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

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?

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.

• Shoot, you are right. I had it as a 2x2 matrix with the same layout. Thank you very much! Oct 23, 2022 at 0:54