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IEEE International Symposium on Information Theory (ISIT)

Finite codelength analysis of the sequential waveform nulling receiver for M-ary PSK

Si-Hui Tan, Zachary Dutton, Ranjith Nair, Saikat Guha

Published Jun 1, 2015• 8 citations
Publisher

Abstract

The conditions for a quantum measurement to discriminate a set of states with the minimum probability of error were specified by Yuen, Kennedy and Lax, and are often termed the YKL conditions [1]. Since light is quantum mechanical, the ultimate limit on minimum-error discrimination of an optical modulation constellation is determined by the YKL bound. Standard optical receivers (i.e., direct, homodyne or heterodyne detection)-even at their respective ideal operation limits-cannot achieve this performance. Recently, it was shown that a `sequential waveform nulling' (SWN) receiver can, not only discriminate an arbitrary M-ary coherent-state (ideal laser-light) constellation asymptotically at the YKL bound in the high-power limit, but that it achieves a factor of 4 better in the asymptotic error-probability exponent compared with heterodyne detection-the only conventional optical receiver that can in principle be employed for detecting an arbitrary phase-and-amplitude modulated constellation [2]. The SWN receiver can be built with standard optical components; i.e., beamsplitters, local-oscillator lasers, delay loops and single-photon detectors. However on the other hand, in the high power regime, heterodyne detection is known to achieve a reliable communication rate that asymptotically approaches the Holevo capacity of a lossy-noisy optical channel (the ultimate limit to the classical capacity of a quantum channel) [3]. In fact, in the high power regime, heterodyne detection was also shown recently to achieve the optimal second-order coding rate, when using the optimal (Gaussian) input distribution [4]. In this paper, we show that when restricted to the M-ary phase-shift keying (PSK) ensemble, that the SWN receiver's superiority over heterodyne detection in its asymptotic error exponent of the demodulation error probability, translates to a slightly higher capacity and a pronouncedly higher finite blocklength reliable-communication rate. We also quantify, via a numerical calculation, the dependence of the SWN receiver's capacity on the order in which the PSK constellation points are nulled. Our results suggest that for short-latency PSK-modulated optical communication in the high spectral efficiency regime-for which heterodyne detection is the conventional receiver choice-that it may be beneficial to employ the SWN receiver, despite the widely-regarded capacity optimality of heterodyne detection in this operating regime.

