A Collection of Pinsker-type Inequalities for Quantum Divergences

TL;DR

This work develops a systematic method to derive Pinsker-type lower bounds relating quantum divergences to the trace distance for divergences that satisfy the data processing inequality. By reducing to two-dimensional binary states, it constructs linear bounds $L_{\mathcal{D}}(\lambda)$ and their convex Legendre transforms $B_{\mathcal{D}}(T)$, and extends them to smoothed divergences. The framework is then applied to Rényi divergences and related quantities, including the Umegaki, collision, and max divergences, deriving analytic, parametric, and numerical bounds. These bounds enable replacing hard divergences with tractable trace-distance bounds in finite-size and operational settings, with potential impact on QKD, Gibbs sampling, and entropic uncertainty relations.

Abstract

Pinsker's inequality sets a lower bound on the Umegaki divergence of two quantum states in terms of their trace distance. In this work, we formulate corresponding estimates for a variety of quantum and classical divergences including $f$-divergences like Hellinger and $χ^2$-divergences as well as Rényi divergences and special cases thereof like the Umegaki divergence, collision divergence, max divergence. We further provide a strategy on how to adapt these bounds to smoothed divergences.

Figures (7)

• Figure 1: Improvement of Pinsker's inequality bound comparing the Umegaki divergence $D$ with the trace distance $T$. The dots depict a random sample from the set ${\Omega_{T,D} = \{(T(\rho, \sigma), D(\midmid{\rho}{\sigma}))\}}$ for a random selection of states $\rho$ and $\sigma$. The red line indicates the optimal convex lower bound $B_D$ on this set while the black line indicates Pinsker's bound.
• Figure 2: The set $\Omega_{T, \mathcal{D}}$ and its linear and convex lower bounds $L_\mathcal{D}$ and $B_\mathcal{D}$. For each trace distance $t \in (0,1)$, the tangent at $t$ with corresponding slope $\lambda$ intercepts the ordinate at $L_\mathcal{D}(\lambda)$. The Legendre transform of the linear lower bound $L_\mathcal{D}$ gives the convex lower bound $B_\mathcal{D}$.
• Figure 3: Convex lower bound for smoothed quantum max divergence. The bound for the smoothed max divergence (solid red line) is obtained by shifting the bound for the unsmoothed max divergence (dashed red line) by $\varepsilon$ to the right and cutting off at $D^\varepsilon_\infty = 0$ and $T=1$.
• Figure 4: $\xi_{\alpha, \lambda}$ on the unit square for $\alpha = 2$ and different values of $\lambda$. The two minima (black dots) move to the vertical boundaries as $\lambda$ increases. For $\lambda \geq \frac{2}{\ln(2)}$ the minima lie at the boundary. Due to the symmetry it is clear that the values of both minima coincide.
• Figure 5: Convex bounds for Rényi divergences (a) and Hellinger divergences (b) each for a selection of $\alpha$. The marked points indicate where the right-side bounds and the left-side bounds meet. Remarkably, the bound for $\alpha=1$ is equal in both cases. Further, note that $B_{D_\frac{1}{2}} \leq B_{D_1} \leq B_{D_\frac{4}{3}} \leq B_{D_2} \leq B_{D_\infty}$.

Theorems & Definitions (6)

• theorem 1
• proof
• theorem 2
• proof
• proof
• proof