Say we have $f \in \operatorname{End}(V^{\otimes n})$ for a vector space $V$ with basis $\{v_1,\dots,v_m\}$. Then, given $i_1,i_2,....i_n \in \{1,2,...,m\}$ we have:
$$f(v_{i_1} \otimes v_{i_2} \otimes ... \otimes v_{i_n}) = \sum_{1 \leq j_1,\dots,j_m \leq m}f^{j_1,\dots,j_{n}}_{i_1,\dots,i_{n}}v_{j_1} \otimes \cdots \otimes v_{j_n}.$$
Is the following $$\operatorname{tr}(f) = \sum_{1 \leq j_1,\dots,j_m \leq m}f^{j_1,\dots,j_{n}}_{j_1,\dots,j_{n}}$$ the correct expression for the trace of $f$? If not, then what is?
Also, is there a difference between the "operator trace", "partial trace", and "quantum trace"? I have not yet noticed a difference. Thanks!