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.
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.
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.
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.
We consider the physical limitations imposed on the information content of an image by the wave and quantum nature of light, when the image is obtained by illuminating a reflecting or transmitting planar object by natural (i.e., fully thermalized) light, or by observation of an object emitting incoherent (thermal) radiation. The discreteness of the degrees of freedom and the statistical properties of thermal radiation are taken into account. We derive the maximum amount of information that can be retrieved from the object. This amount is always finite and is proportional to the area of the object, the solid angle under which the entrance pupil of the receiver is seen from the object, and the time of observation. An explicit expression for the information in the case where the information recorded by the receiver obeys Planck’s spectral distribution is obtained. The amount of information per photon of recorded radiation is a universal numerical constant, independent of the parameters of observation.
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 information content in a thermal photon beam and in a single-mode maser beam is computed, taking account of the limited quantum efficiency of the detector, and extending the results to polychromatic radiation in the thermal case. Formulas for fluctuations of the measured quantity in a polychromatic thermal beam are found, both for a photoelectric and for a thermal type of detector. Numerical examples of the information content are given for three common thermal light sources, using the eye as detector. For all three sources the information is found to be of the order of 104–105 bits/sec for a sample of 1 cm2 at 1 m distance from the the eye.
Quantum information processing leverages the principles of quantum mechanics, utilizing qubits, to improve computational and communicative tasks. In this realm, the quantum channel’s capacity is pivotal in determining the efficiency and accuracy of quantum information handling, with its performance being significantly influenced by channel noise. Our study aims to establish a holistic hybrid quantum noise model to determine the quantum channel capacity. In this paper, we formulated a mathematical expression for this capacity and conducted simulations for both Gaussian and non-Gaussian inputs. A hybrid noise model is constructed by convolution of Poisson-distributed quantum noise with classical additive white Gaussian noise. We characterized the quantum-classical noise and the received signal using Gaussian Mixture Models. The maximum amount of quantum information that can be reliably transmitted over a quantum channel (per use of the channel) is determined by its capacity, and entropy and related quantities like mutual information play a role in calculating this capacity. Our formulation of quantum channel capacity is derived from the mutual information shared between the transmitter and receiver, encompassing the entropies of the signals. The quantum channel presents a higher capacity-to-signal-to-noise ratio for Gaussian inputs than non-Gaussian ones.