It has been well known since the pioneering work of Claude Shannon in the 1940s that a message transmitted with optimal efficiency over a channel of limited bandwidth is indistinguishable from random noise to a receiver who is unfamiliar with the language in which the message is written. We derive some similar results about electromagnetic transmissions. In particular, we show that if electromagnetic radiation is used as a transmission medium, the most information-efficient format for a given message is indistinguishable from blackbody radiation. The characteristic temperature of the radiation is set by the amount of energy used to make the transmission. If information is not encoded in the direction of the radiation, but only in its timing, energy, and polarization, then the most efficient format has the form of a one-dimensional blackbody spectrum.
It has been well known since the pioneering work of Claude Shannon in the 1940s that a message transmitted with optimal efficiency over a channel of limited bandwidth is indistinguishable from random noise to a receiver who is unfamiliar with the language in which the message is written. We derive some similar results about electromagnetic transmissions. In particular, we show that if electromagnetic radiation is used as a transmission medium, the most information-efficient format for a given message is indistinguishable from blackbody radiation. The characteristic temperature of the radiation is set by the amount of energy used to make the transmission. If information is not encoded in the direction of the radiation, but only in its timing, energy, and polarization, then the most efficient format has the form of a one-dimensional blackbody spectrum.
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 literature describing the energy needs for a communications channel has been dominated by analyses of linear electromagnetic transmission, often without awareness that this is a special case. This case leads to the conclusion that an amount of energy equal to kTln 2, where kT is the thermal noise per unit bandwidth, is needed to transmit a bit, and more if quantized channels are used with photon energies hnu > kT. Alternative communication methods are proposed to show that there is no unavoidable minimal energy requirement per transmitted bit. These methods are invoked as part of an analysis of ultimate limits and not as practical procedures.
Based upon the Fluctuation-Dissipation Theorem of Nyquist-Callen-Welton and the spectrum balancing relationship of Kubo-Martin-Schwinger for operator-valued stochastic noise processes, a two-parameter Quantum Noise-Energy Spectral Density (QN-ESD) is constructed and Fourier-Laplace transformed to obtain its associated Quantum Noise-Autocorrelation Function (QN-ACF). By means of a Taylor series expansion, the quantum thermal noise correlations are linked to Number Theory's Reimann-Zeta function. From this perspective, a panoply of spectral components and their QN correlations are characterized for three disjoint regions located across the Electromagnetic Spectrum (EMS). The second-moment characterizations and frequency boundaries for each region are specified as a function of two frequency dependent design parameters, viz., a normalized temperature dependent frequency and a quantum receiver cutoff frequency. These boundaries define where the quadrature noise correlation projections of the QN-ESD become noticeably asymmetric-in-frequency. For the operator-valued white quantum noise region, various communication system performance metrics are characterized and their limits are presented and compared via interconnecting parameter relations to their classical communications (CC) counterparts. These metrics include Shannon's non-additive, but regularized, zero-error capacity in bits/photon-pair, bits/photon, bits/channel-use and bits/sec, the spectral efficiency in bits/sec/Hz and the error-correcting code rate. For long coded messages, Shannon's capacity quadrature approaches 2π√5/ ln(2) ≈ 20 bits/photon-pair or (10 bits/photon) while for single-photon per channel use transmission it approaches 3 bits/photon-pair. Results are graphically illustrated for the quantum noise spectral density, quantum noise correlations, Shannon's Î-Q̂ capacities and spectral efficiency.
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.
Introduction Error-correcting codes were introduced in the 1940s in order to implement a theorem of Shannon's [260] which guaranteed that virtually error-free communication could be obtained even over a noisy channel. The message to be communicated is first “encoded”, i.e. turned into a codeword, by adding "redundancy". The codeword is then sent through the channel and the received message is “decoded” by the receiver into a message resembling, as closely as possible, the original message. The degree of resemblance will depend on how good the code is in relation to the channel. The usual pictorial representation of this communication link is shown in the schematic diagram Figure 2.1. Here a message is first given by the source to the encoder that turns the message into a codeword, i.e. a string of letters from some alphabet, chosen according to the code used; the encoded message is then sent through the channel, where it may be subjected to noise and hence altered; this possibly-altered message, when it arrives at the decoder belonging to the receiver, is, first of all, equated with the most likely codeword, i.e. the one (should that exist) that, in a probabilistic sense depending on the channel, was probably sent; finally this “most likely” codeword is decoded and the message is passed on to the receiver. Example 2.1.1 Suppose we use an alphabet of just two symbols, 0 and 1, and we have only two messages, for example “no” corresponding to 0, and "yes” corresponding to 1. We wish to send the message “no”, and we add redundancy by simply repeating the message five times. Thus we encode the message as the codeword (00000). The channel interferes, perhaps, with the message and could, for example, change it to (01100). The decoder assesses the message and decides that of the two possible codewords, i.e. (00000) and (11111), the former is the more likely, and hence the message is decoded, correctly, as “no”. Notice that we have made several assumptions here: for example we have assumed that the probability of an error at any position in the word is less than, that each codeword is equally likely to be sent, and that the receiver is aware of the code used.