Can Bloch sphere be used to represent mixed states in quantum computation?

[Haorong Wu]

Homework 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]

I would start by showing that $\rho$ possesses 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]

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:
http://akyrillidis.github.io/notes/quant_post_7https://physics.stackexchange.com/questions/204090/understanding-the-bloch-spherehttp://www.vcpc.univie.ac.at/~ian/hotlist/qc/talks/bloch-sphere.pdf

 

[Haorong Wu]

Yes, it is important. Some resources:
http://akyrillidis.github.io/notes/quant_post_7https://physics.stackexchange.com/questions/204090/understanding-the-bloch-spherehttp://www.vcpc.univie.ac.at/~ian/hotlist/qc/talks/bloch-sphere.pdf

Thanks, DrClaude.

Bloch sphere really makes me nervous.

(*_*)

 

[DrClaude]

Thanks, DrClaude.

Bloch sphere really makes me nervous.

(*_*)

It shouldn't . It is simply another way of looking at qubits.

 

[Haorong Wu]

I would start by showing that $\rho$ possesses 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.