Understanding the outer products in density matrices

Question

I don't understand a simple property of the outer product when doing density matrices. I am studying nielsen and chuang's book.

At equation 2.197 they do show the density matrix of the state of quantum teleportation before alice performs her measurements. For $$|\psi\rangle = \frac{1}{2}[|00\rangle(\alpha|0\rangle +\beta |1\rangle)+ |01\rangle(\alpha|1\rangle+\beta|0\rangle)+|10\rangle(\alpha|0\rangle -\beta |1\rangle) + |11\rangle(\alpha|1\rangle -\beta|0\rangle)]$$ The density matrix is just: $$\rho_1= \frac{1}{4}[|00\rangle \langle 00|(\alpha|0\rangle + \beta|1\rangle)(\alpha^*|0\rangle+\beta^*|1\rangle)+\\ |01\rangle\langle01|(\alpha|1\rangle +\beta |0\rangle)(\alpha^*\langle1| +\beta^*\langle0|) +\\ |10\rangle\langle10|(\alpha|0\rangle -\beta |1\rangle)(\alpha^*\langle 0|-\beta^*\langle1|) +\\ |11\rangle\langle11|(\alpha|1\rangle -\beta|0\rangle)(\alpha^*\langle 1| -\beta^*\langle0|) ]$$

But in the example a couple of pages before, they present the density matrix of $|+\rangle$, which is:

$$\rho_2=\frac{1}{2} (|0\rangle+|1\rangle)(\langle0| + \langle 1|) $$ Which it gets "opened up" as $$\rho_2=\frac{1}{2} (|0\rangle\langle0|+ |0\rangle \langle1| + |1\rangle\langle0| + |1\rangle\langle1|) $$

So basically I don't understand why sometimes the outer product you have the cross terms $|0\rangle \langle1|$, and sometimes you dont: I would have expected to have 16 terms in the density matrix of the teleportation, or just 2 in the example of the maximally mixed state.

Specifically, I expect either $\rho_2$ to be (which I understand is plain wrong): $$\rho_2 = \frac{1}{2}(|0\rangle\langle0| + |1\rangle\langle1|)$$ or $\rho_1$ to be all the cross product terms: $$\rho_1= \frac{1}{4}[|00\rangle \langle 00|(\alpha|0\rangle + |01\rangle(\alpha^*|1\rangle+\beta^*|0\rangle)+\\ |01\rangle\langle01|(\alpha|0\rangle +\beta |1\rangle)(\alpha^*\langle0| +\beta^*\langle1|) +\\ |10\rangle\langle10|(\alpha|0\rangle -\beta |1\rangle)(\alpha^*\langle 0|-\beta^*\langle1|) +\\ |11\rangle\langle11|(\alpha|1\rangle -\beta|1\rangle)(\alpha^*\langle 1| -\beta^*\langle1|+\\ |00\rangle \langle 01|(..)()..)+\\ |00\rangle \langle 10|(..)()..)+\\ |00\rangle \langle 11|(..)()..)+\\ |01\rangle \langle 00|(..)()..)+\\ |01\rangle \langle 10|(..)()..)+\\ |01\rangle \langle 11|(..)()..)+\\ etc.. ) ]$$

Can someone tell me what i don get about distributive property of outer products?

Answer 1

You're right, there should be 16 terms. But since they are tracing out Alice's system in the next step only 4 terms are relevant - those that correspond to $|00\rangle \langle 00|$, $|01\rangle \langle 01|$, $|10\rangle \langle 10|$ and $|11\rangle \langle 11|$. Tracing out $|00\rangle \langle 01|\cdot |\phi\rangle\langle\psi|$ will result in 0, because $\text{Tr}(|00\rangle \langle 01|)=\langle 01|00\rangle=0$.