† Corresponding author. E-mail:
Project supported by the National Key Research and Development Program of China (Grant No. 2017YFA0402600), the National Natural Science Foundation of China (Grant Nos. 11690023, 11373014, and 11633001), the Beijing Talents Fund of Organization Department of Beijing Municipal Committee of the CPC, and the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB23000000).
It is well known that the quantum fluctuation of entanglement (QFE) between Unruh–De Witt detector (modeled by a two-level atom) is always investigated in a relativistic setting. However, both of the Unruh radiation and quantum fluctuation effects play an important role in precise measurements of quantum entanglement. In this paper, we have quantitatively analyzed how the relativistic motion affects the QFE for two entangled Unruh–De Witt detectors, one of which is accelerated and interacting with the neighbor external scalar field. Our results show that the QFE, which initially increases by the Unruh thermal noise, will suddenly decay when the acceleration reaches to a considerably large value. Therefore, the relativistic effect will lead to non-negligible QFE effect. We also find that the initial QFE (without acceleration effect) reaches its minimum value at the maximally entangled state and the separable state. More importantly, our analysis demonstrates that although the QFE has a huge decay when the acceleration is greater than ∼0.96, the ratio of ΔE/C is still very large, due to the simultaneous decay of concurrence to a very low value. Finally, enlightened by the well-known equivalence principle, we discuss the possibility of applying the above findings to the dynamics of QFE under the influence of gravitation field.
The difference between classical and quantum theory lies in the fact that physical quantities in quantum mechanics are corresponded to Hermitian operator, which could directly determine the state of the quantum system.[1] Thus, the eigenvalues are the possible values of the Hermitian operator acting on the quantum state via measurements. However, one should note that the outcome of each measurement of a physical quantity should not be identical. For a physical quantity in quantum mechanics, the expected value is usually used to represent the eigenvalue of the quantity, i.e., the expected value stands for the statistical mean of the measurement outcome by repeated several times measurements. As is known, the fluctuations are inevitable in practical measurement processes. More specifically, for a physical quantities A in the quantum state ψ, the fluctuation is defined as
On the other hand, as one of the most important developments in modern physics, quantum entanglement has been extensively investigated in most recent years.[8–10] It is interesting to note that, the importance of quantum entanglement not only embodies the fundamental perspective of quantum information task, but also attributes to its advantages in practical aspects.[11–13] Although quantum entanglement has been achieved in many experiments, however, most of these measurements were carried out without considering the effect of acceleration. Actually, in realistic situation, the preparation of quantum system and the procession of quantum information tasks are always accompanied by accelerated effects.[14–19] In the framework of such accelerated quantum system, the Unruh effect will be generated, which indicates that quantum properties of fields are observer-dependent.[20,21] From theoretical point of view, the Unruh effect will reveal thermal radiation detected by a uniformly accelerated detector in the Minkowski vacuum and that associated with the proper acceleration of the detector. Following this direction, a number of combined analyses involving the dynamics of quantum entanglement and steering between two correlated Unruh–De Witt detectors have been performed in the literature,[10,22,23] which indicated that the type of quantum resource will be reduced by the Unruh effect, while the acceleration effect on quantum systems is non-negligible when using quantum resources to perform quantum information task.[24–28]
Inspire by the above works, in this paper we will investigate the quantum fluctuation of entanglement (QFE) for a two-level atom accelerator, which is modeled by Unruh–De Witt detectors in the relativistic setting. Compared with the global free models extensively used in many papers,[29,30] the Unruh–De Witt detector model applied in this analysis to study the behavior of QFE in a non-inertial system[31] has more advantages. On the one hand, the problems related to single-mode approximation and physically unfeasible detection of quantum correlations in the full space-time can be effectively avoided,[34] which could provide us a better understanding of quantum entanglement. On the other hand, precise measurements of quantum entanglement are highly dependent on the Unruh radiation and quantum fluctuation effects,[35] which supports a quantitative analysis of QFE in relativistic setting. In this paper, we will carry out a quantitative analysis and explore how the Unruh radiation affects the QFE.
