2. Action of unitary operations on the Bloch sphere [12 points; 4 each].

For every 2 x 2 unitary matrix U, the effect of applying U on a qubit can be viewed as a

rotation of the states in the Bloch sphere. For example, it can be shown that

(2)

corresponds to rotation by angle 20 along the axis specified by state 10). In each case

below, give the angle of rotation and also specify the axis of rotation:

[cos

(a) sin

(b)

(c)

cos

i sin

cos

sin

sin 0]

cos

i sin t

cos

sin 0

cos

0

[o]

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