Log-Sobolev inequality, von Neumann entropy and Entanglement of Formation
Abstract
We present two results derived from the sharp log-Sobolev inequality for the uniform measure on a complete graph which concern the von Neumann entropy and the Entanglement of Formation of a state of finite and infinite-dimensional quantum systems. The first result is a sharp Lipschitz lower semicontinuity bound for the von Neumann entropy at any mixed state ρ with uniform positive spectrum (i.e. a state proportional to a projector) w.r.t. the fidelity deficit: the inequality S(ρ)−S(σ)≤ Cρ(1−F(ρ,σ)) valid for any state σ, where Cρ is a constant depending on the rank of ρ. The second result is a sharp Lipschitz lower semicontinuity bound for the Entanglement of Formation at any pure state ρ with uniform positive spectrum of marginal states w.r.t. the fidelity deficit: the inequality EF(ρ)−EF(σ)≤ Cρ(1−Trρσ) valid for any state σ, where Cρ is a constant depending on the Schmidt rank of ρ. In both cases the optimal constant Cρ is equal to the optimal constant Kd in the log-Sobolev inequality for the complete graph with d vertices: in the first case d=rankρ, in the second one d=rankρA=rankρB. The authors are grateful to GPT 5.6 for valuable discussion and technical help in preparing this note.
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BibTeX
@misc{holevo2026logsobolev,
title = {{Log-Sobolev inequality, von Neumann entropy and Entanglement of Formation}},
author = {A. S. Holevo and M. E. Shirokov},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12667},
eprint = {2609.12667},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12667},
}
RIS
TY - UNPB
TI - Log-Sobolev inequality, von Neumann entropy and Entanglement of Formation
AU - A. S. Holevo
AU - M. E. Shirokov
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.12667
UR - https://arxiv.org/abs/2609.12667
AB - We present two results derived from the sharp log-Sobolev inequality for the uniform measure on a complete graph which concern the von Neumann entropy and the Entanglement of Formation of a state of finite and infinite-dimensional quantum systems. The first result is a sharp Lipschitz lower semicontinuity bound for the von Neumann entropy at any mixed state ρ with uniform positive spectrum (i.e. a state proportional to a projector) w.r.t. the fidelity deficit: the inequality S(ρ)−S(σ)≤ C_ρ(1−F(ρ,σ)) valid for any state σ, where C_ρ is a constant depending on the rank of ρ. The second result is a sharp Lipschitz lower semicontinuity bound for the Entanglement of Formation at any pure state ρ with uniform positive spectrum of marginal states w.r.t. the fidelity deficit: the inequality E_F(ρ)−E_F(σ)≤ C_ρ(1−Trρσ) valid for any state σ, where C_ρ is a constant depending on the Schmidt rank of ρ. In both cases the optimal constant C_ρ is equal to the optimal constant K_d in the log-Sobolev inequality for the complete graph with d vertices: in the first case d=rankρ, in the second one d=rankρ_A=rankρ_B. The authors are grateful to GPT 5.6 for valuable discussion and technical help in preparing this note.
ER -