p.1) =[f(p+1/2-2p/2)-f(p)] (23.70) 1-f(p). - 23.4 Exercises P1=0.5 x=maxx"(p,p,) Bit-flip parameter p Figure 23.3 Plots of X (p, p1) as a function of the bit-flipping parameter p in the bit-flip channel, and corresponding channel capacity x(p) = max x'(p,p1) (codewords formed by pure states |0), |+) with probabilities p1, 1 - P1, respectively). Figure 23.3 shows plots of x'(p,p1) for p1 = 0 to p1= 1 through different incre- ments (recalling that x'(p,p1) = x'(p, 1 - p1)), with the top curve X (p) = x'(p, 1/2) representing the channel capacity. Interestingly, the channel capacity is seen to have two maxima, i.e., for p=0 and p=1. These two limiting cases correspond to the noiseless channel, and the "deterministic" bit-flip channel, respectively. A determin- istic bit-flipping is simply a change of code polarity, meaning that the classical cbit codewords from originator to recipient are exactly inverted or complemented, which entails no information degradation or error. The other limiting situation is obtained for p = 1/2, meaning that the qubit has a 50% chance of being flipped and a 50% chance of being conserved in its integrity. The recipient's measurement amounts to a coin-flipping experiment, and all initial information is lost, which is the situation of the useless channel. Exercises 23.1 (B): Given the Pauli matrices X, Y, Z (see definition in Chapter 16), show that for any 2 X 2 operator A with unity trace, and for q = 3p/4, the following relation holds: + + - 23.2 (M): Analyze the effect of the bit-flip quantum channel defined by =e(p)=pXpX+(1-p)p, - with p being a probability distribution and X the Pauli matrix X = -(i ) Quantum channel noise and channel capacity in the case where the density operator p is nondiagonal. Determine the entropy change AS = S(o) - S(p) in the case p = |0) <01 and p = |1) (1). 23.3 (M): Analyze the effect of the phase-flip quantum channel as defined by o=e(p)=pZpZ+(1-p)p, with p being a probability distribution and Z the Pauli matrix in the case where the density operator p is 