*This post describes ideas generated in discussion with Markus Hauru, Curt von Keyserlingk, and Daniel Ranard.*]

An original dream of defining branches based on redundant records (aka redundant classical information, aka GHZ-like correlations) was that it would be possible to decompose the wavefunction of an evolving non-integrable quantum system at each point in time into macroscopically distinguishable branches that individually had bounded amounts of long-range entanglement (i.e., could be efficiently expressed as a matrix product state) even though the amount of long-range entanglement for the overall state diverges in time. If one could numerically perform such a decomposition, and if the branches only “fine-grain in time”, then one could classically sample from the branches to accurately estimate local observables even if the number of branches increases exponentially in time (which we expect them to do).

However, we now think that only a fairly small fraction of all long range entanglement can be attributed to redundantly recorded branches. Thus, even if we found and efficiently handled all such classical information using a decomposition into a number of branches that was increasing exponentially in time (polynomial branch entropy), most branches would nevertheless still have an entanglement entropy across any spatial partition that grew ~linearly in time (i.e., exponentially increasing bond dimension in the MPS representation) until saturating.

In this post I’ll first write down a simple model that suggests the need to generalize the idea of branches in order to account for most long-range entanglement. Then I will give some related reasons to think that this generalized structure will take the form not of a preferred basis, but rather preferred subspaces and subsystems, and together these will combine into a preferred “branch algebra”.… [continue reading]