Bell's inequality and the Schmidt decomposition

A. Mann, Barry C. Sanders, William J. Munro

Given an entanglement of two systems involving nonorthogonal states, we find the Schmidt decomposition for the state. The relation between the Schmidt representation and an ideal measurement of the degree of entanglement of the states is discussed, and a Bell inequality is shown to be violated. The maximal violation of the Bell inequality provides a measurement of the degree of entanglement. The entangled coherent states are provided as a concrete example of the Bell inequality for entangled nonorthogonal states.

B. Stoklasa, L. Motka, Jaroslav Řeháček +3 more

Optical beams with topological singularities have a Schmidt decomposition.Hence, they display features typically associated with bipartite quantum systems; in particular, these classical beams can exhibit entanglement.This classical entanglement can be quantified by a Bell inequality formulated in terms of Wigner functions.We experimentally demonstrate the violation of this inequality for Laguerre-Gauss (LG) beams and confirm that the violation increases with increasing orbital angular momentum.Our experimental scheme, which is designed to give directly the parity of the Wigner function, yields negativity at the origin for LG 10 beams, whereas for LG 20 we always get a positive value.

Keith Hannabuss

Riccardo Manenti, Mário Motta

Abstract This chapter describes entanglement in bipartite systems. We define entanglement for pure states introducing the concept of Schmidt decomposition, and we show that entanglement can be quantified through the von Neumann entropy. We then present dense coding and teleportation, two quantum information protocols that use entanglement as a resource. We present the notions of locality and realism and derive the Clauser-Horne-Shimony-Holt inequality, an extended version of Bell’s inequalities. We then focus on mixed states and we present different techniques to solve the separability problem

Pierre A. Millette