As a fun aside, I am very happy with the continuity bound and its proof. It contains, I think, a very fun and beautiful, but out of context meaningless formula that I want to leave you with. Made me reflect about beauty in maths. And I'd never thought so many different Ms could have real meaning.

✨ We introduce computational versions of the max-divergence (via some beautiful conical structures in QIT) and measured Rényi divergences. We analyze their behavior under efficient operations and show that they from a cohesive framework (for α→∞ they coincide).
Further we consider two applications

We present a ‚quantum’ extension of mixed matrix norms showing hardness results for among other the tasks of computing the minimal output Rényi entropy of entanglement breaking (EB) channels (1->p) and the optimal one-shot distinguishability of a difference of EB channels (1->1).

To go beyond the 1D or the nearest neighbor setting we introduce the Matrix valued quantum Conditional Mutual Information (MCMI) to the Davies mixing setting. Interestingly the MCMI has been studied before, among others in arXiv:1910.09425v2.

However, the question stands: When does it decay?

And lastly, this was me recently presenting a poster at @ IP-Paris about this work. It took up a lot of my first year of PhD and was in the works for a very long time...

Examples of such systems include all local quantum CSS codes on D dimensional lattices at uniformly high temperatures.

To be honest this work is effectively mixing time bounds of the Davies evolution in disguise.