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

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?

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. $$\vert 000\rangle = \vert 0\rangle\otimes\vert 0\rangle\otimes\vert 0\rangle$$
• What's the benefit of using that notation? Is it purely for brevity, or does it allow for neat manipulations (like how the notation $\langle a || b \rangle = \langle a | b \rangle$ always indicates an inner product? Commented Oct 5, 2020 at 20:45