# Bloch sphere for mixed states

#### Haorong Wu

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.

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#### DrClaude

Mentor
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
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:

#### Haorong Wu

Yes, it is important. Some resources:
Thanks, DrClaude.

Bloch sphere really makes me nervous.

(*_*)

#### DrClaude

Mentor
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 $\rho$ 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.

"Bloch sphere for mixed states"

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