Optical pulse position modulation (PPM) places a laser pulse, a.k.a. a coherent state of amplitude
αof mean photon number
N=∣α∣2, in one of
Mconsecutive time slots, Ideal photon detection on each slot achieves a mean probability of error
((M−1)/1M)e−Nof distinguishing the
iMPPM codewords, since
e−Nis 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
∼e−2Nwhen
Me−N≪1. The optimal receiver must make a quantum joint measurement on all
Mslots. Even though receiver algorithms exist that achieve the
∼e−2Nscaling in the high
Nregime, 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
Mslots into
ntiny 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
Mnslices 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.
In optical communication, the transmitter encodes information into a set of light states defined by the modulation format, selected to accommodate specific channel conditions and to remain sufficiently distinguishable at the output. Various receiver architectures have been designed to improve the demodulation performance, ultimately limited by quantum theory. In this work, I introduce a new receiver based on a locally optimal greedy algorithm and apply it to pulse position modulation. The receiver reduces the error probabilities of previously proposed strategies in all signal strength regimes and achieves results comparable with those obtained by numerical optimization of the detection process. In contrast, however, it is conceptually simple and, therefore, can be scaled to arbitrarily high modulation orders for which numerical methods become intractable. In the photon-starved regime characteristic of deep space optical communication, the greedy receiver approaches the quantum-optimal Helstrom bound on state discrimination error probability. In the regime of few-photon pulses, the error reduction offered over the other methods grows up to an order of magnitude.
We introduce a squeezed-state pulse-position modulation (S-PPM) format, where the empty slots are squeezed vacuum states and the pulse slot is a displaced squeezed state. Based on this property, we propose an inverse-squeezing conditional pulse-nulling (IS-CPN) receiver. In the ideal case, inverse squeezing maps S-PPM into an equivalent coherent-state PPM signal with a large pulse energy, leading to a closed-form expression for the receiver error probability. We further analyze IS-CPN under common phase diffusion using a finite-path MAP formulation with phase-averaged likelihoods. Numerical results show that IS-CPN outperforms conventional CPN under the same energy constraint and remains advantageous under phase noise and finite photon-number resolution. These results demonstrate that combining squeezed-state modulation with inverse-squeezing conditional nulling can improve photon-efficient optical communication.
Communications systems builders continue to search for signal formats and receiver architectures that can provide the most efficient utilization of their subsystems, which include power amplifiers as well as transmit and receive apertures. Receivers requiring very small amounts of received power are of particular interest in communications links where transmission distances are very long and losses are large, such as from Deep Space. Helstrom and others ([1],[2],[3]) initiated the study of optimum signal reception using quantum mechanical signal models. They derived the mathematical description and predicted performance of receivers that optimize certain criteria, such as Minimum Probability of Error (MPE). Unfortunately, practical implementation of their proposed receivers has still not been achieved. In parallel, technology has advanced to where noiseless photon counters can be used to achieve quite good performance ([4]). We show here that, when an end-to-end error correction code is added, in fact such a system can out-perform the “optimum” MPE system at low signal powers. In this report, we derive the formulation of a quantum receiver that is shown to be uniformly better than either the MPE or photon-counting receiver.
Coherent states achieve the Holevo capacity of a pure-loss channel when paired with an optimal measurement, but a physical realization of this measurement is unknown, and likely to be of high complexity. In this paper, we focus on the photon-counting measurement and study the photon and dimensional efficiencies attainable with modulations over classical- and nonclassical-state alphabets. We first review state-of-the-art coherent on-off-keying (OOK) and pulse-position modulation (PPM) with a photon-counting measurement, illustrating its asymptotic inefficiency relative to the Holevo limit. Then we analyze two architectures that improve upon the dimensional versus photon efficiency tradeoff achievable with conventional OOK or PPM. We show that at high photon efficiency these architectures achieve an efficiency tradeoff that differs from the best possible tradeoff by only a constant factor. The first architecture is a coherent-state transmitter that relies on feedback from the receiver to control the transmitted energy. The second architecture uses a single-photon number-state source.