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IEEE Conference on Decision and Control (CDC)

Analysis on n-level quantum systems by means of a coordinate transformation

Huilong Xu, Zibo Miao, Qing Zhong Gao

Published Dec 16, 2024• 1 citation
Publisher

Abstract

The structural decomposition of two-level quantum systems has recently been established, related to various applications in quantum information science. As an extension to previous work on two-level quantum systems, this paper aims to investigate the structural properties of n-level quantum systems. We develop algebraic conditions under which the attainment of quantum back-action evading (BAE) measurements, the generation of decoherence-free (DF) subspaces and quantum non-demolition (QND) variables, as well as the existence of steadystate solutions for the Lindblad equations, can be realized. In particular, a specific coordinate transformation is explicitly constructed to facilitate the achievement of above-mentioned objectives, accompanied by an illustrative numerical example.

DOI
10.1109/cdc56724.2024.10886673

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•Jun 18, 2016•0 citations
The structural decomposition of two-level quantum systems has recently been established, related to various applications in quantum information science. As an extension to previous work on two-level quantum systems, this paper aims to investigate the structural properties of n-level quantum systems. We develop algebraic conditions under which the attainment of quantum back-action evading (BAE) measurements, the generation of decoherence-free (DF) subspaces and quantum non-demolition (QND) variables, as well as the existence of steadystate solutions for the Lindblad equations, can be realized. In particular, a specific coordinate transformation is explicitly constructed to facilitate the achievement of above-mentioned objectives, accompanied by an illustrative numerical example.
The purpose of this paper is to conduct a comprehensive structural analysis of quantum finite-level systems in the Heisenberg picture, where the evolution of system observables is modeled by bilinear quantum stochastic differential equations (QSDEs). Firstly, real coordinate transformations that preserve commutation and anticommutation relations are characterized, and relative admissibility between different orthonormal Hermitian bases is formulated. Secondly, strongly aligned subspaces are introduced, based on which decoherence-free subsystems (DFSs) are constructed in general orthonormal Hermitian basis as well as the generalized Gell-Mann basis. Thirdly, conditions for the existence of quantum nondemolition (QND) variables are given in terms of both linear systems theory and spectral separation by output-signal observables. Finally, a sufficient condition for quantum back-action-evading (BAE) measurements is established. A four-qubit nearest-neighbor Heisenberg chain with collective dephasing illustrates these structural properties of quantum finite-level systems.
We establish a framework for realizing back-action-evading (BAE) measurements and quantum non-demolition (QND) variables in linear quantum systems. The key condition, a purely imaginary Hamiltonian with a real or imaginary coupling operator, enables BAE measurements of conjugate observables. Symmetric coupling further yields QND variables. For non-compliant systems, coherent feedback can engineer BAE measurements. Crucially, the QND interaction condition simultaneously ensures BAE measurements and promotes the coupling operator to a QND observable. This work provides a unified structural theory for enhancing precision in quantum metrology and sensing.
This paper studies the Kalman decomposition for linear quantum systems. Contrary to the classical case, the coordinate transformation used for the decomposition must belong to a specific class of transformations as a consequence of the laws of quantum mechanics. We propose a construction method for such transformations that put the system in a Kalman canonical form. Furthermore, we uncover an interesting structure for the obtained decomposition. In the case of passive systems, it is shown that there exist only controllable/observable and uncontrollable/unobservable subsystems. In the general case, controllable/unobservable and uncontrollable/observable subsystems may also be present, but their respective system variables must be conjugate variables of each other. This decomposition naturally exposes decoherence-free modes, quantum-nondemolition modes, quantum-mechanics-free subsystems, and back-action evasion measurements in the quantum system, which are useful resources for quantum information processing, and quantum measurements. The theory developed is applied to physical examples.
We establish a framework for realizing back-action-evading (BAE) measurements and quantum non-demolition (QND) variables in linear quantum systems. The key condition, a purely imaginary Hamiltonian with a real or imaginary coupling operator, enables BAE measurements of conjugate observables. Symmetric coupling further yields QND variables. For non-compliant systems, coherent feedback is designed to engineer BAE measurements. Crucially, the QND interaction condition simultaneously ensures BAE measurements and promotes the coupling operator to a QND observable.
The purpose of this tutorial is to give a brief introduction to linear quantum control systems. The mathematical model of linear quantum control systems is presented first, then some fundamental control-theoretic notions such as stability, controllability and observability are given, which are closely related to several important concepts in quantum information science such as decoherence-free subsystems, quantum non-demolition variables, and back-action evasion measurements. After that, quantum Gaussian states are introduced, in particular, an information-theoretic uncertainty relation is presented which often gives a better bound for mixed Gaussian states than the well-known Heisenberg uncertainty relation. The quantum Kalman filter is presented for quantum linear systems, which is the quantum analogy of the Kalman filter for classical (namely, non-quantum-mechanical) linear systems. The quantum Kalman canonical decomposition for quantum linear systems is recorded, and its application is illustrated by means of a recent experiment. As single- and multi-photon states are useful resources in quantum information technology, the response of quantum linear systems to these types of input is presented. Finally, coherent feedback control of quantum linear systems is briefly introduced, and a recent experiment is used to demonstrate the effectiveness of quantum linear systems and networks theory.
The purpose of this paper is to characterize the structure of quantum linear systems. By means of a scattering representation, a relation between two systems GGG and Σ\SigmaΣ is established, by which we show that GGG is lossless bounded real if and only if Σ\SigmaΣ is lossless positive real. When GGG is a quantum linear system, we show that its fundamental symplectic structure and physical realizability conditions can be expressed in a simpler way in terms of Σ\SigmaΣ. Furthermore, when GGG is a passive quantum linear system, we show that its uncontrollable subspace and unobservable subspace are identical and they are spanned by the eigenvectors of the AAA matrix of GGG, whose corresponding eigenvalues are all on the imaginary axis. For the nonpassive case, several sufficient conditions are given. Finally, in terms of controllability and observability, the problem of quantum linear system decomposition is investigated. In particular, a quantum Kalman decomposition is proposed. This canonical decomposition naturally exposes decoherence-free modes and quantum non-demolished modes, which are useful resources for quantum information science.