Figure 1

(Color online) Sketch of the world lines for the inertial observer Alice and the accelerated observer Rob. The set $(z,t)$ denotes Minkowski coordinates, while the set $(\zeta ,\tau )$ denotes Rindler coordinates. The causally disconnected Rindler regions I and II are evidenced.Reuse & Permissions

Figure 2

(Color online) Plot, as a function of the acceleration parameter $r$, of the bipartite entanglement between the mode described by one observer and the group of modes described by the other two, as expressed by the single-mode determinants $m_{i\left(jk\right)}$ defined in Eq. (16). The inertial entanglement is kept fixed at $s=1$. The solid red line represents $m_{A\mid \left(R\overline{R}\right)}$, the dashed green line corresponds to $m_{R\mid \left(A\overline{R}\right)}$, while the dotted blue line depicts $m_{\overline{R}\mid \left(AR\right)}$.Reuse & Permissions

Figure 3

(Color online) Bipartite entanglement described by Alice and the noninertial observer Rob, who moves with uniform acceleration parametrized by the effective squeezing $r$. From an inertial perspective, the field is in a two-mode squeezed state with squeezing degree $s$. (a) depicts the contangle $\tau \left({\mathbf{\sigma }}_{A\mid R}\right)$, given by Eqs. (5, 18), as a function of $r$ and $s$. In (b) the same quantity is normalized to the original contangle as seen by inertial observers, $\tau \left({\mathbf{\sigma }}_{A\mid R}^P\right)=4s^2$. Notice in (a) how the bipartite contangle is an increasing function of the entanglement $s$, while it decreases with increase in Rob’s acceleration $r$, vanishing in the limit $r\to \infty$. This degradation is faster for higher $s$, as clearly visible in (b).Reuse & Permissions

Figure 4

(Color online) Bipartite entanglement between modes described by Alice and the noninertial Rob moving with uniform acceleration parametrized by $r$. The dotted red curve depicts the dependence of the logarithmic negativity between modes described by Alice and Rob in the instance of a state which corresponds to a two-qubit Bell state from the inertial perspective as computed in Ref. 3. The other solid curves correspond to $\sqrt{\tau \left({\mathbf{\sigma }}_{A\mid R}\right)}$ (the square root of the contangle is taken to provide a fair dimensional comparison) as computed in this paper [see Eq. (18)], in the instance of an entangled two-mode squeezed state described from the perspective of two inertial observers, with different squeezing parameters $s=0.25,0.5,1,2$ (referring to the purple, blue, green, and gold curves, respectively). As a further comparison, the entanglement described by Rob and anti-Rob, given by $\sqrt{\tau \left({\mathbf{\sigma }}_{R\mid \overline{R}}\right)}=2r$ [see Eq. (19)] independently of $s$, is plotted as well (dashed black diagonal line).Reuse & Permissions

Figure 5

(Color online) Genuine tripartite entanglement, as quantified by the residual contangle Eq. (23), among the inertial Alice, Rob in Rindler region I, and anti-Rob in Rindler region II, plotted as a function of the initial squeezing $s$ and of Rob’s acceleration $r$. The tripartite entanglement increases with $r$, and for $r\to \infty$ it approaches the original entanglement content $4s^2$ between modes described by Alice and Rob from the inertial perspective.Reuse & Permissions

Figure 6

(Color online) Total correlations between modes described by Alice and the noninertial observer Rob, moving with acceleration given by the effective squeezing parameter $r$. From an inertial perspective, the field is in a two-mode squeezed state with squeezing degree $s$. (a) depicts the dependence of the mutual information $I\left({\mathbf{\sigma }}_{A\mid R}\right)$, given by Eq. (25), as a function of $r$ and $s$. In (b) the same quantity is normalized to the entropy of entanglement as perceived by inertial observers, $E_V\left({\mathbf{\sigma }}_{A\mid R}^P\right)$, Eq. (26). Notice in (a) how the mutual information is an increasing function of the inertial entanglement $s$; at variance with the entanglement (see Fig. 3), it saturates to a nonzero value in the limit of infinite acceleration. From (b), one clearly sees that this asymptotic value is exactly equal to the entropy of entanglement described by inertial observers.Reuse & Permissions

