Previous work has shown that the capacity of the single-mode photon-counting channel in the region in which quantum effects are most pronounced is1/(kT \ln 2)bit/J. HereTis the quantum noise temperature, andkis Boltzmann's constant. However, to achieve this, the number of photons per second must be very low, so this does not determine the capacity of the average power-limited photon channel in bits per second. Here the previous work is extended and the capacity is determined in bits per second. This includes a determination of the optimum counting time. The capacity in bits per second is for lowTasymptotic toP_{s}/(kT \ln 2), whereP_{s}is the average signal power. This is high enough to suggest that photon counting be seriously considered for applications such as a deep space to near Earth link. Some related problems are also considered including a peak power limitation.
Previous work has shown that the capacity of the single-mode photon-counting channel in the region in which quantum effects are most pronounced is1/(kT \ln 2)bit/J. HereTis the quantum noise temperature, andkis Boltzmann's constant. However, to achieve this, the number of photons per second must be very low, so this does not determine the capacity of the average power-limited photon channel in bits per second. Here the previous work is extended and the capacity is determined in bits per second. This includes a determination of the optimum counting time. The capacity in bits per second is for lowTasymptotic toP_{s}/(kT \ln 2), whereP_{s}is the average signal power. This is high enough to suggest that photon counting be seriously considered for applications such as a deep space to near Earth link. Some related problems are also considered including a peak power limitation.
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.
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 fundamental performance limits and channel capacity of optical commu-nications systems operating over the free-space channel will be examined using quantum detection theory. The performance of the optimum quantum receiver for on{ofi keying and optical binary phase-shift keying is flrst examined as a pure-state (no-noise) problem. The classical capacity of the binary symmetric channel for these two modulation schemes will be evaluated for the optimum quantum receiver by making use of the concept of quantum measurement states. The performance of M-ary pulse-position modulation, which requires a product state representation, will be evaluated along with the performance of certain dense signal sets. Perfor-mance comparisons with classical techniques show an improvement of over 5 dB in some cases when quantum detection is employed. As a further application of the quantum detection theory, the capacity of the binary channel with on{ofi keyed modulation and quantum detection is evaluated and shown to exceed the capacity obtained with classical photon counting. I.
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.
Deep-space optical communication links operate under severely limited signal power, approaching the photon-starved regime that requires a receiver capable of measuring individual incoming photons. This makes the photon information efficiency (PIE), i.e., the number of bits that can be retrieved from a single received photon, a relevant figure of merit to characterize data rates achievable in deep-space scenarios. We review theoretical PIE limits assuming a scalable modulation format, such as pulse position modulation (PPM), combined with a photon counting direct detection receiver. For unrestricted signal bandwidth, the attainable PIE is effectively limited by the background noise acquired by the propagating optical signal. The actual PIE limit depends on the effectiveness of the noise rejection mechanism implemented at the receiver, which can be improved by the nonlinear optical technique of quantum pulse gating. Further enhancement is possible by resorting to photon number resolved detection, which improves discrimination of PPM pulses against weak background noise. The results are compared with the ultimate quantum mechanical PIE limit implied by the Gordon–Holevo capacity bound, which takes into account general modulation formats as well as any physically permitted measurement techniques.
The fundamental performance limits and channel capacity of optical communications systems operating over the free-space channel will be examined using quantum detection theory. The performance of the optimum quantum receiver for on{ofi keying and optical binary phase-shift keying is flrst examined as a pure-state (no-noise) problem. The classical capacity of the binary symmetric channel for these two modulation schemes will be evaluated for the optimum quantum receiver by making use of the concept of quantum measurement states. The performance of M-ary pulse-position modulation, which requires a product state representation, will be evaluated along with the performance of certain dense signal sets. Performance comparisons with classical techniques show an improvement of over 5 dB in some cases when quantum detection is employed. As a further application of the quantum detection theory, the capacity of the binary channel with on{ofi keyed modulation and quantum detection is evaluated and shown to exceed the capacity obtained with classical photon counting.