Quantum probes for minimum length estimation

In this thesis we address the estimation of the minimum length arising from gravitational theories. In particular, we use tools from Quantum Estimation theory to provide bounds on precision and to assess the use of quantum probes to enhance the estimation performances. After a brief introduction to classical estimation theory and to its quantum counterparts, we introduce the concept of minimum length and show how it induces a perturbative term appearing in the Hamiltonian of any quantum system, which is proportional to a parameter depending on the minimum length. We have then systematically studied the effects of this perturbation on different state preparations for several 1-dimensional systems, and evaluated the Quantum Fisher Information in order to find the ultimate bounds to the precision of any estimation procedure. Eventually, we have investigated the role of dimensionality on the precision, studying the 2-dimensional Harmonic oscillator. Our results provide a guideline to design possible future experiments to detect minimal length, and show the potential enhancement achievable using quantum probes.