What is the maximally entangled state in Choi matrix?

Question

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?

Answer 1

For the Choi-Jamiołkowski isomorphism you need the bipartite maximally entangled state of dimension $d$, while the GHZ state is a multipartite entangled state of dimension 2.

The normalization is also relevant. You can use either $$ \sum_{i=0}^{d-1} |ii\rangle$$ or $$ \frac1{\sqrt d}\sum_{i=0}^{d-1} |ii\rangle,$$ both work, but need different formulas. The standard formulas use the first alternative, the non-normalized one.

Also, people talk about the maximally entangled state, because in the bipartite case you have indeed a unique maximally entangled state, modulo local unitaries. In the multipartite case there is no sensible definition of a unique maximally entangle stated.

Answer 2

Definition of Choi–Jamiołkowski isomorphism states that $d$ is the dimension of the hilbert space while your generalized GHZ states is still with hilbert space dimension $2$ for each subsystem. The definition in the link is very clear, the maximally entangled state should be $${\displaystyle |\Phi ^{+}\rangle ={\frac {1}{\sqrt {d}}}\sum _{i=0}^{d-1}|i\rangle \otimes |i\rangle ={\frac {1}{\sqrt {d}}}(|0\rangle \otimes |0\rangle +\cdots +|d-1\rangle \otimes |d-1\rangle )}$$