# Proof that the relative entropy satisfies $S(\rho\|\sigma)=S(T\rho\|T\sigma)$ iff $\hat TT\rho=\rho$, $\hat TT\sigma=\sigma$ for some $\hat T$

To prove the saturation condition for the strong subadditivity of the von Neumann entropy, the authors of [HJPW2004] make use of the following characterisation of when the monotonicity of the relative entropy is saturated: we have $$S(\rho\|\sigma)=S(T\rho\|T\sigma)$$ iff there's a quantum map $$\hat T$$ such that $$\hat TT\rho=\rho$$ and $$\hat TT\sigma=\sigma$$.

I'm actually not entirely sure I fully understand the statement. HJPW talk about $$\hat T$$ being a "quantum operation", but I think it has to be a channel, otherwise the non-fully depolarising channel would be a counterexample: it's an invertible channel (whose inverse is not a channel), but doesn't keep the relative entropy fixed. On the other hand, if we're asking for $$\hat T$$ to be a channel, then we're talking about channels $$T$$ with a left inverse on the image of $$\rho$$ and $$\sigma$$, which is highly reminiscent of a quantum error correction context.

This statement is attributed to [Petz1986] (no arXiv version that I can find). This paper is however not the easiest read, and uses lots of mathematics I'm not very familiar with. Is there another "more modern" proof of this characterisation? Or perhaps a relatively simple argument to show where it comes from?

• I'm not overly familiar with this result but I know that there are some later papers of Petz---which came out around the time of the HJPW paper you cited, cf. here and here---on what's known today as the Petz recovery map (which turns out to be your map $\hat T$) where only finite-dimensional spaces are considered. I hope these can be of use to you! Commented Apr 18 at 13:49
• a recent relevant paper is doi.org/10.1016/j.physleta.2024.129583
– glS
Commented May 17 at 23:03