# What kind of operators should I perform on 4-qubit state $|\psi\rangle_A|GHZ\rangle$ to get expression like $|\Psi\rangle_4$?

Alice wants to send Bob qubit in state $$|\psi_A\rangle = \alpha|0\rangle_A + \beta|1\rangle_A$$ , also she has one qubit from GHZ state $$\frac{1}{\sqrt 2}(|0\rangle_a|00\rangle +|1\rangle_a|11\rangle)$$.

Then she performs Bell basis
($$|\Psi_{\pm}\rangle_{Aa}=\frac{1}{\sqrt 2}(|00\rangle_{Aa}\pm|11\rangle_{Aa})$$, $$|\Phi_{\pm}\rangle_{Aa}=\frac{1}{\sqrt 2}(|01\rangle_{Aa}\pm|10\rangle_{Aa})$$ measurement on qubits $$(A, a)$$.

In terms of projection operators and/or measurement operators how can I get expression for this 4-qubit state: $$|\Psi\rangle_4=\frac{1}{ 2}[|\Psi_{+}\rangle_{Aa}(\alpha|00\rangle +\beta|11\rangle)+|\Psi_{-}\rangle_{Aa}(\alpha|00\rangle -\beta|11\rangle)+|\Phi_{+}\rangle_{Aa}(\beta|00\rangle +\alpha|11\rangle)+|\Phi_{-}\rangle_{Aa}(-\beta|00\rangle +\alpha|11\rangle)]$$

(I understand that if just simplify this exptression I get $$(\alpha|0\rangle_A + \beta|1\rangle_A) \frac{1}{\sqrt 2}(|0\rangle_a|00\rangle +|1\rangle_a|11\rangle)$$ ???
