Continuous Variable Quantum Advantages and Applications in Quantum Optics

This thesis focuses on three main questions in the continuous variable and optical settings: where does a quantum advantage, that is, the ability of quantum machines to outperform classical machines, come from? How to ensure the proper functioning of a quantum machine? What advantages can be gained in practice from the use of quantum information? Quantum advantage in continuous variable comes in particular from the use of so-called non-Gaussian quantum states. We introduce the stellar formalism to characterize these states. We then study the transition from classically simulable models to models which are universal for quantum computing. We show that quantum computational supremacy, the dramatic speedup of quantum computers over their classical counterparts, may be realised with non-Gaussian states and Gaussian measurements. Quantum certification denotes the methods seeking to verify the correct functioning of a quantum machine. We consider certification of quantum states in continuous variable, introducing several protocols according to the assumptions made on the tested state. We develop efficient methods for the verification of a large class of multimode quantum states, including the output states of the Boson Sampling model, enabling the experimental verification of quantum supremacy with photonic quantum computing. We give several new examples of practical applications of quantum information in linear quantum optics. Generalising the swap test, we highlight a connection between the ability to distinguish two quantum states and the ability to perform universal programmable quantum measurements, for which we give various implementations in linear optics, based on the use of single photons or coherent states. Finally, we obtain, thanks to linear optics, the first implementation of a quantum protocol for weak coin flipping, a building block for many cryptographic applications.