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Journal of the Optical Society of America

Information in the time of arrival of a photon packet: capacity of PPM channels*

I. Bar-David

Published Feb 1, 1973• 9 citations
Publisher

Abstract

Expressions for the information rate and capacity of amplitude-modulated photon beams are available in literature. Recent interest in position-modulated laser pulses has motivated the investigation of the following model: A random variable D, which can take on values in the interval (−T, T), is transmitted by centering a narrow, coherent, single-mode light pulse of duration 2D and constant irradiance at t=D. It is assumed that no other than quantum fluctuations, of the otherwise stable source, disturb the transmission. The information that the received photon packet carries about D is registered as the instants {tk} of emission of photoelectrons at the detector. If H denotes the D and Q, the expected number of photoelectrons is 2D, and D≪T, then the mutual information between the D− and {tk} ensembles is, for large Q, approximately H−ln(2D/ρQ). Here, ρ = exp(γ − 1) and γ is Euler’s constant. Under the peak-excursion constraint |D|≤T, this is maximized by H = ln2T, for uniformly distributed D, so that for large Q the capacity C = ln(ρQT/D). The accurate expression, valid for all Q, involves the exponential integral E1(Q). The value of C is used to derive a lower bound on the mean-square error of any estimator of D, by the rate-distortion method. The bound, which decreases as Q2, is compared with the variance of maximum-a-posteriori-probability estimators of delay which, in the case of differentiable pulses, decreases only as Q.

DOI
10.1364/josa.63.000166

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The problems studied in this thesis fall within two different topics in network information theory. The first part of this dissertation will be about information theoretic security and the second about distributed source coding. In the following a brief description of these problems will be given. In information theoretic security, we look at two different types of timing channels and we quantify the maximum rate at which a transmitter can communicate information reliably to a legitimate receiver while keeping an eavesdropper in the dark. The first timing channel that we study is the Poisson channel which is used to model certain direct detection optical communication systems. To transmit a message on this channel, the transmitter encodes information by modulating the intensity of an optical signal while the legitimate receiver and the eavesdropper use the arrival moments of the individual photons to decide which message was transmitted. The second timing channel studied is the exponential server queue. Here the transmitter encodes a message using a chosen sequence of packets inter-arrival times and both the legitimate receiver and the eavesdropper use the corresponding inter-departures from their respective exponential server queues to decode the transmitted message. In distributed source coding, we consider a rate-distortion problem in which a decoder is interested in estimating two correlated Gaussian random variables with mean-square error distortion constraints on each of the reproductions. The variables to be estimated are the roots of a given Gauss-Markov tree and each encoder observes one of the leaves of that tree. We show that a simple compression architecture that performs separate lossy quantization followed by SlepianWolf binning is sum-rate optimal for this problem.

