Log-Sobolev inequality, von Neumann entropy and Entanglement of Formation

• URL: http://arxiv.org/abs/2609.12667v3
• Date: Tue, 15 Sep 2026 17:25:07 GMT
• Title: Log-Sobolev inequality, von Neumann entropy and Entanglement of Formation
• Abstract summary: We present two results derived from the log-Sobolev inequality for the uniform measure on a complete graph.<n>The results concern the von Neumann entropy and the Entanglement of Formation of a state of finite and infinite-dimensional quantum systems.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 $\,S(ρ)-S(σ)\leq C_ρ(1-F(ρ,σ))\,$ 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 $\,E_F(ρ)-E_F(σ)\leq C_ρ(1-\mathrm{Tr}ρσ)\,$ 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 $d=\mathrm{rank}ρ$, in the second one $d=\mathrm{rank}ρ_A=\mathrm{rank}ρ_B$. The authors are grateful to GPT 5.6 for valuable discussion and technical help in preparing this note.

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