y Perror is: (23.36) Quantum channel noise and channel capacity If we assume that the originator uniformly selects the messages from the codeword set {PM}, it is then possible to define an average error probability Perror through (23.37) Define (Perror) as the expectation value of Perror over all possible codewords. The formal demonstration of the HSW theorem then leads to the following result, which I shall here provide "as is:" (Perror) < (23.38) x = S((a)) - (S(o)). (23.39) The result in Eq. (23.38) shows that provided the condition for the code rate R, <S((o)) - ( $(o ) = X, (23.40) and for long message lengths (n 00) the expected error probability is upper-bounded according to (Perror) < 8' = 48 with 8 being any real such that 0<8< 1, i.e., which can be chosen arbitrarily close to zero. The maximization of X, which yields the quantum channel capacity, is an issue at least as complex as in the case of classical communication channels. Suffice it here to infer that under the condition R < max(x) it is possible to find another class of quantum codes for which Perror < 8", with 8" being arbitrarily close to zero. The key conclusion is, therefore: Given a quantum channel E, there exist quantum codes with long codeword size n and code rate R < max(x), for which the error probability Perror (of the recipient failing to identify a codeword OM=E(PM) = given an originator codeword PM) can be made arbitrarily small. This conclusion defines the key condition under which "error-free" transmission of quantum messages through noisy quantum channels can be achieved. It is also possible to show the converse, namely: "for R > X there exists no quantum code of any codeword size able to achieve error-free transmission," meaning that Perror is irremediably bounded away from zero. As stated earlier, there exists a nice parallel between the HSW "capacity" theorem and Shannon's coding theorem for classical, noisy communication channels. To recall from Chapter 13, the classical channel capacity C is defined as the mutual information