# 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

Thus,

$\displaystyle P^2 = \sum_{ij}\left|i\right>\left\left\delta_{ij}\left\left

## 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

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

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

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