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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,
  • $Tr[AB]=Tr[BA]$,
  • $Tr[v_1^Tv_2] = (v_1,v_2)$
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  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: $\text{Tr}(AB) = \text{Tr}(BA)$

Your 3 guesses actually follow from the cyclic property.
$\text{Tr}(|v_1\rangle\langle v_2|) = \text{Tr}(\langle v_2|\cdot|v_1\rangle) = \langle v_2|v_1\rangle$

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