For a list of recent publications see publications. For a list of grants and projects see grants.

- Quantum Thermodynamics
- Topological Quantum Information Systems
- Quantum Walks and Quantum Cellular Automata
- Quantum Memory Channels
- Effective Theories: Renormalization and Decoherence
- Quantum Shannon Theory
- Mathematical Theory of Quantum Operations and Correlations
- From Quantum Spin Systems to Anyons
- Quantum Topological Order and Topological Phases
- Quantum Complexity Theory
- Unbounded Generators of Quantum Dynamical Semigroups
- Quantum Programming Languages

## Quantum Thermodynamics

As thermodynamic machines become smaller and smaller, quantum corrections to classical thermodynamics are of increasing interest. However, there is no well-established and conlusive framework for this new field of Quantum Thermodynamics yet. In fact, the definitions of many basic thermodynamical notions are still disputed: How should one define the work done on a quantum system? What does it mean to perform a process adiabatically? Do quantum analogues of the fundamental laws of thermodynamics exist? How is classical thermodynamics obtained in the large-system limit? The goal of this project is to answer some of these fundamental question.

Contact Persons: Kais Abdelkhalek, Dr. Terry Farrelly, Dr. David Reeb, Dr. Courtney Brell

## Topological Quantum Information Systems

Topologically ordered systems have natural robustness to sources of noise in the laboratory. This makes them ideal candidates to realise quantum computing and information storage technologies. Topological systems are excellent prospects to realise quantum memories for storage of quantum information. In particular, almost all approaches to designing such a memory that does not require active error correction at finite temperature (i.e. a quantum hard-drive) exploit topological properties. Further, topological systems have exotic properties such as anyonic excitations that may be used to implement fault-tolerant quantum computation. Alternatively, if defects in the system such as holes or lattice dislocations can be controlled, these may also be used for quantum computation.

We study and design models for topological systems with a view to implementing quantum information technologies. Protocols for realising computation or error correction in such systems are also of interest. More broadly, we are interested in properties of topological systems such as their stability, relations between different topological orders, and thermalization in such topological systems.

Contact Person: Dr. Courtney Brell

## Quantum Walks and Quantum Cellular Automata

Quantum walks (QW) and quantum cellular automata (QCA) are quantum operations on a discrete lattice. Whereas QW represent a single quantum particle evolving in discrete space - lattice, QCA describe certain evolution of system where a particle is present at each lattice site. Special QCA can be universal programmable quantum computers, while simpler ones can serve as a building block of quantum computation schemes with combined global and local control. QW can serve as a playground for simulating some particle interaction in discrete space-time.

Contact Person: Prof. Dr. Reinhard F. Werner

## Quantum Memory Channels

Quantum channels represent an evolution of an open quantum system with respect to some environment. Quantum memory channels are quantum operations, whose action depends on preceding uses. A central property of memorychannels is the so called “forgetfulness”, which means that the influence of foregoing uses vanishes with an increasing number of channel uses. We are interested in the following question: Given an unknown memory channel. Under which assumptions is it possible, to determine the action of the channel on arbitrary input-states. What would be the protocol for such a tomography.

Contact Person: Prof. Dr. Reinhard F. Werner

## Effective Theories: Renormalization and Decoherence

Our ability to probe the real worlds is always limited by experimental constraints such as the precision of our instruments, or our inability to fully control a system’s environment. It is remarkable that the resulting imperfect data nevertheless contains strong regularities which can be understood in terms of effective laws.

The aim of this project is to produce a systematic formalism for understanding the relationship between various effective descriptions of the same physical system. Our approach naturally uses the language and tools developed in the context of quantum information theory, given that the effective theories describe physical systems under limited knowledge. We focus on two distinct types of experimental limitations: one concerns our inability to resolve small scales, leading to the theory of renormalization, and the other concerns our inability to control boundary conditions, leading to the theory of decoherence.

Decoherence is responsible for the classical appearance of our quantum world. It is caused by the fact that information about the state of a system tends to spread redundantly through its environment due to small uncontrolled interactions. A classical observer indirectly perceives the state of the system by collecting part of the information already contained in the environment. This information is effectively classical due to its redundancy (quantum information cannot be copied). One of our goal is to model the emergence of the classical phase-space structure and dynamics, so as to understand how to best infer the structure of an unknown quantum theory underlying a known classical one, a process known as quantization.

The renormalization group (RG) describes the relationship between effective theories describing physics at different observational scales. It provides the proper language in which interacting quantum field theories are to be understood (such as the standard model of particle physics). Our aim is to understand how the RG relates to our inability to resolve small length scales, and to quantify the resulting loss of information about the true microscopic theory. A parallel project concerns the development of numerical algorithms able to derive large-scale effective theories, in a manner similar to the multiscale entanglement renormalization ansatz (MERA).

