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.
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BibTeX
@misc{thrien2026quantum,
title = {{Quantum tomography on Schwartz space}},
author = {Tobias Thrien and Gabriele Carcassi and David Carfì and Christine A. Aidala},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03521},
eprint = {2611.03521},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03521},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Quantum tomography on Schwartz space
AU - Tobias Thrien
AU - Gabriele Carcassi
AU - David Carfì
AU - Christine A. Aidala
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03521
UR - https://arxiv.org/abs/2611.03521
AB - 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.
ER -