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

- Foundations of Quantum Statistical Mechanics
- From Quantum Spin Systems to Anyons
- Measurement based quantum computing and contextuality
- Quantum Complexity Theory
- Quantum Memory Channels
- Quantum Programming Languages
- Quantum Shannon Theory
- Quantum Topological Order and Topological Phases
- Quantum Walks and Quantum Cellular Automata
- Tensor networks for quantum field theories
- Unbounded Generators of Quantum Dynamical Semigroups

## Foundations of Quantum Statistical Mechanics

Statistical physics is a vast and beautiful branch of physics. It provides us with a framework for understanding physical processes involving far too many degrees of freedom to simply write down an equation of motion and even approximately solve it. The basis of statistical physics is the dynamical laws of the constituent particles (including whether they are classical or quantum, fermions or bosons), as well as some additional general principles. But these principles are somewhat ad hoc. Instead, we ask whether we can do better: can we justify these principles or even remove them entirely? It would be fascinating if we could derive quantum statistical physics from quantum theory and some reasonable assumptions, such as locality of the Hamiltonian, for example.

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## 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

## Measurement based quantum computing and contextuality

Contextuality is one of the key attributes of quantum mechanics and distinguishes quantum mechanics from classical physics. We can understand it, by considering whether it is possible to assign “pre-existing” outcomes to measurements of quantum observables. If this is possible, we can describe the situation with classical mechanics. We assume a model with two different sets C1 and C2 of compatible, i.e. commuting, observables containing a given observable x. It is reasonable to require that the outcome value o attached to the observable x is a property of x alone and thus agrees in C1 and C2. But it turns out not to be the case for all settings in quantum mechanics. We are interested in contextuality because we can see it as a resource for quantum computation. In measurement based quantum computing (MBQC) no quantum speedup can occur without contextuality. MBQC is an alternative to computing with quantum circuit models in which the computation is driven by using only local measurements as computational steps as opposed to unitary gates. Information is processed by a sequence of single-qubit measurements on an entangled multi-qubit resource state, such as a cluster state or graph state.

Contact Person: Kerstin Beer

## 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.

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## 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

## 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

## 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

## 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: Prof. Dr. Reinhard F. Werner

## 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

## Tensor networks for quantum field theories

Tensor networks are a set of powerful variational classes extensively used to model many-body systems where entanglement plays an important role. While there are some impressive theorems concerning how good this approximation classes actually approximate physical values, there are a lot of gaps remaining. How does different tensor networks work for quantum field theories? How should the constituting tensors look like for a particular quantum field theory? What if there is a disorder in system or if the system is open?

Contact Person: Dmytro Bondarenko

## 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