How to project on the $\phi^+$ basis when performing entanglement swapping?

Question

I am working on entanglement swapping. Where there are three nodes, a source node, repeater node and a target node.

Qubits 1 and 2 are entangled and are distributed between source node and repeater node. Qubits 3 and 4 are entangled and are distributed between repeater node and target node. Now I make a measurement on the repeater node between the qubits between 2 and 3 so that they are entangled.

Now, I am tracing out 2 and 3 by projecting to 00 basis. But the ask is to project it to $\phi^{+}$ basis. How can I project it to ?

Answer 1

Given a bipartite pure state $|\Psi\rangle\equiv \sum_{ij} c_{ij} |i,j\rangle$, to project onto a state $|\Phi\rangle\equiv\sum_{ij} d_{ij}|i,j\rangle$ means to compute the quantity $$\langle \Phi|\Psi\rangle = \sum_{ij} \bar c_{ij} d_{ij}.$$