Generic Bell Inequalities for Multipartite Arbitrary Dimensional Systems

Abstract

We present generic Bell inequalities for multipartite arbitrary dimensional systems. The inequalities that any local realistic theories must obey are violated by quantum mechanics for even dimensional systems. A large set of variants are shown to naturally emerge from the generic Bell inequalities. We discuss particular variants of Bell inequalities, that are violated for all the systems including odd dimensional systems.PACS numbers: 03.65. Ud, 03.65.Ta, Quantum nonlocality is the most significant evidence of physical observations that cannot be explained by theories based upon local realism. Local realism is rooted in the classical view of measurement, namely that an observation on one of a pair of subsystems cannot affect the other system faster than the speed of light. In fact, since the advent of quantum mechanics, the nonclassical implications have given rise to fundamental questions on the nature of the act of measurement. In quantum mechanics, a measurement does not provide a preexisting value of the system, rather, it is a manifestation of the state of the probed system and the probing apparatus, as advocated by Mermin [1].As early as 1964, Bell [2] proved that local realism implies constraints on a correlation of measurements between two separate systems. These constraints are incompatible with the quantitative predictions by quantum theory in the case of two coupled spin-1/2 particles. These constraints, expressed as so-called Bell inequalities, are of paramount importance in the conceptual foundations of quantum mechanics. But these are idealised experiments with archetypal non-classical system. A key question remains whether a complex system of high-dimensional quantum subsystems could eventually simulate a pseudo-classical system that does not contradict local realism.Since this startling discovery [2], investigating Bell theorem for a general system has been regarded as one of the most important challenges in quantum mechanics and quantum information science (QIS) [3,4,5,6,7,8,9,10,11,12,13,14,15]. The motivation is obvious from a scientific and technological viewpoint. Firstly, proving Bell theorem for a general system, would show that quantum physics would apply to macroscopic complex systems. Secondly, for QIS to outperform the acquisition, manipulation and transmission of information over its classical counterpart, the property of nonlocality is closely related to its extraordinary power. In fact, the manipulation of a complex quantum system rather than a simple one has practical advantages. Gaining access to the states is easier and more efficient (for example, efficient cluster state QIS [16] and increased security in high-dimensional quantum cryptography [17]). It follows that a nonlocality test for such a complex system is highly desirable. Thirdly, the controllability of quantum operations depends on the nature of the system. Certain logical operations are relatively easy for one system but impossible or difficult for another. It is then essential to understand the nonlocal properties of ...

Entanglement, EPR steering, and Bell-nonlocality criteria for multipartite higher-spin systems

“…This result differs from that of Collins et al [24] for N = 2, who obtained steady violation for increasing d for the maximally entangled states. Cabello [28] also reported a steady violation with increased d with any fixed N , but this effect was not observed by the violations of the inequalities of Son et al [29]. …”

“…Where we have fixed dimensionality, i.e. a spin-J system, the quantum uncertainty relation (17) can be used to derive different entanglement and steering inequalities beyond those of (25)- (29). In particular there will be a quantum uncertainty relation of the form [45] …”

“…More recent work has confirmed that the violation of LHV in the bipartite case increases or is constant with dimensionality d [23][24][25][26][27] and the CGLMP Bell inequalities (of Collins, Gisin, Linden, Massar, and Popescu) [24] have been shown to be tight. Multi-site qudits have been examined by Cabello [28] and Son et al [29], who extended the MABK inequalities, and Chen et al [30], who developed tight inequalities similar to the CGLMP inequalities for bipartite qudits. Results from these authors indicate that the high-dimensional violations are steady, or exponentially increasing, with the number of sites N .…”

“…On the theoretical side, much work has been done for Bell's nonlocality on multi-site qubits [15][16][17], and bipartite qudits [18][19][20][21][22][23][24][25][26][27], though relatively little so far on multipartite systems of higher dimensionality (qudits) [28][29][30]. Our interest here is to explore these different types of "largeness," i.e., many sites, many dimensions, and the combination of both.…”

Bell-inequality test for spatial-mode entanglement of a single massive particle

“…Our results can be generalised to volumes containing N bosons for which multi-dimensional Bell-inequalities exist [30,31]. Indeed, the N particle case has been studied in a different context in [32], where the violation of the CHSH inequality is assigned to the generation of spin entanglement between two condensates due to joint measurements on the condensates by two parties.…”