Almost sure minimality of the set of experiments with classical Kirkwood-Dirac representations

Christopher Langrenez, Wilfred Salmon, Stephan De Bièvre, Jonathan J. Thio, 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

PDF Download¹ © 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.

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. Downloads: PDF, TeX Source
• Thu, 2 Jul 2026 11:36:37 UTC: https://arxiv.org/abs/2405.17557v2. Downloads: PDF, TeX Source

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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

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.

— Chris Long (@Chris_K_Long45) June 16, 2026

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://t.co/llHGFg2D0z.

— Chris Long (@Chris_K_Long45) June 16, 2026

We have just updated the arXiv with the accepted Physical Review A version so there is an open access peer-reviewed version: https://t.co/fTScSTMM8b

— Chris Long (@Chris_K_Long45) July 3, 2026

Footnotes

1. Licence of this file: Third-party all rights reserved; see its entry on the licences page for what each licence covers and the copyright holders. ↩