[Paper Review] A Realization Method for Transfer Functions of Linear Quantum Stochastic Systems Using Static Networks for Input/Output Processing and Feedback

Symeon Grivopoulos, Ian R. Petersen|arXiv (Cornell University)|Nov 14, 2015

Neural Networks and Reservoir Computing32 references3 citations

TL;DR

This paper proposes a novel realization method for transfer functions of Linear Quantum Stochastic Systems (LQSSs) using pre- and post-processing static linear networks to simplify the core system into a reduced LQSS, which is then realized via feedback-connected single-mode cavities. The key contribution is a Krein-space SVD-like decomposition for doubled-up matrices, enabling realization of general LQSSs—passive and non-passive—using only passive optical components in the passive case.

ABSTRACT

The issue of realization of the transfer functions of Linear Quantum Stochastic Systems (LQSSs) is of fundamental importance for the practical applications of such systems, especially as coherent controllers for other quantum systems. So far, most works that addressed this problem have used cascade realizations. In this work, a new method is proposed, where the transfer function of a LQSS is realized by a series of a pre-processing linear static network, a reduced LQSS, and a post-processing linear static network. The introduction of the pre- and post-processing static networks leaves an intermediate reduced LQSS with a simple input/output structure, that is realized by a concatenation of simple cavities. A feedback connection of the cavities through a linear static network is used to produce the correct dynamics for the reduced system. The resulting realization provides a nice structural picture of the system. The key mathematical tool that allows for the construction of this realization, is an SVD-like decomposition for doubled-up matrices in Krein spaces. Illustrative examples are provided for the theory developed.

Motivation & Objective

• To address the fundamental problem of realizing transfer functions of Linear Quantum Stochastic Systems (LQSSs) for practical quantum control applications.
• To overcome limitations of cascade-based realizations, which restrict mode interactions and complicate feedback integration.
• To provide a structurally transparent realization framework using pre- and post-processing static networks to simplify the core system dynamics.
• To extend passive LQSS realization to general (non-passive) LQSSs using a novel matrix decomposition in Krein spaces.
• To establish a mathematically rigorous and experimentally feasible design method for coherent quantum controllers.

Proposed method

• Introduce pre- and post-processing static linear networks to transform the input/output structure of the LQSS, reducing it to a simpler intermediate system.
• Realize the reduced LQSS as a feedback interconnection of single-mode cavities via a static linear network, enabling precise dynamical control.
• Use a Krein-space SVD-like decomposition (Theorem 3) for doubled-up matrices to decompose the system's coupling matrix into Bogoliubov transformations.
• Leverage the structure of the decomposition to ensure that the reduced system can be physically implemented using passive optical components when the original system is passive.
• Apply the decomposition to the Hamiltonian and coupling matrices to derive a canonical form amenable to physical realization with standard quantum optical devices.
• Use the SVD of the coupling matrix in the passive case as a special case of the more general Krein-space decomposition, ensuring consistency and physical realizability.

Experimental results

Research questions

• RQ1 Can a transfer function of a general LQSS be realized using only passive optical components, such as cavities and beam splitters, without measurement?
• RQ2 How can the input/output structure of an LQSS be transformed to simplify its physical realization while preserving its transfer function?
• RQ3 What algebraic tool enables the decomposition of general LQSS coupling matrices into physically realizable components, especially beyond the passive case?
• RQ4 Can a feedback interconnection of single-mode cavities, mediated by a static network, realize any reduced LQSS with a simple input/output structure?
• RQ5 What is the role of the Krein space SVD-like decomposition in enabling the realization of non-passive LQSSs using passive components in the feedback loop?

Key findings

• The proposed method realizes any passive LQSS using only passive optical components—cavities, beam splitters, phase shifters—by reducing the system to a feedback-connected cavity structure.
• For general LQSSs, the method provides a realization using a pre- and post-processing static network, a reduced LQSS, and feedback via a static network, all physically realizable with standard quantum optics.
• The key mathematical tool is a Krein-space SVD-like decomposition (Theorem 3) for doubled-up matrices, which generalizes the classical SVD to indefinite inner product spaces.
• The decomposition ensures that the reduced system's coupling matrix can be realized by a cavity with equal passive and active coupling coefficients when the eigenvalue is zero.
• In the passive case, the realization is always possible and uses only passive components, as demonstrated in Example 8 with a 1-mode, 3-input system.
• The method provides a clear structural picture of the system, with the pre- and post-processors handling input/output transformation and the core cavity network handling the dynamics via feedback.