Quantum nonlocality for all the systems

Abstract

To understand quantum nonlocality for all the systems, we present a generalized Greenberger-Horne-Zeilinger (GHZ) paradox and Bell inequalities. First, we consider a generalized Greenberger-Horne-Zeilinger (GHZ) paradox in a tripartite system with each subsystem of arbitrary even dimension. Contrary to conventional approaches of compatible observables, in order to prove Bell`s theorem, we employ concurrent observables, that are mutually incompatible but still have a common eigenstate such that they are involved in elements of physics reality. It is proved that our formulation of the generalized GHZ paradox is genuinely multi-dimensional. The present approach enables an m-partite system to exhibit the truly d-dimensional GHZ paradox for d even and m odd integers. As a special case it covers previous works which require a (d+1)-partite system. Second, we consider generic Bell inequalities for multipartite multi-dimensional systems. The inequalities that all local realistic theories must obey are violated by quantum mechanics for even-dimensional multipartite 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.