s associated with the uniform mixed states | = (|0) + |1) /V2. In this event, the channel randomly outputs the errored states = 1+), |-), regardless of the input states 1q)in = |0), |1). The conclusion is that there is a probability p that there is no mathematically defined relation between the channel input (|9)in) and output (19)out) qubits. Such an event corresponds to a loss of coherence, a unique quantum effect referred to as decoherence. It is beyond the scope of this book to analyze further the meaning and implication of quantum decoherence, as introduced by random channel noise. The only (but important!) lesson to be learnt here is twofold: (a) the quantum-channel transformation is accurately defined by the density matrix operation = (e (p); (b) without supplemental information, the output states of the quantum channel cannot be accurately or predictably retrieved, because of the decoherence effect. However, it is possible to implement quantum error correction, and hence retrieve the input quantum message (p) and associated qubit symbols (19)in), as will be described in Chapter 24. Let us now look at the VN entropy change through the depolarizing channel. For any pure-state message P, we have, by definition, S(p) = 0 (there is no uncertainty as to what message is being sent). From the recipient's end, and based on the result in Eq. (23.9), we have (23.11) Thus, the quantum channel has an uncertainty of the amount S(o) = f(p/2), which as we know from earlier chapters is maximal for the argument p/2 = 1/2 or p = 1, which gives S(o) = 1 bit. Clearly, this limiting case corresponds to a useless quan- tum channel, where the received message is deterministically o = I/2 and, no mat- ter what the original message information is, the net uncertainty is one classical To conclude this example formally, it is left as a basic exercise to show that the depolarizing channel can be put in the operator-sum representation e(p)=p=+(1-p)p (23.12) where q = 3p/4 and X, Y, Z are the Pauli matrices (see Chapter 16). This expre