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Newton

Analysis, synthesis, and measurement of linear time-invariant quantum systems

J. Ding, Hudson A. Loughlin, Vivishek Sudhir

Published Oct 1, 2026• 0 citations
Publisher

Abstract

Systems theory provides a general framework to analyze and synthesize complex systems. Central to this discipline is the linear time-translation invariant (LTI) abstraction. We extend LTI systems theory into the quantum regime without relying on additional assumptions such as lumped-ness, Markovianity, or knowledge of an internal description. We develop a framework for the quantization of any classical LTI system based solely on its input-output response, bypassing the need to quantize its internal constituents. It describes the minimum additional quantum noise accompanying a given classical input-output response and characterizes the resulting class of quantum LTI systems via the conjugate symplectic Lie group. This algebraic structure enables the synthesis of a quantum LTI system with a prescribed response, the characterization of the invariant structure of its quantum noise, the classification of the specific quantum states it can emit, and the design of measurements optimal for it.

DOI
10.1016/j.newton.2026.100696

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Additional noise in a quantum system can be detrimental to the performance of a quantum coherent feedback control system. This paper proposes a Linear Matrix Inequality (LMI) approach to construct an optimal quantum realization of a given Linear Time Invariant (LTI) system. The quantum realization problem is useful in designing coherent quantum feedback controllers. An optimal method is proposed for solving this problem in terms of a finite horizon quadratic performance index, which is related to the amount of quantum noise appearing at the system’s output. This cost function provides a measure of how much the additional quantum noise in the coherent controller will alter the feedback control system.
Physical Realizability addresses the question of whether it is possible to implement a given LTI system as a quantum system. It is in general not true that a given synthesized quantum controller described by a set of stochastic differential equations is equivalent to some physically meaningful quantum system. However, if additional quantum noises are permitted in the implementation it is always possible to implement an arbitrary LTI system as a quantum system. In this paper we give an expression for the exact number of noises required to implement a given LTI system as a quantum system. Furthermore, we focus our attention on proving that this is a minimum, that is, it is not possible to implement the system as a quantum system with a smaller number of additional quantum noises.
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