I would like to know which property is being used in this example. For $Tr_1$ the partial trace on the first system:
$$Tr_1[(|0\rangle\otimes|0\rangle)(\langle 0| \otimes \langle0|)] =|0\rangle\langle0|\langle 0 | 0 \rangle $$
The property that I can imagine used are:
- the ciclicity of the trace,
- $Tr[AB]=Tr[BA]$,
- $Tr[v_1^Tv_2] = (v_1,v_2)$