Extraction of timelike entanglement from the quantum vacuum

An intriguing property of the massless quantum vacuum state is that it contains entanglement between both spacelike and timelike separated regions of space-time. The implications of timelike entanglement and its connection to standard entanglement, however, are unexplored. Here we show that timelike entanglement can be extracted from the massless Minkowski vacuum and converted into standard entanglement “at a given time” between two inertial, two-state detectors at the same spatial location: one coupled to the field in the past and the other coupled to the field in the future. The procedure used here demonstrates a time correlation as a requirement for extraction; e.g., if the past detector was active at a quarter to 12:00, then the future detector must wait to become active at precisely a quarter past 12:00 in order to achieve entanglement.

Figure 1

Space-time divided into quadrants contained by the future and past light cones (F and P) and the right and left Rindler wedges (R and L). The massless Minkowski vacuum can be expanded in terms of modes restricted to R and L or in terms of modes restricted to F and P.

Figure 2

Three paired sets of window functions, ${\chi }_{xF}\left(\eta \right)=e^{-{(\eta -x)}^2}$ and ${\chi }_{xP}\left(\overline{\eta }\right)=e^{-{(\overline{\eta }-x)}^2}$, symmetrically paired across $t=0$ and plotted in Minkowski time $t$, which produce identical entanglement between the detectors (corresponding to the detector parameter $a=1$ and window function parameters $x=-1.5$, $x=0$, and $x=1.5$). As the window functions are shifted away from $t=0$ (corresponding to larger values of $x$), the detectors remain active for an exponentially longer duration to achieve identical entanglement.

Figure 3

Numerical value of the quantity $I_X-I_A$ for choice of window functions ${\chi }_P=e^{-{\overline{\eta }}^2}$ and ${\chi }_F=e^{-{(\eta -x)}^2}$ for detector parameters $E=1$ and $a=2$. The detectors are entangled at $t=\infty$ for positive values of $I_X-I_A$ only. The position of the window functions must be sufficiently symmetrical about $t=0$ (corresponding to a sufficiently small value of $x$) in order to extract entanglement from the vacuum between the timelike separated regions F and P.