Estimation and reconstruction of quantum states in cavity: Fock and Schrödinger cat states One might want to accumulate statistics about the measurement of a complete set of observables, performed on a large number of realizations of the system. The results constrains the system density operator, which is found by solving a set of equations equating the observed statistics with the theoretical ones. This procedure may lead to difficulties; if the data are noisy, it might happen that the direct constrained density operator is found to be non-physical, e.g. with negative eigenvalues. The number of available copies of the state might be small. The data presenting then large fluctuations. Often, the set of measured observables is incomplete, making it unlikely to constrain the parameters defining the state. In these situations, reconstruction of the state is an estimation problem: How can we find the parameters defining the state from the information provided by incomplete measurements, performed on a finite set of copies and suffering limitations from noise? Inspired by classical estimation theory, we will analyze the general method of maximum likelihood and maximum entropy before describing their application in cavity, illustrated by the reconstruction of Schrödinger cats and Fock states of a field. Note: Serge Haroche
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