Quantitative Research On Contextuality Of Empirical Models

As a generalization of quantum nonlocality,quantum contextuality is an important resource for quantum computing and quantum information process-ing,and so it becomes a hot topic in quantum information theory.The contextuality of empirical models is the mathematical frame of quantum contextuality.Using the theory and method of operator theory,operator algebra,and matrix theory,this dis-sertation systematically researches a series of important theoretical problems,such as the robustness of contextuality of empirical models,the generalized robustness of contextuality of empirical models,the contextuality cost of empirical models,quan-tifying contextuality of empirical models,proposing a more efficient contextuality distillation protocol,and so on.In Chapter 1,we introduce some research background and status on our main contents,list some related notations,definitions and known conclusions,and give the main results of this paper.In Chapter 2,we research the empirical model and the contextuality of it.At first,for a given measurement scenario,some algebraic and topological properties of the set of empirical models,the set of no-disturbance empirical models and the set of non-contextual empirical models are discussed.Then it is proved that the 4-cycle measurement scenario which can reveal contextuality of some two qubit quantum state must be "traceless".Finally,a measure of contextuality of any two qubit quantum state is introduced,which quantifies the maximal contextuality of a two qubit quantum state for 4-cycle measurement scenarios.Moreover,it is obtained that the measure of contextuality of a two qubit quantum state is zero if and only if any 4-cycle measurement scenario can not reveal contextuality of the quantum state.What is worth noting is that the measure of contextuality of a two qubit quantum state is entirely determined by the spectrum of the state,which says that if and only if eigenvalues of the quantum state satisfy the constraint condition,it follows that there exists a 4-cycle measurement scenario such that the quantum empirical model induced by the state and the measurement scenario is contextual.In Chapter 3,we study the robustness of contextuality and the contextuality cost of an empirical model,and discuss the relationship between the two.Firstly,motivated by the robustness of entanglement and the robustness of quantum cor-relations,the robustness of contextuality of an empirical model is introduced,and then it is proved that the robustness of contextuality can be used to distinguish the non-contextual empirical models from contextual ones.Based on the fact that empirical models are a family of probability distributions and the outcomes of mea-surements may have errors,the boundedness and continuity of the robustness of contextuality are proved over the set of all no-disturbance empirical models.For n-cycle boxes which are special empirical models,the robustness of contextuality of them is computed,and then the relationship between the robustness of contex-tuality and the violation level of non-contextual inequalities is established.Finally,the cost of contextuality is introduced,and it is shown that the measure can be used to distinguish the non-contextual empirical models from contextual ones,the empirical model is strongly contextual if and only if its contextuality cost is the maximal value 1,and the relationship between the robustness of contextuality and the contextuality cost is discussed.In Chapter 4,we investigate the generalized robustness of contextuality.At first,based on the research on the generalized robustness of entanglement,two mea-sures are introduced,which are the generalized relative robustness of contextuality of an empirical model e with respect to an empirical model f and the generalized robustness of contextuality of e.It is obtained that the generalized robustness of contextuality can be used to distinguish the non-contextual empirical models from contextual ones.Different from the robustness of contextuality,the generalized ro-bustness of contextuality is finite for any empirical model.Above all,as long as the generalized relative robustness of contextuality of e with respect to an empirical model f is finite,the product of the generalized robustness of contextuality of the two empirical models is less than or equal to 1,this shows that the generalized ro-bustness of contextuality of the two empirical models can not be large at the same time.Finally,for n-cycle boxes,the relationship between the generalized robustness of contextuality and the robustness of contextuality is established.In Chapter 5,we present a quantity of contextuality of empirical models to quantify contextuality of them.Firstly,a family of classical correlated quantum states associated with an empirical model e is introduced,and then it is shown that it is non-contextual if and only if there exists a product state in these classical correlated quantum states.In terms of trace distance,a quantity of contextual-ity of an empirical model is introduced,and many properties of the quantity of contextuality are shown,such as it can be used to distinguish non-contextual em-pirical models from contextual ones,it is convex and continuous,and so on.For the Peres-Mermin-isotropic boxes PM?,Mermin-isotropic boxes M?,and CHn-isotropic boxes CHn?,based on the sufficient conditions for contextuality of them,the necessary and sufficient conditions of them being contextual are obtained,and the quantity of contextuality of any two of them can be easily compared.In Chapter 6,we discuss a more efficient contextuality distillation protocol.At first,focused on the generalized correlated n-cycle boxes en,?,?,it is proved that the contextuality of them can be quantified by the parameter ?.Then a contex-tuality distillation protocol*is proposed,which can increase ? and so can distill every contextual n-cycle box en,?,?.The most important thing is that the con-textuality distillation protocol*can distill any contextual en??,? to the maximally contextual n-cycle box CHn as the times of iteration goes to infinity.Compared with the known distillation protocols for non-locality and contextuality,the con-textuality distillation protocol*is optical in terms of its distillation rate,has the strongest appropriate ability,and can witness more 4-cycle boxes which can make communication complexity trivial.