Mathematical structures in quantum information theory: tensors, correlations and state estimation

This thesis is devoted to different aspects of quantum information science as well as the closely related topic of multilinear algebra. We present several results in the fields of quantum measurement theory, the theory of bipartite Bell inequalities, the activation of nonlocal quantum correlations, the identification of resourceful multipartite quantum states as well as the characterization of the eigenstructure of highly symmetric real-valued tensors. Gearing towards applications of quantum technology, we develop scalable methods that allow for the simultaneous prediction of many observables of a multi-qubit system.
First, we investigate how quantum measurements with many outcomes can be simulated by measurements with fewer outcomes. In particular, we present a minimal scheme that certifies this simulability based on correlations. Further, we analyze this minimal scheme with respect to noise robustness. Afterwards, we pick up the quantum measurement problem and discuss whether the realization of a partially observed measurement is compatible with the universality of the unitary time evolution.
Second, we introduce a family of bipartite Bell inequalities and associated quantum correlations that allow for an extremely low detection efficiency as well as high robustness to noise. Further, we discuss how these inequalities can be optimized by means of symmetry considerations. Subsequently, we examine the phenomenon of activation of quantum correlations. We develop methods that allow for rigorous statements on the statistical significance of such an experimental demonstration. These methods include the construction of a suitable confidence polytope as well as an algorithm to determine the correlation class of a quantum state.
Third, we present an algorithm that allows for finding maximally resourceful multipartite quantum states. We provide a rigorous proof of convergence and apply it to multiple quantifiers of quantum resources, e.g., the geometric measure of entanglement. This reveals an interesting connection to so-called absolutely maximally entangled states.
Then, we discuss the eigenstructure of certain highly symmetric tensors, whose construction is based on simplex frames. We provide a full characterization of the eigenvectors for an arbitrary number of parties and local dimension two. Further, we discuss whether the eigenvectors can be obtained by the power iteration method.
The last part of this thesis is concerned with scalable methods that allow for simultaneously predicting many expectation values of a multi-qubit system with high accuracy.
For this purpose, we extend the technique of classical shadows, originally based on projective measurements, to generalized measurements. This yields a simple formulation, allowing for the incorporation of symmetries and the possibility of optimizing the measurement directions towards a set of targeted observables. Moreover, we combine classical shadows with error mitigation techniques, rendering the incorporation of preparation errors in the estimation of many expectation values possible.