Sudden death of entanglement and teleportation fidelity loss via the Unruh effect

We use the Unruh effect to investigate how the teleportation of quantum states is affected when one of the entangled qubits used in the process is under the influence of some external force. In order to reach a comprehensive understanding, a detailed analysis of the acceleration effect on such entangled qubit system is performed. In particular, we calculate the mutual information and concurrence between the two qubits and show that the latter has a “sudden death” at a finite acceleration, whose value will depend on the time interval along which the detector is accelerated.

Figure 1

(Color online) The graph exhibits the mutual information $I\left(A:R\right)$ for a singlet initial state as a function of acceleration $a/\Omega$, with ${ϵ}^2=8{\pi }^2\cdot {10}^{-6}$, $\Omega =100$, $\Delta =1000$, and $\kappa =0.02$. The most interesting feature is related with the fact that the curve is not monotonic, acquiring its minimum value at $a_0/\Omega \approx 545.75$. We note that the dimensionless quantity $a/\Omega$ reflects the temperature of the Unruh thermal bath as experienced by Rob’s detector per its energy gap (up to a $1/\left(2\pi \right)$ factor).

Figure 2

(Color online) The graph shows the entanglement, $E^{AR\varphi }$, between the qubit system and field as a function of acceleration $a/\Omega$ assuming the same initial state and $ϵ$, $\Omega$, $\Delta$, and $\kappa$ parameters as in Fig. 1. We note that the entanglement takes its maximum value, $E_{\text{max}}^{AR\varphi }\approx 1.58$, at $a_0/\Omega \approx 545.75$, precisely where $I\left(A:R\right)$ has its minimum. This is interesting to note that because the normalized $|{\Psi }_{\infty }^{AR\varphi }⟩$ is Schimidt decomposed [14] [see Eq. (3.16)], the corresponding Schimidt number is 3 and the maximum entanglement $E_{\text{max}}^{AR\varphi }=\text{log}3\approx 1.58$.

Figure 3

(Color online) The graph follows the behavior of the entanglement, $E^{AR\varphi }$, along the time for three distinct proper accelerations: $a/\Omega =100$ (full line), $a/\Omega =a_0=545.75$ (dashed line), and $a/\Omega =2000$ (dotted line) assuming the same initial state and $ϵ$, $\Omega$, $\Delta$, and $\kappa$ parameters as in Fig. 1. We note that for $a\le a_0$, $E^{AR\varphi }$ increases monotonically as a function of time. However, for $a>a_0$, the qubit system and field get maximally entangled at some time $\tau ={\tau }_e<\Delta$, after which the qubit system recovers back part of its correlations from the total system. Although not being visually evident, the graph is plotted in the acceleration time interval $\tau =\left[1,\Delta \right]$, which respects the constraint $\Omega \tau ⪢1$ [see discussion above Eq. (2.11)].

Figure 4

(Color online) The concurrence $C\left({\rho }_{\infty }^{AR}\right)$ is plotted as a function of the acceleration $a/\Omega$ assuming the same initial state and $ϵ$, $\Omega$, $\Delta$, and $\kappa$ parameters as in Fig. 1. The sudden death of the entanglement between the two qubits is observed at $a_{\text{sd}}/\Omega \approx 273.00$.

Figure 5

(Color online) The teleportation fidelity $F$ is plotted as a function of the acceleration $a/\Omega$ with the values of $ϵ$, $\Omega$, $\Delta$, and $\kappa$ being the same as in Fig. 1.