2 research outputs found
The Hilbertian Tensor Norm and Entangled Two-Prover Games
We study tensor norms over Banach spaces and their relations to quantum
information theory, in particular their connection with two-prover games. We
consider a version of the Hilbertian tensor norm and its dual
that allow us to consider games with arbitrary output alphabet
sizes. We establish direct-product theorems and prove a generalized
Grothendieck inequality for these tensor norms. Furthermore, we investigate the
connection between the Hilbertian tensor norm and the set of quantum
probability distributions, and show two applications to quantum information
theory: firstly, we give an alternative proof of the perfect parallel
repetition theorem for entangled XOR games; and secondly, we prove a new upper
bound on the ratio between the entangled and the classical value of two-prover
games.Comment: 33 pages, some of the results have been obtained independently in
arXiv:1007.3043v2, v2: an error in Theorem 4 has been corrected; Section 6
rewritten, v3: completely rewritten in order to improve readability; title
changed; references added; published versio
A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations
We are interested in the problem of characterizing the correlations that
arise when performing local measurements on separate quantum systems. In a
previous work [Phys. Rev. Lett. 98, 010401 (2007)], we introduced an infinite
hierarchy of conditions necessarily satisfied by any set of quantum
correlations. Each of these conditions could be tested using semidefinite
programming. We present here new results concerning this hierarchy. We prove in
particular that it is complete, in the sense that any set of correlations
satisfying every condition in the hierarchy has a quantum representation in
terms of commuting measurements. Although our tests are conceived to rule out
non-quantum correlations, and can in principle certify that a set of
correlations is quantum only in the asymptotic limit where all tests are
satisfied, we show that in some cases it is possible to conclude that a given
set of correlations is quantum after performing only a finite number of tests.
We provide a criterion to detect when such a situation arises, and we explain
how to reconstruct the quantum states and measurement operators reproducing the
given correlations. Finally, we present several applications of our approach.
We use it in particular to bound the quantum violation of various Bell
inequalities.Comment: 33 pages, 2 figure