Typical and Atypical structures in quantum theory

Quantum theory, confirmed in numerous sophisticated experiments, is widely believed to describe our world at the micro scale. It is thus legitimate to investigate, which structures are allowed by quantum theory, and which of them can potentially be relevant for developments of quantum technologies.The basic notion of a quantum state a mathematical tool used to compute probabilities, characterizing the outcomes of a quantum measurement is of a primary importance. Furthermore, one analyses quantum maps, which describe how quantum states evolve in time, and quantum supermaps, representing evolution in the space of quantum maps. Assuming that the number of outcomes is finite, all these sets form convex bodies embedded in a real space of a suitable dimension. The case, if the physical system is composed of several subsystems, is of a special interest, as one can analyse correlations and entanglement between subsystems. Main goal of this project is to investigate properties of typical quantum states, maps and supermaps, and to identify distinguished, atypical structures with extreme properties, useful for processing of quantum information. We will search for new constructions of absolutely maximally entangled multipartite states, which imply existence of quantum error correcting codes, novel schemes of mutually unbiased bases and symmetric informationally complete generalized quantum measurements, which offer optimal measurement accuracy. Moreover, we plan to analyze quantum supermaps with distinguished properties, and study, how these structures behave under decoherence, as quantum features become gradually suppressed.To put all these structures of the same footing we are going to use generalizations of the Choi-Jamiokowski isomorphism, which relates quantum maps with quantum states of the extended system, and apply the theory of random matrices to elucidate differences between typical objects with generic features and the atypical ones with desired properties.