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.
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.
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.
Directional optical noise makes photons pay for both squeezing and signal modulation. This paper determines the optimal transmitter and the ultimate classical communication rate, even when arbitrary entangled and non-Gaussian encodings across many channel uses are allowed. For a finite collection of independent squeezed thermal attenuators with different losses and noise covariances, the complete quantum optimization reduces exactly to a scalar allocation of the total mean photon budget. The result covers every input energy, including the low-energy regime in which only one quadrature should be modulated, channels receiving zero photons, and pure squeezed environments. 1. Exact capacity and all-block additivity. Let Φ=⨂j=1dΦj, let E≥0 be the total average input photon number per block, and let Cj(Ej) be the explicitly attained one-channel rate specified in the paper. Then C(Φ,E)=χΦ(E)=maxEj≥0,∑jEj≤E∑j=1dCj(Ej),χΦ⊗n(nE)=nC(Φ,E)(n≥1). The optimization includes arbitrary joint input ensembles, with an average physical photon constraint and unrestricted decoding. Independent Gaussian modulation of pure squeezed states attains the answer. Thus neither non-Gaussian codewords nor entanglement across the specified independent channels or repeated blocks improves the capacity. Information quantities use natural logarithms; division by ln2 gives bits. 2. From measured channel noise to a transmitter design. In quadratures with vacuum covariance I/2, each channel acts as V↦ηV+diag(yq,yp), where 0<η<1, yq≥yp>0, and yqyp≥(1−η)2/4. Define ue=21ypyq,Eth=ue−21+2ηyq−yp. Below this threshold, a bounded one-dimensional maximization determines the optimal seed squeezing and the modulation of the quieter quadrature. Above it, both quadratures can be modulated and the average output is isotropic. Writing g(x)=(x+1)ln(x+1)−xlnx, the high-energy rate is Cj(E)=g(ηE+2η+yq+yp−1)−g(2η+yqyp−21),E≥Eth. Both seed squeezing and signal modulation are charged to the same physical input budget. The manuscript gives the low-energy variational formula and the attaining covariances explicitly. 3. When should another channel receive photons? The one-channel capacity curves are increasing and concave. For strictly mixed environments, the optimal allocation equalizes marginal information gains: Cj′(Ej∗)=λ(Ej∗>0),Cj′(0+)≤λ(Ej∗=0). A previously unused channel turns on when its initial information gain becomes competitive with the declining gain of the channels already in use. Pure-environment endpoints follow by continuity of the attained capacity. In the two-channel example in Figure 1, the second channel activates at total budget approximately E=0.436 photons. The figures evaluate the analytic allocation law and display how photons divide between squeezing, modulation and different links. 4. A converse for the complete correlated input. A joint quadratic output-entropy support excludes all competing encodings. Its proof develops a sharp thermal-reference contraction using both Bogoliubov orientations and independently adjusted thermal references. The appendix supplies the regularized minimizer, Gibbs identity, full-state Hessian comparison, interpolation domains, residual propagation and limiting arguments. This connects a computable Gaussian transmitter design to a bound on every allowed quantum code. Use and scope. Communication theorists and optical-source designers can use measured loss and quadrature noise to allocate a common photon budget, choose the seed squeezing and modulation, identify inactive channels, and calculate an attained capacity benchmark for these memoryless channels. The all-energy result addresses the low-energy Holevo additivity question for a squeezed thermal attenuator, and extends it to heterogeneous finite products under one total budget. Version 1.0, 13 October 2026. The 16-page PDF contains the article, two vector figures, complete supplementary proofs and references.
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 N and allowed average decoding error ε equals the vanishing-error capacity at mean photon number N/(1−ε). Together, these results identify the ultimate limits of classical communication for some of the most fundamental models of optical communication.
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 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.