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arXiv:1401.5240 · Physical Review A

Identifiability of open quantum systems

Daniel Klaus Burgarth, Kazuya Yuasa

Published Mar 12, 2014• 8 citations
arXivHTMLPDF

Abstract

We provide a general framework for the identification of open quantum systems. By looking at the input-output behavior, we try to identify the system inside a black box in which some Markovian time evolution takes place. Due to the generally irreversible nature of the dynamics, it is difficult to assure full controllability over the system. Still, we show that the system is identifiable up to similarity under a certain rank condition. The framework also covers situations relevant to standard quantum process tomography, where we do not have enough control over the system but have a tomographically complete set of initial states and observables. Remarkably, the similarity cannot in general be reduced to unitarity even for unitary systems, and the spectra of Hamiltonians are not identifiable without additional knowledge.

DOI
10.1103/physreva.89.030302

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We provide a general framework for the identification of open quantum systems. By looking at the input-output behavior, we try to identify the system inside a black box in which some Markovian time evolution takes place. Due to the generally irreversible nature of the dynamics, it is difficult to assure full controllability over the system. Still, we show that the system is identifiable up to similarity under a certain rank condition. The framework also covers situations relevant to standard quantum process tomography, where we do not have enough control over the system but have a tomographically complete set of initial states and observables. Remarkably, the similarity cannot in general be reduced to unitarity even for unitary systems, and the spectra of Hamiltonians are not identifiable without additional knowledge.
The aim of quantum system identification is to estimate the ingredients inside a black box, in which some quantum-mechanical unitary process takes place, by just looking at its input-output behavior. Here we establish a basic and general framework for quantum system identification, that allows us to classify how much knowledge about the quantum system is attainable, in principle, from a given experimental setup. We show that controllable closed quantum systems can be estimated up to unitary conjugation. Prior knowledge on some elements of the black box helps the system identification. We present an example in which a Bell measurement is more efficient to identify the system. When the topology of the system is known, the framework enables us to establish a general criterion for the estimability of the coupling constants in its Hamiltonian.
The dynamics of an unknown closed quantum system can be characterized by a Hamiltonian. We discuss the identification of this Hamiltonian from quantum process tomography data at several time points for systems with a time-independent Hamiltonian. Quantum process tomography (QPT) yields a set of unitary operators and allows us to convert the identification problem into an estimation problem for the eigenfrequencies of the system. Spectral decomposition of Hermitian and unitary operators shows that process tomography needs to be performed at least at two different time points and the identifiability of the Hamiltonian depends on the choice of these points. Sufficient conditions on the timing to ensure complete identifiability of the Hamiltonian are presented under the assumption that unitary matrices obtained from QPT are exact without numerical errors.
We develop a general approach for monitoring and controlling evolution of open quantum systems. In contrast to the master equations describing time evolution of density operators, here, we formulate a dynamical equation for the evolution of the process matrix acting on a system. This equation is applicable to non-Markovian and/or strong-coupling regimes. We propose two distinct applications for this dynamical equation. We first demonstrate identification of quantum Hamiltonians generating dynamics of closed or open systems via performing process tomography. In particular, we argue how one can efficiently estimate certain classes of sparse Hamiltonians by performing partial tomography schemes. In addition, we introduce an optimal control theoretic setting for manipulating quantum dynamics of Hamiltonian systems, specifically for the task of decoherence suppression.
Abstract Established methods for characterizing quantum information processes do not capture non-Markovian (history-dependent) behaviors that occur in real systems. These methods model a quantum process as a fixed map on the state space of a predefined system of interest. Such a map averages over the system’s environment, which may retain some effect of its past interactions with the system and thus have a history-dependent influence on the system. Although the theory of non-Markovian quantum dynamics is currently an active area of research, a systematic characterization method based on a general representation of non-Markovian dynamics has been lacking. In this article we present a systematic method for experimentally characterizing the dynamics of open quantum systems. Our method, which we call quantum process identification (QPI), is based on a general theoretical framework which relates the (non-Markovian) evolution of a system over an extended period of time to a time-local (Markovian) process involving the system and an effective environment. In practical terms, QPI uses time-resolved tomographic measurements of a quantum system to construct a dynamical model with as many dynamical variables as are necessary to reproduce the evolution of the system. Through numerical simulations, we demonstrate that QPI can be used to characterize qubit operations with non-Markovian errors arising from realistic dynamics including control drift, coherent leakage, and coherent interaction with material impurities.
A central aim of physics is to describe the dynamics of physical sys-tems. Schrödinger’s equation does this for isolated quantum systems. Describing the time evolution of a quantum system that interacts with its environment, in its most general form, has proved to be difficult because the dynamics is dependent on the state of the environment and the corre-lations with it. For discrete processes, such as quantum gates or chemical reactions, quantum process tomography provides the complete descrip-tion of the dynamics, provided that the initial states of the system and the environment are independent of each other. However, many physical systems are correlated with the environment at the beginning of the ex-periment. Here, we give a prescription of quantum process tomography that yields the complete description of the dynamics of the system even when the initial correlations are present. Surprisingly, our method also gives quantitative expressions for the initial correlation. There is a rich history to the studies of decoherence of quantum systems due to the interactions with the surrounding degrees of freedom. When the dynamics of the system (S) is Markovian it can be described by a master equation [1, 2, 3]. Nowadays many researchers are interested in systems that are non-Markovian, as there is mounting evidence that some natural systems of importance may be non-Markovian [4] and such features may allow to manipulate and control quantum systems in desired ways. There is also a great deal of interest in systems that are initially correlated with their environments (E) because non-Markovianity and initial system-environment (SE) correlations are intimately related [5, 6, 7]. 1 ar
Open quantum systems are a rich area of research in the intersection of quantum mechanics and stochastic analysis. By considering a variety of master equations, we unify multiple views of autonomous and controlled open quantum systems and, through considering their measurement dynamics, connect them to classical linear and bilinear system identification theory. This allows us to formulate corresponding notions of quantum state identifiability for these systems which, in particular, applies to quantum state tomography, providing conditions under which the probed quantum system is reconstructible. Interestingly, the dynamical representation of the system lends itself to considering two types of identifiability: the full master equation recovery and the recovery of the corresponding system matrices of the linear and bilinear systems. These concepts are discussed in detail, and conditions under which reconstruction is possible are given. We set the groundwork for a number of constructive approaches to the identification of open quantum systems.