Kavli Affiliate: David Gross
| First 5 Authors: Laurens T. Ligthart, David Gross, , ,
| Summary:
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.
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