nondiagonal. Determine the entropy change AS = S(o) - S(p) - in the four cases p = |0) <0|, p = |1) (1), p = and p=|-> X-1. 23.4 (M): Analyze the effect of the bit-phase-flip quantum channel as defined by o=e(p)=pYpY+(1-p)p, with p being a probability distribution and Y the Pauli matrix in the case where the density operator p is nondiagonal. Determine the entropy change AS = S(o) - S(p) in the four cases p = |0) (0|, p = |1) (1), p = 1+)(+1, and p = -> <-|. 23.5 (M): Determine the quantum channel capacity corresponding to the depolarizing channel assuming the three-symbol alphabet P1=0)(0,P2=1)(1,p3=2)(21 = associated with probabilities p1, P2, P3. Provide a plot of the capacity as a function of the depolarizing parameter p. 23.6 (T): Determine the quantum channel capacity corresponding to the depolarizing channel e(p)=pZpZ+(1-p)p assuming the two-symbol alphabet P1 = |0) 1+) (+1 - 10) + 11 associated with probabilities p1, P2. 23.4 Exercises Hint: plot x((p)=S((e))-(S(8)) as a function of p, for different values of p1 and justify that the capacity X = max(x') is achieved for p1 = p2 = 1/2; also show that X (p) < X (0) = S(p), where X (0) is the capacity of the constant or noiseless quantum channel and S(p) is the VN entropy of the originator source. Quantum error correction This chapter deals with the subject of quantum error correction and the related codes (QECC), which can be applied to noisy quantum channels and quantum memories with the purpose of preserving or protecting the information integrity. I first describe the basics of quantum repetition codes, as applicable to bit-flip and phase-flip quantum channels. Then I consider the 9-qubit Shor code, which has the capability of diagnosing and correcting any combination of bit-flip and phase-flip errors, up to one error of each type. Furthermore, it is shown that the Shor code is, in fact, capable of fully restoring qubit integrity under a continuum of bit or phase errors, a property that has no counterpart in the classical world of error-correction codes. 