The Transfer Function of Generic Linear Quantum Stochastic Systems Has a Pure Cascade Realization

TL;DR

This paper proves that generic linear quantum stochastic systems can be realized as pure cascades, simplifying their implementation for control and filtering applications, by developing algorithms for symplectic matrix decompositions.

Contribution

It extends the cascade realization result from passive to active quantum systems and introduces algorithms for symplectic QR and Schur decompositions.

Findings

Cascade realization exists for generic active systems.

Algorithms for symplectic QR and Schur decompositions are developed.

Numerical examples demonstrate the realization of quantum amplifiers.

Abstract

This paper establishes that generic linear quantum stochastic systems have a pure cascade realization of their transfer function, generalizing an earlier result established only for the special class of completely passive linear quantum stochastic systems. In particular, a cascade realization therefore exists for generic active linear quantum stochastic systems that require an external source of quanta to operate. The results facilitate a simplified realization of generic linear quantum stochastic systems for applications such as coherent feedback control and optical filtering. The key tools that are developed are algorithms for symplectic QR and Schur decompositions. It is shown that generic real square matrices of even dimension can be transformed into a lower $2 \times 2$ block triangular form by a symplectic similarity transformation. The linear algebraic results herein may be of independent interest for applications beyond the problem of transfer function realization for quantum systems. Numerical examples are included to illustrate the main results. In particular, one example describes an equivalent realization of the transfer function of a nondegenerate parametric amplifier as the cascade interconnection of two degenerate parametric amplifiers with an additional outcoupling mirror.