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

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 $\rho$ with uniform positive spectrum (i.e. a state proportional to a projector) w.r.t. the fidelity deficit: the inequality $S(\rho )-S(\sigma )\le C_{\rho }(1-F(\rho ,\sigma ))$ valid for any state $\sigma$, where $C_{\rho }$ is a constant depending on the rank of $\rho$. The second result is a sharp Lipschitz lower semicontinuity bound for the Entanglement of Formation at any pure state $\rho$ with uniform positive spectrum of marginal states w.r.t. the fidelity deficit: the inequality $E_F(\rho )-E_F(\sigma )\le C_{\rho }(1-\mathrm{T}\mathrm{r}\rho \sigma )$ valid for any state $\sigma$, where $C_{\rho }$ is a constant depending on the Schmidt rank of $\rho$. In both cases the optimal constant $C_{\rho }$ is equal to the optimal constant $K_d$ in the log-Sobolev inequality for the complete graph with $d$ vertices: in the first case $d=\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\rho$, in the second one $d=\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}{\rho }_A=\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}{\rho }_B$. The authors are grateful to GPT 5.6 for valuable discussion and technical help in preparing this note.