Quantifying Multipartite Nonlocality
The nonlocal correlations of multipartite entangled states can be reproduced by a classical model if sufficiently many parties join together or if sufficiently many parties broadcast their measurement inputs. The maximal number \(m\) of groups and the min
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Quantifying Multipartite Nonlocality
By performing local measurements on an n-partite entangled state one obtains outcomes that may be nonlocal, in the sense that they violate a Bell inequality [1]. Since the seminal work of Bell, nonlocality has been a central subject of study in the foundations of quantum theory and has been supported by many experiments [2, 3]. More recently, it has also been realized that it plays a key role in various quantum information applications [4, 5], where it represents a resource different from entanglement.1 While nonlocality has been extensively studied in the bipartite (n = 2) and to a lesser extent in the tripartite (n = 3) case, the general n-partite case remains much unexplored. The physics of many-particle systems, however, is well known to differ fundamentally from the one of a few particles and to give rise to new interesting phenomena, such as phase transitions or quantum computing. Entanglement theory, in particular, appears to have a much more complex and richer structure in the n-partite case than it has in the bipartite setting [6, 7]. This is reflected by the fact that multipartite entanglement is a very active field of research that has led to important insights into our understanding of many-particle physics (see, e.g., [8, 9]). In view of this, it seems worthy to investigate also how nonlocality manifests itself in a multipartite scenario. What new features emerge in this context and what are their fundamental implications? How to characterize the nonlocality of experimentally realizable multi-qubit states, such as W states for instance? What role do n-partite nonlocal correlations play in quantum information protocols, e.g., in measurementbased computation [10]?. The vision behind the present paper is that in order to answer such questions and make further progress on our understanding of multipartite nonlocality, one should first find ways to quantify it. Motivated by this idea, we introduce two simple measures that quantify the multipartite extent of nonlocality.
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This chapter appeared as: “J.-D. Bancal, et al., Quantifying Multipartite Nonlocality, Phys. Rev. Lett. 103, 090503 (2009).” J.-D. Bancal, On the Device-Independent Approach to Quantum Physics, Springer Theses, DOI: 10.1007/978-3-319-01183-7_5, © Springer International Publishing Switzerland 2014
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5 Quantifying Multipartite Nonlocality
(a)
(b)
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Fig. 5.1 Different groupings of n = 4 parties into m groups. Within each group, every party can communicate to any other party, as indicated by the arrows. a If all parties join into one group (m = 1), they can achieve any correlations. b, c If they split into m = 2 groups they can realize some non-local correlations but not all. d If they are all separated (m = n), they can only reproduce local correlations
A natural way to characterize nonlocality is to attempt to replicate it using models where some non-local interactions (such as communication) are allowed between some parties. The first measure that we consider is based on classical
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