##### Exploring the Limits of Controlled Markovian Quantum Dynamics with Thermal Resources

F. vom Ende, E. Malvetti, G. Dirr, T. Schulte-Herbrüggen

Open Systems & Information Dynamics 30 (01), 2350005 (2023).

"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."

##### Quantum optimal control in quantum technologies. Strategic report on current status, visions and goals for research in Europe

C. P. Koch, U. Boscain, T. Calarco, G. Dirr, S. Filipp, S. J. Glaser, R. Kosloff, S. Montangero, T. Schulte-Herbrüggen, D. Sugny, F. K. Wilhelm

Epj Quantum Technology 9 (1), 19 (2022).

Quantum optimal control, a toolbox for devising and implementing the shapes of external fields that accomplish given tasks in the operation of a quantum device in the best way possible, has evolved into one of the cornerstones for enabling quantum technologies. The last few years have seen a rapid evolution and expansion of the field. We review here recent progress in our understanding of the controllability of open quantum systems and in the development and application of quantum control techniques to quantum technologies. We also address key challenges and sketch a roadmap for future developments.

##### Strict positivity and D-majorization

F. vom Ende

Linear & Multilinear Algebra 26 (2020).

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.

##### On the Alberti-Uhlmann Condition for Unital Channels

S. Chakraborty, D. Chruscinski, G. Sarbick, F. vom Ende

Quantum 4, 360 (2020).

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.

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

F. vom Ende

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-d-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.

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

T. Schulte-Herbrüggen, F. vom Ende, G. Dirr

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<∞ using methods of d-majorisation.

##### Reachable Sets from Toy Models to Controlled Markovian Quantum Systems

G. Dirr, F. vom Ende, T. Schulte-Herbrüggen, Ieee

58th IEEE Conference on Decision and Control (CDC) 2322-2329 (2019).

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 >= 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.

##### Unitary dilations of discrete-time quantum-dynamical semigroups

F. vom Ende, G. Dirr

Journal of Mathematical Physics 60 (12), 122702 (2019).

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 is an element of N0). 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. Published under license by AIP Publishing.

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

F. vom Ende, G. Dirr, M. Keyl, T. Schulte-Herbrüggen

Open Systems & Information Dynamics 26 (3), 1950014 (2019).

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) is an element of R plus a bang-bang (i.e., on-off) switchable noise term in Kossakowski-Lindblad form. Generalizing standard majorization results from finite Gamma(V) 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 it only has been known in finite dimensional analogues. The proof of the result is currently limited to the bounded control Hamiltonians H-j and for noise terms Gamma(V) with compact normal V.

##### Optimal control of hybrid optomechanical systems for generating non-classical states of mechanical motion

V. Bergholm, W. Wieczorek, T. Schulte-Herbrüggen, M. Keyl

Quantum Science and Technology 4 (3), 34001 (2019).

Cavity optomechanical systems are one of the leading experimental platforms for controlling mechanical motion in the quantum regime. We exemplify that the control over cavity optomechanical systems greatly increases by coupling the cavity also to a two-level system, thereby creating a hybrid optomechanical system. If the two-level system can be driven largely independently of the cavity, we show that the nonlinearity thus introduced enables us to steer the extended system to non-classical target states of the mechanical oscillator with Wigner functions exhibiting significant negative regions. We illustrate how to use optimal control techniques beyond the linear regime to drive the hybrid system from the near ground state into a Fock target state of the mechanical oscillator. We base our numerical optimization on realistic experimental parameters for exemplifying how optimal control enables the preparation of decidedly non-classical target states, where naive control schemes fail. Our results thus pave the way for applying the toolbox of optimal control in hybrid optomechanical systems for generating non-classical mechanical states.