Christandl–Ferrara–Lancien continuity conjecture for filtered relative entropy

Let $\mathcal{F}\subseteq\mathcal{D}(\mathcal{H})$ be a set of states star-shaped around the maximally mixed state, where $D=\dim(\mathcal{H})<\infty$, and let $\mathcal{L}\subseteq\operatorname{CPTP}(\mathcal{H}\to\mathcal{H}')$ be any set of channels. Let $D^{\mathcal{L}}(\rho\|\mathcal{F})$ denote the filtered relative entropy associated with $\mathcal{L}$ and $\mathcal{F}$, and let $\|\rho-\sigma\|_{\mathcal{L}}$ be the corresponding filtered distance.

Christandl–Ferrara–Lancien's conjecture. There should exist a universal constant $\kappa$ and a function $g:[0,1]\to\mathbb{R}_+$ with

$$\lim_{\varepsilon\to0^+}g(\varepsilon)=0$$

such that, for every $\rho,\sigma\in\mathcal{D}(\mathcal{H})$ satisfying

$$\frac{1}{2}\|\rho-\sigma\|_{\mathcal{L}}\leq\varepsilon,$$

one has

$$\left|D^{\mathcal{L}}(\rho\|\mathcal{F})-D^{\mathcal{L}}(\sigma\|\mathcal{F})\right|\leq\kappa\varepsilon\log D+g(\varepsilon).$$

The conjecture proposes dimension-controlled continuity for filtered relative entropy uniformly over the specified state and channel sets; the paper states it as an open problem and develops results aimed at proving it.