h associated probability distribution {pi = p(pi)}, the capacity X of a noisy quantum channel E providing the operation Oi=E(pi) is given by the maximum difference: x=max[S(o)-S(o))] = x{S[E((pi))]-S[e(pi)])}, (23.24) Flipping complex amplitudes leave 1+) = (|0) + |1)) /V2 invariant, while resulting in a global phase factor eir = -1, - which does not modify the definition of |->. Quantum channel noise and channel capacity where the brackets < > indicate an averaging over the distribution {pi}. This definition can be put into the more explicit or complete formulation: (23.25) As with Shannon's channel coding theorem, the capacity X represents the maximum code rate for which the probability of transmission error can be made arbitrarily small, assuming sufficiently long message lengths. The following provides a background and a basic interpretation for the HSW theorem, showing how it nicely parallels Shannon's channel coding theorem. As we have seen in Chapter 22, a quantum message of length n can be conceived as an n-qubit "block" represented by the tensor state |M)=1x1x2 Xn), with each qubit /xk) being randomly selected with an occurrence probability Pi, from a given symbol alphabet {|ai)}i=1...N of size N. We assume that the originator chooses among 2nR message possibilities, where R > 0. This limited set of "quantum codewords" of length n can be referred to as a "code," and R as a code rate. A symbol density operator Pmk can be associated with any message qubit in position k, where Mk is an index selected from the alphabet i = 1, 2, N. Hence, the originator's message block or codeword is characterized by a density operator PM constructed from the n-tensor product:2 PM=Pm10 Pm2 X X Pmn. (23.26) We note that since the message block is defined as an n-tensor product, there is no entanglement between any of the qubits therein. The recipient message, or codeword is received as M=m2Omn (23.27) where (23.28) Following the description in Chapter 22 (but not being concerned here about compres- sion), it is clear that a s