Quantum tomography on Schwartz space

Abstract

In this work we show that standard results in quantum tomography, which are typically proven in the context of Hilbert spaces, can analogously be established in the theory of Schwartz spaces. In particular, we will see that quantum states are equivalently characterized by their Wigner function, the characteristic of their Wigner function, and the probability distributions for a family of linear combinations of position and momentum. Physically, the latter can be understood as the evolution of its probability distribution in position over time under, for example, free or harmonic evolution. The result is obtained with techniques that are more accessible to a physicist audience, and shows that, at least for tomography results, the full structure of Hilbert spaces is not necessary.

The paper asks whether standard continuous-variable quantum tomography can be formulated entirely on Schwartz space $\mathcal S(\mathbb R)$, without invoking the Hilbert-space completion $L^2$. It considers pure states represented by normalized smooth, rapidly decaying wavefunctions and establishes their characterization, up to global phase, by Wigner functions, Weyl–Heisenberg characteristic functions, and complete families of quadrature probability distributions. The reconstruction proceeds through explicit transformations. Fourier inversion recovers the rank-one kernel $K_\psi(x,y)=\psi(x)\overline{\psi(y)}$ from the Wigner function; fixing the phase at any point where the wavefunction is nonzero then recovers $\psi$. The characteristic function $\chi_\psi(a,b)=\langle\psi|e^{i(aX+bP)/\hbar}|\psi\rangle$ is the two-dimensional Fourier transform of the Wigner function. For quadratures $X\cos\alpha+P\sin\alpha$, the Fourier transform of the measured probability density gives $\chi_\psi(s\cos\alpha,s\sin\alpha)$. Consequently, distributions for all $\alpha\in[0,\pi)$ determine the state. The argument uses Fourier-transform closure and regularity properties of Schwartz functions. The quadrature family is interpreted through position measurements during harmonic evolution, covering the required angles in finite time, or free evolution, whose full angular coverage requires unbounded times. The paper explicitly presents these as Schwartz-space versions of existing tomography results rather than new informational-completeness claims. Its scope is pure Schwartz wavefunctions, not arbitrary mixed or $L^2$ states, and it gives no finite-sample or noise-stability guarantees. It also acknowledges that positional degrees of freedom alone do not provide a general foundation for quantum theory.