† Corresponding author. E-mail:
Herein, we present an approach to look for the best phenomenon to measure quantum correlation. The system of two isolated qubits each interacting with a single-mode cavity was theoretically created to study the quantum correlation. Some of the phenomena, such as the quantum discord and concurrence, were generated through such a system. The influences of initial state purity, qubit motion, and detuning parameters were discussed for the phenomena. These parameters for a specific value show that the behavior of phenomena are analogous. It is interesting to mention that some values of detuning undergo a sudden death of phenomena, and the quantum discord still captures the qubits quantum correlation. We predict that the quantum discord may be a better measure of quantum correlation than concurrence.
In recent years, the main resource for tasks of quantum information processing was quantum correlations (QCs). One of the important kinds of QCs was the quantum entanglement. Entanglement played a main role in quantum information and computation,[1–7] quantum state teleportation,[8] and quantum cryptography.[9] Entanglement was observed for some theoretical works[10] and experimental researches.[11] Some of these works were mixed separable states and allowed us to show the quantum algorithms advantages over their classical analogies. The classical algorithms were observed for some systems without entanglement.[12,13]
Besides, another important type of QCs was firstly defined by Ollivir[14] called quantum discord (QD). QD was used to quantify all nonclassical correlation in a system. Relative to the definition of mutual information, QD explains that two classical equivalent ways turn out to be inequivalent in the quantum domain. It has been previously mentioned that QD is an independent one and a more general measure of QCs that may include entanglement. Quantum states with nonclassical correlations were provided by a larger region through QD, and a nonzero QD but not entanglement may be responsible for the efficiency of quantum computing.[15–17] Such observations were suggested that QD could be another asset for quantum information processing.
Recently, some works were studied for QD[18–20] but the analytical calculation of QD was comparatively difficult. On the other hand, the authors in Ref. [21] obtained the analytical formulas of QD for a family of two-qubit states. For more general quantum states, the authors in Refs. [22], [23], [24], and [25] derived some straightforward expressions to obtain QD, and concluded that QD, classic correlation, and entanglement were independent measures of correlations.
This paper is organized as follows. Section
In the present work, we employ the interaction system of two identical moving qubits (A and B) or atoms with two single-mode SU(1, 1) cavities (a and b). We assume that the system of each qubit–cavity is isolated. This means that the atom “A” at a constant velocity only interacted with a field “a”, and the atom “B” is only interacted with a field “b”, see Fig.
The time-depended Hamiltonian for such a system is given by
![](cpb_27_10_100201/cpb_27_10_100201_ieqn1.gif)
![](cpb_27_10_100201/cpb_27_10_100201_ieqn2.gif)
![](cpb_27_10_100201/cpb_27_10_100201_ieqn3.gif)
![](cpb_27_10_100201/cpb_27_10_100201_ieqn4.gif)
![](cpb_27_10_100201/cpb_27_10_100201_ieqn5.gif)
![](cpb_27_10_100201/cpb_27_10_100201_ieqn6.gif)
![](cpb_27_10_100201/cpb_27_10_100201_ieqn7.gif)
![](cpb_27_10_100201/cpb_27_10_100201_ieqn8.gif)
![](cpb_27_10_100201/cpb_27_10_100201_ieqn9.gif)
To compute the density operator analytically, which will employ searching for the best measure of quantum correlation, we start by calculating the time evolution of system (
![](cpb_27_10_100201/cpb_27_10_100201_ieqn10.gif)
Now, to find the better measure of quantum correlation between the two moving and non-moving qubits of the present system, we study some of the phenomena, such as quantum discord and entanglement.
Here, to investigate the entanglement of two qubits, we study the concurrence (C) measure,[26] which is considered as an effective approach to quantify entanglement and is defined as follows:
![](cpb_27_10_100201/cpb_27_10_100201_ieqn11.gif)
Quantum discord is considered to be one of the important measures of quantum correlation, which is defined as a difference between the total correlation measured by quantum mutual information and the classical correlation[27,28] as follows:
The symbol S(ρj) of Eq. (
The effect of parameter r on the time evolution of concurrence is shown in Fig.
On the other hand, to explore the effects of r on the quantum discord, we plot in Fig.
Now, we give fixed values for the parameter r to visualize the difference of QD from C. The time evolution of QD and C is shown in Fig.
To visualize the influences of atomic motion and field-mode structure on quantum discord and concurrence C, the behavior of the phenomena with r = 1, ka = kb = 1/2, and ma = mb = 1 are displayed in Fig.
In addition, in Fig.
In summary, some of the phenomena, such as the quantum discord and concurrence, have been demonstrated between two isolated moving qubits each separately interacting with a single-mode SU(1, 1) cavity. The effects of some parameters, such as the field-mode structure, purity of the initial state, atomic motion, and detuning have been studied for the current phenomena. These phenomena are strongly influenced due to these parameters. It is observed that in the absence of qubit motion and a specific r for the resonance and off-resonance cases the entanglement completely disappears while the quantum discord remains unchanged. Also, a sudden death via the detuning effect for these phenomena appears. Besides, in the presence of qubit motion and field-mode structure for r = 1, the phenomena exhibit the same behavior for the resonance and off-resonance cases. It is concluded that a minimum value of quantum discord is always greater than that of concurrence, and then when concurrence is zero the quantum discord can still capture the quantum correlation between the two qubits. Hence, the quantum discord is considered as a best measure of quantum correlation.
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