Worked Problems in Physics

Nielsen and Chuang Exercise 2.16

Posted in Nielsen/Chuang by rpmuller on March 1, 2010

Exercise 2.16 Show that any projector satisfies {P^2=P}.

Any projector may be written

\displaystyle  P = \sum_i\left|i\right>\left<i\right|. \ \ \ \ \ (25)

Thus,

\displaystyle  P^2 = \sum_{ij}\left|i\right>\left<i|j\right>\left<j\right| = \sum_{ij}\left|i\right>\delta_{ij}\left<j\right| = \sum_i \left|i\right>\left<i\right| = P. \ \ \ \ \ (26)

Nielsen and Chuang Exercise 2.15

Posted in Nielsen/Chuang by rpmuller on March 1, 2010

Exercise 2.15 Show that {(A^\dagger)^\dagger=A}

(Getting a little lazy with the equation typesetting here…) The adjoint is a combination of two operations, the transpose, and the complex conjugate, both of which are identity when applied twice. Thus, the combination of them applied twice also yields the identity operation.

Nielsen and Chuang Exercise 2.14

Posted in Nielsen/Chuang by rpmuller on March 1, 2010

Exercise 2.14 Show that the adjoint is antilinear

\displaystyle  \left(\sum_ia_iA_i\right)^\dagger = \sum_ia_i^*A_i^\dagger. \ \ \ \ \ (22)

\displaystyle  \left(\sum_ia_iA_i\right)^\dagger = \left[ \begin{array}{rrr} \sum_ia_iA^i_{11} & \dots & \sum_ia_iA^i_{1m} \\ \vdots & \ddots & \vdots \\ \sum_ia_iA^i_{n1} & \dots & \sum_ia_iA^i_{nm} \\ \end{array} \right]^\dagger \ \ \ \ \ (23)

\displaystyle  = \left[ \begin{array}{rrr} \sum_ia_i^*A^i_{11} & \dots & \sum_ia_i^*A^i_{n1} \\ \vdots & \ddots & \vdots \\ \sum_ia_i^*A^i_{1m} & \dots & \sum_ia_i^*A^i_{nm} \\ \end{array} \right] = \sum_ia_i^*A_i^\dagger \ \ \ \ \ (24)