Unbounded Violation of Tripartite Bell Inequalities

Abstract

Abstract:We prove that there are tripartite quantum states (constructed from random unitaries) that can lead to arbitrarily large violations of Bell inequalities for dichotomic observables. As a consequence these states can withstand an arbitrary amount of white noise before they admit a description within a local hidden variable model. This is in sharp contrast with the bipartite case, where all violations are bounded by Grothendieck's constant. We will discuss the possibility of determining the Hilbert space dimension from the obtained violation and comment on implications for communication complexity theory. Moreover, we show that the violation obtained from generalized Greenberger-Horne-Zeilinger (GHZ) states is always bounded so that, in contrast to many other contexts, GHZ states do not lead to extremal quantum correlations in this case. In order to derive all these physical consequences, we will have to obtain new mathematical results in the theories of operator spaces and tensor norms. In particular, we will prove the existence of bounded but not completely bounded trilinear forms from commutative C*-algebras. Finally, we will relate the existence of diagonal states leading to unbounded violations with a long-standing open problem in the context of Banach algebras.

Maximal violation of a bipartite three-setting, two-outcome Bell inequality using infinite-dimensional quantum systems

“…In a broader sense, the aim of this concept is to put a lower bound on the Hilbert space dimension needed to reproduce the statistics arising from a Bell experiment. Dimension witnesses have been found or conjectured in many different scenarios, such as for the bipartite setting [13,14,17,[31][32][33] for the multipartite setting [34] and even for the case of a single system [35].…”

Operator Space Theory: A Natural Framework for Bell Inequalities

“…Nowadays, they are at the heart of the modern development of Quantum Information, with applications in a wide variety of areas: quantum key distribution [4,5], entanglement detection, multipartite interactive proof systems [6,7], communication complexity [8], Hilbert space dimension estimation [9,10,11,12], etc.…”

“…One reason for that is that so far there was no suitable mathematical tool for them. In [12], we showed how for the special case of correlation Bell inequalities, Operator Space Theory-a modern field in mathematical analysis-provides exactly the right language and tools to tackle some of the more difficult problems. There we used operator space techniques to solve an old question of Tsirelson [15]: the existence of unbounded violations for tripartite correlation Bell inequalities.…”

New concise upper bounds on quantum violation of general multipartite Bell inequalities

“…But this is not already the case for quantum violation of bipartite Bell inequalities on joint probabilities and last years bounds on the maximal quantum violation of Bell inequalities were intensively discussed in the literature via different mathematical tools [7][8][9][10][11][12][13][14][15] .…”