In classical information theory the major conceptual step, done by C. Shannon, was to abstract the intuitive notion of a communication channel, which lead to the development of classical information theory. With the development of quantum physics and the new possibilities it provides for communication and computation the mathematical concept of a quantum channel, describing general input-output relations was developed in the same way. Quantum channels are therefore the main building blocks of quantum information theory and unlike their classical counterpart their basic properties are not yet understood. Of main concern for quantum information theory are the properties of quantum channels under sequential and parallel composition. Such compositions arise for example in quantum algorithms, when the same channel is applied multiple times, or in quantum communication tasks, when a channel acts on different parts of an encoded state.

In physical experiments and applications of quantum information theory it is an important question, how close the current state of the physical system is to a given target state. This is of particular importance for state preparation, but also for the adiabatic implementation of quantum algorithms. Using Wiener‐algebra functional calculus, we derived improved convergence bounds for the sequential application of a quantum channel, which improve even their classical counterparts. For the first time such methods were used to derive such a bound.

For quantum communication the notion of a quantum capacity, which quantifies the maximal achievable rate for information transmission in the limit of arbitrarily many parallel channel uses assisted by certain resources, is of great importance. The study of these quantities often requires different tools, than their classical counterparts, and another goal of this project is devoted to their study.

How many uses of some given quantum channel do we need to simulate the behaviour of another one? Channel simulation protocols exactly answer these sorts of questions, having impact on the design of communication protocols and the study of their performance. Of particular importance is the quantification of different resources needed to perform simulation tasks. This allows us to characterize the power of quantum channels with respect to communication protocols where the availability of certain resources is limited. For instance, since long‐distance entanglement is experimentally hard to implement, it makes sense to ask for the amount of entanglement necessary to implement a quantum channel. We answered this and related channel simulation questions in great detail, allowing for precise characterizations of quantum channels and new applications to quantum cryptographic protocols.

Quantum channels are abstract objects, modeling input-output relations independent from the physical implementation. Reversely, tools developed for the study of quantum channels should be applicable to concrete physical questions. A particular example where this thinking turned out to be quite fruitful was the study of one-dimensional arrays of quantum particles, also known as spin chains. There, it was an open question for quite some time whether the knowledge of the strength of correlation between distant particles allows for a quantification of the amount of entanglement present in the system. Using tools from the study of quantum channels, this question could be answered to the positive.

We introduced a new line of research by identifying decision problems in quantum information which are algorithmically undecidable, in the sense, that there is no algorithm to decide a general instance of the problem. By reduction to the Post's correspondence problem, we identified certain decision problems arising in quantum control theory to be undecidable, which shows that there cannot exist a general solution to those questions, which one might hope to find.

Start date: (36 months)

Funding support: 600 000 €

### Project partners

- Technical University of Munich (TUM) - Department of Mathematics (Germany)
- ETH Zurich - Department of Physics (Switzerland)
- UCM - Department of Mathematical Analysis (Spain)