Page 69 - Introduction to Information Optics

54 1. Entropy Information and Optics

The radiant energy provided can be written as

^-7-. (1.186)
A. 2Ax'

for which we have
he
(1.187)

Theoretically speaking, there is no lower limit to Ax, as long as AE is able to
increase. However, in reality, as AE increases, the perturbation of the observa-
tion cannot be ignored. Therefore in practice, when AE reaches a certain
quantity, the precise observation of Ax is obstructed and the observation of
smaller and smaller particles presents ever-increasing difficulty.
Finally, let us emphasize that Heisenberg's principle of uncertainty observa-
tion is restricted to the ensemble point of view; that is, for a special observation,
the uncertainty may be violated. However, we have never been able to predict
when this observation will occur. Therefore, a meaningful answer to the
Heisenberg uncertainty principle, is only true under the statistical ensemble.

1.8. QUANTUM MECHANICAL CHANNEL

In the preceding sections we have presented an information channel from a
many particles point of view. Intuitively, the formulation of the channel as we
assumed is quite correct. However, when we deal with a communication
channel in quantum mechanical regime, the results we have evaluated may lead
to erroneous consequences. For instance, the capacity of a continuous additive
Gaussian channel is given by

C = Avlog 2 1 1 -I — I bits/sec,

in which we see that, if the average noise power N approaches zero, the channel
capacity approaches infinity. This is obviously contradictory to the basic
physical constraints. Therefore, as the information transmission moves to
high-frequency regime, where the quantum effect takes place, the communica-
tion channel naturally leads to a discrete model. This is where the quantum
theory of radiation replaces the classical wave theory.
We consider a quantum mechanical channel, for which the information
source represents an optical signal (e.g., temporal signal). The signal