# Which simple property of partial trace are we using here?

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,
• $$Tr[AB]=Tr[BA]$$,
• $$Tr[v_1^Tv_2] = (v_1,v_2)$$

1. mixed-product property of the Kronecker product: $$(A \otimes B)(C \otimes D) = (AC \otimes BD)$$
2. definition of the partial trace: $$\text{Tr}_1(A\otimes B) = \text{Tr}(A)B$$
3. cyclic property of the trace: $$\text{Tr}(AB) = \text{Tr}(BA)$$
$$\text{Tr}(|v_1\rangle\langle v_2|) = \text{Tr}(\langle v_2|\cdot|v_1\rangle) = \langle v_2|v_1\rangle$$