I was recently invited to give an overview talk on Petz’s “transpose” or “recovery” map at the workshop Workshop on Quantum Information and Quantum Black Holes, organized by Norihiro Iizuka (Osaka) and Tomonori Ugajin (Kyoto). I’ve put together what I came to learn about the properties of Petz’s transpose map, its uses is various areas of information theory and physics, and (most importantly!) its conceptual meaning. Slides are available here. Mark Wilde’s textbook contains a chapter entirely devoted to the technical aspects of Petz’s map and the theory of approximate recoverability.

2025/4/22 Update: in these four years, we’ve learned a lot more about the Petz transpose map, especially its interpretation and relationship to Bayesian inversion. I’ve thus updated the slides to include at least the pointer to these new papers. The most important (in my opinion) is the derivation of the Petz transpose map from a principle of minimum change. The original slides presented in the video are still available at this link.