Figure 7

(Color online) Sketch of the world lines for the two noninertial observers Leo and Nadia. The set $(z,t)$ denotes Minkowski coordinates, while the set $(\zeta ,\tau )$ denotes Rindler coordinates. The causally disconnected Rindler regions I and II are shown.Reuse & Permissions

Figure 8

(Color online) Entanglement condition (33) for different frequency modes, assuming that Leo and Nadia have the same acceleration $\aleph =2\pi$ (a) and (b) $10\pi$. Entanglement is present only in the frequency range where the plotted surfaces assume negative values, and vanishes for frequencies where the plots become positive; the threshold [saturation of (33)] is highlighted with a black line. Only modes whose frequencies are sufficiently high exhibit bipartite entanglement. For higher accelerations of the observers, the range of entangled frequency modes gets narrower, and in the infinite-acceleration limit the bipartite entanglement between all frequency modes vanishes.Reuse & Permissions

Figure 9

(Color online) Entanglement between different frequency modes assuming that Leo and Nadia have the same acceleration $\aleph =2\pi$.Reuse & Permissions

Figure 10

(Color online) Bipartite entanglement between the modes described by the two noninertial observers Leo and Nadia, both traveling with uniform acceleration given by the effective squeezing parameter $a$. From an inertial perspective, the field is in a two-mode squeezed state with squeezing degree $s$. (a) depicts the contangle $\tau \left({\mathbf{\sigma }}_{L\mid N}\right)$, given by Eqs. (5, 37), as a function of $a$ and $s$. In (b) the same quantity is normalized to the inertial contangle, $\tau \left({\mathbf{\sigma }}_{L\mid N}^P\right)=4s^2$. Notice in (a) how the bipartite contangle is an increasing function of the inertial entanglement $s$, while it decreases with increasing acceleration $a$. This degradation is faster for higher $s$, as clearly visible in (b). At variance with the case of only one accelerated observer (Fig. 3), in this case the bipartite entanglement can be completely destroyed at finite acceleration. The black line depicts the threshold acceleration $a^{*}\left(s\right)$, Eq. (36), such that for $a⩾a^{*}\left(s\right)$ the bipartite entanglement described by the two noninertial observers is exactly zero.Reuse & Permissions

Figure 11

(Color online) Residual contangle, Eq. (39), not stored in bipartite form, distributed among the modes described by the noninertial observers Rob and Nadia in Rindler region I and their virtual counterparts anti-Leo and anti-Nadia in Rindler region II, as quantified by the residual contangle Eq. (23), among the inertial Alice, plotted as a function of the initial squeezing $s$ and of the acceleration $a$ of both observers. In the regime of high acceleration $(a\to \infty )$, the displayed residual entanglement is completely distributed in the form of genuine four-partite quantum correlations. This four-partite entanglement is monotonically increasing with increasing acceleration $a$, and diverges as $a$ approaches infinity.Reuse & Permissions

Figure 12

(Color online) Total correlations between the modes described by the two noninertial observers Leo and Nadia, traveling with equal, uniform acceleration given by the effective squeezing parameter $a$. From the inertial perspective, the field is in a two-mode squeezed state with squeezing degree $s$. (a) shows the mutual information $I\left({\mathbf{\sigma }}_{L\mid N}\right)$, given by Eq. (41), as a function of $a$ and $s$. In (b) the same quantity is normalized to the entropy of entanglement perceived by inertial observers, $E_V\left({\mathbf{\sigma }}_{L\mid N}^P\right)$, Eq. (42). Notice in (a) how the mutual information is an increasing function of the squeezing parameter $s$ and saturates to a nonzero value in the limit of infinite acceleration; in contrast, the entanglement vanishes at finite acceleration (see Fig. 10). (b) shows that this asymptotic value is smaller than the entropy of entanglement described by inertial observers (which is equal to the classical correlations described by the inertial observers). Therefore, classical correlations are also degraded when both observers are accelerated, in contrast to the case where only one observer is in uniform acceleration (see Fig. 10).Reuse & Permissions

Figure 13

(Color online) Plot, as a function of the acceleration parameter $a$ and the squeezing parameter $s$, of the difference between the mutual information described by the inertial Alice and the noninertial Rob, and the mutual information described by the uniformly accelerating Leo and Nadia, as given by Eq. (43).Reuse & Permissions