setup designed for small to intermediate size optical telescopes and thus lower photon rates (diameters < some meters) is compared to a setup capable of measuring high photon rates, which is planned to be installed at Cherenkov telescopes with dish diameters of > 10 m. We use a Xenon lamp as a common light source simulating starlight. Both setups measure the expected correlation signal and work at the expected shot-noise limit of statistical uncertainties for measurement times between 10 min and 23 h. We discuss the quantitative differences in the measurement results and give an overview of suitable operation regimes for each of the interferometer concepts.

astro-ph.IM↗