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

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 $C^{\ast }$-tensor products.

Presentation improved and inaccuracy in lemma 11 removed

References in corpus (8)

• Device-independent security of quantum cryptography against collective attacks
• A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations
• Bell nonlocality in networks
• Connes' embedding problem and Tsirelson's problem
• Information-Theoretic Implications of Quantum Causal Structures
• Full network nonlocality
• A convergent inflation hierarchy for quantum causal structures
• Two convergent NPA-like hierarchies for the quantum bilocal scenario

Cited by in corpus (9)

• Semidefinite programming relaxations for quantum correlations
• Proposals for ruling out the real quantum theories in an entanglement-swapping quantum network with causally independent sources
• State polynomials: positivity, optimization and nonlinear Bell inequalities
• Uncertainty relations from state polynomial optimization
• Two convergent NPA-like hierarchies for the quantum bilocal scenario
• Refuting spectral compatibility of quantum marginals
• Symmetric observations without symmetric causal explanations
• Positivity of state, trace, and moment polynomials, and applications in quantum information
• Fully quantum inflation: quantum marginal problem constraints in the service of causal inference