ADVANCES IN QUANTUM PARAMETER ESTIMATION AND OTHER TOPICS

The first half of this thesis deals with the problem of parameter estimation in a quantum system. In quantum parameter estimation theory, the quantum Fisher information is usually considered as the ultimate precision limit, beyond which no further improvement is possible due to the inherent stochasticity of quantum measurements. On the other hand, in this thesis we will show that a better precision is achievable than predicted by a quantum Fisher information analysis. This is true if some regularity assumptions about the underlying quantum parametric model are relaxed. In such situations, the quantum Fisher information does not completely capture the best possible performance of quantum measurements, and a different approach must be followed. In the second part of the thesis, we will focus on some applications of orthogonal array theory to two notable quantum information problems: the problem of multipartite entanglement classification and the quantum marginal problem. Introduced by Rao in 1947, orthogonal arrays have been usefully applied to different fields, from cryptography and coding theory to the statistical design of experiments, software testing and quality control. Remarkably, orthogonal arrays have also found application in quantum information and, in particular, in the study of quantum entanglement. We will employ tools from orthogonal array theory to study a toy version of the multipartite entanglement classification problem. Finally, we will show how orthogonal arrays can be employed to build constructive solutions to low-dimensional quantum marginal problems.