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arXiv:2410.00014

Linear quantum systems: poles, zeros, invertibility and sensitivity

Zhiyuan Dong, Guofeng Zhang, Joseph H. W. Lee, Ian R. Petersen

Published Sep 14, 2024• 0 citations
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Abstract

The non-commutative nature of quantum mechanics imposes fundamental constraints on system dynamics, which, in the linear realm, are manifested through the physical realizability conditions on system matrices. These restrictions give system matrices a unique structure. This paper aims to study this structure by investigating the zeros and poles of linear quantum systems. Firstly, it is shown that $-s_0$ is a transmission zero if and only if $s_0$ is a pole of the transfer function, and $-s_0$ is an invariant zero if and only if $s_0$ is an eigenvalue of the $A$-matrix, of a linear quantum system. Moreover, $s_0$ is an output-decoupling zero if and only if $-s_0$ is an input-decoupling zero. Secondly, based on these pole-zero correspondences and inspired by a recent work on the stable inversion of classical linear systems \cite{DD2023}, we show that a linear quantum system must be Hurwitz unstable if it is strongly asymptotically left invertible. Two types of stable input observers are constructed for unstable linear quantum systems. Finally, the sensitivity of a coherent feedback network is investigated; in particular, the fundamental tradeoff between ideal squeezing and system robustness is studied on the basis of system sensitivity analysis.

