Optimal Quantum Logarithmic Trace Inequality

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.

• The paper establishes an optimal quantum logarithmic trace inequality by deriving a universal constant Gₛ expressed in closed form using the Lambert W function.
• It introduces an iterative integration-by-parts method and a layer-cake representation to lift classical scalar inequalities to the quantum operator setting without prefactor loss.
• The results yield tighter bounds for finite-resource quantum protocols, enhancing one-shot coding theorems and converse results in quantum information theory.

Optimal Quantum Logarithmic Trace Inequality

Introduction

The paper "Optimal Quantum Logarithmic Trace Inequality" (2604.14617) addresses the derivation and analysis of a sharp quantum logarithmic trace inequality. This result provides a strictly tighter bound than the recent operator-level inequalities of Cheng et al., replacing the standard prefactor $c_s/s$ by an optimal universal constant $G_s$ expressible in closed form via the Lambert $W$ function. These trace inequalities play a foundational role in one-shot quantum information theoretic primitives such as decoupling, convex-splitting, and quantum covering lemmas, directly impacting achievable bounds for finite resource scenarios.

Main Results

The core of the work is the establishment of the operator inequality:

$Q(\rho\|\sigma) \leq G_s Q_{1+s}(\rho\|\sigma)$

where $Q(\rho\|\sigma) = [\rho((\rho+\sigma)-\sigma)]$ relates to a sandwiched difference of Umegaki relative entropies, and $Q_{1+s}(\rho\|\sigma)$ is the layer-cake or sandwiched Rényi divergence at order $1+s$. $G_s$ is defined as the supremum,

$G_s = \sup_{r>0} \frac{1+r}{r^s},$

which is shown to admit an explicit formula involving $W_{-1}$, the negative branch of the Lambert $G_s$0 function. Notably, the approach leverages an iterative integration-by-parts transformation of the layer-cake representation, avoiding the loss in prefactors incurred by pinching or reduction to commuting scenarios.

The work establishes that $G_s$1 is indeed the smallest possible universal constant in this inequality: no $G_s$2 can satisfy the inequality for all strictly positive operators. The optimality persists for density matrices for all $G_s$3. For $G_s$4, the constant drops to $G_s$5 in the commuting (classical) case, while the fully noncommuting quantum scenario remains unresolved.

Furthermore, in the physically significant regime $G_s$6 (the Umegaki relative entropy limit), the improvement over the Cheng et al. bound $G_s$7 is quantitatively maximized, with a reduction factor of $G_s$8 as $G_s$9. The work also clarifies the precise points of transition between different operational regimes, showcasing threshold phenomena tied to normalization and commutativity.

Technical Approach

The proof architecture draws upon:

• Layer-Cake Representation: The trace functional is rewritten using a layer-cake (or operator-valued Stieltjes) representation, connecting $W$0 with projective measurements $W$1.
• Iterative Integration by Parts: The functional is transformed iteratively, lifting optimal classical scalar inequalities $W$2 to the fully quantum operator domain without loss.
• Structural Reduction: The commutative case is treated as a maximization over classical probability vectors, enabling explicit analysis of the supremum and demonstrating that in many regimes, the commutative situation realizes the quantum optimum.
• Majorization Techniques: For the normalized case and analysis under the commutativity constraint, majorization methods justify restriction to uniform quantum states.

The analysis further extends to a stronger family of inequalities for the quantum collision divergence, delineating regimes of optimality and relating these sharper inequalities to improved finite-resource constants in key quantum information primitives.

Numerical Improvements and Claims

A critical numerical claim is that the ratio $W$3 varies from 1 at $W$4 to $W$5 as $W$6. This enhancement, evidenced in both commutative and noncommutative settings (up to threshold $W$7), yields uniformly tighter bounds for one-shot protocols. The new prefactors immediately apply to operational statements for decoupling, convex-splitting, and covering, thus improving existing one-shot coding theorems and second-order converse results in quantum Shannon theory.

The paper also makes a sharp assertion that optimality of the constant $W$8 is reached without the need for pinching or regularization, directly within the quantum setting via the derived iterative framework. The implication is that quantum-to-classical reductions are unnecessary for achieving the best constants, except beyond the identified threshold in the normalized case.

Implications and Directions for Future Research

On the practical side, the paper’s results imply systematically improved error bounds and tighter achievable rates in a range of quantum information protocols involving finite resources. Theoretically, the approach demonstrates that operator-level inequalities previously accessible only with lossy classical reductions and prefactor blow-up can now be reached “losslessly” via functional analytic techniques directly at the quantum level.

A key open problem is the determination of the optimal universal constant for noncommuting density matrices beyond $W$9, where current methods only resolve the commuting (classical) case. The possibility that measured Rényi divergences might replace layer-cake divergences in these inequalities is noted, since they are pointwise smaller and—if proven to admit a similar lifting—could further tighten operational bounds.

Finally, the framework established here offers a blueprint for systematically lifting classical sharp constants for trace inequalities to genuinely quantum analogues, opening avenues for analogous improvements in other contexts where trace inequalities play a central analytic role.

Conclusion

This paper establishes the optimal logarithmic trace inequality for positive operators in quantum information, identifies the sharp prefactor $Q(\rho\|\sigma) \leq G_s Q_{1+s}(\rho\|\sigma)$0 (determined via the Lambert $Q(\rho\|\sigma) \leq G_s Q_{1+s}(\rho\|\sigma)$1 function), and demonstrates that operator-level sharpness can be retained through iterative analytic transformation without recourse to pinching or classicalization. The result uniformly refines prior bounds in foundational one-shot quantum protocols, precipitating both immediate quantitative refinements and laying groundwork for further theoretical and operational advances in quantum finite-resource information theory.