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- Question : EX1 - The polarization of a light wave is described by two complex parameters ? = cos? ei?x? ? = sin ? ei?y satisfying ???2 + ???2 = 1. More explicitly, the electric field is Ex?t? = E0 cos? cos??t ??x? = E0 Re ? cos?ei?x e?i?t ? ? Ey?t? = E0 sin ? cos??t ??y? = E0 Re ? sin?ei?y e?i?t ? ? Determine the axes of the ellipse traced by the tip of the electric field vector and the direction in which it is traced. 2. This light wave is made to pass through a polarizing filter whose axis is parallel to Ox. Show that measurement of the intensity at the exit of the filter allows ? to be determined. 3. Now the filter is oriented such that its axis makes an angle of ?/4 with Ox. What is the reduction of the intensity at the exit from the filter? Show that this second measurement permits determination of the phase difference ? = ?y ? ?x.
- Question : EX2 - Vectors of ? which are not normalized, ??? and ????, are defined as ??? = ?x?x?+ ?y?y?? ???? = ?x? ?x?+ ?y? ?y?? Show that the operation ??? ? ???? is a projection: ???? = ?? ???? where ?? is the projector onto the vector ??? = ??x?+??y??
- Question : EX3 - Show that a photon with state vector ??? is transmitted by the ??? ?? polarizer with unit probability, and that a photon of state vector ???? = ????x?+???y? is stopped by this polarizer.
- Question : EX1 - Justify the following expressions for the states ?R? and ?L? respectively repre- senting right- and left-handed polarized photons: ?R? = 1 ?2??x?+i?y??? ?L? = 1 ?2??x??i?y???
- Question : EX2 - e define the states ??? and ???? (2.19) representing photons linearly polarized along directions making an angle ? with Ox and Oy, respectively, and also the states ?R?? = ?12????+i?????? ?L?? = ?12?????i?????? How are ?R?? and L?? related to ?R? and ?L?? Do these state ve
- Question : EX3 - We construct the Hermitian operator ? = ?R??L? What is the action of ? on the vectors ?R? and ?L?? Determine the action of exp??i??? on these vectors.
- Question : EX4 - Write the matrix representing ? in the basis ??x?? ?y??. Show that ?2 = I and recover exp??i???. By comparing with question 2, give the physical interpretation of the operator exp??i???.
- Question : EX1 - Let us suppose that Eve analyzes the polarization of a photon sent by Alice using an analyzer oriented as . If Alice orients her polarizer as , the probability that Eve measures the value of the qubit as +1 is 100% when Alice sends a qubit +1, but only 50% when Alice uses a polarizer. The probability that Eve measures +1 when Alice sends +1 then is p = 1 2 + 1 2 ?1 2 ? = 3 4 ? Let us suppose that Eve orients her analyzer in a direction making an angle ? with Ox. Show that the probability p??? for Eve to measure +1 when Alice sends +1 is now p??? = 1 4 ?2+cos2?+sin 2??? Show that for the optimal choice ? = ?0 = ?/8 p??0? ? 0?854? a larger value than before. Would it have been possible to predict without calcu- lation that the optimal value must be ? = ?0 = ?/8? However, as explained in Section 7.2, the information gain of Eve is less than with the naive strategy.
- Question : EX2 - Suppose that instead of using a basis ?
- Question : EX1 - Let us take two Hermitian operators A and B. Show that their commutator ?A? B? is anti-Hermitian, ?A?B? ?= AB?BA = iC? where C is Hermitian: C = C
- Question : EX2 - The expectation values of A and B are defined as ?A?? = ???A??? ?B?? = ???B??? and the dispersions ??A and ??B in the state ??? as ???A?2 = ?A2?? ???A???2 = ??A??A??I?2??? ???B?2 = ?B2?? ???B???2 = ??B ??B??I?2??? Finally, we define Hermitian operators of zero expectation value (which are a priori specific to the state ???) as A0 = A??A??I? B0 = B??B??I? What is their commutator? The norm of the vector ?A0 +i?B0????? where ? is chosen to be real, must be positive: ???A0 +i?B0?????? ? 0? Derive the Heisenberg inequality ???A????B? ? 1 2 ???C????? Care must be taken in interpreting this inequality. It implies that if a large number of quantum systems are prepared in the state ???, and if their expectation values and dispersions ??A??? ??A?? ??B??? ??B?, and ?C?? are measured in independent experiments, then these expectation values will obey the Heisenberg inequality. In contrast to what is sometimes found in the literature, the dispersions ??A and ??B are not at all associated with the experimental errors. There is 32 What is a qubit? nothing which a priori prevents ?A??, for example, from being measured with an accuracy better than ??A.
- Question : EX3 - The position and momentum operators X and P (in one dimension) obey the commutation relation ?X?P? = i?I? where ? is the Planck constant, ? = 1?054

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