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 5
 Problem Statement

From Nielsen's QC exercise 2.72:
Show that an arbitrary density matrix for a mixed state qubit may be written as
##\rho = \frac {I+ \vec r \cdot \vec \sigma} 2##,
where ##\vec r## is a real threedimensional vector such that ##\ \vec r \ \leq 1##. This vector is known as the Bloch vector for the state ##\rho##
 Relevant Equations

##\vec \sigma## are Pauli matrices
##\rho \equiv \sum_i p_i \left  \psi _i \right > \left < \psi_i \right ##
Well, I have no clues for this problem.
Since I can get nothing from the definition of ##\rho##, I tried from the right part.
Also, I know that ##\left ( \vec r \cdot \vec \sigma \right ) ^2={r_1}^2 {\sigma _1}^2+{r_2}^2 {\sigma _2}^2+{r_3}^2 {\sigma _3}^2##.
Plus, ##\rho## is positive; then I only need to show that ##\rho ^2=I^2+2 \vec r \cdot \vec \sigma +{r_1}^2 {\sigma _1}^2+{r_2}^2 {\sigma _2}^2+{r_3}^2 {\sigma _3}^2##.
Well, I'm stuck again.
Maybe I went the wrong direction?
Also, Are there any places I can find the solutions for Nielsen's book? I feel nervous that I can not check my solutions to see whether I'm right or not.
Oh, another question. I'm wondering, is Bloch sphere important in quantum computation? Maybe some references I should read? Nielsen's book doesn't introduce it comprehensively.
Thank you for reading!
Since I can get nothing from the definition of ##\rho##, I tried from the right part.
Also, I know that ##\left ( \vec r \cdot \vec \sigma \right ) ^2={r_1}^2 {\sigma _1}^2+{r_2}^2 {\sigma _2}^2+{r_3}^2 {\sigma _3}^2##.
Plus, ##\rho## is positive; then I only need to show that ##\rho ^2=I^2+2 \vec r \cdot \vec \sigma +{r_1}^2 {\sigma _1}^2+{r_2}^2 {\sigma _2}^2+{r_3}^2 {\sigma _3}^2##.
Well, I'm stuck again.
Maybe I went the wrong direction?
Also, Are there any places I can find the solutions for Nielsen's book? I feel nervous that I can not check my solutions to see whether I'm right or not.
Oh, another question. I'm wondering, is Bloch sphere important in quantum computation? Maybe some references I should read? Nielsen's book doesn't introduce it comprehensively.
Thank you for reading!