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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Quantum-statistical restrictions on information transmission in finite-frequency-band photonic and electronic channels

Fedor V. Karpushko, Mikhail A. Khodasevich

Published May 6, 1996• 0 citations
Publisher

Abstract

Hence photons and electrons obey different statistics, information features of noisy photonic and electronic channels are essentially different in quantum limit. For one dimensional channels quantum effects start to influence the information capacity when the signal power is comparable with the equilibrium noise power in a channel. Finite frequency band fermionic channels are limited in respect to the information rate growth. Bosonic channels are free of this limitation.

DOI
10.1117/12.239841

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Hence photons and electrons obey different statistics, information features of noisy photonic and electronic channels are essentially different in quantum limit. For one dimensional channels quantum effects start to influence the information capacity when the signal power is comparable with the equilibrium noise power in a channel. Finite frequency band fermionic channels are limited in respect to the information rate growth. Bosonic channels are free of this limitation.
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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.
We will begin with a simple type of stationary stochastic1 systems of quantum physics using them within a frame of the Shannon Information Theory and Thermodynamics but starting with their algebraic representation. Based on this algebraic description a model of information transmission in those systems by defining the Shannon information will be stated in terms of variable about the system state. Measuring on these system is then defined as a spectral decomposition of measured quantities operators. The information capacity formulas, now of the narrow-band nature, are derived consequently, for the simple system governed by the Bose–Einstein (B–E) Law [bosonic (photonic) channel] and that one governed by the Fermi-Dirac (F–D) Law [fermionic (electron) channel]. The not-zero value for the average input energy needed for information transmission existence in F–D systems is stated [11, 12].
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.
The classical capacity of a noisy optical channel is the maximum rate at which information can be transmitted reliably, given a constraint on the mean input energy. For the ubiquitous bosonic Gaussian channels, which include standard models of loss and noise in fibre and free-space optical links, determining this rate requires addressing whether entanglement between channel uses or non-Gaussian signals can outperform Gaussian encodings. More precisely, the question is whether the energy-constrained Holevo information is additive across arbitrarily many channel uses and coincides with its Gaussian counterpart. We answer these fundamental questions in the affirmative for every single-mode bosonic Gaussian channel at every input energy. Namely, we prove that independently prepared Gaussian states with Gaussian-distributed random displacements achieve the capacity, including in the low-energy regime where squeezing the signals competes with encoding information in their displacement. Our proof connects an entanglement inequality to a sharp, additive lower bound on output entropy. We then characterise communication above capacity: under a strict total photon-number cutoff, the probability of correct decoding decreases exponentially with the number of channel uses, and we derive explicit bounds on the decay exponent. When only the mean photon number is constrained, a simple energy-rescaling formula determines exactly the entire asymptotic rate--error trade-off: the capacity at mean photon number NNN and allowed average decoding error εεε equals the vanishing-error capacity at mean photon number N/(1−ε)N/(1-ε)N/(1−ε). Together, these results identify the ultimate limits of classical communication for some of the most fundamental models of optical communication.
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.