When the density of a Poisson process is integrable various expressions published in the literature are incorrect. This is especially the case of the probability distribution of the distance between an origin and the following points of the process. The first purpose of this paper is to explain why the integrability of the density changes the situation. The second is to discuss various consequences of this fact on the probability distribution of random variables extracted from the process. Computer experiments are presented and are in excellent agreement with theoretical results. Some extensions of the same problem concerning renewal processes are discussed

Several relations between the rate distortion function and the probability of error in detection are studied in the context of quantum communication theory. Both of these criteria are associated with the same fidelity function.

Quantum communication with a nonsymmetrical set of three states is considered. Some results obtained from numerical computation are presented. A model for quantum detection called "near optimum", similar to the binary case, is studied. Calculations of error probability in detection schemes subject to information criteria are performed and discussed. Further, comparison of the error probability in detection between these different schemes is analyzed.>

Quantum communication is studied under the cutoff-rate criterion for transmission of M coherent states of real amplitude (M=2, . . ., 7). Comparison with other measurement operators of interest, such as the number and the quasi-classical operators, is made. The effect of quantization of the decision level on the cutoff rate is considered.>

The cutoff rate R/sub 0/ of an optical channel transmitting M-ary phase-shift-keyed coherent states is calculated for a receiver using the optimum quantum detector. This is shown to be nearly equivalent, when M>>1, to a system that measures the phase of the received signal. A closed-form expression yielding a good approximation to R/sub 0/ is derived.>

A new method for bounding the cutoff rateRis presented. As an application, the case of ideal laser input signal with thermal noise on the photon counting PPM channel with soft decision decoding is given. The main result is that the cutoff rate efficiency (defined aslim_{s\rightarrow + \infty} R/s, wheresis the average signal photon number) is given by (1- \gamma)/(1 + \gamma)(where\gamma = n_{0}/(1 + n_{0}), n_{0}being the average thermal noise photons). This value is much smaller than the capacity efficiency,log (1/gamma), given by Pierce et al. in [1].

Calculation of the capacity for a Poisson channel with a source noise as modeled by Pierce is done by bounding the error Probability. Based on Chernoff's bound properties, a more general method yielding a formula for the channel capacity is outlined.

Results of receiver performance for the detection of a coherent light signal in the presence of background radiation are obtained and compared for two different types of background noise: a Gaussian field and a Gaussian field whose complex analytic signal is not circularly symmetric. It is pointed out that the randomness in the noise radiation does not significantly diminish the probability of correct detection.