arXiv subjects

Frederik vom Ende

Publications and source records attributed to Frederik vom Ende.

34 records · Page 2Linked to original sources

arXiv2023

Exploring the Limits of Controlled Markovian Quantum Dynamics with Thermal Resources

Our aim is twofold: First, we rigorously analyse the generators of quantum-dynamical semigroups of thermodynamic processes. We characterise a wide class of GKSL-generators for quantum maps within thermal operations and argue that every infinitesimal generator of (a one-parameter semigroup of) Markovian thermal operations belongs to this class. We completely classify and visualise them and their non-Markovian counterparts for the case of a single qubit. Second, we use this description in the framework of bilinear control systems to characterise reachable sets of coherently controllable quantum systems with switchable coupling to a thermal bath. The core problem reduces to studying a hybrid control system ("toy model") on the standard simplex allowing for two types of evolution: (i) instantaneous permutations and (ii) a one-parameter semigroup of $d$-stochastic maps. We generalise upper bounds of the reachable set of this toy model invoking new results on thermomajorisation. Using tools of control theory we fully characterise these reachable sets as well as the set of stabilisable states as exemplified by exact results in qutrit systems.

quant-ph↗

arXiv2023

Exploring the Limits of Open Quantum Dynamics II: Gibbs-Preserving Maps from the Perspective of Majorization

Motivated by reachability questions in coherently controlled open quantum systems coupled to a thermal bath, as well as recent progress in the field of thermo-/vector-majorization we generalize classical majorization from unital quantum channels to channels with an arbitrary fixed point $D$ of full rank. Such channels preserve some Gibbs-state and thus play an important role in the resource theory of quantum thermodynamics, in particular in thermo-majorization. Based on this we investigate $D$-majorization on matrices in terms of its topological and order properties, such as existence of unique maximal and minimal elements, etc. Moreover we characterize $D$-majorization in the qubit case via the trace norm and elaborate on why this is a challenging task when going beyond two dimensions.

quant-ph↗

arXiv2023

Exploring the Limits of Open Quantum Dynamics I: Motivation, New Results from Toy Models to Applications

Which quantum states can be reached by controlling open Markovian $n$-level quantum systems? Here, we address reachable sets of coherently controllable quantum systems with switchable coupling to a thermal bath of temperature $T$. The core problem reduces to a toy model of studying points in the standard simplex allowing for two types of controls: (i) permutations within the simplex, (ii) contractions by a dissipative semigroup. By illustration, we put the problem into context and show how toy-model solutions pertain to the reachable set of the original controlled Markovian quantum system. Beyond the case $T=0$ (amplitude damping) we present new results for $0 <T < \infty$ using methods of $d$-majorisation.

quant-ph↗

arXiv2023

Quantum-Dynamical Semigroups and the Church of the Larger Hilbert Space

In this work we investigate Stinespring dilations of quantum-dynamical semigroups, which are known to exist by means of a constructive proof given by Davies in the early 70s. We show that if the semigroup describes an open system, that is, if it does not consist of only unitary channels, then the evolution of the dilated closed system has to be generated by an unbounded Hamiltonian; subsequently the environment has to correspond to an infinite-dimensional Hilbert space, regardless of the original system. Moreover, we prove that the second derivative of Stinespring dilations with a bounded total Hamiltonian yields the dissipative part of some quantum-dynamical semigroup -- and vice versa. In particular this characterizes the generators of quantum-dynamical semigroups via Stinespring dilations.

quant-ph↗

arXiv2022

Which Bath-Hamiltonians Matter for Thermal Operations?

In this article we explore the set of thermal operations from a mathematical and topological point of view. First we introduce the concept of Hamiltonians with resonant spectrum with respect to some reference Hamiltonian, followed by proving that when defining thermal operations it suffices to only consider bath Hamiltonians which satisfy this resonance property. Next we investigate continuity of the set of thermal operations in certain parameters, such as energies of the system and temperature of the bath. We will see that the set of thermal operations changes discontinuously with respect to the Hausdorff metric at any Hamiltonian which has so-called degenerate Bohr spectrum, regardless of the temperature. Finally we find a semigroup representation of the (enhanced) thermal operations in two dimensions by characterizing any such operation via three real parameters, thus allowing for a visualization of this set. Using this, in the qubit case we show commutativity of the (enhanced) thermal operations as well as convexity of the thermal operations without the closure. The latter is done by specifying the elements of this set exactly.

