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Kay Schwieger

Publications and source records attributed to Kay Schwieger.

15 recordsLinked to original sources
arXiv2024

Realizations of free actions via their fixed point algebras

Let $G$ be a compact group, let $\mathcal{B}$ be a unital C$^*$-algebra, and let $(\mathcal{A},G,α)$ be a free C$^*$-dynamical system, in the sense of Ellwood, with fixed point algebra $\mathcal{B}$. We prove that $(\mathcal{A},G,α)$ can be realized as the invariants of an equivariant coaction of $G$ on a corner of $\mathcal{B} \otimes \mathcal{K}(\mathfrak{H})$ for a certain Hilbert space $\mathfrak{H}$ that arises from the freeness of the action. This extends a result by Wassermann for free C$^*$-dynamical systems with trivial fixed point algebras. As an application, we show that any faithful \Star-representation of $\mathcal{B}$ on a Hilbert space $\mathfrak{H}_{\mathcal{B}}$ gives rise to a faithful covariant representation of $(\mathcal{A},G,α)$ on some truncation of $\mathfrak{H}_{\mathcal{B}} \otimes \mathfrak{H}$.

math.OA↗
arXiv2022

An Atiyah Sequence for Noncommutative Principal Bundles

We present a derivation-based Atiyah sequence for noncommutative principal bundles. Along the way we treat the problem of deciding when a given $^*$-automorphism on the quantum base space lifts to a $^*$-automorphism on the quantum total space that commutes with the underlying structure group.

math.OA↗
arXiv2021

E. Schrödinger's 1931 paper "On the Reversal of the Laws of Nature" ["Über die Umkehrung der Naturgesetze",Sitzungsberichte der preussischen Akademie der Wissenschaften, physikalische mathematische Klasse, 8 N9 144-153]

We present an English translation of Erwin Schrödinger's paper on "On the Reversal of the Laws of Nature". In this paper Schrödinger analyses the idea of time reversal of a diffusion process. Schrödinger's paper acted as a prominent source of inspiration for the works of Bernstein on reciprocal processes and of Kolmogorov on time reversal properties of Markov processes and detailed balance. The ideas outlined by Schrödinger also inspired the development of probabilistic interpretations of quantum mechanics by Fényes, Nelson and others as well as the notion of "Euclidean Quantum Mechanics" as probabilistic analogue of quantization. In the second part of the paper Schrödinger discusses the relation between time reversal and statistical laws of physics. We emphasize in our commentary the relevance of Schrödinger's intuitions for contemporary developments in statistical nano-physics.

physics.hist-ph↗
arXiv2021

Lifting spectral triples to noncommutative principal bundles

Given a free action of a compact Lie group $G$ on a unital C*-algebra $\mathcal{A}$ and a spectral triple on the corresponding fixed point algebra $\mathcal{A}^G$, we present a systematic and in-depth construction of a spectral triple on $\mathcal{A}$ that is build upon the geometry of $\mathcal{A}^G$ and $G$. We compare our construction with a selection of established examples.

math.OA↗
arXiv2018

A model for calorimetric measurements in an open quantum system

We investigate the experimental setup proposed in [New J. Phys., 15, 115006 (2013)] for calorimetric measurements of thermodynamic indicators in an open quantum system. As theoretical model we consider a periodically driven qubit coupled with a large yet finite electron reservoir, the calorimeter. The calorimeter is initially at equilibrium with an infinite phonon bath. As time elapses, the temperature of the calorimeter varies in consequence of energy exchanges with the qubit and the phonon bath. We show how under weak coupling assumptions, the evolution of the qubit-calorimeter system can be described by a generalized quantum jump process including as dynamical variable the temperature of the calorimeter. We study the jump process by numeric and analytic methods. Asymptotically with the duration of the drive, the qubit-calorimeter attains a steady state. In this same limit, we use multiscale perturbation theory to derive a Fokker--Planck equation governing the calorimeter temperature distribution. We inquire the properties of the temperature probability distribution close and at the steady state. In particular, we predict the behavior of measurable statistical indicators versus the qubit-calorimeter coupling constant.

quant-ph↗
arXiv2017

Noncommutative Coverings of Quantum Tori

We introduce a framework for coverings of noncommutative spaces. Moreover, we study noncommutative coverings of irrational quantum tori and characterize all such coverings that are connected in a reasonable sense.

