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Quantum Mechanics I. 8 March, 2014 INDIVIDUAL ASSIGNMENT Assignment 3: Spin States and Logic Gates ⟩ ⟩ 1. A spin 1/2 system is in the state | + z where | ± z are eigenstates of the Ŝz spin operator. An experimentalist intends to apply a rotation operator e−iπŜy /~ , desiring ⟩ ⟩ the convert the state | + z to | − z . (a) Show that e−iπŜy /~ indeed does the job. (b) Show that the inner product between the final state and the desired state | − z ⟩ is 1. We will use the square of this inner product is a figure of merit for the inversion operation. (c) Now comes the practical implementation of this rotation! It appears that applying a rotation about the Ŝy axis in the lab is prone to errors. The instrument’s y axis is mis-calibrated: when the experimenter applies a pulse nominally rotating the state through θ about the Ŝy axis, she is in fact applying θ(1 + ε) rotation where ε is a small number (|ε| ≪ 1). What is the final state if this erroneous inversion pulse is applied (with a nominal rotation of π radians)? (d) What is the figure of merit for this erroneous pulse? (e) Another problem is that ε is not known, neither can it be removed. So the experimenter applies a series of rotation operators whose nominal rotations are (in the same order) 90◦ about Ŝy , 180◦ about Ŝx followed by 90◦ about Ŝy . The rotation about Ŝx is perfect since there is no error associated with this channel. Show mathematically that this sequence of rotations does a better job ⟩ ⟩ of converting | + z to | − z . (f) Show the trajectory of the state on the Bloch sphere for the erroneous as well as for the composite pulse. 2. A measurement of Sz is performed on an electron prepared in state √ ⟩ 1 ⟩ ⟩ 3 |ψ = | + z +i |−z . 2 2 A measurement of Sx is subsequently performed on the same electron. What is the probability that the first measurement returns −~/2, and the second returns ~/2? Due Date: March. 17, 2014, 5:00 pm 1 Quantum Mechanics I. 8 March, 2014 ⟩ ⟩ 3. A two-dimensional Hilbert space is spanned by the basis states |α and |β . Three ⟩ quantum logic gates act on a state initialized as |α (a) first a Hadamard gate, (b) then a phase gate and (c) finally another Hadamard gate. The quantum circuit is shown here. α φ UΗ UΗ ψout The action of the gates on the basis states is given by these prescriptions. ⟩ ⟩) 1 ( ⟩ ÛH |α = √ |α +|β 2 ⟩ ⟩) 1 ( ⟩ ÛH |β = √ |α −|β 2 ⟩ ⟩ Ûϕ |α = |α ⟩ ⟩ Ûϕ |β = eiϕ |β . ⟩ Find the output state |ψout . What is the probability that upon measurement in the ⟩ ⟩ ⟩ {|α , |β } basis, the output state is |β ? Due Date: March. 17, 2014, 5:00 pm 2