Partial trace over the first tensor factor

Let $H_G,H_Q$ be finite-dimensional complex inner-product spaces, and fix the factor order $H_G\otimes H_Q$. For ordered orthonormal bases $(g_s)$ and $(q_{\alpha })$, define ${\mathrm{Tr}}_G:\mathcal{B}(H_G\otimes H_Q)\to \mathcal{B}(H_Q),({\mathrm{Tr}}_{G}X{)}_{\alpha \beta }={\sum }_s{X}_{(s,\alpha ),(s,\beta )},$ where $X_{(s,\alpha ),(t,\beta )}=⟨X(g_t\otimes q_{\beta }),g_s\otimes q_{\alpha }⟩$ under the linear-first convention. Equivalently, this is the partial trace over the second factor after swapping the two wires. The partial-trace characterization and wire-permutation unitarity in The partial trace is characterized by the trace pairing and is basis-independent and Finite wire permutations are unitary basis permutations show that the map is independent of the chosen orthonormal bases. In particular, the same definition applies when $H_Q=H_L\otimes H_R$, with that residual factor order left unchanged. For an operator on $(H_G\otimes H_L)\otimes H_R$, use the canonical associator of the ordered tensor-register construction to view it on $H_G\otimes (H_L\otimes H_R)$ before taking this partial trace.