On the Energy of a Single-Bit Communication in the Poisson Channel with Feedback

The transmission of a single bit over the Poisson channel is considered when a noiseless feedback is available. We consider two cases: with and without peak power constraint. In the absence of peak power constraint, we show that sending one single bit reliably through a continuous-time Poisson channel requires at least half the energy of a photon. On the other hand, when a peak power constraint is imposed, we give a simple variable-length coding scheme allowing the error probability to be $\epsilon$.