How Does a Mixed Quantum State Relate to Bloch Sphere Representation?

[bowlbase]

Homework Statement

What is reduced density matrix $\rho_A$ and the Bloch vector representation for a state that is 50% $|0 \rangle$ and 50% $\frac{1}{\sqrt{2}}(|0 \rangle + |1 \rangle)$

Homework Equations

The Attempt at a Solution

[/B]
I haven't seen many (any?) examples of this so I'm trying to feel my way through it. So first matrix should be
$\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}$
and the second:
$\frac{1}{2}\begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}$
Both have a 50% chance so $\frac{1}{4}\begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}+\frac{1}{2}\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}$
Giving:
$\rho_A=\begin{bmatrix} \frac{3}{4} & \frac{1}{4} \\ \frac{1}{4} & \frac{1}{4} \end{bmatrix}$

I think this is correct.

I'm not sure about the Bloch sphere portion. Can anyone give me some direction?

 

[jfizzix]

I would look up the Bloch vector representation of spin-1/2 systems. In short, any

Homework Statement

What is reduced density matrix $\rho_A$ and the Bloch vector representation for a state that is 50% $|0 \rangle$ and 50% $\frac{1}{\sqrt{2}}(|0 \rangle + |1 \rangle)$

Homework Equations

The Attempt at a Solution

[/B]
I haven't seen many (any?) examples of this so I'm trying to feel my way through it. So first matrix should be
$\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}$
and the second:
$\frac{1}{2}\begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}$
Both have a 50% chance so $\frac{1}{4}\begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}+\frac{1}{2}\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}$
Giving:
$\rho_A=\begin{bmatrix} \frac{3}{4} & \frac{1}{4} \\ \frac{1}{4} & \frac{1}{4} \end{bmatrix}$

I think this is correct.

I'm not sure about the Bloch sphere portion. Can anyone give me some direction?

Any $2\otimes 2$ matrix can be expressed as a sum over the 4 basis matrices $I,\sigma_{x},\sigma_{y}$ and $\sigma_{z}$. For convenience, we'll define $\sigma_{0}=I$ as the $2\otimes 2$ identity matrix.

These matrices form an orthogonal basis with the inner product between two matrices defined as the trace of the product of the two matrices:
$Tr[\sigma_{i}\sigma_{j}]=2\delta_{ij}$

As an example, we can have a $2\otimes 2$ matrix $A$ expressed as
$A=a_{0} \sigma_{0} +a_{1}\sigma_{x}+a_{2}\sigma_{y}+a_{3}\sigma_{z}$

We can use the orthogonality of the basis matrices to find $a_{0}$ through $a_{3}$.

For example:
$a_{2}= \frac{Tr[A\sigma_{2}]}{Tr[\sigma_{2}\sigma_{2}]}$

Now for a density matrix, $A=\rho$, and $(a_{1},a_{2},a_{3})$ form a vector $\vec{a}$ on the Bloch sphere. Once you know the components of $\vec{a}$, you will have the Bloch vector of the spin-1/2 system.