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

arXiv:2609.12667v1 Announce Type: new 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 $\rho$ with uniform positive spectrum (i.e. a state proportional to a projector) w.r.t. the fidelity deficit: the inequality $\,S(\rho)-S(\sigma)\leq C_\rho(1-F(\rho,\sigma))\,$ 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: the inequality $\,E_F(\rho)-E_F(\sigma)\leq \frac{1}{2}\,C_\rho\|\rho-\sigma\|_1\,$ 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 $d=\mathrm{rank}\rho$, in the second one $d=\mathrm{rank}\rho_A=\mathrm{rank}\rho_B$. The authors are grateful to GPT 5.6 for valuable discussion and technical help in preparing this note.

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 $\rho$ with uniform positive spectrum (i.e. a state proportional to a projector) w.r.t. the fidelity deficit: the inequality $\,S(\rho)-S(\sigma)\leq C_\rho(1-F(\rho,\sigma))\,$ 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: the inequality $\,E_F(\rho)-E_F(\sigma)\leq \frac{1}{2}\,C_\rho\|\rho-\sigma\|_1\,$ 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 $d=\mathrm{rank}\rho$, in the second one $d=\mathrm{rank}\rho_A=\mathrm{rank}\rho_B$. The authors are grateful to GPT 5.6 for valuable discussion and technical help in preparing this note.

Quantum Physics (quant-ph); Information Theory (cs.IT); Mathematical Physics (math-ph)

arXiv:2609.12667 [quant-ph]

(or arXiv:2609.12667v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2609.12667 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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

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