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arXiv:1609.07595 · IEEE Transactions on Automatic Control

A Modified Frequency Domain Condition for the Physical Realizability of Linear Quantum Stochastic Systems

Arash Kh. Sichani, Ian R. Petersen

Published Jul 12, 2017• 9 citations
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Abstract

This paper is concerned with a modified version of the frequency domain physical realizability (PR) condition for linear quantum systems. We consider open quantum systems whose dynamic variables satisfy the canonical commutation relations of an open quantum harmonic oscillator and are governed by linear quantum stochastic differential equations (QSDEs). In order to correspond to physical quantum systems, these QSDEs must satisfy PR conditions. We provide a relatively simple proof that the PR condition is equivalent to the frequency domain (J,J)-unitarity of the input-output transfer function and orthogonality of the feedthrough matrix of the system without the technical spectral assumptions required in previous work. We also show that the poles and transmission zeros associated with the transfer function of PR linear quantum systems are the mirror reflections of each other about the imaginary axis. An example is provided to illustrate the results.

DOI
10.1109/tac.2017.2726340

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This paper is concerned with a modified version of the frequency domain physical realizability (PR) condition for linear quantum systems. We consider open quantum systems whose dynamic variables satisfy the canonical commutation relations of an open quantum harmonic oscillator and are governed by linear quantum stochastic differential equations (QSDEs). In order to correspond to physical quantum systems, these QSDEs must satisfy PR conditions. We provide a relatively simple proof that the PR condition is equivalent to the frequency domain (J,J)-unitarity of the input-output transfer function and orthogonality of the feedthrough matrix of the system without the technical spectral assumptions required in previous work. We also show that the poles and transmission zeros associated with the transfer function of PR linear quantum systems are the mirror reflections of each other about the imaginary axis. An example is provided to illustrate the results.
This paper aims to provide conditions under which a quantum stochastic differential equation can serve as a model for interconnection of a bilinear system evolving on an operator group SU(2) and a linear quantum system representing a quantum harmonic oscillator. To answer this question we derive algebraic conditions for the preservation of canonical commutation relations (CCRs) of quantum stochastic differential equations (QSDE) having a subset of system variables satisfying the harmonic oscillator CCRs, and the remaining variables obeying the CCRs of SU(2). Then, it is shown that from the physical realizability point of view such QSDEs correspond to bilinear-linear quantum cascades.
The paper is concerned with a problem of coherent (measurement-free) filtering for physically realizable (PR) linear quantum plants. The state variables of such systems satisfy canonical commutation relations and are governed by linear quantum stochastic differential equations, dynamically equivalent to those of an open quantum harmonic oscillator. The problem is to design another PR quantum system, connected unilaterally to the output of the plant and playing the role of a quantum filter, so as to minimize a mean square discrepancy between the dynamic variables of the plant and the output of the filter. This coherent quantum filtering (CQF) formulation is a simplified feedback-free version of the coherent quantum LQG control problem which remains open despite recent studies. The CQF problem is transformed into a constrained covariance control problem which is treated by using the Frechet differentiation of an appropriate Lagrange function with respect to the matrices of the filter.
This paper is concerned with the physical realizability of linear quantum stochastic differential equations (QSDEs). In a recent paper, James, Nurdin and Petersen gave necessary and sufficient conditions for the physical realizability of linear QSDEs with real coefficients. These conditions were algebraic conditions on the system state space matrices. In this paper, we first study general linear QSDEs with complex coefficients and explain how to relate them via unitary transformations to the class of linear QSDEs with real coefficients considered by James, Nurdin Petersen. We use this relation between linear QSDEs with real and complex coefficients to derive necessary and sufficient algebraic conditions for the physical realizability of the general complex linear QSDEs being considered. Then we prove the equivalence between these algebraic conditions and a condition that an associated linear system is (J, J')-unitary.
Coherent feedback control considers purely quantum controllers in order to overcome disadvantages such as the acquisition of suitable quantum information, quantum error correction, etc. These approaches lack a systematic characterization of quantum realizability. Recently, a condition characterizing when a system described as a linear stochastic differential equation is quantum was developed. Such condition was named physical realizability, and it was developed for linear quantum systems satisfying the quantum harmonic oscillator canonical commutation relations. In this context, open two-level quantum systems escape the realm of the current known condition. When compared to linear quantum system, the challenges in obtaining such condition for such systems radicate in that the evolution equation is now a bilinear quantum stochastic differential equation and that the commutation relations for such systems are dependent on the system variables. The goal of this paper is to provide a necessary and sufficient condition for the preservation of the Pauli commutation relations, as well as to make explicit the relationship between this condition and physical realizability.
A recently emerging approach to the feedback control of linear quantum systems involves the use of a controller which itself is a quantum linear system. This approach to quantum feedback control, referred to as coherent quantum feedback control, has the advantage that it does not destroy quantum information, is fast, and has the potential for efficient implementation. An important issue which arises both in the synthesis of linear coherent quantum controllers and in the modeling of linear quantum systems, is the issue of physical realizability. This issue relates to the property of whether a given set of linear quantum stochastic differential equations corresponds to a physical quantum system satisfying the laws of quantum mechanics. Under suitable assumptions, the paper shows that the question of physical realizability is equivalent to a frequency domain (J,J) -unitary condition. This is important in controller synthesis since it is the transfer function matrix of the controller which determines the closed loop system behavior.
This paper is concerned with translation invariant networks of linear quantum stochastic systems with nearest neighbour interaction mediated by boson fields. The systems are associated with sites of a one-dimensional chain or a multidimensional lattice and are governed by coupled linear quantum stochastic differential equations (QSDEs). Such interconnections of open quantum systems are relevant, for example, to the phonon theory of crystalline solids, atom trapping in optical lattices and quantum metamaterials. In order to represent a large-scale open quantum harmonic oscillator, the coefficients of the coupled QSDEs must satisfy certain physical realizability conditions. These are established in the form of matrix algebraic equations for the parameters of an individual building block of the network and its interaction with the neighbours and external fields. We also discuss the computation of mean square performance functionals with block Toeplitz weighting matrices for such systems in the thermodynamic limit per site for unboundedly increasing fragments of the lattice.