Page 70 - Introduction to Information Optics

1.8. Quantum Mechanical Channel 55

propagated through the quantum mechanical channel is assumed perturbed by
an additive thermal noise. For simplicity, we assume that the signal is one
dimensional (i.e., the photon fluctuation is restricted to only one polarized
state), in which the propagation occurs in the same direction as the wave
vectors. Note that the corresponding occupation number of the quantum levels
can be fully described. Obviously, these occupation quantum levels correspond
to the microsignal structure of the information source. We can therefore assume
that these occupation numbers can be uniquely determined for those represen-
ting an input signal ensemble. We further assume that an ideal receiver (an
ideal photon counter) is used at the output end of the channel; that is. the
receiver is capable of detecting the photon signal. We stress that the interaction
of the photon signal and the detector are assumed statistical; that is, a certain
amount of information loss is expected at the output end of the receiver. The
idealized model of the receiver we have proposed mainly simplifies the method
by which the quantum effect on the channel can be calculated. Let us now
assume that the input photon signal can be quantized as represented by a set
of microstate signals {a,-}, and each a ( is capable of channeling to the state of
bj, where the set {bj}, j — 1 = 2,..., n, represents the macroscopic states of the
channel. Note that each macroscopic state bj is an ensemble of various
microscopic states within the channel. Thus by denoting P(bjla {] corresponding
transitional probability, for each applied signal a, the corresponding condi-
tional entropy H(B/a^ can be written as

H(B/a t) = - £ P(bj/
./=!
Thus, the entropy equivocation of the channel is

n
H(B/A) = £ P(a i)H(B/a i). (1.188)

Since the (output) entropy can be written as

H(B)= I P(b,)logP(bj). (1.189)
,/=!

The mutual information provided by the quantum mechanical channel can be
determined by

I(A; B) = H(B] - H(B/A), (1.190)