Unruh effect as a noisy quantum channel

We studied the change of the nonlocal correlation of the entanglement in Rindler space-time by showing that the Unruh effect can be interpreted as a noisy quantum channel having a complete positive and trace preserving map with an “operator sum representation.” It is shown that the entanglement fidelity is obtained in analytic form from the operator sum representation, which agrees well numerically with the entanglement monotone and the entanglement measure obtained previously. Nonzero entropy exchange between the system $Q$ and region II of the Rindler wedge indicates the nonlocal correlation between causally disconnected regions. We have also shown the subadditivity of entropies numerically.

Figure 1

Rindler space-time. In regions I and II, time coordinates $\eta =\mathrm{const}.$ are straight lines through the origin. Space coordinates $\zeta =\mathrm{const}.$ are hyperbolas with null asymptotes $H_{+}$ and $H_{-}$, which act as event horizons. The Minkowski coordinates $t,z$ and Rindler coordinates $\eta ,\zeta$ are given by $t=a^{-1}exp\left(a\zeta \right)sinha\eta$ and $z=a^{-1}exp\left(a\zeta \right)cosha\eta$, where $a$ is a uniform acceleration. Alice and Rob initially share a two-mode squeezed state at the event $P$. We consider the case of Alice in stationary and Rob (green hyperbola) under uniform acceleration.

Figure 2

Entanglement fidelity $F_e$ versus acceleration $r$. This measure of entanglement is obtained in analytical form, i.e., $F_e=\frac{1}{4{cosh}^{2}r}{\left(1+\frac{1}{{cosh}^{2}r}\right)}^2$, as a function of the acceleration $r$ from the operator sum representation [18]. The results agree well with the entanglement monotone [13, 14].

Figure 3

Comparison of mutual information $I\left({\rho }_{AR}\right)$ and entropy exchange $S_e$. The mutual information is a measure of total correlation between Alice and Rob in their entangled state while the entropy exchange is a common entropy for two initially uncorrelated systems. A maximally mixed state of maximally entangled states has mutual information equal to 1 [13].

Figure 4

Numerical proof of subadditivity for the entropy $S\left({\rho }_{AR}\right)\le S\left({\rho }_A\right)+S\left({\rho }_R\right)$.