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