---
tags: KD-Distribution, Theory
sort-date: 2026-06-16
description: We show that if two d-dimensional observables are picked at random, the set of classical states of the resulting KD distribution is a minimal polytope of dimension 2(d−1).
---
Christopher Langrenez,
Wilfred Salmon,
Stephan De Bièvre,
Jonathan J. Thio,
[[/index|Christopher K. Long]],
and David R. M. Arvidsson-Shukur
*All authors contributed equally. The order of authors was randomized.*
Published: 16 June 2026
DOI: [10.1103/v7z4-qsz8](https://doi.org/10.1103/v7z4-qsz8)
[[PDFs/Almost sure minimality of the set of experiments with classical Kirkwood-Dirac representations.pdf|PDF Download]] © 2026 American Physical Society.
# Abstract
> A central problem in quantum information is determining quantum-classical boundaries. In the quasiprobability framework, a state is called classical if it is represented by a quasiprobability distribution that is positive, and thus a probability distribution. In recent years, the Kirkwood-Dirac (KD) distributions have gained much interest due to their numerous applications in modern quantum-information research. A particular advantage of the KD distributions is that they can be defined with respect to arbitrary observables. Here, we show that if two $d$-dimensional observables are picked at random, the set of classical (positive) states of the resulting KD distribution is a minimal polytope of dimension $2(d−1)$ with $2d$ explicitly known vertices. This implies minimality of the sets of KD-real observables, of KD-positive measurement elements, and of KD-positivity-preserving unitaries. We show how these results have implications on robust observations of nonclassical phenomena, on classical simulations of quantum circuits, and on foundations of quantum theory.
# Citation
Christopher Langrenez, Wilfred Salmon, Stephan De Bièvre, Jonathan J. Thio, Christopher K. Long, and David R. M. Arvidsson-Shukur. Almost sure minimality of the set of experiments with classical Kirkwood-Dirac representations, *Phys. Rev. A* **113**, 062215 (2026), DOI: [10.1103/v7z4-qsz8](https://doi.org/10.1103/v7z4-qsz8).
## BibTeX
```bibtex
@article{Langrenez2026,
title = {Almost sure minimality of the set of experiments with classical Kirkwood-Dirac representations},
author = {Langrenez, Christopher and Salmon, Wilfred and De Bi\`evre, Stephan and Thio, Jonathan J. and Long, Christopher K. and Arvidsson-Shukur, David R. M.},
journal = {Phys. Rev. A},
volume = {113},
issue = {6},
pages = {062215},
numpages = {11},
year = {2026},
month = {Jun},
publisher = {American Physical Society},
doi = {10.1103/v7z4-qsz8},
url = {https://link.aps.org/doi/10.1103/v7z4-qsz8}
}
```
# Other versions
- Mon, 27 May 2024 18:00:04 UTC: [*https://arxiv.org/abs/2405.17557v1*](https://arxiv.org/abs/2405.17557v1). Downloads: [[PDFs/2405.17557v1.pdf|PDF]], [[TeX_Source/2405.17557v1.tar.gz|TeX Source]]
- Thu, 2 Jul 2026 11:36:37 UTC: [*https://arxiv.org/abs/2405.17557v2*](https://arxiv.org/abs/2405.17557v2). Downloads: [[PDFs/2405.17557v2.pdf|PDF]], [[TeX_Source/2405.17557v2.tar.gz|TeX Source]]
# Analytics
- [Google Scholar](https://scholar.google.com/citations?view_op=view_citation&citation_for_view=GRSIcsEAAAAJ:hqOjcs7Dif8C)
- [SciRate](https://scirate.com/arxiv/2405.17557)
- [INSPIRE-HEP](https://inspirehep.net/literature/2790865)
# Social media posts
## Bluesky
> [!Thread]
> For many quasiprobability distributions, the set of positive (“classical”) states is difficult to characterize. However, for a random Kirkwood–Dirac distribution, we completely characterized not only the positive states, but also the positive measurements and unitaries.
— Christopher K. Long (@christopher-k-long.bsky.social) 16 June 2026 at 17:11
> The generalization from states to a complete fragment of quantum mechanics delayed our publication in PRA. Now, I am very happy with the more fleshed-out picture our article now provides: doi.org/10.1103/v7z4....
[image or embed]
— Christopher K. Long (@christopher-k-long.bsky.social) 16 June 2026 at 17:11
> We have just updated the arXiv with the accepted Physical Review A version so there is an open access peer-reviewed version: arxiv.org/abs/2405.17557
[image or embed]
— Christopher K. Long (@christopher-k-long.bsky.social) 3 July 2026 at 06:06
## LinkedIn
For many quasiprobability distributions, the set of positive (“classical”) states is difficult to characterize. However, for a random Kirkwood–Dirac distribution, Christopher Langrenez, Wilfred Salmon, Stephan De Bievre, JJ Thio, I, and David Arvidsson-Shukur completely characterized not only the positive states, but also the positive measurements and unitaries. The generalization from states to a complete fragment of quantum mechanics delayed our publication in PRA. Now, I am very happy with the more fleshed-out picture our article now provides: https://lnkd.in/eSN-yxKm.
— Christopher K. Long (LinkedIn) 16 June 2026
## X
> [!Thread]
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