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International Journal of Advanced Mathematical Sciences

Review on investigating the possibility of local hidden variable theories-quantum teleportation

Dorcas Addo, Steven Adam Abel, Richard Kwame Ansah, Isaac Nkrumah

Published Jan 14, 2018• 1 citation
Publisher

Abstract

The core of the paper was to investigate the possibility of local hidden variable theory and its application in quantum teleportation. We reviewed literature on the Bell's inequality which is necessary for quantum teleportation. Quantum teleportation utilises a single-particle entangled state which can be successfully achieved by the application of the locality assumption which leads to Bell's inequality. A violation of the Bell's inequality signifies the nonlocal nature of a single particle useful for quantum teleportation.

DOI
10.14419/ijams.v6i1.8755

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The core of the paper was to investigate the possibility of local hidden variable theory and its application in quantum teleportation. We reviewed literature on the Bell's inequality which is necessary for quantum teleportation. Quantum teleportation utilises a single-particle entangled state which can be successfully achieved by the application of the locality assumption which leads to Bell's inequality. A violation of the Bell's inequality signifies the nonlocal nature of a single particle useful for quantum teleportation.
Entangled states whose Wigner functions are non-negative may be viewed as being accounted for by local hidden variables (LHV). Recently, there were studies of Bell's inequality violation (BIQV) for such states in conjunction with the well known theorem of Bell that precludes BIQV for theories that have LHV underpinning. We extend these studies to teleportation which is also based on entanglement. We investigate if, to what extent, and under what conditions may teleportation be accounted for via LHV theory. Our study allows us to expose the role of various quantum requirements. These are, e.g., the uncertainty relation among non-commuting operators, and the no-cloning theorem which forces the complete elimination of the teleported state at its initial port.
Recently, Palazuelos presented the first example of super-activation of quantum nonlocality. More precisely, by combining several copies of a state rho admitting a local hidden variable model, it becomes possible to violate a Bell inequality, in a deterministic way. Here we show that super-activation of quantum nonlocality is in fact a rather generic phenomenon. In particular, we show that any d x d entangled state with entanglement fraction larger than 1/d can be super-activated. Remarkably this implies that all entangled states which are useful for teleportation are nonlocal resources, hence providing a direct link between teleportation and nonlocality.
The violation of Bell’s inequality in quantum mechanics implies that there exist nonlocality and entanglement. When the density matrix of a composite system cannot be written as a convex combination of the product of the density matrices of its subsystems, we say there exists entanglement. For pure states, the existence of entanglement always leads to the violation of Bell’s inequality. However, in the case of the mixed states, there may be entanglement, but Bell's inequality is not violated and in other words, the nonlocality is not manifested. In addition to Bell's inequality, quantum teleportation is also a manifestation of nonlocality. Quantum teleportation using entangled states is more successful than quantum teleportation with separable states. Therefore, the corresponding fidelity of teleported state with the initial state (in short, the fidelity) of the former is always greater than the fidelity of the latter. In this paper, for Werner's state, we will show that in a range of the related parameter, while the CHSH inequality is violated, the fidelity, which indicates the amount of success of quantum teleportation, is lower than the upper bound of the corresponding fidelity for states that can be simulated with a local hidden variable theory. Meanwhile, we will see that for Gisin's state with hidden nonlocality, filtering, which leads to the appearance of nonlocality and more specifically leads to the violation of the CHSH inequality, also increases the fidelity.
While entanglement and violation of Bell inequalities were initially thought to be equivalent quantum phenomena, we now have different examples of entangled states whose correlations can be described by local hidden--variable models and, therefore, do not violate any Bell inequality. We provide an up to date overview of the existing local hidden--variable models for entangled quantum states, both in the bipartite and multipartite case, and discuss some of the most relevant open questions in this context. Our review covers twenty five years of this line of research since the seminal work by Werner [R. F. Werner, Phys. Rev. A 40, 8 (1989)] providing the first example of an entangled state with a local model, which in turn appeared twenty five years after the seminal work by Bell [J. S. Bell, Physics 1, 195 (1964)], about the impossibility of recovering the predictions of quantum mechanics using a local hidden--variables theory.