(This post is vague, and sheer speculation.)

Following a great conversation with Miles Stoudenmire here at PI, I went back and read a paper I forgot about: “Entanglement and the foundations of statistical mechanics” by Popescu et al.^{a }. This is one of those papers that has a great simple idea, where you’re not sure if it’s profound or trivial, and whether it’s well known or it’s novel. (They cite references 3-6 as “Significant results along similar lines”; let me know if you’ve read any of these and think they’re more useful.) Anyways, here’s some background on how I think about this.

If a pure quantum state is drawn at random (according to the Haar measure) from a -dimensional vector space , then the entanglement entropy

across a tensor decomposition into system and environment is highly likely to be almost the maximum

for any such choice of decomposition . More precisely, if we fix and let , then the fraction of the Haar volume of states that have entanglement entropy more than an exponentially small (in ) amount away from the maximum is suppressed exponentially (in ). This was known as Page’s conjecture^{b }, and was later proved^{c }^{d }; it is a straightforward consequence of the concentration of measure phenomenon.… [continue reading]