Optimal Quantum Logarithmic Trace Inequality

• URL: http://arxiv.org/abs/2604.14617v1
• Date: Thu, 16 Apr 2026 04:57:25 GMT
• Title: Optimal Quantum Logarithmic Trace Inequality
• Abstract summary: We establish a sharp logarithmic trace inequality that strengthens recent bounds of Cheng et al.(arXiv:2507.07961)<n>We prove that $G_s$ is the optimal universal constant, in the sense that no smaller constant satisfies the inequality for all positive operators.<n>These sharper inequalities enhance key primitives in quantum information theory, including decoupling, convex-splitting, and covering lemmas.
• Score: 2.7074235008521246
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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