Abstract For the well-known relative entropy of coherence (REC), a quantum state attains maximum coherence only in optimal bases, which are mutually unbiased bases to the eigenvectors of the given state. In this paper, we explore the conditions of obtaining the maximum of REC under noisy channels and explain the geometric significance of optimal basis form the viewpoint of the Bloch sphere. We also show that the eigenvectors of the state will remain constant throughout the whole process of evolution, as long as the effects of channels induce a synchronous change for the components of the initial state Bloch vector. Furthermore, as a special example, we investigate the dynamics of the maximum REC for any finite dimensional single quantum system under a depolarizing (DP) channel, and obtain an analytical form of maximum coherence for a d -dimensional maximally coherent state. Meanwhile, a concise analytical expression of maximum coherence for two classes of an N -qubit system under the DP channel is also provided.
The uncertainty relation reveals the intrinsic difference that distinguishes the quantum world from the classical world. In this paper, the quantum uncertainty relations of the sandwiched R\'enyi relative entropy of coherence (SRREOC) and unified $(\ensuremath{\alpha},\ensuremath{\beta})$-relative entropy of coherence (UREOC) are studied. First, the uncertainty relation of the SRREOC for pure states, with extension to mixed states by convex-roof construction, is given in a
d-dimensional quantum system. Second, the analytical formula of the UREOC is presented; it is shown that the UREOC is a good candidate for a quantum coherence measure. Furthermore, the quantum uncertainty relation of the UREOC for pure states is obtained in a
d-dimensional quantum system, then it is generalized to mixed states. In addition, our results cover the quantum uncertainty relations of the other three coherence measures. Third, two methods are given to compare the lower bounds of the SRREOC and UREOC for single-qubit pure states. By comparison, we find that the results in this paper not only can be applied to single-qubit states but also can be extended to a
d-dimensional quantum system.
The dynamics of entanglement of two superconducting charge qubit system interacting with a squeezed coherent state were examined by means of linear entropy. The influence of the relative phase of the two qubits' levels and squeezing parameter on the time evolution of linear entropy was discussed. The results showed that the linear entropy almost tends to zero at half the revival time when the two qubits were initially in the maximally entangled state. It was also showed that a greater number of oscillations of the linear entropy were observed with the increasing of squeezing parameter and the sustainment time of the maximum entropy was increased. There was no change in the maximum value of the linear entropy. Moreover,the relative phase of the two qubits' levels decreased the degree of entanglement between the two qubits and the field.