Abstract
The study of positivity properties of trace class operators is essential in the theory of quantum mechanical density matrices; the latter describe the “mixed states” of quantum mechanics and are essential in information theory. While a general theory for these positivity results is still lacking, we present some new results we have recently obtained and which generalize and extend the well-known conditions given in the 1970s by Kastler, Loupias, and Miracle-Sole, generalizing Bochner’s theorem on the Fourier transform of a probability measure. The tools we use are the theory of pseudodifferential operators, symplectic geometry, and Gabor frame theory. We also speculate about some consequences of a possibly varying Planck’s constant for the early universe.
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Acknowledgement
Maurice de Gosson has been financed by the grant P27773-N23 of the Austrian Research Foundation FWF.
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Nicola, E., de Gosson, M., Nicola, F. (2017). Quantum Harmonic Analysis and the Positivity of Trace Class Operators; Applications to Quantum Mechanics. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2017. Lecture Notes in Computer Science(), vol 10589. Springer, Cham. https://doi.org/10.1007/978-3-319-68445-1_46
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DOI: https://doi.org/10.1007/978-3-319-68445-1_46
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