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Proceedings of the IRE

Quantum Effects in Communications Systems

J. Gordon

Published Sep 1, 1962• 196 citations
Publisher

Abstract

The information capacity of various communications systems is considered. Quantum effects are taken fully into account. The entropy of an electromagnetic wave having the quantum statistical properties of white noise in a single transmission mode is found, and from it the information efficiency of various possible systems may be derived. The receiving systems considered include amplifiers, heterodyne and homodyne converters and quantum counters. In the limit of high signal or noise power (compared to hνB, where h is Planck's constant and ν and B are, respectively, the center frequency and bandwidth of the channel) the information efficiency of an amplifier can approach unity. In the limit of low powers the amplifier becomes inefficient, while the efficiency of the quantum counter can approach unity. The amount of information that can be incorporated in a wave drops off rather rapidly when the power drops below hνB.

DOI
10.1109/jrproc.1962.288169

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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•May 6, 1996•0 citations
The information capacity of various communications systems is considered. Quantum effects are taken fully into account. The entropy of an electromagnetic wave having the quantum statistical properties of white noise in a single transmission mode is found, and from it the information efficiency of various possible systems may be derived. The receiving systems considered include amplifiers, heterodyne and homodyne converters and quantum counters. In the limit of high signal or noise power (compared to hνB, where h is Planck's constant and ν and B are, respectively, the center frequency and bandwidth of the channel) the information efficiency of an amplifier can approach unity. In the limit of low powers the amplifier becomes inefficient, while the efficiency of the quantum counter can approach unity. The amount of information that can be incorporated in a wave drops off rather rapidly when the power drops below hνB.
We consider a communication system consisting of an information source of electromagnetic radiation. It is assumed that the information source radiates an electromagnetic signal with an average power S, and the signal propagated through the electromagnetic channel is assumed to be perturbed by additive thermal noise at temperature T. For simplicity, we assume that the electromagnetic signal is one-dimensional (i.e., the photon fluctuation is restricted to only one polarized state), in which the propagation occurs in the same direction of the wave vectors. It is emphasized that the corresponding occupation number of the quantum levels can be fully described. Obviously, these occupation quantum levels correspond to the microsignal structure of the information source. It is therefore assumed that these occupation numbers can be uniquely determined for those representing an input signal ensemble. We further assume that an ideal receiver (an ideal photon counter) is used at the output end of the channel. That is, the receiver is able to detect the specified frequencies of the electromagnetic signal. It is also emphasized that in practice the interaction of the electromagnetic signal and the detector are strictly statistical, that is, a certain amount of information loss is expected at the output end of the receiver. The idealized model of the receiver we have proposed mainly simplifies the method by which the quantum effect on the communication channel can be easily calculated.
An upper bound for the information rate in a communication link is set by the finite number of orthogonal states available to the electromagnetic field, is the link is subject to constraints on the average power and bandwidth. This bound may not be attainable in a free-space link since the transmitter does not have complete control over what is received. Some hypothetical systems are examined to see how close they get. At high photon rates it is possible with difficulty to improve slightly on an 'x-p' system, and at low photon rates on a photon-detecting system, but it seems that these systems are nearly the best, and certainly they are not far below the upper bound, in the sense that the ratios of the information rates to the maximum are close to unity. In all these systems the normal modes (Planck oscillators) are treated as independent channels. It is suggested that a beam splitter can provide a simple model for a free-space link.
The problem of including quantum effects within existing methods of communication system analysis is investigated.A mathematical model is postulated for describing narrow bandwidth systems.Utilizing the Glauber P representation, the model permits analysis of both real field detection (measurement of electric and magnetic field magnitudes) and photon detection.The resulting equation which describes photon detection is shown to be consistent with previous work which has been proven experimentally.Unique aspects of the postulated model are its ability to describe both random and deterministic modulation functions in conventional terms, and its ability to demonstrate the inapplicability of the "photon channel" model for describing present-day systems.An analysis of quantum effects at the transmitter shows that modulation of the radiated field must take place by one of two distinctly different processes.Wave modulation is defined as a process which conveys information in the amplitude and phase of the electromagnetic field.Photon modulation is defined as a process which conveys information in the exact number of photons per pulse of the transmitted field (i.e., the "photon channel" of Stern, Gordon, ani othfrs).A wave modulated signal is shown to describe present-day modulation schemes and, moreover, to yield random photon fluctuations.It is concluded a wave modulated signal is not capable of describing a photon modulated signal.In contrast, a photon modulated signal is shown to possess a completely random phase fluctuation thus demonstrating the inability of the "photon channel" to describe presently used wave modulation schemes.For wave modulation the Glauber P function is shown to be identical (with a change of variable) to the classical joint probability density of the quadrature components of a narrowband sinusoid used in conventional analysis.Equations are derived which "transform" classical probability distributions into distributions which include quantum effects.The "partitioning noise" studied by Hagfors and Bowen is found to constitute a source of noise only in the "photon channel."The "partitioning effect" is shown to merely preserve the Poisson character of photon counts in wave modulated fields and therefore does not introduce a noise in present-day systems.For a wave modulated system with real field detection it is shown that photon noise can be included in conventional analysis by adding to the normal input noise, a noise density of hf c /2 Watts/Hertz where h is Planck's constant and f is the system carrier frequency.Use of photon heterodyne detection requires the addition of an input noise density of hf c Watts/Hertz.The photon noise is exactly additive Gaussian for real field detection and approximately additive Gaussian for photon heterodyne detection.Detection by counting the received photons is shown to yield neither Gaussian nor Poissonian statistics.However, with an "ideal measurement process," when the effective received noise density n c is much less than hf c , the variance of the counts are found to approach that of a Poisson distribution.Conversely, when nc is much larger than hf c , the .variance approaches that derivable from a classical analysis.A channel capacity equation for a wave modulated system is found which is equal to those derived by Lachs and Jelsma.The equation is shown to give capacities less than that derived by Gordon.
Abstract : Attempt is made to establish an upper bound on the information capacity of an electromagnetic wave propagation circuit between two apertures when quantum effects come into play. We first discuss the specifications of a signal, and find that we can associate approximately BT quantities with a signal of bandwidth B and duration T, and that the energy associated with these quantities must differ by at least one quantum of energy. We then consider radiative attenuation loss within the framework of quantum theory and find that the radiative attenuation introduces a random noise in the circuit which is analogous to partition tube noise. With the channel matrix determined, we discuss the problem of evaluating the channel capacity under average and peak power constraints. Due to mathematical difficulties we are only able to establish the channel capacity explicitly when the radiative losses become extreme. In this limiting case-as a byproduct-we also find that the best encoding procedure is a binary on-off system.
We analyze the information-capacity limitations of the optical communication channel, as determined by noise accumulation from optical amplification and nonlinear wave-mixing. We review the concepts of signal-to-noise ratio and entropy for binary-coded and continuous communications, leading to a definition of ultimate capacity for the optically-amplified channel. A unified quantum model, describing both amplification and nonlinearity limitations, makes possible to determine the power transmission window within which the channel capacity can be maximized.