I am reading 2013, Horsman D. et al. - Surface code quantum computing by lattice surgery. I have been trying to understand how the outcome of rough lattice merging, which is discussed using both stabilizer formalism and logical qubit formalism, can be retrieved by carrying out operations on physical qubits. My current understanding is that, while logically lattice merging is equivalent to measuring the product of logical operators $X_1X_2$, this measurement procedure would not involve the merge qubits introduced at the boundary at all, effectively leaving them in their initial $|0\rangle$ state.
I am therefore assuming that either:
a) one physically measures each $A_{V}$ separately and shows that, depending on the sign of the product of these $A_V$, you can identify the output state with the logical state of the extended code as shown in eq. (4).
b) one measures $X_1X_2$ (independent of the merge qubits) and that a subsequent decoding procedure is required to fix the potential syndrome along the line of merging qubits.
I know the stabilizer analysis in the appendix A is much clearer and is supporting the former hypothesis, but then why does the main text use the latter and, most importantly, why is it correct?
