Gunther Dirr
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Gunther Dirr has a total of 43 co-authors. Here are the top co-authors by number of shared publications:
- Frederik Vom Ende
Dahlem Center for Complex Quantum Systems Freie Universität Berlin Arnimallee 14 14195 Berlin Germany
9 shared publications U. Helmke
Institute of Mathematics, University of Würzburg, Germany
9 shared publicationsUwe Helmke
5 shared publicationsThomas Schulte-Herbrüggen
School of Natural Sciences, Technische Universität München, 85737 Garching, Germany, and Munich Center for Quantum Science and Technology (MCQST) and Munich Quantum Valley (MQV) Munich, Germany
4 shared publications- Uwe Helmke
Institute of Mathematics, University of Würzburg, Würzburg, Germany
4 shared publications - Emanuel Malvetti
Department Chemie, TUM School of Natural Sciences, Technische Universität München, Garching, Germany
2 shared publications
Latest Publications
- 5 December 2024
- Published by Society for Industrial & Applied Mathematics (SIAM) in SIAM Journal on Control and Optimization
- Vol. 63 (1) , S53-S81
Reachability, Coolability, and Stabilizability of Open Markovian Quantum Systems with Fast Unitary Control
Open Markovian quantum systems with fast unitary control can be reduced to an equivalent control system on the standard simplex modeling the dynamics of the eigenvalues of the density matrix describing the quantum state. We explore this reduced control system for answering questions on reachability and stabilizability with immediate applications to the cooling of Markovian quantum systems. We show that for certain tasks of interest, the control Hamiltonian can be chosen time-independent.
- 5 December 2024
- Published by Society for Industrial & Applied Mathematics (SIAM) in SIAM Journal on Control and Optimization
- Vol. 63 (1) , S53-S81
Reachability, Coolability, and Stabilizability of Open Markovian Quantum Systems with Fast Unitary Control
Open Markovian quantum systems with fast unitary control can be reduced to an equivalent control system on the standard simplex modeling the dynamics of the eigenvalues of the density matrix describing the quantum state. We explore this reduced control system for answering questions on reachability and stabilizability with immediate applications to the cooling of Markovian quantum systems. We show that for certain tasks of interest, the control Hamiltonian can be chosen time-independent.
- Open Access
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.
Open AccessExploring 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.
- 17 April 2023
- Published by Society for Industrial & Applied Mathematics (SIAM) in SIAM Journal on Control and Optimization
- Vol. 61 (2) , 560-585
Compactness of Fixed Point Maps and the Ball-Marsden-Slemrod Conjecture
Given a parameter dependent fixed point equation [Formula: see text], we derive an abstract compactness principle for the fixed point map [Formula: see text] under the assumptions that (i) the fixed point equation can be solved by the contraction principle and (ii) the map [Formula: see text] is compact for fixed [Formula: see text]. This result is applied to infinite-dimensional, semilinear control systems and their reachable sets. More precisely, we extend a noncontrollability result of Ball, Marsden, and Slemrod [ SIAM J. Control Optim., 20 (1982), pp. 575–597] to semilinear systems. First we consider [Formula: see text]-controls, [Formula: see text]. Subsequently we analyze the case [Formula: see text].
- 17 April 2023
- Published by Society for Industrial & Applied Mathematics (SIAM) in SIAM Journal on Control and Optimization
- Vol. 61 (2) , 560-585
Compactness of Fixed Point Maps and the Ball-Marsden-Slemrod Conjecture
Given a parameter dependent fixed point equation [Formula: see text], we derive an abstract compactness principle for the fixed point map [Formula: see text] under the assumptions that (i) the fixed point equation can be solved by the contraction principle and (ii) the map [Formula: see text] is compact for fixed [Formula: see text]. This result is applied to infinite-dimensional, semilinear control systems and their reachable sets. More precisely, we extend a noncontrollability result of Ball, Marsden, and Slemrod [ SIAM J. Control Optim., 20 (1982), pp. 575–597] to semilinear systems. First we consider [Formula: see text]-controls, [Formula: see text]. Subsequently we analyze the case [Formula: see text].
