TSIRELSON'S PROBLEM AND KIRCHBERG'S CONJECTURE

By Tobias Fritz. 2010

158 citations recorded by the source corpus.

Tsirelson's problem asks whether the set of nonlocal quantum correlations with a tensor product structure for the Hilbert space coincides with the one where only commutativity between observables located at different sites is assumed. Here it is shown that Kirchberg's QWEP conjecture on tensor products of C*-algebras would imply a positive answer to this question for all bipartite scenarios. This remains true also if one considers not only spatial correlations, but also spatiotemporal correlations, where each party is allowed to apply their measurements in temporal succession. We provide an example of a state together with observables such that ordinary spatial correlations are local, while the spatiotemporal correlations reveal nonlocality. Moreover, we find an extended version of Tsirelson's problem which, for each nontrivial Bell scenario, is equivalent to the QWEP conjecture. This extended version can be conveniently formulated in terms of steering the system of a third party. Finally, a comprehensive...

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/bab63167240f76ced4c98eddfdf01f52c38214cc

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Citations in the corpus

About the Connes Embedding Conjecture---Algebraic approaches--- (2012, background): “ compactness, one has Qc= \ n { h ϕf(p (k) i q (l) j ) i k,l i,j : fpositive type on En} 30 NARUTAKAOZAWA = \ n { " 1 m2 Xm v,w=1 ω −iv jw m f(s v kt w l ) # k,l i,j : fpositive type on En}. See [Fr] for more information. Next, we deal with Qs. Although the description Qs= closure [ n { h h(P”
Characterisations of the weak expectation property (2013, background): “The group ∗j=1Z2 contains F2 (see, for example, [12]) and hence, by Proposition 3.”
Grothendieck's Theorem, past and present (2011, background): “The Connes-Kirchberg problem As it turns out (see [68, 37]) Tsirelson’s problem is (essentially) equivalent to one of the most famous ones in von Neumann algebra theory going back to Connes’s paper [26].”
Introduction to Sofic and Hyperlinear Groups and Connes' Embedding Conjecture (2013, background): “Other very recent discoveries include the fact that Connes’ embedding conjecture is related to Tsirelson’s problem, a major open problem in Quantum Information Theory (see [49], [66], and [93]) and that it is connected to the “minimal” and “commuting” tensor products [73] of some group operator syst”
Tensor Products of C*-Algebras and Operator Spaces (2020, methodology): “The proof of this equivalence was completed in several steps, in the papers [95] and [138], the final step being taken by Ozawa in [191].”
A Combinatorial Approach to Nonlocality and Contextuality (2012, background): “Now it is known that Kirchberg’s QWEP conjecture and Connes’ embedding problem are equivalent to the residual finite-dimensionality of some of these C∗-algebras, for example for C∗(B2,3,2) [43, 45].”
THE SET OF QUANTUM CORRELATIONS IS NOT CLOSED (2017, background): “A two-player non-local game G consists of finite question sets IA and IB, finite output sets OA and OB, and a function V : OA×OB×IA×IB → {0, 1}.”
Nuclearity Related Properties in Operator Systems (2011, background): “rs in a quantum experiment can be described by relativistic approach. The proper deﬁnitions and basic result in this question are beyond the scope of this paper and we refer the reader to [38], [20], [14]. In [20] it was shown that when the actors are Alice and Bob (that is, in the bipartite scenari”
Connes' embedding problem and Tsirelson's problem (2010, methodology): “After this work was completed, we learned that one direction (Connes ⇒ Tsirelson) was independently obtained by [9].”