The infinitesimal minimum output entropy conjecture for single-mode gauge-covariant Gaussian channels

Let

be a single-mode gauge-covariant bosonic Gaussian channel represented by $=e^{t}$. For any input state

with entropy $S=S_0>0$, let $^{\operatorname{th}}_{g^{-1}(S_0)}$ denote the thermal state with the same entropy. Infinitesimal minimum output entropy conjecture. The entropy production at $t=0$ is minimized by this thermal input:

$$\left.\frac{d}{dt}S(\Phi_t(\rho))\right|_{t=0}\geq \left.\frac{d}{dt}S\!\left(\Phi_t\!\left(\rho^{\operatorname{th}}_{g^{-1}(S_0)}\right)\right)\right|_{t=0}.$$

This infinitesimal statement is presented as implying the finite-time minimum output entropy conjecture for all $t\geq 0$. The paper discusses a proof attempt and acknowledges that a previously claimed proof had a major problem; the status of the conjecture is therefore unresolved in the supplied text.