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

TL;DR

It is proved that the quantum inflation hierarchy is complete for the bilocal scenario in the commuting observable model of locality, and a bilocal version of an observation by Tsirelson is given, namely, in finite dimensions, the commuting observables model and the tensor product model of geography coincide.

Abstract

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.

Figures (3)

• Figure 1: The bilocal scenario. Alice and Bob share a bipartite quantum state $\sigma_{A B_A}$ and Bob and Charlie share a bipartite quantum state $\sigma_{B_C C}$. Alice performs a measurement with the POVM $\{A_{\alpha|x}\}_\alpha$ based on the setting measurement setting $x$. Bob and Charlie perform a similar measurement. The conditional probabilities $p(\alpha\beta\gamma|xyz)$ that can arise in this way are called bilocal correlations.
• Figure 2: Logical structure of the proof given in Sec. \ref{['sec:fac to biloc']}. The equivalences claimed in Sec. \ref{['sec:equivalences']} follow from this chain of implications among the various models of bilocal quantum correlations.
• Figure 3: The level 2 inflation of the bilocal scenario. Each of the states $\sigma_{AB_A}$ and $\sigma_{B_C C}$ has been copied. The total state of the system is permutation symmetric under the exchange of each of these copies. The inflation technique builds on this observation.

Theorems & Definitions (23)

• Definition 1: Two-party correlations, tensor product model
• Definition 2: Two-party quantum correlations, commuting observables model
• Definition 3: Two-party quantum correlations, commuting operator model: algebraic formulation
• Definition 4: Tensor product model
• Definition 5: Commuting observables model
• Corollary 6: Reduced model
• Corollary 7: Mixed model
• Corollary 8: Renou-Xu model renou2022two
• Corollary 9
• Theorem 10