- Open Access
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.Open AccessExploring 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. - Open Access
Compactness of Fixed Point Maps and the Ball-Marsden-Slemrod Conjecture
Given a parameter dependent fixed point equation $x = F(x,u)$, we derive an abstract compactness principle for the fixed point map $u \mapsto x^*(u)$ under the assumptions that (i) the fixed point equation can be solved by the contraction principle and (ii) the map $u \mapsto F(x,u)$ is compact for fixed $x$. This result is applied to infinite-dimensional, semi-linear control systems and their reachable sets. More precisely, we extend a non-controllability result of Ball, Marsden, and Slemrod [1] to semi-linear systems. First we consider $L^p$-controls, $p>1$. Subsequently we analyze the case $p=1$.Open AccessCompactness of Fixed Point Maps and the Ball-Marsden-Slemrod Conjecture
Given a parameter dependent fixed point equation $x = F(x,u)$, we derive an abstract compactness principle for the fixed point map $u \mapsto x^*(u)$ under the assumptions that (i) the fixed point equation can be solved by the contraction principle and (ii) the map $u \mapsto F(x,u)$ is compact for fixed $x$. This result is applied to infinite-dimensional, semi-linear control systems and their reachable sets. More precisely, we extend a non-controllability result of Ball, Marsden, and Slemrod [1] to semi-linear systems. First we consider $L^p$-controls, $p>1$. Subsequently we analyze the case $p=1$. - 1 May 2021
- Published by Elsevier in Journal of Differential Equations
- Vol. 283, 216-262
Uniform and L-ensemble reachability of parameter-dependent linear systems
- 1 May 2021
- Published by Elsevier in Journal of Differential Equations
- Vol. 283, 216-262
Uniform and L-ensemble reachability of parameter-dependent linear systems
- Open Access
Uniform and $L^q$-Ensemble Reachability of Parameter-dependent Linear Systems
In this paper, we consider families of linear systems (linear ensembles) defined by matrix pairs $\big( A(\theta),B(\theta) \big)$ depending on a parameter $\theta \in \p$ that is varying over a compact subset $\p$ of the complex plane. In particular, we investigate the following control task: Find an open-loop control which is {\it independent} of the parameter $\theta \in \p$ and steers a given family of initial states $x_0(\theta)$ arbitrarily close to a desired family of terminal states $f(\theta)$ in finite time. Here, the maps $\theta \mapsto x_0(\theta)$ and $\theta \mapsto f(\theta)$ are assumed to lie in a common appropriately chosen {Banach space $X_n(\p)$ of $\C^n$-valued functions}. If this task is solvable for all initial and terminal states, the pair $\big( A(\theta),B(\theta) \big)$ is called {(completely)} ensemble controllable with respect to $X_n(\p)$. Using a well-known infinite-dimensional version of the Kalman rank condition for systems on Banach spaces, we derive sufficient conditions for cascade and parallel connections linear ensembles. Moreover, we prove an abstract decomposition theorem which results from a spectral splitting of the matrix family $A(\theta)$. Based on thses findings as well as approximation theory and cyclicity conditions of multiplications operators, we obtain necessary and sufficient conditions for ensemble controllability (reachability) with respect to the Banach spaces of continuous functions and $L^q$-functions. In the last section, results on {averaged} controllability (reachability) for linear families $\big( A(\theta),B(\theta),C(\theta) \big)$ are presented.Open AccessUniform and $L^q$-Ensemble Reachability of Parameter-dependent Linear Systems
In this paper, we consider families of linear systems (linear ensembles) defined by matrix pairs $\big( A(\theta),B(\theta) \big)$ depending on a parameter $\theta \in \p$ that is varying over a compact subset $\p$ of the complex plane. In particular, we investigate the following control task: Find an open-loop control which is {\it independent} of the parameter $\theta \in \p$ and steers a given family of initial states $x_0(\theta)$ arbitrarily close to a desired family of terminal states $f(\theta)$ in finite time. Here, the maps $\theta \mapsto x_0(\theta)$ and $\theta \mapsto f(\theta)$ are assumed to lie in a common appropriately chosen {Banach space $X_n(\p)$ of $\C^n$-valued functions}. If this task is solvable for all initial and terminal states, the pair $\big( A(\theta),B(\theta) \big)$ is called {(completely)} ensemble controllable with respect to $X_n(\p)$. Using a well-known infinite-dimensional version of the Kalman rank condition for systems on Banach spaces, we derive sufficient conditions for cascade and parallel connections linear ensembles. Moreover, we prove an abstract decomposition theorem which results from a spectral splitting of the matrix family $A(\theta)$. Based on thses findings as well as approximation theory and cyclicity conditions of multiplications operators, we obtain necessary and sufficient conditions for ensemble controllability (reachability) with respect to the Banach spaces of continuous functions and $L^q$-functions. In the last section, results on {averaged} controllability (reachability) for linear families $\big( A(\theta),B(\theta),C(\theta) \big)$ are presented. - 10 August 2020
- Published by Theta Foundation in Journal of Operator Theory
- Vol. 84 (2) , 323-338
Von Neumann type trace inequalities for Schatten-class operators
- 10 August 2020
- Published by Theta Foundation in Journal of Operator Theory
- Vol. 84 (2) , 323-338
Von Neumann type trace inequalities for Schatten-class operators
- Open Access
- 1 December 2019
- Published by AIP Publishing in Journal of Mathematical Physics
- Vol. 60 (12) , 122702
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 . 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.- 1 December 2019
- Published by AIP Publishing in Journal of Mathematical Physics
- Vol. 60 (12) , 122702
Open AccessUnitary 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 . 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. - Open Access
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 $\mathbf{\Gamma}_V$ 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 $\mathbf{\Gamma}_V$ with compact normal $V$.Open AccessReachability 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 $\mathbf{\Gamma}_V$ 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 $\mathbf{\Gamma}_V$ with compact normal $V$.