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arXiv:1303.3320 · Proceedings of the ... American Control Conference/Proceedings of the American Control Conference

On the preservation of commutation and anticommutation relations of n-level quantum systems

Luis A. Duffaut Espinosa, Zibo Miao, Ian R. Petersen, Valery Ugrinovskii, M. R. James

Published Jun 1, 2013• 8 citations
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Abstract

The goal of this paper is to provide conditions under which a quantum stochastic differential equation (QSDE) preserves the commutation and anticommutation relations of the SU(n) algebra, and thus describes the evolution of an open n-level quantum system. One of the challenges in the approach lies in the handling of the so-called anomaly coefficients of SU(n). Then, it is shown that the physical realizability conditions recently developed by the authors for open n-level quantum systems also imply preservation of commutation and anticommutation relations.

DOI
10.1109/acc.2013.6580216

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The goal of this paper is to provide conditions under which a quantum stochastic differential equation (QSDE) preserves the commutation and anticommutation relations of the SU(n) algebra, and thus describes the evolution of an open n-level quantum system. One of the challenges in the approach lies in the handling of the so-called anomaly coefficients of SU(n). Then, it is shown that the physical realizability conditions recently developed by the authors for open n-level quantum systems also imply preservation of commutation and anticommutation relations.
Physical Realizability and Preservation of Commutation and Anticommutation Relations for nnn-Level Quantum Systems
The purpose of this paper is to address the problem of physical realizability for nnn-level quantum systems. We provide necessary and sufficient conditions for quantum stochastic differential equations that ensure the existence of physical parameters characterizing the unitary evolution required by the laws of quantum mechanics. Also, these conditions guarantee the preservation of the commutation and anticommutation relations of the underlying Lie algebra su(n)su(n)su(n).
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.
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.
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.
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.
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.