Statistical Topics and Stochastic Models for Dependent Data with Applications

4Estimation of Piecewise-deterministic Trajectories in a Quantum Optics Scenario

The manipulation of individual copies of quantum systems is one of the most groundbreaking experimental discoveries in the field of quantum physics. On both an experimental and a theoretical level, it has been shown that the dynamics of a single copy of an open quantum system is a trajectory of a piecewise-deterministic process (PDP). To the best of our knowledge, this application field has not been explored by the literature in applied mathematics, from both probabilistic and statistical perspectives. The objective of this chapter is to provide a self-contained presentation of this kind of model, as well as its specificities in terms of observations scheme of the system, and a first attempt to deal with a statistical issue that arises in the quantum world.

4.1. Introduction

4.1.1. The postulates of quantum mechanics

Quantum mechanics is one of the most successful theoretical frameworks in physics, its predictions having been tested and verified with remarkable precision in countless scenarios. Roughly speaking, and with the due exceptions, quantum mechanics deals with phenomena happening at the subatomic, atomic and molecular scales, and with how these systems interact with one another. Several new and counterintuitive phenomena arise at these scales, such as non-locality (Brunner et al. 2014), non-determinism (Bera et al. 2017), wave-particle duality (Wheeler and Zurek 1986), quantum entanglement ...