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=====Introducing the Density Operator (20 minutes)=====
  * Start by asking, "What is the Oven?  Why is the oven strange in our experiments?" The oven is odd because if you send it through any preparation device, you'll get out a fifty percent probability of finding it in either the spin up or spin down direction.  
  * The answer to the weirdness is that there is no state coming out of the oven, that is, there is no way to represent the oven using bras or kets.  The solution to this problem is to represent the oven using density matrices.
  * To help introduce density matrices, let's analyze the case where we find the probability that state $\vert\psi \rangle$ is in the $\vert+ \rangle$ state.  This operation is written as
  
$$P_{+} \, = \, \left\vert\langle +\vert\psi \rangle\right\vert^{2}=\langle \psi\vert+ \rangle\langle +\vert\psi \rangle \; \; .$$

So, the rightmost expression is some number its complex conjugate.  But, we don't //have// to think of the expression this way.  Notice that in the middle there's a projection operator hidden between the $\psi$ terms.

  * We can think of the rightmost expression as three different matrices, where $\langle \psi\vert$ is a row, $\vert+ \rangle\langle +\vert$ is a 2x2 matrix, and $\vert\psi \rangle$ is a column.  This would look like
$$ \left(\begin{array}{cc}
a & b\\
\end{array}\right) \left(\begin{array}{cc}
c & d\\
e & f\\
\end{array}\right) \left(\begin{array}{c}
g \\
h \\
\end{array}\right) \; \; . $$

  * We haven't talked much about the trace yet; what do we know about the trace?  The trace is the sum of the diagonal elements of a matrix.  Given a matrix

$$A\,=\,\left(\begin{array}{ccc}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33} \\
\end{array}\right) \; \; , $$
$$tr(A)\;=\;a_{11}+a_{22}+a_{33} \; \; .$$

The reason we are interested in the trace is that it has the unique property that

$$tr(ABCDE)=tr(BCDEA) \; \; .$$

Now, if we multiply out the three matrices that represent $\langle \psi\vert+ \rangle\langle +\vert\psi \rangle$, we are left with a single number that is defined as the trace of the remaining 1x1 matrix.  
  * Why is this result important?  The property we have for the trace of multiple matrices allows for the rearrangement of $\langle \psi\vert+ \rangle\langle +\vert\psi \rangle$ as long as we take the trace when we are finished.  So, let us rearrange the expression as such:

$$\langle \psi\vert+ \rangle\langle +\vert\psi \rangle \; \; -> \; \; \vert+ \rangle\langle +\vert\psi \rangle\langle \psi\vert \; \; $$.

  * This gives us the outer product of $\vert+ \rangle$ with itself and the outer product of $\vert\psi \rangle$ with itself.  Performing the matrix multiplication leaves a 2x2 matrix remaining rather than a 1x1 matrix, and the trace of this 2x2 matrix will give a number equal to the trace of the 1x1 matrix.  We can now explicitly see the term

$$\vert\psi \rangle\langle \psi\vert \; \; .$$

This is the density matrix for the state $\vert\psi \rangle$.

  * Now, we have effectively found a new way to do the same probability operation we've been doing for the past couple weeks, but we have to move the bras and kets around, do several outer products, and take the trace.  Why should we care?  Let's investigate some properties of the density matrix.

  * This is a good time to perform the [[swbq:qmsw:spswdensopprop|Practice With Outer Product Matrix Properties]] activity.

  * Following up with the [[..:..:activities:spact:spoven|Describing the Sample Oven in the Stern-Gerlach Experiment Mathematically]] activity is also wise.


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