Quantum Optics, as the child of Optics and Quantum Mechanics, has inherited a double linearity: that of Maxwell equations, which use optical modes as a basis of solutions, and that of the Schrödinger equation, which uses quantum state bases. Considering these two bases on an equal footing and tailoring quantum fields not only in given modes, but also optimizing the spatio-temporal shapes of the modes in which the state is defined, opens wide perspectives for treating complex quantum states. Our aim is to explore and characterize theoretically the quantum states that span on many optical modes (from several tens to several thousands) and many Hilbert space basis states, to unravel their intrinsic properties and to find optimized witnesses of different properties such as multi-entanglement. We also investigate the use of optimized multimode states for pushing the quantum limits of the multiplexed estimation of physical parameters and for increasing the channel capacity of optical communications.

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