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Bloch sphere for mixed states

  • Thread starter Haorong Wu
  • Start date
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 three-dimensional 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!
 

DrClaude

Mentor
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I would start by showing that ##\rho## possess all the properties of a density matrix, namely that the diagonal elements are all real, positive, and ##\leq 1## and
$$
\rho = \rho^\dagger
$$
$$
\mathrm{tr}(\rho) = 1
$$
$$
\mathrm{tr}(\rho^2) \leq 1
$$

I would then show that any 2x2 density matrix can be written by specifying ##\mathbf{r}## only.
 

DrClaude

Mentor
6,867
3,007
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.
Yes, it is important. Some resources:
 
Yes, it is important. Some resources:
Thanks, DrClaude.

Bloch sphere really makes me nervous.

(*_*)
 

DrClaude

Mentor
6,867
3,007
Thanks, DrClaude.

Bloch sphere really makes me nervous.

(*_*)
It shouldn't :smile:. It is simply another way of looking at qubits.
 
I would start by showing that ##\rho## possess all the properties of a density matrix, namely that the diagonal elements are all real, positive, and ##\leq 1## and
$$
\rho = \rho^\dagger
$$
$$
\mathrm{tr}(\rho) = 1
$$
$$
\mathrm{tr}(\rho^2) \leq 1
$$

I would then show that any 2x2 density matrix can be written by specifying ##\mathbf{r}## only.
Great! I solved the problem with your hint. Thanks, DrClaude. I can sleep well tonight.
 

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