Decomposing outer products of tensor products into tensor products of outer products

Question

In Nielsen and Chuang we define the partial trace operator, defined as

$$\operatorname{tr}_2(|a_1\rangle\langle a_2| \otimes |b_1\rangle\langle b_2|) = |a_1\rangle\langle a_2| \operatorname{tr}(|b_1\rangle\langle b_2|)$$

They go on to say that $\operatorname{tr}_2(|11\rangle\langle00|) = |1\rangle\langle0|\langle0|1\rangle$, which presumably means that $|11\rangle\langle00| = |1\rangle\langle0|\otimes|1\rangle\langle0|$, but how did we arrive at this? Can a similar expression be derived for multi-qubit states?

Answer 1

This is just notation. We write $\vert 1\rangle\langle 0\vert \otimes \vert 1\rangle\langle 0\vert$ as $\vert 11\rangle\langle 00\vert$.

It generalizes exactly as you would expect for multiple qubits e.g. $\vert 000\rangle = \vert 0\rangle\otimes\vert 0\rangle\otimes\vert 0\rangle$