error probability can be the same for the two symbols (binary symmetric channel), which gives, for instance, (23.2) The above recall from the classical information theory will now help to conceive of the effect of noise in a quantum communication channel. As we have seen in Chapter 22, a quantum message M is a "block" represented by the tensor state |M> =19192...9n), with each qubit lqi) being randomly selected from a given symbol alphabet {|xk)}k=1...N of size N. Given the fact that each symbol /xk) is associated with an occurrence probability Pk, it is possible to define a symbol density operator according to: (23.3) Consistently, the full symbol block of length n is characterized by the density operator: = pon, (23.4) which represents the originator's message M. As with the definition of a density operator, we have tr(pM) = 1. Consequently, some message OM may be received at the recipient's end. The correspondence between the originator's and the recipient's messages may be defined through a transformation, or quantum operation, E, such that OM=E(PM). (23.5) Such a quantum operation E must be trace preserving, SO that OM is a density operator with tr(om) = 1. Since the message transformation from originator to recipient is fully defined by the operation E, it is customary to refer to E as the quantum channel itself. In the ideal or "noiseless" case, one would expect that the quantum channel E corre- sponds to some unitary (trace-preserving) transformation U, which gives (23.6) Thus, according to the above operation the message is not invariant by transmission through the quantum channel (OM # PM), but its integrity is fully conserved. Indeed, it only takes the recipient to apply the inverse operation E-1 defined by U-1 U+ to the received message OM, to and, hence, fully retrieve the originator's message. A trivial case of ideal or noiseless channel is given by U = I where I is the 2 X 2 identity matrix. A more general definition for any 23.1 Noisy quantum channels quantum channel E is (23.7) The definition in 