Randomized measurements as a tool in quantum information processing

With limited control over quantum systems, how can we extract information about them and characterize their properties? This thesis addresses this question from several perspectives. After explaining the basic concepts of quantum information, we investigate various methods for the analysis of quantum states.
In the first part of the thesis, we develop abstract theories of the so-called randomized measurements. The idea is to randomly rotate measurement directions and examine quantum correlations based on the statistical moments of the resulting probability distribution. This method eliminates the need for calibration of measurement devices or a common reference frame between spatially-separated parties. First, we present several criteria for detecting multipartite entanglement and bound entanglement from randomized measurements. Next, we propose hierarchies of multiqubit entanglement criteria and analyze the statistical significance using large deviation bounds. Finally, we provide the complete characterization of two-qubit entanglement from randomized measurements.
In the second part, we explore several applications of randomized measurements. First, we probe the dimensionality of entanglement from work fluctuations in randomized two-point energy measurement protocols in thermodynamic systems. Second, we advance reference-frame-independent quantum metrology to estimate precision under nonlinear Hamiltonian dynamics. Third, we introduce different approaches to characterize spin squeezing by establishing the scheme of collective randomized measurements. Lastly, we advance methods to certify entanglement based on the moments of the partially transposed density matrix.
In the last part, we deepen the understanding of quantum correlations from geometrical viewpoints. First, we study various constraints on three-qubit states in terms of two-body correlations. Second, we offer quantum speed limits describing the divergence of a perturbed open system from its unperturbed trajectory. Finally, we provide a formulation to discuss the sensitivity of multiparticle entanglement under the classicalization of one particle.