Accessible and inaccessible quantum coherence in relativistic quantum systems

The quantum coherence of a multipartite system is investigated when some of the parties are moving with uniform acceleration and the analysis is carried out using the single-mode approximation. Due to acceleration the quantum coherence is divided into two parts as accessible and inaccessible coherence. First we investigate tripartite systems, considering both GHZ and $W$ states. We find that the quantum coherence of these states does not vanish in the limit of infinite acceleration, rather asymptoting to a nonzero value. These results hold for both single- and two-qubit acceleration. In the GHZ and $W$ states the coherence is distributed as correlations between the qubits and is known as global coherence. However, quantum coherence can also exist due to the superposition within a qubit, the local coherence. To study the properties of local coherence we investigate a separable state. The GHZ state, $W$ state, and separable states contain only one type of coherence. Next we consider the $W\overline{W}$ and star states in which both local and global coherences coexist. We find that under uniform acceleration both local and global coherence show similar qualitative behavior. Finally, we derive analytic expressions for the quantum coherence of $N$-partite GHZ and $W$ states for $n<N$ accelerating qubits. We find that the quantum coherence of a multipartite GHZ state falls exponentially with the number of accelerated qubits, whereas for multipartite $W$ states the quantum coherence decreases only polynomially. We conclude that $W$ states are more robust to Unruh decoherence and discuss some potential applications in satellite-based quantum communication and black-hole physics.

Figure 1

The $(z,t)$ plane of the Minkowski coordinates is divided into four regions of the Rindler coordinates. The solid lines are the future ($F$) and past ($P$) event horizons and the dashed lines are the trajectories of the uniformly accelerated observers. Here $H$ represents the horizon.

Figure 2

A schematic diagram of coherence in tripartite is shown where the circles labeled $A, B$, and $C$ represent the qubit and the blue arrow between them represents the coherence. Panel (a) contains the coherence in a tripartite system when all the qubits are in inertial frame. Panel (b) contains the coherence in the tripartite system when qubit $C$ is under acceleration. Here qubit $C$ enclosed by the red ellipse with dashed boundary line is split into two Rindler modes I and II. Consequently, the coherence shared by qubit $C$ also is split into two parts. The coherence shared with $C$ of Rindler mode I is the accessible coherence and the coherence shared with $C$ of Rindler mode II is the inaccessible coherence. The lighter shades of blue for the accessible and inaccessible coherence represents their relative strength to the coherence initially present with $C$ and shown in panel (a).

Figure 3

The variation of coherence in a noninertial frame of reference is studied for a GHZ state under the variation of (a) the acceleration parameter $r$ and (b) the generalization parameter $\theta$.

Figure 4

The variation of quantum coherence of GHZ state with both $r_1$ and $r_2$ is shown for (a) $\theta =\pi /4$, (b) $\theta =\pi /5$, and (c) $\theta =\pi /6$, respectively.

Figure 5

The variation of quantum coherence as a function of $r$ for fixed values of $\theta$ and $\varphi$ is given for (a) the tripartite generalized $W$ state and (b) the two-qubit reduced systems ${\rho }_{AB}, {\rho }_{BC}$, and ${\rho }_{AC}$. Here the notation $W\left({\rho }_{AB}\right)$ means the reduced state ${\rho }_{AB}$ corresponding to the symmetric $W$ state.

Figure 6

Contour plot of the quantum coherence $C\left(\rho \right)$ as a function of $\theta$ and $\varphi$ of the generalized $W$ state when one qubit is in a noninertial frame is shown for (a) $r=0.01$ and (b) $r=4.0$.

Figure 7

In the generalized $W$ state when both Bob's and Charlie's qubits are accelerated, the variation of quantum coherence as a function of the acceleration parameters $r_1$ and $r_2$ is given for (a) $W$ state; (b) $\theta =\pi /5, \varphi =\pi /5$; (c) $\theta =\pi /6, \varphi =\pi /6$; and (d) $\theta =\pi /3, \varphi =\pi /6$.

Figure 8

A plot of the total coherence $\left(C_T\right)$, global coherence $\left(C_G\right)$, and local coherence $\left(C_L\right)$ for (a) the $W\overline{W}$ state, (b) star state when the central qubit is accelerating and (c) star state when the peripheral qubit is accelerating.

Figure 9

A plot of the variation of the normalized decoherence (a) with the acceleration parameter $r$ and (b) with the number of accelerated qubits $n$ for a fixed values of $r=1.5$ and $r=2.0$. The total number of qubits in the system is $N=11$.

Figure 10

A schematic illustration of the bifurcation of the Bell state, which displays linear loss of coherence into two extreme classes of states viz. GHZ class with exponential loss of coherence and $W$ class with polynomial loss of coherence.