here it is possible to start addressing the issues that are central to information theory, namely, "How efficiently can we code information in a quantum communication channel?" both in terms of economy of means- - the concept of data compression and accuracy of transmission - the concept of message integrity or minimal data error, referred to here as fidelity. In QIT, the concepts of compression and fidelity are, in fact, intimately and nicely related, as described in this section. As described in the previous chapter, a quantum communication channel requires an alphabet of pure states, which are used to encode the classical information con- tained in message bits, letters, or codewords. In the classical case, a message like m = 1001110101000110 can be viewed either as a sequence of 16 individual bits that are transmitted one at a time, or as a single "block" of two-byte size (one byte is an eight-bit sequence). Similarly, we may conceive of a full quantum message not as a time sequence of n qubit symbols, but as a "block" represented by the tensor state |M> = /x1x2 with each qubit /xi) being randomly selected from a given symbol alphabet {|ak)}k=1...N of size N. Since each symbol |ak) is associated with an occurrence Quantum data compression probability Pk, we can define a symbol density operator as: (22.1) Consistently, the full symbol block is characterized by the message density operator: (22.2) I shall now provide a basic illustration of the above, which will also enable me to introduce the concept of fidelity. Assume a quantum communication channel with messages based on a two-symbol alphabet of pure states {|a), |b)}, defined as: (1)=10)=(6) (22.3) The classical message source, X, is assumed to be uniformly distributed, SO the two quantum symbols have equal probabilities Pa = pb = 1/2. Thus, the symbol density operator is given by p = pala)(al+ palb)(b) (22.4) -1(ii) The eigenvalues of p are easily found to be da = (1+1//2)/2 s2(/8) and Na = (1 - 1/2)/2 = sin2(8), respectively. The corresponding eigenvecto