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IEEE Transactions on Information Theory

Some quantum effects in information channels

Thomas Stern

Published Sep 1, 1960• 57 citations
Publisher

Abstract

In this paper the quantum nature of electromagnetic radiation is used as a basis for a mathematical model of a continuous channel. It is shown that this "photon channel" model leads to more realistic conclusions regarding information transmission. Among the results obtained are: 1) the maximum entropy for a narrow-band source under an average power limitation, 2) the frequency distribution (Bose-Einstein) for a wide-band power-limited source, 3) the transmission rate through a Poisson channel with additive Poisson noise.

DOI
10.1109/tit.1960.1057578

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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.

•Jun 1, 1968•0 citations
In this paper the quantum nature of electromagnetic radiation is used as a basis for a mathematical model of a continuous channel. It is shown that this "photon channel" model leads to more realistic conclusions regarding information transmission. Among the results obtained are: 1) the maximum entropy for a narrow-band source under an average power limitation, 2) the frequency distribution (Bose-Einstein) for a wide-band power-limited source, 3) the transmission rate through a Poisson channel with additive Poisson noise.
On the base of entropy defect principle the models are considered of narrow-band quantum communication channels and compared with the classic description of information channels. The classic model is shown to be applicable in a wide range of channels' parameters. But in the case of fermionic channels it is necessary to take into account that values of signal power and channel capacity are limited. These maximum values can be approximated by the simple expressions. Bosonic channels are free of these limitations. The rough quantum model of narrow-band bosonic channels gives too high values of capacity at low communication rate as compared with accurate one.
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.
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.
An expression is derived for the capacity of a noiseless photon channel as a function of bandwidth, carrier frequency, and average signal power. This expression is an extension of a formula derived by Gordon which is applicable in the limit of narrow bandwidth. By comparison of the two results, it is shown that Gordon's formula is highly accurate over a large range of bandwidth, and that significant deviations occur only when the average signal power is extremely small. The derivation is based on a model of transmitting photons at different frequencies, distinguishable to the limit of the Heisenberg principle, which allows distinguishable states to have the same energy. This degeneracy, which in the narrow-band limit is simply related to the time-bandwidth product, plays a role in the (maximum entropy) energy distributions. These distributions are shown to vary from noise-like (where the rms fluctuations exceed the mean value) to Poisson as the time-bandwidth product varies from unity (or nondegenerate) to infinity. For any degeneracy, the distribution is the discrete analog of its continuous (classical) counterpart. For infinite degeneracy, when photon frequencies may take a continuum of values, the channel capacity is achieved for a signal power spectral density which has the same form as the average energy of a quantized oscillator in thermal equilibrium. As a concluding result, it is shown that for a noiseless channel with fixed average power but without any frequency restrictions, there exists an upper bound to channel capacity which is proportional to the square root of the average signal power.
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.