Receiver Algorithms to Approach the Quantum Limit of Demodulating Pulse Position Modulation

Abstract

Optical pulse position modulation (PPM) places a laser pulse, a.k.a. a coherent state of amplitude$\alpha$of mean photon number$N=\vert \alpha\vert ^{2}$, in one of$M$consecutive time slots, Ideal photon detection on each slot achieves a mean probability of error$((M-1)/1M)e^{-N}$of distinguishing the$iM$PPM codewords, since$e^{-N}$is the probability the pulse-containing slot does not produce a click, per Poisson-shot-noise photo-detection theory. The quantum (Helstrom) limit of the minimum probability of error is lower than above, has a closed-form expression, and scales as$\sim e^{-2N}$when$Me^{-N}\ll 1$. The optimal receiver must make a quantum joint measurement on all$M$slots. Even though receiver algorithms exist that achieve the$\sim e^{-2N}$scaling in the high$N$regime, none are known that bridge the classical-quantum gap for small$N$, the primary regime of interest for optical PPM. It is also not known how close to the Helstrom limit can one get using LOCC. (local operations and classical communications), i.e., a receiver that slices each of the$M$slots into$n$tiny slices, makes a measurement on the first slice, and based on the measurement result picks a measurement to apply to the next slice, etc., until all the$Mn$slices have been measured. In this paper, we propose an LOCC receiver for demodulating PPM that uses semiclassical coherent feedback control and photon detection, which outperforms all known PPM receivers, including one that employed squeezing, a non-classical operation. To bridge the remaining gap to the Helstrom limit, one might need truly quantum operations within a joint (non-LOCC) receiver.