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 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 observable model of locality. We also give a bilocal version of an observation by Tsirelson, namely, in finite dimensions, the commuting observable model and the tensor product model of locality coincide. These results answer questions recently posed by Renou and Xu [arXiv:2210.09065v2 (2022)]. Finally, we point out that our techniques can be interpreted more generally as giving rise to a semidefinite programming 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.
Skip Nav Destination
Article navigation
28 July 2023
Research Article|
July 11 2023
The inflation hierarchy and the polarization hierarchy are complete for the quantum bilocal scenario
Laurens T. Ligthart
;
Laurens T. Ligthart
a)
(Conceptualization, Formal analysis, Writing – original draft)
Institute for Theoretical Physics, University of Cologne
, Köln, Germany
a)Author to whom correspondence should be addressed: [email protected]
Search for other works by this author on:
David Gross
David Gross
(Conceptualization, Formal analysis, Writing – original draft)
Institute for Theoretical Physics, University of Cologne
, Köln, Germany
Search for other works by this author on:
a)Author to whom correspondence should be addressed: [email protected]
J. Math. Phys. 64, 072201 (2023)
Article history
Received:
January 26 2023
Accepted:
June 21 2023
Citation
Laurens T. Ligthart, David Gross; The inflation hierarchy and the polarization hierarchy are complete for the quantum bilocal scenario. J. Math. Phys. 28 July 2023; 64 (7): 072201. https://doi.org/10.1063/5.0143792
Download citation file:
Pay-Per-View Access
$40.00
Sign In
You could not be signed in. Please check your credentials and make sure you have an active account and try again.
Citing articles via
Mathematical models of human memory
Mikhail Katkov, Michelangelo Naim, et al.
Graded Poisson and graded Dirac structures
Manuel de León, Rubén Izquierdo-López
Stochastic dynamics of particle systems on unbounded degree graphs
Georgy Chargaziya, Alexei Daletskii
Related Content
A fermionic de Finetti theorem
J. Math. Phys. (December 2017)
Lower bounds on the entanglement needed to play XOR non-local games
J. Math. Phys. (October 2011)
Tsirelson's problem and asymptotically commuting unitary matrices
J. Math. Phys. (March 2013)
de Finetti reductions for correlations
J. Math. Phys. (May 2015)
SU(p,q) coherent states and a Gaussian de Finetti theorem
J. Math. Phys. (April 2018)