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 [itex]2\otimes 2[/itex] matrix can be expressed as a sum over the 4 basis matrices [itex]I,\sigma_{x},\sigma_{y}[/itex] and [itex]\sigma_{z}[/itex]. For convenience, we'll define [itex]\sigma_{0}=I[/itex] as the [itex]2\otimes 2[/itex] 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:
[itex]Tr[\sigma_{i}\sigma_{j}]=2\delta_{ij}[/itex]

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

We can use the orthogonality of the basis matrices to find [itex]a_{0}[/itex] through [itex]a_{3}[/itex].

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

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