On the Alleged Locality in the Schr\"odinger Picture

Vedral claims that the Schr\"odinger picture can describe quantum systems as locally as the Heisenberg picture, relying on a product notation for the density matrix. Here, I refute that claim. I show that the so-called `local factors' in the product notation do not correspond to individual systems and therefore fail to satisfy Einsteinian locality. Furthermore, the product notation does not track where local gates are applied. Finally, I expose internal inconsistencies in the argument: if, as is also stated, Schr\"odinger-picture locality ultimately depends on explicit bookkeeping of all operations, then the explanatory power of the product notation is de facto undermined.