| In recent years, many breakthroughs have happened in quantum information, whatever in theory or experiment. It is gaining much attention and widely researches as a kind of interdisciplinary field combined with basic of quantum mechanics, classical information and computer science. Quantum entanglement, as a very useful resource in quantum information tasks, has been showing a huge advantage and potential application prospect in these fields. Recently, however, it was found that quantum entanglement does not represent all non-canonical correlations, so a new measurement system called quantum discord walks into people’s horizons and is has been considered a more useful quantum resource. It has been proven that quantum discord plays a crucial role in quantum computation, quantum phase transition and other related areas. The geometric quantum discord (QGD) as a widely applied measurement system of quantum correlations has gotten a lot of research results. The QGD employs the Hilbert-Schmidt distance, which is defined as Schatten two-norm. Such a distance is well-known, not satisfy the property of being contractive under trace-preserving quantum channels, which is indeed the reason behind the aforementioned drawback of GD. This naturally leads to a redefinition of the GD in terms of a metric that obeys the constriction property. Such a metric is the trace distance, which employs the Schatten one-norm. Here we use trace distance discord (TDD) to measure QGD. Hence, the investigation of TDD dynamics in the various physical systems not only can deepen our understanding for the concept of quantum correlations, but also is valuable for the unique feature of quantum correlations beyond quantum entanglement. This dissertation is committed to studying the performance of quantum correlations in the XXZ model with Dzyaloshinskii-Moriya interaction, by employing the quantum renormalization group (QRG) method. The major results are listed as the following:(1) The dynamical behaviors of the TDD in the quantum phase transition have been studied by exploiting the quantum renormalization group method. From the phase diagram, the anisotropy and DM parameters are the two main roles which govern the trend of the figure line. With importing the DM interaction, the spoiled correlation is restored when compared with the simple model consisting of only three qubits. The reason can be understood that the DM interaction contributes the strong planar quantum fluctuations.(2) With implementing the QRG operation, we can realize the thermodynamic limit of the model. This method not only helps us to get the critical points from the phase diagram of the model but also allows us to keep attention the variation in the correlation as the size of the system become large. The singularity line corresponds to the phase boundary separating the antiferromagnetic from the spin-fluid phases. The scaling behavior characterizes how the critical point of the model is touched as the system is increased. We have also verified that the gapped or gapless nature of a phase is relevant to the crossing behavior close to the quantum phase transition (QPT). The singularity will become more pronounced and touch the quantum critical line from the gapped phase when the size of the system is enlarged. To sum up, as the TDD is a special choice to measure quantum correlation with geometrical measuring, we prove that the QPT of quantum correlation also can be studied in the geometric discord (GD) territory faultlessly.On the other hand, although the study about quantum entanglement is very deep and broad, but nobody cares entanglement in noninertial system, especially in the dissipative environment with temperature parameters. We consider a more general decoherence environment, generalized amplitude damping (GAD) channel, and explore the dynamic of quantum entanglement for Dirac fields under non-zero temperature environment in a noninertial frames. We assume that there are two observers, Alice and Bob, share an initial entangled state at the same point in flat Minkowski spacetime. Alice stays stationary while Bob moves with uniform acceleration. Let Bob move in the noisy environment and discuss the dynamic of quantum entanglement influenced by Unruh effect and the GAD channel. Here we want to know how the entanglement undergoes a sudden death or it disappears as some channel parameters tend to a certain value. With the calculation and graphics drawing, we could get the related results:We find that the concurrence decreases with the increase of acceleration and channel parameter r has a powerful impact on the entanglement. Especially, the entanglement is affected by the dissipation environment sharply when the channel parameter p=0.5. Astonishingly, our result manifests the inequivalence of the quantization for a Dirac field with this given system. We not only expand the investigation to a more general finite temperature environment, but also probe the behavior of quantum entanglement in the noninertial frames. |
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