Form Of Quantum Entropy Associated Metric

Entanglement plays a key role in quantum computation and quantum infor-mation, which is a kind of nonclassical correlation not available in classical world. Recently, it had been shown that there are also quantum tasks that display the quantum advantage without entanglement, for example, the deterministic quantum computation with one qubit. This shows that entanglement does not account for all the nonclassical properties of quantum states. Some different background mea-sures of nonclassical correlation were given, for example, quantum discord, quantum deficit, measurement-induced nonlocality, and so on. We study the fundamental differences and similarities between these measures. We identify and discuss some fundamental properties of these measures. We also study the dynamics of quantum discord of two qubits independently coupled to an Ising spin chain in a transverse field and the communication networks.The main contributions in this thesis are listed as follows:(1) In Chapter2, we revisit the upper bound of quantum discord given by the von Neumann entropy of the measured subsystem. Using the Koashi-Winter rela-tion, we obtain a trade-off between the amount of classical correlation and quantum discord in the tripartite pure states. The difference between the quantum discord and its upper bound is interpreted as a measure on the classical correlative capacity. Further, we give the explicit characterization of the quantum states saturating the upper bound of quantum discord, through the equality condition for the Araki-Lieb inequality. We analyze the optimal measurements to access classical correlation in arbitrary two-qubit system. Then, we define the minimum quantum discord on the correlation direction, and demonstrated that it could be a good approximation of quantum discord.(2) In Chapter3, we present the concept of the one-way unlocalizable quantum discord and investigate its properties, for example, the lower bound, the upper bound and the additivity. We provide a polygamy inequality for it in tripartite pure quantum system of arbitrary dimension. Several trade-offs between the one-way unlocalizable quantum discord and other correlations are given. Finally, we also provide another lower bound for bipartite shareability of quantum correlation beyond entanglement in a tripartite system.(3) In Chapter4, we quantify the measurement-induced nonlocality from the perspective of the relative entropy. This quantification leads to an operational interpretation for the measurement-induced nonlocality, namely it is the maximal entropy increase after the locally invariant measurement. The relative entropy of nonlocality is upper bounded by the entropy of measured subsystem. Based on quantum side information, we show that the sum between the relative entropy of nonlocality and the minimal quantum side information is equal to quantum mutual information. Several trade-off can be obtained by quantum side information and missing information. We also show that relative entropy of nonlocality is equal to the maximal distillable entanglement between the measurement apparatus and the system. We establish relationship between the relative entropy of nonlocality and the geometric nonlocality based on the Hilbert-Shcmidt norm. We also obtain analytical results of the relative entropy of nonlocality for Bell-diagonal states.(4) In Chapter5, we investigate the so-called one-way quantum deficit, which is a measure of quantum correlation emerging from a thermodynamical approach. We show that the one-way quantum deficit is the upper bound of the von Neumann entropy of the measured system, and give a sufficient conditions for saturating upper bound. Then, we compare the one-way quantum deficit with quantum discord, and give an example to explain their difference. We revisit quantum measurement when the apparatus is initially in a mixed state. We find that the amount of entanglement between the system and the apparatus is given by the increasing entropy of the system under the measurement transformation. We show that the information gained is equal to the amount of entanglement under performing perfect measurement. Finally, we introduce various correlations in terms of the relative entropy.(5) In Chapter6, we study the dynamics of quantum discord of two qubits independently coupled to an Ising spin chain in a transverse field, which exhibits a quantum phase transition. In the vicinity of the critical point, the evolution of quantum discord is shown to be more complicated than that of entanglement. Furthermore, we find that separable states can also be used to reflect the quantum criticality of the environment. Second, we also study the dynamics of quantum discord of two qubits independently coupled to the communication network channel, which may enhance quantum correlation. Finally, we discuss the dynamics process of the quantum discord for general initial states under several typical quantum channels.