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Expressions for the information rate and capacity of amplitude-modulated photon beams are available in literature. Recent interest in position-modulated laser pulses has motivated the investigation of the following model: A random variable D, which can take on values in the interval (−T, T), is transmitted by centering a narrow, coherent, single-mode light pulse of duration 2D and constant irradiance at t=D. It is assumed that no other than quantum fluctuations, of the otherwise stable source, disturb the transmission. The information that the received photon packet carries about D is registered as the instants {tk} of emission of photoelectrons at the detector. If H denotes the D and Q, the expected number of photoelectrons is 2D, and D≪T, then the mutual information between the D− and {tk} ensembles is, for large Q, approximately H−ln(2D/ρQ). Here, ρ = exp(γ − 1) and γ is Euler’s constant. Under the peak-excursion constraint |D|≤T, this is maximized by H = ln2T, for uniformly distributed D, so that for large Q the capacity C = ln(ρQT/D). The accurate expression, valid for all Q, involves the exponential integral E1(Q). The value of C is used to derive a lower bound on the mean-square error of any estimator of D, by the rate-distortion method. The bound, which decreases as Q2, is compared with the variance of maximum-a-posteriori-probability estimators of delay which, in the case of differentiable pulses, decreases only as Q.
The process of attenuation of electromagnetic energy when transmitted over some communication channel is stochastic in nature. Hence one can expect that, since equivocation is imposed in the transmission, in addition to reduction of average signal power, there will be a consequent degradation in channel capacity. We have attempted to calculate the effect of this phenomenon on an otherwise noiseless photon channel and when the source itself is assumed to have no undesired fluctuations. In order to do so, we have employed a straightforward variational technique and introduced the partition function method of statistical mechanics to obtain an expression for channel capacity. Using approximate methods, we find that the maximum information transfer is achieved when the probability of receiving exactlykphotons goes down, asymptotically with largek, ase^{-ck}/ \sqrt{k}, wherecis a positive constant. For very large received average energyU(in photon number), the channel capacity behaves approximately asC = \frac{1}{2} \ln U, one-half the capacity of an ideal noiseless photon channel (given by Gordon's formula for the capacity of an electromagnetic wave). For very small received average energy, we find that the (optimum) received distribution is very nearly Bose-Einstein, and the channel capacity is not degraded below its ideal value. The results are applicable to an idealized laser oscillator and also to photoelectric detection. That is, both devices by themselves may be regarded as "channels" for which the input may be modulated as desired and the output is stochastically dependent on the input.
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.
The chief emphasis of the paper is given to the calculation of the information capacity of a beam of light for on-off signal modulation. A preliminary and approximate calculation in Sec. 2 is based on Gaussian modulation. On-off modulation permits an exact calculation of the information capacity; the calculation is based on the Poisson statistics of nondegenerate ensembles of Bose entities.Most of the results are given in terms of the information capacity per transmitted photon, measured in bits per photon, and called the information efficiency It. When there is no ambient light, and when the probability p that the gate is open is equal to one-half, the maximum possible information efficiency is just one bit per transmitted photon; when, however, the probability p of an open gate is less than one-half, the information efficiency It may be higher, with the upper bound log2(1/p).The calculation of the information efficiency in Sec. 3 supposes that no light reaches the receiver except that from the gated source. In Sec. 4 ambient light also is permitted to reach the receiver; extensive numerical results were obtained with a high-speed digital computer and are presented in tables and graphs.The term light is used throughout the paper instead of radiation, in order to emphasize that the results hold only for nondegenerate beams of radiation. The output of laser oscillators is therefore excluded.
Abstract We derive fundamental lower bounds on the performance of optical metrology and communication systems in a Bayesian framework. The derivation uses classical rate-distortion theory in conjunction with bounds on the capacity to transmit classical information of various optical channels specified by the system design. The bounds are expressed in terms of the system parameters, the prior probability distribution of the parameter, and the average energy, i.e. number of photons E in the probe state. For phase estimation, the bounds pertain to a cyclic mean squared error criterion incorporating the cyclic nature of the phase. In the absence of optical loss, our bounds are applicable to multimode linear phase modulation schemes (including ancilla-assisted ones), and to nonlinear modulations on a single mode. The bounds display inverse-quadratic Heisenberg-limit scaling of the cyclic mean square error with respect to E . In the presence of any finite amount of loss, a lower bound on ancilla-assisted phase estimation with standard quantum limit scaling is derived, which is shown to be little different from a similar bound for coherent-state probes. For systems involving a single optical mode, we also obtain lower bounds on the mean squared error of estimating any classical parameter, and on the average error probability of any M -ary communication system under an average energy constraint. The bounds are valid for arbitrary quantum measurements, for any prior probability distribution, and do not rely on unbiasedness assumptions.
The amount of information transmitted via a photon beam that is modulated in s equal, discrete steps from a given maximum expected number of pulses kmax and down to a given kmin is computed. The numbers of pulses refer to the counts of a photoemissive detector. The information i reaches a maximum im for a certain number s = sm for which the difference δk of the expected number of detected pulses from successive levels is about 2.7 times the rms fluctuation of the kmax level. The reliability (ratio of the information obtained in the presence of noise to the information that would be obtained without noise) of this maximum information im is about 0.8. im is very nearly a function of the single parameter c=(kmax−kmin)/kmin12. These results hold, provided that the rms fluctuation of the kmax level is not too different from that of the kmin level. For kmin = 0, maximum information is obtained with a somewhat smaller spacing δk, and its reliability is about 0.61 for kmax ≥ 20.
The problems studied in this thesis fall within two different topics in network information theory. The first part of this dissertation will be about information theoretic security and the second about distributed source coding. In the following a brief description of these problems will be given. In information theoretic security, we look at two different types of timing channels and we quantify the maximum rate at which a transmitter can communicate information reliably to a legitimate receiver while keeping an eavesdropper in the dark. The first timing channel that we study is the Poisson channel which is used to model certain direct detection optical communication systems. To transmit a message on this channel, the transmitter encodes information by modulating the intensity of an optical signal while the legitimate receiver and the eavesdropper use the arrival moments of the individual photons to decide which message was transmitted. The second timing channel studied is the exponential server queue. Here the transmitter encodes a message using a chosen sequence of packets inter-arrival times and both the legitimate receiver and the eavesdropper use the corresponding inter-departures from their respective exponential server queues to decode the transmitted message. In distributed source coding, we consider a rate-distortion problem in which a decoder is interested in estimating two correlated Gaussian random variables with mean-square error distortion constraints on each of the reproductions. The variables to be estimated are the roots of a given Gauss-Markov tree and each encoder observes one of the leaves of that tree. We show that a simple compression architecture that performs separate lossy quantization followed by SlepianWolf binning is sum-rate optimal for this problem.