Contact Person: Dr. Cedric Beny

## Quantum Shannon Theory

The field of quantum Shannon theory is dedicated to the generalisation and extension of concepts owed to C. E. Shannon on the mathematical foundations of classical information processing and communication to the quantum world. Obtaining tight bounds on the rate at which information can be transmitted fatihfully over quantum channels is of paricular interest. However, unlike classical channels, there is a variety of different types of quantum channel capacities that correspond to the type of information the user wants to communicate, such as classical or quantum information. This diversity of information communication together with the possibility of using entanglement as a resource makes the analysis of quantum channel capacities a challenge. Current projects include questioning the optimality of regularised coherent information as a measure of the quantum communcation capacity for certain classes of quantum channels, and the investigation of the information-carrying capacities of finite-size systems, as opposed to Shannon’s infinite-size limit, by utilizing the recently developed one-shot quantum information techniques.

Contact Person: Prof. Dr. Reinhard F. Werner, Dr. David Reeb

## Mathematical Theory of Quantum Operations and Correlations

For quantum systems, the space of allowed states is much more complex than for classical systems, which is why quantum protocols can be more powerful than classical ones. The mathematical reasons for this complexity are the non-commutativity of the underlying algebras and the tensor product structure behind the composition of individual quantum systems. The goal of this research is to mathematically understand the structure of the quantum mechanical states and operations, and to use these results to investigate the power of quantum protocols, as has for example been achieved in witnessing entanglement and in bounding quantum distillation and communication capacities. Current projects include the investigation of tensor powers of positive maps, and of the monotonicity of distance measures on quantum state space. An essential tool in this field is the use of convex geometry and operator algebras along with recent semidefinite approximation hierarchies for polynomial optimization problems.

Contact Person: Dr. David Reeb

## From Quantum Spin Systems to Anyons

Anyons are (quasi)particles or excitations that behave differently from the more familiar bosons or fermions. In some cases this behaviour can be exploited to do quantum computations. This, in a nutshell, is topological quantum computing. In this project we are interested in the mathematical description of quantum spin systems which have anyonic excitations, for example Kitaev’s quantum double models. From this mathematical description one can recover all relevant properties of these excitations. The ultimate aim is to develop a unified framework to describe these excitations, and to study the characterizing properties of such systems from an operator algebraic point of view.

Contact Person: Prof. Dr. Reinhard F. Werner

## Quantum Topological Order and Topological Phases

Beside the well-understood phase transitions described by Landauer’s theory of symmetry breaking, quantum spin systems on a lattice can exhibit phase transitions that cannot be described by local order parameters. The general understanding is that the phases involved in such transitions should be stable agains local perturbations of the system and just depend on global aspects of the system like the topology of the manifold in which the lattice is embedded. However there is still no general framework which treats this so-called topological order in a unified manner. For instance there are different notions of topological phases each suitable for the class of models under consideration. We are interested in the relation between these different notions and properties of such systems especially also in the thermodynamic limit.

Contact Person: Leander Fiedler

## Quantum Complexity Theory

Quantum complexity theory categorizes computational problems into complexity classes which are defined by the computational resources needed for problem solving. In contrast to classical complexity classes which are mainly based on Turing machines of different computational power, quantum complexity classes are formulated by protocols involving quantum circuits.

So far research has brought up a vast variety of quantum complexity classes, but the relationships between them are still widely unexplored. Advances on this topic are made by finding alternative definitions for complexity classes, proving the membership of important problems and developing the tools and subroutines that allow for the reformulation of protocols.

Contact Person: Friederike Dziemba

## Unbounded Generators of Quantum Dynamical Semigroups

Quantum dynamical semigroups (QDS) describe the evolution of quantum states in open quantum processes. From the mathematical point of view, a QDS on a Hilbert space is a strongly continuous one-parameter semigroup of completely positive operators on the trace class. Almost all known QDS in the physical literature have generators of so-called GKS-Lindblad form. Thus the question now arises of whether every generator of QDS has this form. For a finite-dimensional Hilbert space the answer is in the affirmative. In the case of an infinite-dimensional Hilbert space the problem is more complicated since a generator can be unbounded, i.e. not defined on the whole trace class. We may consider a generator with the GKS-Lindblad form on some ket-bra domain, where the set of ket-vectors is dense in a Hilbert space. As a consequence, the extensions of such a generator may produce more then one QDS under some conditions. It is known that among these QDS there exists a minimal semigroup in the sense of completely positivity. The minimal QDS are sometimes called standard. So we can restate the above question: Is every QDS a standard? It turns out that this is not true. We investigate the mechanism of non-standardness and the physical nature behind it, and aim to construct wide classes of non-standard QDS.

Contact Person: Prof. Dr. Reinhard F. Werner, Dr. David Reeb, Dr. Timur Tlyachev

## Quantum Programming Languages

Quantum Programming Languages (QPLs) are formal systems which serve as a means to formulate quantum algorithms and communication processes, in particular quantum cryptographic protocols, in a more accurate way than is possible with verbatim texts or informal pseudocode. The goal is to transfer and extend concepts of classical programming to quantum programming and quantum communication. In order to experiment with a QPL the language system should be installed on top of a classical simulator. Therefore, in addition to its close connection to quantum algorithms, research on QPLs is also closely related to the subject of classical simulatability of quantum systems. For details and basic references look at this survey.

Contact Person: Prof. Dr. Roland Rüdiger