quant-ph↗

arXiv2022

Phase Spaces, Parity Operators, and the Born-Jordan Distribution

Phase spaces as given by the Wigner distribution function provide a natural description of infinite-dimensional quantum systems. They are an important tool in quantum optics and have been widely applied in the context of time-frequency analysis and pseudo-differential operators. Phase-space distribution functions are usually specified via integral transformations or convolutions which can be averted and subsumed by (displaced) parity operators proposed in this work. Building on earlier work for Wigner distribution functions [A. Grossmann, Comm. Math. Phys. 48(3), 191 (1976)], parity operators give rise to a general class of distribution functions in the form of quantum-mechanical expectation values. This enables us to precisely characterize the mathematical existence of general phase-space distribution functions. We then relate these distribution functions to the so-called Cohen class [L. Cohen, J. Math. Phys. 7(5), 781 (1966)] and recover various quantization schemes and distribution functions from the literature. The parity-operator approach is also applied to the Born-Jordan distribution which originates from the Born-Jordan quantization [M. Born, P. Jordan, Z. Phys. 34(1), 858 (1925)]. The corresponding parity operator is written as a weighted average of both displacements and squeezing operators and we determine its generalized spectral decomposition. This leads to an efficient computation of the Born-Jordan parity operator in the number-state basis and example quantum states reveal unique features of the Born-Jordan distribution.

quant-ph↗

arXiv2022

The $d$-Majorization Polytope

We investigate geometric and topological properties of $d$-majorization -- a generalization of classical majorization to positive weight vectors $d \in \mathbb{R}^n$. In particular, we derive a new, simplified characterization of $d$-majorization which allows us to work out a halfspace description of the corresponding $d$-majorization polytopes. That is, we write the set of all vectors which are $d$-majorized by some given vector $y \in \mathbb{R}^n$ as an intersection of finitely many half spaces, i.e. as solutions to an inequality of the type $Mx\leq b$. Here $b$ depends on $y$ while $M$ can be chosen independently of $y$. This description lets us prove continuity of the $d$-majorization polytope (jointly with respect to $d$ and $y$) and, furthermore, lets us fully characterize its extreme points. Interestingly, for $y\geq 0$ one of these extreme points classically majorizes every other element of the $d$-majorization polytope. Moreover, we show that the induced preorder structure on $\mathbb{R}^n$ admits minimal and maximal elements. While the former are always unique the latter are unique if and only if they correspond to the unique minimal entry of the $d$-vector.

math.CO↗

arXiv2020

Reachability in Controlled Markovian Quantum Systems: An Operator-Theoretic Approach

In quantum systems theory one of the fundamental problems boils down to: Given an initial state, which final states can be reached by the dynamic system in question? Formulated in the framework of bilinear control systems, the evolution shall be governed by an inevitable Hamiltonian drift term, finitely many control Hamiltonians allowing for (at least) piecewise constant control amplitudes, plus a (possibly bang-bang switchable) noise term in Kossakowski-Lindblad form. Now assuming switchable coupling of finite-dimensional systems to a thermal bath of arbitrary temperature, the core problem of reachability boils down to studying points in the standard simplex amenable to two types of controls that can be used interleaved: Permutations within the simplex, and contractions by a dissipative one-parameter semigroup. We illustrate how the solutions of the core problem pertain to the reachable set of the original controlled Markovian quantum system. This allows us to show that for global as well as local switchable coupling to a temperature-zero bath one can generate every quantum state from every initial state up to arbitrary precision. Moreover we present an inclusion for non-zero temperatures as a consequence of our results on d-majorization. Then we consider infinite-dimensional open quantum-dynamical systems following a unital Kossakowski-Lindblad master equation extended by controls. Here the drift Hamiltonian can be arbitrary, the finitely many control Hamiltonians are bounded, and the switchable noise term is generated by a single compact normal operator. Via new majorization results of ours, we show that such bilinear quantum control systems allow to approximately reach any target state majorized by the initial one, as up to now only has been known in finite-dimensional analogues.

quant-ph↗

arXiv2020

On the Alberti-Uhlmann Condition for Unital Channels

We address the problem of existence of completely positive trace preserving (CPTP) maps between two sets of density matrices. We refine the result of Alberti and Uhlmann and derive a necessary and sufficient condition for the existence of a unital channel between two pairs of qubit states which ultimately boils down to three simple inequalities.

quant-ph↗

arXiv2020

Strict Positivity and $D$-Majorization

Motivated by quantum thermodynamics we first investigate the notion of strict positivity, that is, linear maps which map positive definite states to something positive definite again. We show that strict positivity is decided by the action on any full-rank state, and that the image of non-strictly positive maps lives inside a lower-dimensional subalgebra. This implies that the distance of such maps to the identity channel is lower bounded by one. The notion of strict positivity comes in handy when generalizing the majorization ordering on real vectors with respect to a positive vector $d$ to majorization on square matrices with respect to a positive definite matrix $D$. For the two-dimensional case we give a characterization of this ordering via finitely many trace norm inequalities and, moreover, investigate some of its order properties. In particular it admits a unique minimal and a maximal element. The latter is unique as well if and only if minimal eigenvalue of $D$ has multiplicity one.

