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arXiv:2606.31541

Inverse-squeezing receivers for squeezed-state pulse-position modulation under ideal and phase-diffusion conditions

Enhao Bai, Fengkai Sun, Tianyi Wu, Huankai Zhang, Jian Peng, Chen Dong, Zhenrong Zhang

quant-phSubmitted Jun 30, 2026• Announced Jul 1, 2026• 0 citations
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Abstract

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.

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

Leo Bia, Christos N. Gagatsos, Saikat Guha

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.

IEEE International Symposium on Information Theory (ISIT)•Jul 7, 2024•0 citations
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.
We propose an inverse-squeezing Kennedy receiver for discriminating binary phase-shift-keyed displaced squeezed vacuum states. The receiver combines a Kennedy-type nulling displacement, an orthogonally oriented inverse-squeezing operation and photon-number-resolving detection with a maximum-a-\emph{posteriori} threshold rule. Its key mechanism is that the inverse-squeezing stage converts transmitter-side squeezing into enhanced photon-number contrast, or equivalently an effective coherent-state energy gain, that can be directly exploited at the measurement stage. Under ideal equal-prior conditions, the receiver surpasses the standard quantum limit for squeezed-state binary phase-shift keying at approximately N≈0.3N\approx 0.3N≈0.3, outperforms the Helstrom bound of coherent-state binary phase-shift keying at approximately N≈0.4N\approx 0.4N≈0.4, and reaches the 1\% error level near N≈0.6N\approx 0.6N≈0.6. We further analyze its performance under realistic imperfections, including finite detector efficiency, dark counts, channel phase diffusion, receiver thermal noise and transmission loss. The results show that adaptive thresholding preserves robust performance against detector and noise imperfections over practical parameter ranges, whereas transmission loss progressively suppresses the squeezing-enabled advantage. These findings indicate that, for the fixed source parametrization adopted in this work, the proposed receiver is most advantageous in the low-loss regime, especially at low source energies.
Optical pulse-position modulation (PPM) is one of the primary modulation formats being investigated for use in deep-space communications, as well as terrestrial fiber optical communications. We consider the problem of demodulating M-ary optical PPM waveforms, and propose a structured receiver whose mean probability of symbol error is smaller than all known receivers, and approaches the Helstrom (quantum) limit of the minimum probability of error (MPE) of discriminating between the coherent-state PPM signals. The receiver uses photodetection coupled with optimized phase-coherent optical feedback control through the PPM pulse slots and a phase-sensitive parametric amplifier. We present a general framework of optical receivers known as the ‘conditional pulse nulling receiver’, and present new results on ultimate limits and achievable regions of the trade-off space between the spectral versus photon efficiency of PPM, for the single-spatial-mode far-field pure-loss optical communication channel.
Conditional pulse nulling (CPN) receiver is proposed by Dolinar to discriminate pulse position modulation (PPM) signals. The receiver, which uses a beam splitter and an on-off photon detector can outperform the standard quantum limit (SQL) for PPM signals. In this paper, this receiver is applied to multi-pulse PPM (MPPM) and binary quantum coding signals, and a dynamic programming algorithm is proposed to optimize the control strategy. The MPPM signals are used to improve the band-utilization efficiency, and the binary quantum coding enables the communication system to correct the error. Numerical simulation results show that the CPN receiver with optimized strategy outperform SQL for both MPPM and binary quantum coding signals.
The discrimination of quantum states is fundamental to quantum information processing. Inverse-squeezing Kennedy (IS-Kennedy) receivers can outperform the coherent-state BPSK Helstrom benchmark at the same energy by converting transmitter-side squeezing into an effective coherent-state separation gain, without violating the Helstrom bound for the squeezed-state alphabet. This work investigates how squeezing mismatch degrades this mechanism. We show that imperfect inverse squeezing transforms the ideally nulled output into a residually squeezed state, thereby altering the photon-number statistics before detection. This residual-squeezing picture reveals a strong physical asymmetry between squeezing-magnitude and squeezing-phase mismatches. Magnitude mismatch produces an energy-independent error floor in the high-signal-energy regime, whereas phase mismatch generates a residual squeezing term that grows with signal energy. In the small-residual-squeezing regime, this leads to a polynomial growth of the leading error contribution and a rapid collapse of the SQL advantage. We also identify a parity-step effect in photon-number-resolving detection: because the nulled residual squeezed vacuum contains only even photon numbers, increasing detector resolution improves the high-energy robustness only when the effective saturation threshold crosses the next even photon number. These results identify phase locking as the dominant bottleneck for IS-Kennedy-type non-Gaussian receivers under unitary squeezing mismatch and provide design guidelines for robust squeezed-state quantum receivers.
Pulse position modulation (PPM) is widely adopted in free space optical transmission, and is a candidate for deep space transmission, also in quantum form. The change of signal photon number probability distribution is caused by thermal noise in quantum PPM format. We prove that squeezing improves state discrimination at fixed channel energy, and also in the presence of thermal noise. Then we compare the difference between coherent state and squeezed state in thermal noise and reveal that the squeezed state has better robustness. It is shown that for the fixed channel energy, the larger the squeezed parameter is not the better. For some given total channel energies, we further study and get the optimal squeezed parameter. Finally, combined with the actual situation, the simulation results show that, as a resource for noisy quantum communication channels, the squeezing operation can improve the quality of deep space quantum communication system with PPM.
Optical pulse position modulation (PPM) places a laser pulse, a.k.a. a coherent state of amplitudeα\alphaαof mean photon numberN=∣α∣2N=\vert \alpha\vert ^{2}N=∣α∣2, in one ofMMMconsecutive time slots, Ideal photon detection on each slot achieves a mean probability of error((M−1)/1M)e−N((M-1)/1M)e^{-N}((M−1)/1M)e−Nof distinguishing theiMiMiMPPM codewords, sincee−Ne^{-N}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−2N\sim e^{-2N}∼e−2NwhenMe−N≪1Me^{-N}\ll 1Me−N≪1. The optimal receiver must make a quantum joint measurement on allMMMslots. Even though receiver algorithms exist that achieve the∼e−2N\sim e^{-2N}∼e−2Nscaling in the highNNNregime, none are known that bridge the classical-quantum gap for smallNNN, 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 theMMMslots intonnntiny 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 theMnMnMnslices 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.