Bell Inequalities Based On Group Actions

With the rapid rise of quantum information theory, quantum information theory has a great potential application in science, technology, production and life. it has become a frontier interdiscipline in mathematics, physics, especially the application of Bell in-equalities. It not only can be used to test quantum states whether or not are quantum entanglement, but also ensures the security and stability in quantum cryptography. So considering Bell inequalities has a huge and profound significance. Bell inequalities are a key issue in quantum information. Exploring the connection between Bell inequali-ties and the basic interpretation of quantum mechanics has a profound understanding in some basic problems of quantum entanglement, quantum nonlocality, and quantum information extraction and so on.The problems of the thesis researching are Bell inequalities based on group actions. The main results are showed in the following:1. Cd (?) Cd. First, we use the same idea of the paper [29] to construct a Bell inequality. Then, we want to transfer the maximum value which violated Bell inequality into the largest eigenvalue of the matrix A to calculate it. Next, we consider the two case of C2d’(?) C2d(d=2d’) and C2d’-1(?)C2d’-1(d=2d’-1,d’>2), then we have given and showed that a general form of the maximum value which violated Bell inequality, and a general form of a quantum state which maximally violated Bell inequality, respectively. Finally, we give some examples to illustrate that these results agree with the numerical results presented by Mark Hilleryet al. if measuring the quantum state is the eigenvector correspoding to the largest eigenvalue of F, Bell inequality can reach the the maximum value which violated Bell inequality, that is the largest eigenvalue of F. This indicates that Bell inequality is maximally violated by the quantum state, that is to say the quantum state has a definite quantum correlation.2. Cd(?)Cd(?)Cd. First, we use the same idea of chapter III in this paper to study Bell inequality based on group actions, but only considering the sets of the probabilities of four orbits is a little different. Then we use the condition that if the probabilities of these four orbits was considered by quantum mechanics, and the sum of the probabilities of these four orbits will have an upper bound. Next, we want to transfer the upper bound into the largest eigenvalue of the matrix F to calculate it, and then which is transfered into the largest eigenvalue of the matrix 2dM(λ). Finally,the general form of M(λ)is given in both C2d’-1(?)C2d-1(?)C2d’-1(d=2d’-1,d’≥2)and C2d’(?)C2d’(?)C2d’(d=2d’), respectively. Using the exampleC3(?)C3(?)C3 illstrates that the upper bound violates Bell inequality.