Wigner-positive quantum states can have lower entropy than the vacuum

Abstract

The Wigner entropy conjecture posits that pure Gaussian states minimize the Shannon entropy of non-negative Wigner functions, known as the Wigner entropy. We show that mixing a specific pure Wigner-negative state with another suitably chosen quantum state can restore Wigner positivity while retaining a Wigner entropy that is slightly - but strictly - below the vacuum entropy. As a consequence, we disprove the Wigner entropy conjecture, as well as the stronger Wigner majorization conjecture, by deriving simple analytical counterexamples with violations of the entropy bound no larger than 10⁻³. Such states with sub-vacuum Wigner entropy can be found arbitrarily close to the vacuum and can also exhibit sub-vacuum Wigner-Rényi α-entropy for 0<α<2. We identify the physical mechanism behind the existence of such states, which arises from a subtle interplay between the uncertainty principle and non-Gaussianity. The key idea of this paper was developed with the help of AI tools, building on the characterization of extreme non-negative Wigner functions.

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BibTeX RIS

BibTeX

@misc{herstraeten2026wignerpositive,
  title = {{Wigner-positive quantum states can have lower entropy than the vacuum}},
  author = {Zacharie Van Herstraeten and Nicolas J. Cerf and Ulysse Chabaud},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.24670},
  eprint = {2609.24670},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.24670},
}

RIS

TY  - UNPB
TI  - Wigner-positive quantum states can have lower entropy than the vacuum
AU  - Zacharie Van Herstraeten
AU  - Nicolas J. Cerf
AU  - Ulysse Chabaud
PY  - 2026
DA  - 2026/09/21
PB  - arXiv
DO  - 10.48550/arxiv.2609.24670
UR  - https://arxiv.org/abs/2609.24670
AB  - The Wigner entropy conjecture posits that pure Gaussian states minimize the Shannon entropy of non-negative Wigner functions, known as the Wigner entropy. We show that mixing a specific pure Wigner-negative state with another suitably chosen quantum state can restore Wigner positivity while retaining a Wigner entropy that is slightly - but strictly - below the vacuum entropy. As a consequence, we disprove the Wigner entropy conjecture, as well as the stronger Wigner majorization conjecture, by deriving simple analytical counterexamples with violations of the entropy bound no larger than 10^−3. Such states with sub-vacuum Wigner entropy can be found arbitrarily close to the vacuum and can also exhibit sub-vacuum Wigner-Rényi α-entropy for 0<α<2. We identify the physical mechanism behind the existence of such states, which arises from a subtle interplay between the uncertainty principle and non-Gaussianity. The key idea of this paper was developed with the help of AI tools, building on the characterization of extreme non-negative Wigner functions.
ER  -

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