DOI
10.1109/isit.2015.7282739

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The conditions for a quantum measurement to discriminate a set of states with the minimum probability of error were specified by Yuen, Kennedy and Lax, and are often termed the YKL conditions [1]. Since light is quantum mechanical, the ultimate limit on minimum-error discrimination of an optical modulation constellation is determined by the YKL bound. Standard optical receivers (i.e., direct, homodyne or heterodyne detection)-even at their respective ideal operation limits-cannot achieve this performance. Recently, it was shown that a `sequential waveform nulling' (SWN) receiver can, not only discriminate an arbitrary M-ary coherent-state (ideal laser-light) constellation asymptotically at the YKL bound in the high-power limit, but that it achieves a factor of 4 better in the asymptotic error-probability exponent compared with heterodyne detection-the only conventional optical receiver that can in principle be employed for detecting an arbitrary phase-and-amplitude modulated constellation [2]. The SWN receiver can be built with standard optical components; i.e., beamsplitters, local-oscillator lasers, delay loops and single-photon detectors. However on the other hand, in the high power regime, heterodyne detection is known to achieve a reliable communication rate that asymptotically approaches the Holevo capacity of a lossy-noisy optical channel (the ultimate limit to the classical capacity of a quantum channel) [3]. In fact, in the high power regime, heterodyne detection was also shown recently to achieve the optimal second-order coding rate, when using the optimal (Gaussian) input distribution [4]. In this paper, we show that when restricted to the M-ary phase-shift keying (PSK) ensemble, that the SWN receiver's superiority over heterodyne detection in its asymptotic error exponent of the demodulation error probability, translates to a slightly higher capacity and a pronouncedly higher finite blocklength reliable-communication rate. We also quantify, via a numerical calculation, the dependence of the SWN receiver's capacity on the order in which the PSK constellation points are nulled. Our results suggest that for short-latency PSK-modulated optical communication in the high spectral efficiency regime-for which heterodyne detection is the conventional receiver choice-that it may be beneficial to employ the SWN receiver, despite the widely-regarded capacity optimality of heterodyne detection in this operating regime.
Discriminating coherent states of light is an important instance of quantum state discrimination that is central to all applications of laser light. We obtain the ultimate quantum limit on the error probability exponent for discriminating among any M multimode coherent-state signals via the recently developed theory of the quantum Chernoff exponent in M-ary multi-copy state discrimination. A receiver called the Sequential Waveform Nulling (SWN) receiver is proposed for discriminating an arbitrary coherent-state ensemble using only auxiliary coherent-state fields, beam splitters, and non-number-resolving single photon detectors. An explicit error probability analysis of the SWN receiver is used to show that it achieves the quantum limit on the error probability exponent, which is shown to be a factor of four greater than the error probability exponent of an ideal heterodyne-detection receiver on the same ensemble. Apart from being of fundamental interest, these results are relevant to communication, sensing, imaging, and quantum information processing systems that use laser light.
Laser-light (coherent-state) modulation is sufficient to achieve the ultimate (Holevo) capacity of classical communication over a lossy and noisy optical channel, but requires a receiver that jointly detects long modulated codewords with highly nonlinear quantum operations, which are near-impossible to realize using current technology. We analyze the capacity of the lossy-noisy optical channel when the transmitter uses coherent state modulation but the receiver is restricted to a general quantum-limited Gaussian receiver, i.e., one that may involve arbitrary combinations of Gaussian operations (passive linear optics: beamsplitters and phase-shifters, second order nonlinear process (squeezers), along with homodyne or heterodyne detection measurements) and any amount of classical feedforward within the receiver. Under these assumptions, we show that the Gaussian receiver that attains the maximum mutual information is either homodyne detection, heterodyne detection, or time sharing between the two, depending upon the received power level. In other words, our result shows that to exceed the theoretical limit of conventional coherent optical communications, one has to incorporate non-Gaussian (i.e. higher order nonlinear) operations in the receiver.
Laser-light (coherent-state) modulation is sufficient to achieve the ultimate (Holevo) capacity of classical communication over a lossy and noisy optical channel, but requires a receiver that jointly detects long modulated code words with highly nonlinear quantum operations, which are near-impossible to realize using current technology. We analyze the capacity of the lossy-noisy optical channel when the transmitter uses coherent-state modulation but the receiver is restricted to a general quantum-limited Gaussian receiver, i.e., one that may involve arbitrary combinations of Gaussian operations [passive linear optics: beam splitters and phase shifters; second-order nonlinear optics (or active linear optics): squeezers, along with homodyne or heterodyne detection measurements] and any amount of classical feedforward within the receiver. Under these assumptions, we show that the Gaussian receiver that attains the maximum mutual information is either homodyne detection, heterodyne detection, or time sharing between the two, depending upon the received power level. In other words, our result shows that to exceed the theoretical limit of conventional coherent optical communication, one has to incorporate non-Gaussian, i.e., third- or higher-order nonlinear operations in the receiver. Finally we compare our Gaussian receiver limit with experimentally feasible non-Gaussian receivers and show that in the regime of low received photon flux, it is possible to overcome the Gaussian receiver limit by relatively simple non-Gaussian receivers based on photon counting.
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.
We study the problem of designing optical receivers to discriminate between multiple coherent states using coherent processing receivers---i.e., one that uses arbitrary coherent feedback control and quantum-noise-limited direct detection---which was shown by Dolinar to achieve the minimum error probability in discriminating any two coherent states. We first derive and reinterpret Dolinar's binary-hypothesis minimum-probability-of-error receiver as the one that optimizes the information efficiency at each time instant, based on recursive Bayesian updates within the receiver. Using this viewpoint, we propose a natural generalization of Dolinar's receiver design to discriminate MMM coherent states, each of which could now be a codeword, i.e., a sequence of NNN coherent states, each drawn from a modulation alphabet. We analyze the channel capacity of the pure-loss optical channel with a general coherent-processing receiver in the low-photon number regime and compare it with the capacity achievable with direct detection and the Holevo limit (achieving the latter would require a quantum joint-detection receiver). We show compelling evidence that despite the optimal performance of Dolinar's receiver for the binary coherent-state hypothesis test (either in error probability or mutual information), the asymptotic communication rate achievable by such a coherent-processing receiver is only as good as direct detection. This suggests that in the infinitely long codeword limit, all potential benefits of coherent processing at the receiver can be obtained by designing a good code and direct detection, with no feedback within the receiver.
For space-based laser communications, when the mean photon number per received optical pulse is much smaller than one, there is a large gap between communications capacity---measured in bits communicated per pulse---achievable with a receiver that detects (converts from optical to electrical domain) each modulated pulse one at a time, versus with the quantum-optimal joint-detection receiver that acts on a long codeword comprised of nnn modulated pulses; an effect often termed superadditive capacity. The action of this receiver cannot be described as the detection of each individual pulse, interspersed with classical feedforward and soft-information post-processing. In this paper, we consider the simplest scenario where a large superadditive capacity is known: a pure-loss channel with a coherent-state binary phase-shift keyed (BPSK) modulation. The two BPSK states can be mapped conceptually to two non-orthogonal states of a single qubit, described by an inner product that is a function of the mean photon number of each BPSK pulse. Using this map, we derive an explicit construction of the quantum circuit of a joint-detection receiver based on a recent idea of belief-propagation with quantum messages (BPQM) [arXiv:1607.04833]. We analyze this scheme rigorously and show that it achieves the quantum limit of minimum average error probability in discriminating 888 (BPSK) codewords of a length-555 binary linear code with a tree factor graph. We quantify its performance improvement over the (Dolinar) receiver that optimally detects one pulse at a time. Our result suggests that a BPQM-receiver might attain the Holevo capacity, the quantum limit of classical communication capacity, of this BPSK-modulated pure-loss channel. This suggests a new application for a small, special-purpose, photonic quantum computer capable of so-called cat-basis universal qubit logic.