Geodesic
Quantum Bochner theorems and incompatible observables
Kybernetika, Tome 46 (2010) no. 6, pp. 1061-1068

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A quantum version of Bochner's theorem characterising Fourier transforms of probability measures on locally compact Abelian groups gives a characterisation of the Fourier transforms of Wigner quasi-joint distributions of position and momentum. An analogous quantum Bochner theorem characterises quasi-joint distributions of components of spin. In both cases quantum states in which a true distribution exists are characterised by the intersection of two convex sets. This may be described explicitly in the spin case as the intersection of the Bloch sphere with a regular tetrahedron whose edges touch the sphere.
Classification : 60B15, 81S30
Keywords: Bochner's Theorem; multiplier-nonnegative-definiteness; Wigner quasidensities; Pauli matrices 
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     author = {Hudson, Robin L.},
     title = {Quantum {Bochner} theorems and incompatible observables},
     journal = {Kybernetika},
     pages = {1061--1068},
     publisher = {mathdoc},
     volume = {46},
     number = {6},
     year = {2010},
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     zbl = {1219.81175},
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     url = {http://geodesic.mathdoc.fr/item/KYB_2010__46_6_a10/}
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Hudson, Robin L. Quantum Bochner theorems and incompatible observables. Kybernetika, Tome 46 (2010) no. 6, pp. 1061-1068. http://geodesic.mathdoc.fr/item/KYB_2010__46_6_a10/