quant-ph↗

arXiv2019

Reachable Sets from Toy Models to Controlled Markovian Quantum Systems

In the framework of bilinear control systems, we present reachable sets of coherently controllable open quantum systems with switchable coupling to a thermal bath of arbitrary temperature $T geq 0$. The core problem boils down to studying points in the standard simplex amenable to two types of controls that can be used interleaved: (i) permutations within the simplex, (ii) contractions by a dissipative one-parameter semigroup. Our work illustrates how the solutions of the core problem pertain to the reachable set of the original controlled Markovian quantum system. We completely characterize the case $T=0$ and present inclusions for $T>0$.

math.OC↗

arXiv2019

Von Neumann Type of Trace Inequalities for Schatten-Class Operators

We generalize von Neumann's well-known trace inequality, as well as related eigenvalue inequalities for hermitian matrices, to Schatten-class operators between complex Hilbert spaces of infinite dimension. To this end, we exploit some recent results on the $C$-numerical range of Schatten-class operators. For the readers' convenience, we sketched the proof of these results in the Appendix.

math.FA↗

arXiv2019

Reachability in Infinite Dimensional Unital Open Quantum Systems with Switchable GKS-Lindblad Generators

In quantum systems theory one of the fundamental problems boils down to: given an initial state, which final states can be reached by the dynamic system in question. Here we consider infinite dimensional open quantum dynamical systems following a unital Kossakowski-Lindblad master equation extended by controls. More precisely, their time evolution shall be governed by an inevitable potentially unbounded Hamiltonian drift term $H_0$, finitely many bounded control Hamiltonians $H_j$ allowing for (at least) piecewise constant control amplitudes $u_j(t)\in{\mathbb R}$ plus a bang-bang (i.e. on-off) switchable noise term $\mathbfΓ_V$ in Kossakowski-Lindblad form. Generalizing standard majorization results from finite to infinite dimensions, we show that such bilinear quantum control systems allow to approximately reach any target state majorized by the initial one, as up to now only has been known in finite dimensional analogues.---The proof of the result is currently limited to the control Hamiltonians $ H_j$ being bounded and noise terms $\mathbfΓ_V$ with compact normal $V$.

quant-ph↗

arXiv2019

The C-Numerical Range for Schatten-Class Operators

We generalize the $C$-numerical range $W_C(T)$ from trace-class to Schatten-class operators, i.e. to $C\in\mathcal B^p(\mathcal H)$ and $T\in\mathcal B^q(\mathcal H)$ with $1/p + 1/q = 1$, and show that its closure is always star-shaped with respect to the origin. For $q \in (1,\infty]$, this is equivalent to saying that the closure of the image of the unitary orbit of $T\in\mathcal B^q(\mathcal H)$ under any continous linear functional $L\in(\mathcal B^q(\mathcal H))'$ is star-shaped with respect to the origin. For $q=1$, one has star-shapedness with respect to $\operatorname{tr}(T)W_e(L)$, where $W_e(L)$ denotes the essential range of $L$. Moreover, the closure of $W_C(T)$ is convex if $C$ or $T$ is normal with collinear eigenvalues. If $C$ and $T$ are both normal, then the $C$-spectrum of $T$ is a subset of the $C$-numerical range, which itself is a subset of the closure of the convex hull of the $C$-spectrum. This closure coincides with the closure of the $C$-numerical range if, in addition, the eigenvalues of $C$ or $T$ are collinear.

math.FA↗

arXiv2019

Unitary Dilations of Discrete-Time Quantum-Dynamical Semigroups

We show that the discrete-time evolution of an open quantum system generated by a single quantum channel $T$ can be embedded in the discrete-time evolution of an enlarged closed quantum system, i.e. we construct a unitary dilation of the discrete-time quantum-dynamical semigroup $(T^n)_{n \in \mathbb N_0}$. In the case of a cyclic channel $T$, the auxiliary space may be chosen (partially) finite-dimensional. We further investigate discrete-time quantum control systems generated by finitely many commuting quantum channels and prove a similar unitary dilation result as in the case of a single channel.

math-ph↗

arXiv2018

The C-Numerical Range in Infinite Dimensions

In infinite dimensions and on the level of trace-class operators $C$ rather than matrices, we show that the closure of the $C$-numerical range $W_C(T)$ is always star-shaped with respect to the set $\operatorname{tr}(C)W_e(T)$, where $W_e(T)$ denotes the essential numerical range of the bounded operator $T$. Moreover, the closure of $W_C(T)$ is convex if either $C$ is normal with collinear eigenvalues or if $T$ is essentially self-adjoint. In the case of compact normal operators, the $C$-spectrum of $T$ is a subset of the $C$-numerical range, which itself is a subset of the convex hull of the closure of the $C$-spectrum. This convex hull coincides with the closure of the $C$-numerical range if, in addition, the eigenvalues of $C$ or $T$ are collinear.

math.FA↗