Hermite-Gaussian model for quantum states

Abstract In order to characterize quantum states within the context of information geometry, we propose a generalization of the Gaussian model, which we called the Hermite–Gaussian model. We obtain the Fisher–Rao metric and the scalar curvature for this model, and we show its relation with the one-dimensional quantum harmonic oscillator. Using this model we characterize some families of states of the quantum harmonic oscillator. We find that for eigenstates of the Hamiltonian, mixtures of eigenstates and even or odd superpositions of eigenstates, the associated Fisher–Rao metrics – which are relevant in the context of quantum parameter estimation theory – are diagonal. Finally, we consider the action of the amplitude damping channel and we discuss the relationship between the quantum decay and the different geometric indicators.

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