A new paper shows that standard results in quantum tomography, usually proved in the framework of Hilbert spaces, can be established analogously within the theory of Schwartz spaces. In particular, the authors show that a quantum state is equivalently characterized by its Wigner function, by the characteristic of its Wigner function, and by the probability distributions of a family of linear combinations of position and momentum.

Physically, the latter can be interpreted as the time evolution of the position probability distribution under a given dynamics, for example free or harmonic evolution. That is, measuring how the probability of finding the particle at different positions redistributes over time contains, in principle, the same information about the state as the full Wigner function.

The result is obtained with techniques more accessible to a physics audience, according to the authors, and shows that, at least for tomography results, the full structure of Hilbert spaces is not necessary. Schwartz spaces, commonly used in functional analysis to treat smoothly decaying functions, are enough to support these equivalences.

The conclusion opens a route to reformulate part of quantum information theory in a lighter analytic language, without giving up known results. It remains to be seen how far other tools of quantum theory — for example those related to open dynamics or measurement — admit a similar reduction in mathematical structure.