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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 don't 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?

glS
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I believe in HJPW by 'quantum map' they mean 'quantum channel'. The map $\tilde T$ is something called Petz recovery map. To get some intuition about it you might try with proving it with classical random variables, in which case the Petz recovery map reduces to the Bayesian estimator.

user166061
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