Optimal Quantum Logarithmic Trace Inequality

Quantum Physics [Submitted on 16 Apr 2026] Optimal Quantum Logarithmic Trace Inequality Gilad Gour We establish a sharp logarithmic trace inequality that strengthens recent bounds of Cheng et al.(arXiv:2507.07961) by replacing their prefactor c_s/s with the strictly smaller constant G_s. The constant G_s is defined via the scalar inequality \log(1+r)\le G_s r^s and admits a closed-form expression in terms of the Lambert W function. Our approach introduces an iterative integration-by-parts procedure that lifts optimal scalar bounds to the operator level without loss. We prove that G_s is the optimal universal constant, in the sense that no smaller constant satisfies the inequality for all positive operators. For density matrices, this optimality persists up to s\le \frac{1}{2\log(2)}, while beyond this threshold the commuting case exhibits a strictly smaller optimal constant and the noncommuting case remains open. In the regime s\to0, our result improves the prefactor c_s/s of Cheng et al. by a factor of 1/e. These sharper inequalities enhance key primitives in quantum information theory, including decoupling, convex-splitting, and covering lemmas, leading to tighter finite-resource bounds. Comments: 4+5 pages, 1 figure, comments are welcome Subjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph) Cite as: arXiv:2604.14617 [quant-ph]   (or arXiv:2604.14617v1 [quant-ph] for this version)   https://doi.org/10.48550/arXiv.2604.14617 Focus to learn more Submission history From: Gilad Gour [view email] [v1] Thu, 16 Apr 2026 04:57:25 UTC (62 KB) Access Paper: HTML (experimental) view license Current browse context: quant-ph < prev   |   next > new | recent | 2026-04 Change to browse by: math math-ph math.MP References & Citations INSPIRE HEP NASA ADS Google Scholar Semantic Scholar Export BibTeX Citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Demos Related Papers About arXivLabs Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)