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arXiv:1210.4300 · Physical Review A

Fine-grained uncertainty relation and biased nonlocal games in bipartite and tripartite systems

Ansuman Dey, Tanumoy Pramanik, A. S. Majumdar

Published Jan 18, 2013• 21 citations
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Abstract

The fine-grained uncertainty relation can be used to discriminate among classical, quantum, and superquantum correlations based on their strength of nonlocality, as has been shown for bipartite and tripartite systems with unbiased measurement settings. Here we consider the situations when two and three parties choose settings with bias for playing certain nonlocal games. We show analytically that while the fine-grained uncertainty principle is still able to distinguish classical, quantum, and superquantum correlations for biased settings corresponding to certain ranges of the biasing parameters, the above-mentioned discrimination is not manifested for all biasing.

DOI
10.1103/physreva.87.012120

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The fine-grained uncertainty relation can be used to discriminate among classical, quantum, and superquantum correlations based on their strength of nonlocality, as has been shown for bipartite and tripartite systems with unbiased measurement settings. Here we consider the situations when two and three parties choose settings with bias for playing certain nonlocal games. We show analytically that while the fine-grained uncertainty principle is still able to distinguish classical, quantum, and superquantum correlations for biased settings corresponding to certain ranges of the biasing parameters, the above-mentioned discrimination is not manifested for all biasing.
The upper bound of the fine-grained uncertainty relation is different for classical physics, quantum physics, and no-signaling theories with maximal nonlocality (superquantum correlation), as was shown in the case of bipartite systems [J. Oppenheim and S. Wehner, Science 330, 1072 (2010)]. Here, we extend the fine-grained uncertainty relation to the case of tripartite systems. We show that the fine-grained uncertainty relation determines the nonlocality of tripartite systems as manifested by the Svetlichny inequality, discriminating between classical physics, quantum physics, and superquantum correlations.
Quantum nonlocality and the uncertainty principle are two fundamental cornerstones of quantum theory, reflecting totally different aspects that distinguish it from classical theory. However, Oppenheim and Wehner introduced fine-grained uncertainty relations (FGURs) to account for the degree of quantum nonlocality. In the simplest Bell scenario, they showed that the quantum nonlocality measured by the maximal violation of some Bell inequality was determined by the FGUR. Along this line of research, we first derive explicit FGURs with general weights about probability distributions of two general projective measurements on local systems. Then we split those joint probabilities of Bell inequalities into a single-party conditional probability and the remaining joint probabilities so that the weighted FGURs can be applied straightforwardly to obtain the maximal violation in various scenarios ranging from bipartite, tripartite, to a complete set of multipartite Bell inequalities. Furthermore, in the bipartite scenario, the exact correlation boundary can be shown to be solely determined by our weighted local FGURs.
Abstract Nonlocality, which is the key feature of quantum theory, has been linked with the uncertainty principle by fine-grained uncertainty relations, by considering combinations of outcomes for different measurements. However, this approach assumes that information about the system to be fine-grained is local, and does not present an explicitly computable bound. Here, we generalize above approach to general quasi-fine-grained uncertainty relations (QFGURs) which applies in the presence of quantum memory and provides conspicuously computable bounds to quantitatively link the uncertainty to entanglement and Einstein–Podolsky–Rosen (EPR) steering, respectively. Moreover, our QFGURs provide a framework to unify three important forms of uncertainty relations, i.e., universal uncertainty relations, the uncertainty principle in the presence of quantum memory, and fine-grained uncertainty relation. This result gives a direct significance to uncertainty principle, and allows us to determine whether a quantum measurement exhibits typical quantum correlations, meanwhile, it reveals a fundamental connection between basic elements of quantum theory, specifically, uncertainty measures, combined outcomes for different measurements, quantum memory, entanglement and EPR steering.
Abstract Multipartite information principles are needed to understand nonlocal quantum correlations. Towards that end, we provide optimal bounds on genuine multipartite nonlocality for classes of THRESHOLD games using the LOCCG (Local Operations and Classical Communication with Grouping) model. Our proof develops a reduction between multipartite nonlocal and biased bipartite nonlocal games. Generalizing this reduction to a larger class of games may build a bridge from multipartite to bipartite principles.
It is well known that measurements performed on spatially separated entangled quantum systems can give rise to correlations that are nonlocal, in the sense that a Bell inequality is violated. They cannot, however, be used for superluminal signaling. It is also known that it is possible to write down sets of ``superquantum'' correlations that are more nonlocal than is allowed by quantum mechanics, yet are still nonsignaling. Viewed as an information-theoretic resource, superquantum correlations are very powerful at reducing the amount of communication needed for distributed computational tasks. An intriguing question is why quantum mechanics does not allow these more powerful correlations. We aim to shed light on the range of quantum possibilities by placing them within a wider context. With this in mind, we investigate the set of correlations that are constrained only by the no-signaling principle. These correlations form a polytope, which contains the quantum correlations as a (proper) subset. We determine the vertices of the no-signaling polytope in the case that two observers each choose from two possible measurements with ddd outcomes. We then consider how interconversions between different sorts of correlations may be achieved. Finally, we consider some multipartite examples.
The multipartite correlations derived from local measurements on some composite quantum systems are inconsistent with those reproduced classically. This inconsistency is known as the quantum nonlocality and shows a milestone in the foundations of quantum theory. Still, it is NP hard to decide a nonlocal quantum state. We investigate an extended question as to how to characterize the nonlocal properties of quantum states that are distributed and measured in networks. We first prove the generic tripartite nonlocality of chain-shaped quantum networks using semiquantum nonlocal games. We then introduce an approach to prove the generic activated nonlocality as a result of entanglement swapping for all the bipartite entangled states. The result is further applied to show the multipartite nonlocality and activated nonlocality for all the nontrivial quantum networks consisting of any entangled states. Our results provide the nonlocality witnesses and quantum superiorities of all the connected quantum networks or nontrivial hybrid networks in contrast to classical networks.