Weekly update (6/14/2026)

Quantum ensemble as “biggest” equivalence classes of preparations. I added a new open problem in the quantum section (which is currently still a mess... and I also realized it is not indexed by the script, because I am so behind setting up the infrastructure). In classical mechanics, every probability measure over states is a distinct ensemble (it is a simplex). In quantum mechanics, multiple probability measure over states correspond to the same ensemble: a quantum ensemble is effectively an equivalence class of distinct preparations. But how big is that equivalence class? The conjecture is that it is the biggest possible given some minimal consistency conditions. But which ones? Who knows! That’s why it’s an open problem!

Better bound on the generalized JSD/Mixing entropy. Demitri, one of our undergrads, noticed that we can get a better bound on the mixing entropy on lines than I previously thought. This week, we integrated the cleaned up proof, and my previous results becomes a corollary of the more general bound.

Finalized notion of boundedness. Another think Demitri helped this week with was to finalize the notion of boundedness for topological convex spaces. He also provided a couple of figure that can be found after the definition and the proof of an equivalent condition.

As for the rest, was busy preparing for next week conference and for the summer school next week. I was able to squeeze in preliminary work for cleaning up the idea that unitary evolution can be seen as an infinite sequence of collapses. The statement of the theorem of the sketch of the proof is ready. “Just need” to clean up the proof!