Sharp continuity bounds link quantum entropy and entanglement measures
Log-Sobolev inequality, von Neumann entropy and Entanglement of Formation
Information Theory
Summary
This paper provides new mathematical bounds that tell us how much certain quantum properties, like entropy and entanglement, can change when a quantum state changes a little bit. The authors use a known inequality from graph theory to find exact constants that govern these changes. This helps to understand and measure quantum information reliably, even for complex quantum systems with many parts. These bounds work for both simple and very large quantum systems.
What this means in practice
- •For quantum algorithm developers: Use precise continuity bounds to estimate how small errors impact quantum entropy and entanglement in algorithms.
- •For quantum communication engineers: Improve design of quantum communication protocols by quantifying stability of entanglement under state changes.
A theory result. No direct application yet.
Authors
A. S. Holevo, M. E. Shirokov
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: the inequality $\,E_F(ρ)-E_F(σ)\leq \frac{1}{2}\,C_ρ\|ρ-σ\|_1\,$ 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.