Engineering arbitrary two-mode CV gates with fixed two-mode interfaces

We explore the possibility to engineer two-mode continuous-variable (CV) gates by cascading multiple fixed two-mode linear interfaces. We first categorize every gate by its transmission strength, which is invariant under local operations. We show that combining any two interfaces with appropriate single-mode operations in between can arbitrarily alter the transmission strength, thus can engineer arbitrary two-mode CV gate. Our protocol requires squeezing on only one of the modes, so it is applicable to platforms with limited controllability. Additionally, we prove that it is generally insufficient to engineer a perfect transducer with two arbitrary interfaces, but always possible with three. It is the theoretical scheme with the minimum resources.