I came across this circuit diagram for CV quantum state teleportation:
It was accompanied by a brief description, but I still feel as though I am lacking in some intuition as to why this circuit works in teleporting the state in qumode 0 to qumode 2. What would the full procedure to teleport some coherent state across a communication channel using CV quantum computing protocols be?
For more context, the beam-splitter gates are $50/50$, and the two qumodes $q_1$ and $q_2$ have been squeezed by a factor of $-2$ and $2$ respectively (although infinite squeezing would be ideal for this circuit). The measurements are the homodyne-x and homodyne-p measurement gates respectively (as shown on the circuit). This then leads to $q_2$ being condition displaced in the $x$ and $p$ quadratures, based on the previous measurements.
The part of the circuit I'm struggling to understand is the role of the second beam-splitter.