Quantum communication in the presence of a horizon

Based on homodyne detection, we discuss how the presence of an event horizon affects quantum communication between an inertial partner, Alice, and a uniformly accelerated partner, Rob. We show that there exists a low frequency cutoff for Rob’s homodyne detector that maximizes the signal to noise ratio and it approximately corresponds to the Unruh frequency. In addition, the low frequency cutoff which minimizes the conditional variance between Alice’s input state and Rob’s output state is also approximately equal to the Unruh frequency. Thus the Unruh frequency provides a natural low frequency cutoff in order to optimize quantum communication of both classical and quantum information between Alice and Rob.

Figure 1

Alice (static) sends Rob (accelerated) a Gaussian wave packet which straddles Rob’s future horizon.

Figure 2

Strength of local oscillator for various low frequency cutoffs: $k_{\text{cut}}=0.00001$ (top), 0.001 (middle), 0.1, (bottom) $\delta =\frac{k_{so}}{\sigma }=10$.

Figure 3

Normalized variance for various low frequency cutoffs: $k_{\text{cut}}=0.01$ top, 0.05 (middle), 0.1 (bottom), $\delta =\frac{k_{so}}{\sigma }=10$.

Figure 4

Signal to noise ratio versus low frequency cutoff for $k_{so}(t-x)=n\pi$, $\delta =\frac{k_{so}}{\sigma }=10$. The signal to noise ratio decreases when the low frequency cutoff become smaller and larger. The low frequency cutoff that maximizes the signal to noise ratio is between 0.1 and 0.2.

Figure 5

Signal to noise ratio versus low frequency cutoff for $k_{so}(t-x)=(\frac{1}{2}+n)\pi$, $\delta =\frac{k_{so}}{\sigma }=10$. The signal to noise ratio first increases and then tends to be a constant when the low frequency cutoff becomes smaller.

Figure 6

Conditional variance versus low frequency cutoff, $\delta =\frac{k_{so}}{\sigma }=10$.