Kybernetika 46 no. 6, 1061-1068, 2010

Quantum Bochner theorems and incompatible observables

Robin L. Hudson


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.


Bochner's Theorem, multiplier-nonnegative-definiteness, Wigner quasidensities, Pauli matrices


60B15, 81S30


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