The Theoretical Study Of State-independent Quantum Contextuality In Four-dimensional Space

Quantum correlations,such as quantum entanglement and quantum discord,are the basic resources of quantum computation and quantum communication.In recent years,They have attracted much attention of many reseachers.Quantum contextuality is also an important quantum resource which is a characteristic property of quantum physics and means it is impossible that the measuring results of a quantum system are independent of the context.In contrast,the noncontextual hidden variable theorem holds that the results are independent of context.Contextuality is first revealed by Kochen-Specker(KS)the-orem.The theorem shows that in a Hilbert space of d?3,it is impossible to assign a definite value of 1 or 0 to each projection operator?P_m.The proof of KS theorem involves117 three-dimensional vectors(rays)and the number of vectors is gradually reduced to 18four dimensional(4D)vectors.Yu and Oh prove quantum contextuality by using 13 three dimensional vectors.However,in the testing contextual experiments by using Yu-Oh set,four extra vectors have to be added to ensure that each measurement vector is in a set of orthogonal complete basis.In this paper,we first show a state-independent proof of KS theorem by constructing a set of 40 vectors in 4D space and then derive the optimal inequality.Peres set of 24 vectors is a subset of our 40-vector set which can present a state-independent proof of KS theorem.We also derive the optimal inequality of the Peres set.The maximal ratio between the prediction of quantum mechanics and the classical bound with our set is greater than that with 13 vectors in three-dimensional space.Compared with the Yu-Oh set and Peres set the ratio corresponding to our set is the largest one which implies a larger deviation from the classical bound.Our optimal inequality is more robust-ness against possible imperfections of experiments and suitable to test quantum violation in contextual experiments.A hypergraph is a set of vertices(points)and a set of edges(linked points with seg-ments).The Greechie graph is a special case of it.In order to graphically represent the contextuality we show the contextual set of orthogonal bases in the hypergraph,the vec-tors of an orthogonal base are linked with maximal smooth curves or line segments.There are many publications about experimentally testing contextuality.As we know,in the contextual experiments of the sequential measurement,a critical assumption is that the probabilities of the subsequent measurement outcomes will not be affected by previous measurements.If the no-signaling condition is not satisfied,the probability measured is dependent of the context,which is unreasonable because one need to consider the influ-ence of sequential measurement.In other words,the bound of noncontextual inequal-ity has no physical meaning unless the no-signaling conditon is satisfied in the measure-ments.But it is difficult in satisfying the no-signaling condition in the contextual experi-ments.Here,we provide the no-signaling condition between the successive measurements within the range of experimental accuracy for our inequality and the orthogonal basis sets{1,2,3,4},{5,6,7,8},{9,10,11,12},{13,14,15,16},{17,18,19,20},{21,22,23,24},{25,26,27,28},{29,30,31,32},{33,34,35,36},{37,38,39,40}are listed for experimen-tally testing.Our work has a theoretical foundation for the future work.