Detection of quantum entanglement across the event horizon

Abstract

We investigate the problem of distinguishing between separable and entangled states of two quantum wave packets, one of which falls into a black hole. Intuitively, one might expect the two scenarios to be indistinguishable, since the information carried by one wave packet is hidden beyond the event horizon. We show, however, that fundamental limitations on the localizability of quantum states render the two scenarios, in principle, distinguishable. Employing tools from quantum state discrimination theory, we analyze a concrete realization and discuss the configurations that maximize the probability of successfully distinguishing between the two cases.

Figures (6)

• Figure 1: (a) A star undergoes gravitational collapse, forming a black hole in the asymptotic future. The dashed cut line indicates the spatial cross-section shown in the right panel. (b) During the collapse, at the location corresponding to the Schwarzschild radius (marked by the dashed circle), a quantum state is prepared using two orthogonal wave packets. In the asymptotic future, an observer Rob aboard a rocket measures the state of the field outside the event horizon.
• Figure 2: Minkowski diagram with Rindler regions I and II. Region I is covered with conformal Rindler coordinates $(c\tau, \xi)$. The world line $\xi = 0$ corresponds to a uniformly accelerated observer, Rob.
• Figure 3: Two inertial observers, Alice and Bob, have access to orthogonal wave packets $\phi_\mathrm{A}$ and $\phi_\mathrm{B}$, respectively. At time $t = 0$, the envelopes of the wave packets are centered at positions $\pm c^2/a$ in Minkowski coordinates $(ct,x)$. Using these wave packets, either a two-mode squeezed state or two thermal states are prepared. An observer Rob, accelerating with constant proper acceleration $a$ in the $x$-direction, measures the state of the field by coupling to a wave packet $\psi_\mathrm{R}$ at $t = 0$, when his instantaneous velocity is zero.
• Figure 4: Fidelity $F$ as a function of the dimensionless acceleration parameter $\frac{aL}{c^2}$ for $N = 6$, $\Lambda = \frac{c}{2L}$ and different values of $s$.
• Figure 5: Average number of particles seen by an accelerated detector in the vacuum $\langle \hat{n} \rangle_\mathrm{U}$ as a function of the dimensionless acceleration parameter $\frac{aL}{c^2}$ for $N = 6$ and $\Lambda = \frac{c}{2L}$.