(a) Why quantum correlations are weird: Hardy’s nonlocality paradox, (b) Brief recap of quantum kinematics, (c) The relevance of group C∗-algebras, (d) Results for two iterated binary measurements, (e) Outlook: From Kirchberg’s conjectures to Tsirelson’s problem. Hardy’s nonlocality paradox as a possibilistic Bell inequality. The most basic scenario for demonstrating what quantum correlations are is the following: there are two parties, com-monly dubbed Alice and Bob, experimenting in distant labs. They each receive a physical system prepared in some way, e.g. a photon, from a common source:
Abstract Two or more particles which have interacted in the past, may in general maintain nontrivial quantum mechanical correlations, described by non-factorized wave functions. These correlations, under optimal conditions, can manifest themselves no matter how far they travel away from each other and even when the separate parts cannot dynamically influence each other, leading to apparently paradoxical situations. These correlations are known as quantum entanglement. However counter-intuitive and surprising it may sometimes look, it is rather one of the more beautiful and experimentally well established predictions of quantum mechanics.
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 mathematical appendix offers background material on complete positivity, tensor products of C*-algebras, group C*-algebras, and some simple reformulations of the QWEP conjecture.
Network Bell experiments reveal a form of nonlocality conceptually different from standard Bell nonlocality. Standard multiparty Bell experiments involve a single source shared by a set of observers. In contrast network Bell experiments feature multiple independent sources and each of them may distribute physical systems to a set of observers who perform randomly chosen measurements. The
n-locality scenario in star-network configuration is an instance of network Bell experiment. Such a scenario involves
n number of edge observers (Alices) a central observer (Bob) and
n number of independent sources having no prior correlation. Each Alice shares an independent state with the central observer Bob. Usually in network Bell experiments one considers that each party measures only two observables. In this work we propose a nontrivial generalization of
n-locality scenario in star-network configuration where each Alice performs some integer
m number of binary-outcome measurements and the central party Bob performs ${2}^{m\ensuremath{-}1}$ measurements. We derive a family of generalized
n-locality inequalities for any arbitrary
m. Using an elegant sum-of-squares approach we derive the optimal quantum violation of the aforementioned inequalities when each and every Alice measures
m number of mutually anticommuting observables. For
m=2 and 3 one obtains the optimal quantum value with a two-qubit entangled state shared between each Alice and Bob. We further demonstrate that the optimal quantum violation of
n-locality inequality for any arbitrary
m can be obtained when every Alice shares $\ensuremath{\lfloor}m/2\ensuremath{\rfloor}$ copies of two-qubit maximally entangled state with the central party Bob. We also argue that for
m>3 a single copy of a two-qubit-entangled state may not be enough to exhibit the violation of
n-locality inequality but multiple copies of it can activate the quantum violation. We discuss the implications of our study and raise some open questions.
Nonlocality can be witnessed either through Bell-inequality violations or through logical contradictions such as Hardy's paradox. In the bipartite two input two outcome scenario, these two routes have distinct geometric behavior: CHSH-maximal correlations are exposed points of the quantum set, whereas known Hardy-type self-testing correlations on the no-signaling boundary are non-exposed. Here we show that this bipartite intuition fails in the tripartite two input two outcome scenario. We study the tripartite instance of a multipartite Hardy-type paradox and prove that the correlation attaining the maximal Hardy success probability self-tests the Greenberger--Horne--Zeilinger state and the associated measurements. Although this correlation lies on the no-signaling boundary, we show that it is an extremal and exposed point of the quantum correlation set. Moreover, it coincides with the correlation attaining the maximal violation of the Mermin inequality. Thus, in the tripartite GHZ scenario, the logical-paradox and Bell-inequality routes to nonlocality select the same exposed quantum boundary point. We also establish a robust version of the self-test, showing that small deviations from the ideal Hardy constraints imply quantitative closeness to the target state and measurements. Our results reveal a qualitative geometric difference between bipartite and tripartite Hardy-type nonlocality and suggest a broader investigation of exposedness for multipartite Hardy correlations in the multiparty setting.
The purpose of this lecture is to give a self-contained demonstration of a version of Bell's theorem and a discussion of the significance of the theorem and the experiments which it inspired. The lecture should be comprehensible to people who have had no previous acquaintance with the literature on Bell's theorem, but I hope that explicitness about premisses and consequences will make it useful even to those who are familiar with the literature. All versions of Bell's theorem are variations, and usually generalizations, of the pioneering paper of J. S. Bell of 1964, entitled “On the Einstein–Podolsky–Rosen Paradox.” All of them consider an ensemble of pairs of particles prepared in a uniform manner, so that statistical correlations may be expected between outcomes of tests performed on the particles of each pair. If each pair in the ensemble is characterized by the same quantum state φ, then the quantum mechanical predictions for correlations of the outcomes can in principle be calculated when the tests are specified. On the other hand, if it is assumed that the statistical behavior of the pairs is governed by a theory which satisfies certain independence conditions (always similar to the Parameter and Outcome Independence conditions stated below, though the exact details vary from version to version of Bell's theorem), then it is possible to derive a restriction upon the statistical correlations of the outcomes of tests upon the two particles.