Violations Of Bell Inequalities Under Random Pure States

Bell inequality is an important tool to distinguish classical correlations and quantum correlations.It has extensive applications in communication complexity,information theory and quantum cryptography.On the other hand,the involving of random ensembles provides new tools and approaches in dealing with many problems in quantum information theory.In this thesis,we will study the violation of Bell inequalities under some given random pure states.More precisely,we will study the following four types of Bell inequalities: CHSH-Bell inequality,Parameter CHSH inequality,Chained CHSH inequality and 3322-Bell inequality.To study the violation of given Bell inequalities,we will use two kinds of random ensembles,which are widely used in random matrix theory.One is the Hilbert-Schmidt ensemble,another is the structured ensemble,which depends on superposition multiplicity of the maximally entangled state.For CHSH-Bell inequality,Parameter CHSH-Bell inequality and Chained CHSH-Bell inequality,fixing the optimal measurement operators(in this sense,the Bell operators are fixed),we will consider the expectation values of the optimal Bell operators under the given random ensembles.The above three kinds of Bell inequalities are violated when the pure states are randomlized by the Hilbert-Schmidt ensemble.For the structured ensemble,the expectations of Bell operators,decrease as superposition multiplicity of the maximally entangled state goes to large.For Parameter CHSH-Bell inequality,we find violations,while for CHSH-Bell inequality and Chained CHSH-Bell inequality,we do not.For 3322-Bell inequality,the situation becomes complex since the optimal measurement operators are still unknown.By using the approach in [31],we will fix the dimension,and numerically we will find the measurement operators which might provide large violations under random ensembles.Fortunately,we find violations both by using the Hilbert-Schmidt ensemble and the structured one.Moreover,in the case of the structured ensemble,the violations increase as superposition multiplicity of the maximally entangled state goes to small.Finally,we note that the expectation value(in two random ensembles)of Bell operators increases as the dimension goes to large.It is interesting since people believe that optimal violation of 3322-Bell inequality is obtained in infinite dimension.