math.OA↗
arXiv2017

Part I, Free Actions of Compact Abelian Groups on C*-Algebras

We study free and compact group actions on unital C*-algebras. In particular, we provide a complete classification theory of these actions for compact Abelian groups and explain its relation to the classical classification theory of principal bundles.

math.OA↗
arXiv2017

Part II, Free Actions of Compact Groups on C*-Algebras

We study a simple subclass of free actions of non-Abelian groups on unital C*-algebras, namely cleft actions. These are characterized by the fact that the associated noncommutative vector bundles are trivial. In particular, we provide a complete classification theory for these actions and describe its relations to classical principal bundles.

math.OA↗
arXiv2017

An application of Pontryagin's principle to Brownian particle engineered equilibration

We present a stylized model of controlled equilibration of a small system in a fluctuating environment. We derive the equations governing the optimal control steering \emph{in finite time} the system between two equilibrium states. The corresponding thermodynamic transition is optimal in the sense that occurs at minimum entropy if the set of admissible controls is restricted by certain bounds on the time derivatives of the protocols. We apply our equations to the engineered equilibration of an optical trap considered in a recent proof of principle experiment. We also analyze an elementary model of nucleation previously considered by Landauer to discuss the thermodynamic cost of one bit of information erasure. We expect our model to be a useful benchmark for experiment design as it exhibits the same integrability properties of well known models of optimal mass transport by a compressible velocity field.

cond-mat.mes-hall↗
arXiv2016

Fluctuation Relation for Qubit-Calorimetry

Motivated by proposed thermometry measurement on an open quantum system, we present a simple model of an externally driven qubit interacting with a finite sized, fermion environment acting as calorimeter. The derived dynamics is governed by a stochastic Schrödinger equation coupled to the temperature change of the calorimeter. We prove a fluctuation relation and deduce from it a notion of entropy production. Finally, we discuss the first and second law associated to the dynamics.

quant-ph↗
arXiv2015

On the efficiency of heat engines at the micro-scale and below

We investigate the thermodynamic efficiency of sub-micro-scale heat engines operating under the conditions described by over-damped stochastic thermodynamics. We prove that at maximum power the efficiency obeys for constant isotropic mobility the universal law $η=2\,η_{C}/(4-η_{C})$ where $η_{C}$ is the efficiency of an ideal Carnot cycle. The corresponding power optimizing protocol is specified by the solution of an optimal mass transport problem. Such solution can be determined explicitly using well known Monge--Ampère--Kantorovich reconstruction algorithms. Furthermore, we show that the same law describes the efficiency of heat engines operating at maximum work over short time periods. Finally, we illustrate the straightforward extension of these results to cases when the mobility is anisotropic and temperature dependent.

cond-mat.stat-mech↗
arXiv2014

On how nanomechanical systems can minimize dissipation

Information processing machines at the nanoscales are unavoidably affected by thermal fluctuations. Efficient design requires understanding how nanomachines can operate at minimal energy dissipation. In this letter we focus on mechanical systems controlled by smoothly varying potential forces. We show that optimal control equations come about in a natural way if the energy cost to manipulate the potential is taken into account. When such cost becomes negligible, the optimal control strategy can be constructed by transparent geometrical methods and recovers the solution of optimal mass transport equations in the overdamped limit. Our equations are equivalent to hierarchies of kinetic equations of a form well-known in the theory of dilute gases. From our results, optimal strategies for energy efficient nanosystems may be devised

cond-mat.stat-mech↗
arXiv2014

Diagonal Couplings of Quantum Markov Chains

In this article we extend the coupling method from classical probability theory to quantum Markov chains on atomic von Neumann algebras. In particular, we establish a coupling inequality, which allow us to estimate convergence rates by analyzing couplings. For a given tensor dilation we construct a self-coupling of a Markov operator. It turns out that on a dense subset the coupling is a dual version of the extended dual transition operator previously studied by Gohm etal. We deduce that this coupling is successful if and only if the dilation is asymptotically complete.

math.OA↗
arXiv2013

Ergodic Properties of Quantum Birth and Death Chains

We study a class of quantum Markov processes that, on the one hand, is inspired by the micromaser experiment in quantum optics and, on the other hand, by classical birth and death processes. We prove some general geometric properties and irreducibility for non-degenerated parameters. Furthermore, we analyze ergodic properties of the corresponding transition operators. For homogeneous birth and death rates we show how these can be fully determined by explicit calculation. As for classical birth and death chains we obtain a rich yet simple class of quantum Markov chains on an infinite space, which allow only local transitions while having divers ergodic properties.

math.OA↗