Page 72 - Introduction to Information Optics

1.8. Quantum Mechanical Channel 57

the noise energy per unit time (noise power) can be calculated by

N = E,(vXlv = -r- r * = , (1.194)

where £ is an arbitrarily small positive constant.
Thus, the minimum required entropy for the signal radiation is

r dr T (1.195)

where E(T) is signal energy density per unit time as a function of temperature
T, and T is the temperature of the blackbody radiation. Thus, in the presence
of a signal the output radiation energy per unit time (the power) can be written

F - S + N,

where S and N are the signal and the noise power, respectively. Since the signal
is assumed to be deterministic (i.e., the microstate signal), the signal entropy
can be considered zero. Remember that the validity of this assumption is
mainly based on the independent statistical nature between the signal and the
noise, for which the photon statistics follow Bose-Einstein distribution. How-
ever, the Bose-Einstein distribution cannot be used for the case of fermions,
because, owing to the Pauli exclusion principle, the microstates of the noise are
restricted by the occupational states of the signal, or vice versa. In other words,
in the case of fermions, the signal and the noise can never be assumed to be
statistically independent. For the case of Bose-Einstein statistics, we see that
the amount of entropy transfer by radiation remains unchanged:

Since the mutual information is I(A; B) = H(B) — H(B/A), we see that
I (A; B) reaches its maximum when H(B) is maximum. Thus, for maximum
information transfer, the photon signal should be chosen randomly. But the
maximum value of entropy H(B) occurs when the ensemble of the microstates
of the total radiation (the ensemble B) corresponds to Gibbs's distribution,
which reaches to the thermal equilibrium. Thus, the corresponding mean
occupational quantum number of the total radiation also follows Bose-Einstein