This paper is organized as follows. In Section
In this section, from the view point of quantum information, we will give a brief description of the Unruh–De Witt detectors[22] and furthermore discuss the dynamics of a pair of detectors (considering the relativistic motion of one detector). Based on the well known two-level atom system, the simplest model for one quantum system, our Unruh–De Witt detectors are modeled by a point-like two-level atoms (each atom represents a detector). However, in our analysis we also try to extend the interaction term and investigate the interaction between this two-level system and its nearby fields. Note that the detector is semiclassical, because it possesses a classical world line but its internal degree of freedom is treated quantum mechanically. In order to investigate the behavior of quantum properties, one usually assume that the detectors are initially sharing some quantum correlations between the Minkowski spacetime and observed by two observers called Alice and Rob, respectively. Alice and Rob’s detectors initially are prepared for the inertial frame, then we let that Alice still keeps inertial and always be switched off, while Rob’s detector interacts with the scalar field, and moves with uniform acceleration for a time duration Δ. The Rob’s detector will move with its world line as
Considering the interaction between Rob’s detector and the field, the initial state of the total system can be expressed as
![](cpb_29_5_050402/cpb_29_5_050402_ieqn2.gif)
![](cpb_29_5_050402/cpb_29_5_050402_ieqn3.gif)
Now working with the interaction term, we turn the state into the interaction representation labeled by I and rewrite the final state
![](cpb_29_5_050402/cpb_29_5_050402_ieqn5.gif)
![](cpb_29_5_050402/cpb_29_5_050402_ieqn6.gif)
![](cpb_29_5_050402/cpb_29_5_050402_ieqn7.gif)
![](cpb_29_5_050402/cpb_29_5_050402_ieqn8.gif)
![](cpb_29_5_050402/cpb_29_5_050402_ieqn9.gif)
![](cpb_29_5_050402/cpb_29_5_050402_ieqn10.gif)
With the aim of investigating the evolution of the detectors’ states after interacting with the field, the part of the external field ϕ(f) should be traced. Then we obtain final matrix between Alice and Rob detectors
It is worth noting that, quantum fluctuation determined in terms of the von Neumann entropy operator is a stochastic quantity, the fluctuation of which will be taken int account and discussed in this section. In general, one can use entropy operator
![](cpb_29_5_050402/cpb_29_5_050402_ieqn15.gif)
![](cpb_29_5_050402/cpb_29_5_050402_ieqn16.gif)
![](cpb_29_5_050402/cpb_29_5_050402_ieqn17.gif)
Moreover, in the following analysis we also focus on the concurrence (C), which has been widely applied in entanglement measurements in arbitrary bipartite systems,[39] to quantify quantum entanglement without statistical fluctuation (since concurrence is not dependent on entropy). Thus, the calculation of the entanglement fluctuation will reduce to the determination of the concurrence. Now the fluctuation of quantum entanglement with the arbitrary bipartite system is quantified as[40,41]
![](cpb_29_5_050402/cpb_29_5_050402_ieqn18.gif)
![](cpb_29_5_050402/cpb_29_5_050402_ieqn19.gif)
In this section we will study the behaviors of QFE under the influence of the Unruh radiation. With the initial state of the system given in Eq. (
It is interesting to understand whether the presence of the detector’s acceleration will change the quantum fluctuation of entanglement (QFE). In Fig.
Here, we are also interested in the dynamics of QFE for varying initial parameters, which determines the degree of quantum entanglement in our analysis (θ = π/4 corresponds to the maximally entangled state). In Fig.
In order to obtain a better understanding of the effect of statistical fluctuation in entanglement, we illustrate in Fig.
The subject of quantum entanglement continues to be one of great importance in modern physics. Over the past decades, many of the studies in this field have concentrated on the realization of quantum entanglement without considering the effect of acceleration. However, in realistic situation, the preparation of quantum system and the procession of quantum information tasks are always accompanied by accelerated effects. In the framework of such accelerated quantum system, the Unruh effect will be generated, which indicates that quantum properties of fields are observer-dependent. Focusing on one of the most general quantum resources, quantum entanglement, it can be quantified by one statistical quantity or operator, von Neumann entropy. However, quantum entanglement has fluctuation under the description of von Neumann entropy. In this paper, we have investigated the dynamic of quantum fluctuation of entanglement (QFE) with two entangled Unruh-De Witt detectors(modeled by a two-level atom), one of which is accelerated and interacting with the neighbor external scalar field. Here we summarize our main conclusions in more details.
Firstly, we find that the QFE initially increases with the Unruh thermal radiation and then suddenly decays when the acceleration reaches to q ∼ 0.96, which indicates that QFE cannot be ignored when the relativistic motion is taken into consideration. It is found that the initial QFE (q = 0) is highly dependent on the initial parameter. More specifically, the initial QFE of the system (without acceleration effect) will reach its minimum value at the maximally entangled state and the separable state, which means that maximally entangled state will bring less QFE. Such findings strongly indicate the possibility of making use of quantum entanglement to achieve quantum information tasks. Focusing on the dynamics of QFE for varying initial parameters, our results show that the QFE reaches its maximum at the initial parameter θ=π/8, 3π/8, while its minimum value is respectively determined at θ = 0, π/4, π/2. The investigation of ΔE as a function of θ indicates that, the QFE between Alice and Rob’s detectors will exhibit apparent oscillatory behavior with a period of π/2, in terms of different value for the initial parameters, In order to quantify the the effect of statistical fluctuation on entanglement, we employ the ratio between QFE and quantum entanglement to describe this effect, in which quantum entanglement is described here in terms of concurrence. Our findings indicate that the ratio of ΔE/C gradually increases with increasing Rob’s acceleration, which suggests that the Unruh thermal radiation will inevitably makes quantum entanglement degeneration and concurrently induces quantum fluctuation of entanglement. In addition, our analysis demonstrates that although QFE has a huge decay when the acceleration q is larger than ∼ 0.96, the ratio of ΔE/C is still very large, due to the simultaneous decay of concurrence to a very low value. Finally, with the rapid developments in both quantum technology and quantum communication, it is possible to achieve quantum entanglement by implementing quantum tasks with relativistic motion in the near future. Considering the fact that realistic quantum systems always exhibit gravitational and relativistic features, our analyses in this paper can be extended to the investigation of the dynamics of QFE, under the influence of gravitation field. This is supported by General Relativity, due to the equivalence principle that states all accelerated reference frames possess a gravitational field.
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