Quantum Correlation And Its Dynamics

Quantum information has attracted much attention since it has many advan-tages over classical information. Quantum correlation as an important resource in quantum information has important applications in quantum computation, quan-tum phase transition, and broadcasting of quantum states. However, any real quan-tum system inevitably interacts to some extent with its environment, then quantum decoherence can occur if the process is irreversible. Therefore, investigation of the dynamics of quantum correlation is important and necessary.In this thesis, we investigate quantum correlation and its dynamics. The main results are as follows:First, we investigate completely positive maps for an open system interacting with its environment in the presence of initial correlations. In this situation, we have to restrict the set of initial states. Therefore, we consider the set of initial states that share one common completely positive map within the framework of direct sum decomposition of state space and the framework of assignment maps. In the framework of direct sum decomposition of state space, a general expression of the initial states are explicitly given, and we prove that the reduced dynamics of the open system can be described by a completely positive map for arbitrary unitary operator USE(t) as long as the initial states of the combined system are with the structure which we have given. The set of initial states includes not only separable states with vanishing or nonvanishing quantum discord but also entangled states. It significantly extends the previous results which can be taken as special cases of our results. In the framework of assignment maps, we prove that the set of initial states given in the framework of direct sum decomposition of state space is a necessary and sufficient condition for2×N quantum systems of which the reduced dynamics is a completely positive map for arbitrary unitary operator USE(t). We also give the expression of assignment maps which are completely positive for the initial states of M x N quantum systems. These results mean that the framework of direct sum decomposition of state space and the framework of assignment maps are consistent.Second, we investigate the dynamics of quantum correlations for bipartite quan-tum systems. We investigate the dynamics of geometric measure of quantum discord of two qubits in two independent reservoirs and one common reservoir, respectively, where the initial states of the two qubits are Bell-diagonal states. When the two qubits are in two independent reservoirs, the dynamics of geometric measure of quantum discord for the reservoirs with sub-Ohmic (0<s<1) and Ohmic (s=1) spectral densities are similar to each other, and the dynamics of geometric measure of quantum discord may be regarded as three classes:a monotonic decreasing func-tion of time t; a piecewise monotonic decreasing function with one turning point, i.e., having a sudden transition phenomenon; a constant in a finite time interval be-fore becoming a monotonic decreasing function, i.e., having a frozen phenomenon. However, for the super-Ohmic reservoirs with1<s<2and T=0, there is one more class of the dynamics of geometric measure of quantum discord. That is, the dynamics of geometric measure of quantum discord may keep being a constant for all the time, i.e., being frozen forever. The conditions satisfied by the initial states are given for all the four classes. When the two qubits are in one common reser-voirs, there exists a decoherence-free subspace, and then the dynamics of geometric measure of quantum discord is frozen forever. Besides, the dynamics of geometric measure of quantum discoid for the reservoirs with sub-Ohmic (0<s<1), Ohmic (s=1) spectral densities, and super-Ohmic (1<s<2) spectral densities at the temperature T=0are similar to one another, and may be regarded as five classes: a monotonic decreasing function of time t; a monotonic increasing function of time t; a monotonic decreasing function of time t at first and at some time turning to a monotonic increasing function of time t; a monotonic increasing function of time t at first and at some time turning to a monotonic decreasing function of time t and also having a sudden transition phenomenon; a piecewise function with one turning point and having a sudden transition phenomenon, in the first piece, it is a monotonic decreasing function of time t at first and at some time turning to a monotonic increasing function of time t, and in the second piece, it is a monotonic decreasing function of time t. The conditions satisfied by the initial states are given for all the five classes.Third, we investigate the dynamics of quantum correlations for multipartite quantum systems. We consider the dynamics of q-global quantum discord and ge-ometric global quantum discord of a class of n-qubt states which undergo a local bit flip channel, a bit flip and phase flip channel, a phase flip channel, and a de-polarizing channel, respectively. Our results show that the dynamics of q-global quantum discord and geometric global quantum discord are symmetric with respec-t to p=0.5, and they also obtain the minimum value zero at p=0.5. The sudden transition phenomenon of q-global quantum discord and geometric global quantum discord occurs at the same conditions, q-global quantum discord may have a frozen phenomenon if q=1and n is even. However, geometric measure of quantum dis-cord may have a frozen phenomenon for all n. When the time parameter p remains the same value, q-global quantum discord at first is a monotonic increasing function of parameter q and then becomes a monotonic decreasing function of parameter q. The dynamics of q-global quantum discord and geometric global quantum discord for the states which undergo a local bit flip channel or a local bit flip and phase flip channel or a local phase flip channel are similar to each other. In the depolarizing channel, the dynamics of q-global quantum discord and geometric global quantum discord obtain the minimum value zero at p=0.75, and they are a monotonic de-creasing function of parameter p at first and then become a monotonic increasing function, but in this channel there is no sudden transition phenomenon or frozen phenomenon.