The inflation hierarchy and the polarization hierarchy are complete for the quantum bilocal scenario

It is a fundamental but difficult problem to characterize the set of correlations that can be obtained by performing measurements on quantum mechanical systems. The particularly challenging when preparation procedure for states assumed comply with given causal structure. Recently, first completeness result this compatibility has been given, based so-called inflation technique. However, was achieved imposing additional technical constraints, such as an upper bound Schmidt rank observables. Here, we show these complications are unnecessary in bilocal scenario, much-studied abstract model entanglement swapping experiments. We prove hierarchy complete scenario commuting observables locality. also give version observation Tsirelson, namely finite dimensions, and tensor product locality coincide. These results answer questions recently posed Renou Xu. Finally, point out our techniques interpreted more generally giving rise SDP optimizing polynomial functions operator algebras defined generators relations. polarization follows from de Finetti theorem maximal $C^*$-tensor products.