DOI
10.48550/arxiv.2410.00014

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The non-commutative nature of quantum mechanics imposes fundamental constraints on system dynamics, which, in the linear realm, are manifested through the physical realizability conditions on system matrices. These restrictions give system matrices a unique structure. This paper aims to study this structure by investigating the zeros and poles of linear quantum systems. Firstly, it is shown that −s0-s_0−s0​ is a transmission zero if and only if s0s_0s0​ is a pole of the transfer function, and −s0-s_0−s0​ is an invariant zero if and only if s0s_0s0​ is an eigenvalue of the AAA-matrix, of a linear quantum system. Moreover, s0s_0s0​ is an output-decoupling zero if and only if −s0-s_0−s0​ is an input-decoupling zero. Secondly, based on these pole-zero correspondences and inspired by a recent work on the stable inversion of classical linear systems \cite{DD2023}, we show that a linear quantum system must be Hurwitz unstable if it is strongly asymptotically left invertible. Two types of stable input observers are constructed for unstable linear quantum systems. Finally, the sensitivity of a coherent feedback network is investigated; in particular, the fundamental tradeoff between ideal squeezing and system robustness is studied on the basis of system sensitivity analysis.
The non-commutative nature of quantum mechanics imposes fundamental constraints on system dynamics, which, in the linear realm, are manifested through the physical realizability conditions on system matrices. These restrictions give system matrices a unique structure. This paper aims to study this structure by investigating the zeros and poles of linear quantum systems. Firstly, it is shown that−s0-s_{0}−s0​is a transmission zero if and only ifs0s_{0}s0​is a pole of the transfer function, and−s0-s_{0}−s0​is an invariant zero if and only ifs0s_{0}s0​is an eigenvalue of theAAA-matrix, of a linear quantum system. Moreover,s0s_{0}s0​is an output-decoupling zero if and only if−s0-s_{0}−s0​is an input-decoupling zero. Secondly, based on these pole-zero correspondences and inspired by a recent work on the stable inversion of classical linear systems [1], we show that a linear quantum system must be Hurwitz unstable if it is strongly asymptotically left invertible. Two types of stable input observers are constructed for unstable linear quantum systems. Finally, the sensitivity of a coherent feedback network is investigated; in particular, the fundamental tradeoff between ideal squeezing and system robustness is studied on the basis of system sensitivity analysis.
The non-commutative nature of quantum mechanics imposes fundamental constraints on system dynamics, which in the linear realm are manifested by the physical realizability conditions on system matrices. These restrictions endow system matrices with special structure. The purpose of this paper is to study such structure by investigating zeros and poses of linear quantum systems. In particular, we show that −s0∗-s_{0}^{*}−s0∗​ is a transmission zero if and only if s0s_{0}s0​ is a pole, and which is further generalized to the relationship between system eigenvalues and invariant zeros. Additionally, we study left-invertibility and fundamental tradeoff for linear quantum systems in terms of their zeros and poles.
New fundamental limits on design and control of linear quantum systems are presented. A main difference between quantum and classical systems comes from the fact that quantum signals are represented by operator-valued vectors. The transfer function representation is useful to describe a system in the quantum setting for the same reason as it is useful classically. It is shown that the noncommutative property of quantum signals can be characterized by a specific relation between zeros and poles of a quantum transfer function. Then, two quantum mechanical tradeoffs follows from the transfer function constraint. One is a well known tradeoff, Heisenberg’s uncertainty relation, and the other is a tradeoff between the ability of noise reduction and sensitivity to modeling error. While these tradeoffs hold at each frequency, quantum systems have another constraint between different frequencies based on Bode integral as classical systems usually do. In the quantum case, however, this constraint is also dependent on quantum mechanical parameters.
Abstract : The proposed work attempted to establish a theoretical framework for the analysis and control of quantum linear systems and their interactions with non-classical quantum fields by developing control theoretic concepts exploiting special features of quantum systems. The research aimed to continue developing a new mathematical framework for robust control of quantum linear systems and extend results to quantum linear systems subject to non-classical quantum fields. The major outcomes of this project are (i) derivation of quantum filtering equations for systems non-classical input states including single photon states, (ii) determination of how linear quantum systems respond to multichannel non-classical states, (iii) linear quantum models for quantum memories and the zero dynamics principle for perfect information transfer, (iv) development of new structured uncertainty methods that ensure robust stability of quantum systems based on nominal linear models, and (v) physical realizability results for finite level quantum systems.
Determining whether every trajectory of the linear system x˙=Ax\dot{x}=Axx˙=Ax converges to zero (or, equivalently, whether AAA is Hurwitz stable) is a fundamental problem in control theory. It arises, for example, in verifying the stability of feedback controllers and state observers, as well as of linearized dynamical models. We study the quantum query complexity of deciding whether a matrix A∈Cn×nA\in\mathbb{C}^{n\times n}A∈Cn×n is Hurwitz stable, given an exact normalized block-encoding of AAA. The query complexity is characterized in terms of the system dimension nnn and a supplied lower bound 0<ν≤ωA0<ν\leq ω_A0<ν≤ωA​ on the imaginary-axis distance ωA:=min⁡t∈Rσmin⁡(A−itI)ω_A:=\min_{t\in\mathbb{R}}σ_{\min}(A-\mathrm{i}tI)ωA​:=mint∈R​σmin​(A−itI), where σmin⁡(A)σ_{\min}(A)σmin​(A) is the smallest singular value of AAA. We prove a query-complexity lower bound of Ω(n/ν)Ω(\sqrt{n}/ν)Ω(n​/ν), which holds even when AAA is restricted to be real and diagonal. For general matrices with nonzero ωAω_AωA​, we encode an approximation of the right-half-plane spectral projector sign⁡+(A):=(I+sign⁡(A))/2\operatorname{sign}^+(A):=(I+\operatorname{sign}(A))/2sign+(A):=(I+sign(A))/2 as a block of a matrix inverse, where AAA is Hurwitz if and only if the projector is zero. To test this, we produce a quantum algorithm which determines if a (β,b,1/(4n))(β,b,1/(4\sqrt{n}))(β,b,1/(4n​))-block-encoded projector matrix (like sign⁡+(A)\operatorname{sign}^+(A)sign+(A)) is zero or not; for constant success probability, it has complexity Θ(βn)Θ(β\sqrt{n})Θ(βn​). When the zero test is applied to specific approximations of sign⁡+(A)\operatorname{sign}^+(A)sign+(A), this yields an upper bound of O(nν−2log⁡(n/ν))O(\sqrt{n}ν^{-2}\log(\sqrt{n}/ν))O(n​ν−2log(n​/ν)) queries in the general case, and O(nν−1log⁡n)O(\sqrt{n}ν^{-1}\log n)O(n​ν−1logn) when AAA is Hermitian or normal. The latter upper bound matches the lower bound up to a logarithmic factor. Thus the dimension dependence of quantum Hurwitz testing is nearly optimal, while the remaining factor in the non-normal case isolates an open conditioning gap.
Abstract In this paper, the synthesis of robust memory modes for linear quantum passive systems in the presence of unknown inputs has been studied, aimed at facilitating secure storage and communication of quantum information. In particular, we can switch on decoherence-free (DF) modes in the storage stage by placing the poles on the imaginary axis via a coherent feedback control scheme, and these memory modes can further be simultaneously made robust against perturbations to the system parameters by minimizing the condition number associated with imaginary poles. The DF modes can also be switched off by tuning the controller parameters to place the poles in the left half of the complex plane in the writing/reading stage. We develop explicit algebraic conditions guiding the design of such a coherent quantum controller, which involves employing an augmented system model to counter the influence of unknown inputs. Examples are provided to illustrate the procedure of synthesizing robust memory modes for linear optical quantum systems.