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Quantization of Gaussians
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Our paper is devoted to the oscillator semigroup, which can be defined as the set of operators whose kernels are centered Gaussian. Equivalently, they can be defined as the the Weyl quantization of centered Gaussians. We use the Weyl symbol as the main parametrization of this semigroup. We derive formulas for the tracial and operator norm of the Weyl quantization of Gaussians. We identify the subset of Gaussians, which we call quantum degenerate, where these norms have a singularity.
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From classical probability densities to quantum states: quantization of Gaussians for arbitrary orderings
Under s-ordered quantization, a Gaussian maps to a valid quantum state for lambda <= (1+s)^{-1}, and for antinormal ordering s=-1 even the delta function becomes the vacuum state.
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