How to compute the maximum possible coherence of a two-particle Bell state?

Question

I am reading through some notes and am stuck on a bit of math that shows the max possible coherence. Our wave function is $|\psi\rangle =\frac{|01\rangle+|10\rangle}{2}$ and doing $|\psi\rangle \langle\psi|$ gives us

$\frac{|01\rangle \langle01| + |01\rangle \langle10| + |10\rangle \langle01| + |10\rangle \langle10|}{2}$

Multiplying these and adding them together gives me

$\frac{1}{2} (\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} + \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix})$

which is

$\frac{1}{2}\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$

but the given solution is

$\frac{1}{2}\begin{pmatrix} 0&0&0&0\\0&1&1&0\\0&1&1&0\\0&0&0&0 \end{pmatrix}$

Can I get some help pointing out where my math is wrong here?

Answer 1

Welcome to QCSE. It doesn't look like you took the outer product properly, for example, you might have gotten lost in thinking that $|01\rangle$ is a two-dimensional ket, but it's actually four-dimensional.

In more detail, $|01\rangle$ corresponds to 1 written in binary, and has a 1 in the second row of the vector, while $|10\rangle$ corresponds to binary 2, and has a 1 in the third row - the first row would correspond to $|00\rangle$ and the fourth row is $|11\rangle$.

That is, we have: $$|01\rangle=\begin{pmatrix} 0 \\ 1 \\ 0 \\ 0\end{pmatrix}\:\:|10\rangle=\begin{pmatrix} 0 \\ 0 \\ 1 \\ 0\end{pmatrix}$$

But it looks like your matrices are written as if you are taking the outer product of:

$$|01\rangle=\begin{pmatrix} 1 \\ 0 \end{pmatrix}\:\: |10\rangle=\begin{pmatrix} 0 \\ 1\end{pmatrix}$$

This is improper.