On synthesis of linear quantum stochastic systems by pure cascading

Recently, it has been demonstrated that an arbitrary linear quantum stochastic system can be realized as a cascade connection of simpler one degree of freedom quantum harmonic oscillators together with a direct interaction Hamiltonian which is bilinear in the canonical operators of the oscillators. However, from an experimental point of view, realizations by pure cascading, without a direct interaction Hamiltonian, would be much simpler to implement and this raises the natural question of what class of linear quantum stochastic systems are realizable by cascading alone. This technical note gives a precise characterization of this class of linear quantum stochastic systems and then it is proved that, in the weaker sense of transfer function realizability, all passive linear quantum stochastic systems belong to this class. A constructive example is given to show the transfer function realization of a two degrees of freedom passive linear quantum stochastic system by pure cascading.