This should have gone out yesterday… but I was on a plane and then forgot!
New essay. Last Monday I published a new essay on the relationship between the Born rule and entropy. It’s a less technical version of this brief.
Foundations 2026. Last week I was in California, Irvine, for the Foundations of Physics 2026 conference. It was the first time it was organized in the US. You can see a few pictures here on substack… when I remembered to take them.
IPMU 2026. In the same week, Tobias went to the Information Processing and Management of Uncertainty in Knowledge-Based Systems conference. Unfortunately, it was on the same week, so we had to divide and conquer.
Upcoming Summer School. This Wednesday we’ll have the summer school… which reminds me I have to send the final information to those who registered.
Upcoming workshop. In two weeks, instead, we’ll have a workshop on non-additive measures and generalized versions of probability in physics. This will be part of our grant from the John Templeton Foundation.
The conferences pretty much took the whole week, so little technical progress to report… though I managed to prove a theorem on the plane.
The essay and brief on the relationship between the Born rule and entropy sharpen the point nicely: the Born rule is not just compatible with von Neumann entropy; over the density-operator ensemble space, it is tied to it by the same mutual-exclusivity structure.
The most useful move, to me, is the equal-mixture case. For two pure states, the entropy of ρ = ½ρψ + ½ρφ is a strictly decreasing function of the Born overlap p(φ|ψ). So distinguishability, entropy, and overlap are not three unrelated notions. In the two-state slice, they are different readings of the same geometry. Orthogonality then becomes the special case where the equal mixture has one bit of entropy and the alternatives are mutually exclusive.
From my perspective, this suggests a strong constraint on any deeper account of the Born rule: it is not enough to recover outcome frequencies. The underlying preparation-and-record dynamics must also recover the entropy geometry of mixtures. In other words, the same basin-weight measure that gives p(k|preparation) should make mutually exclusive record channels behave like Shannon alternatives, while nonorthogonal preparations carry reduced entropy because their record basins are not fully exclusive.
That gives a concrete diagnostic target: derive the Born weights and von Neumann mixture entropy from one finite-window preparation measure, not from two separate postulates. The interesting stress tests would be unequal mixtures, more than two preparations, and imperfect record formation, where the entropy/Born equivalence should expose exactly which assumptions are doing the work.