For a quantum channel $\mathcal{E}$, Choi matrix is defined as follows:
$C(\mathcal{E}) = (\mathcal{E} \otimes I) (\sum_{i}\lvert ii\rangle \sum_j \langle jj\lvert) $, where $\sum_{i}\lvert ii\rangle$ is the maximally entangled state.
It is clear that for 2 qubit case, this maximally entangled state is just the Bell state, i.e. $\sum_i \lvert ii \rangle = \frac{1}{\sqrt{2}}(\lvert 00 \rangle + \lvert 11 \rangle)$. But I thought the generalized Bell state, which is generalized GHZ-state, is the maximally entangled state so that $\sum_i \lvert ii \rangle =$ generalized GHZ state for 3+ qubits. But the notation from many other references gives $\sum_{i=1}^d \lvert ii \rangle$ where $d$ is the dimension of Hilbert space that this maximally entangled state lives in. This confuses me since the generalized GHZ state will always have 2 terms. What am I missing here?