Wigner entropy below vacuum: physical counterexamples and stability limits
Abstract
Positive Wigner functions can have less Shannon entropy than the vacuum, even arbitrarily close to the vacuum state. We construct physical counterexamples and identify the competition that controls their entropy: the relative entropy to the vacuum phase-space density can exceed twice the mean photon number. A two-level family exhibits a finite window in which coherence lowers entropy while preserving global Wigner positivity. A complementary construction repairs a remote negative tail with an exponentially small thermal admixture; an explicit mixing weight of 2×10−21 preserves a rigorously established entropy decrease. Optimisation over all one-mode Wigner-nonnegative states gives the sharp low-energy scale Eγ/[ln(1/E)]β, with γ≃0.7412 and β≃0.5861. Because γ<1, tensor products can have vanishing total energy and trace distance from vacuum while their entropy deficit diverges. We derive the exact half-transmission threshold for universal Shannon-entropy recovery and connect it to companion results showing residual non-Gaussian structure. The counterexamples reveal distinct controls on entropy: coherence sets the local descent, mode number amplifies it, and loss restores the vacuum bound.
Related
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BibTeX
@misc{he2026wigner,
title = {{Wigner entropy below vacuum: physical counterexamples and stability limits}},
author = {Zixuan He},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13312},
eprint = {2609.13312},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13312},
}
RIS
TY - UNPB
TI - Wigner entropy below vacuum: physical counterexamples and stability limits
AU - Zixuan He
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.13312
UR - https://arxiv.org/abs/2609.13312
AB - Positive Wigner functions can have less Shannon entropy than the vacuum, even arbitrarily close to the vacuum state. We construct physical counterexamples and identify the competition that controls their entropy: the relative entropy to the vacuum phase-space density can exceed twice the mean photon number. A two-level family exhibits a finite window in which coherence lowers entropy while preserving global Wigner positivity. A complementary construction repairs a remote negative tail with an exponentially small thermal admixture; an explicit mixing weight of 2×10^−21 preserves a rigorously established entropy decrease. Optimisation over all one-mode Wigner-nonnegative states gives the sharp low-energy scale E^γ/[ln(1/E)]^β, with γ≃0.7412 and β≃0.5861. Because γ<1, tensor products can have vanishing total energy and trace distance from vacuum while their entropy deficit diverges. We derive the exact half-transmission threshold for universal Shannon-entropy recovery and connect it to companion results showing residual non-Gaussian structure. The counterexamples reveal distinct controls on entropy: coherence sets the local descent, mode number amplifies it, and loss restores the vacuum bound.
ER -