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arXiv:quant-ph/9706020 · Physical Review A

Perturbative expansions for the fidelities and spatially correlated dissipation of quantum bits

L.-M. Duan, Guang‐Can Guo

Published Dec 1, 1997• 28 citations
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Abstract

We construct generally applicable short-time perturbative expansions for some fidelities, such as the input-output fidelity, the entanglement fidelity, and the average fidelity. Successive terms of these expansions yield characteristic times for the damping of the fidelities involving successive powers of the Hamiltonian. The second-order results, which represent the damping rates of the fidelities, are extensively discussed. As an interesting application of these expansions, we use them to study spatially correlated dissipation of quantum bits. Spatial correlations in the dissipation are described by a correlation function. Explicit conditions are derived for independent decoherence and for collective decoherence.

DOI
10.1103/physreva.56.4466

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We construct generally applicable short-time perturbative expansions for some fidelities, such as the input-output fidelity, the entanglement fidelity, and the average fidelity. Successive terms of these expansions yield characteristic times for the damping of the fidelities involving successive powers of the Hamiltonian. The second-order results, which represent the damping rates of the fidelities, are extensively discussed. As an interesting application of these expansions, we use them to study spatially correlated dissipation of quantum bits. Spatial correlations in the dissipation are described by a correlation function. Explicit conditions are derived for independent decoherence and for collective decoherence.
We construct a generally applicable short-time perturbative expansion for coherence loss. Successive terms of this expansion yield characteristic times for decorrelation processes involving successive powers of the Hamiltonian. The second order results are sufficient to precisely reproduce expressions for ``decoherence times'' obtained in the literature by much more involved and indirect methods. Examples illustrating the influence of initial conditions and the need to evaluate higher order terms are given in the context of the Jaynes-Cummings model. It is shown that, in this case, the short-time decoherence behavior can probe the importance of antiresonant contributions.
We construct generally applicable small-loss rate expansions for the density operator of an open system. Successive terms of those expansions yield characteristic loss rates for dissipation processes. Three applications are presented in order to give further insight into the context of those expansions. The first application, of a two-level atom coupling to a bosonic environment, shows the procedure and the advantage of the expansion, whereas the second application that consists of a single mode field in a cavity with linewidth $\ensuremath{\kappa}$ due to partial transmission through one mirror illustrates a practical use of those expansions in quantum measurements, and the third one, for an atom coupled to modes of a lossy cavity shows another use of the perturbative expansion.
The stability of quantum systems to perturbations of the Hamiltonian is studied. This stability is quantified by the fidelity. Dependence of fidelity on the initial state as well as on the dynamical properties of the system is considered. In particular, systems having a chaotic or regular classical limit are analysed. The fidelity decay rate is given by an integral of the correlation function of the perturbation and is thus smaller the faster correlation function decays. If the perturbation can be written as a time derivative of another operator, meaning that the time averaged perturbation vanishes, fidelity freezes at a constant value and starts to decay only after a long time inversely proportional to the perturbation strength. In composite systems stability of entanglement to perturbations of the Hamiltonian is analysed in terms of purity. For regular systems purity decay is shown to be independent of Planck's constant for coherent initial states in the semiclassical limit. The accelerated decoherence of macroscopic superpositions is also explained. The theory of fidelity decay is applied to the stability of quantum computation and an improved quantum Fourier transform algorithm is designed.
Entanglement fidelity quantifies how well a quantum channel preserves the correlations between a transmitted system and an inaccessible reference system. We derive closed-form expressions for the entanglement fidelity associated with several standard quantum noise models, including the random Pauli-X, dephasing, depolarizing, Werner-Holevo, generalized Pauli (Weyl), and amplitude-damping channels. For each model, we express the entanglement fidelity in terms of a general input density operator ρρρ, using Schumacher's Kraus-operator approach, which provides a channel-agnostic recipe applicable to any completely positive trace-preserving (CPTP) map with a finite Kraus representation. We then specialize to a communication scenario in which the source emits a two-letter parametric alphabet, thereby making explicit the dependence of entanglement preservation on both channel and source parameters. The resulting expressions enable direct comparisons of channel performance and rankings for representative families of input states, including common qubit states.
Quantum dissipation involves both energy relaxation and decoherence, leading toward quantum thermal equilibrium. There are several theoretical prescriptions of quantum dissipation but none of them is simple enough to be treated exactly in real applications. As a result, formulations in different prescriptions are practically used with different approximation schemes. This review examines both theoretical and application aspects on various perturbative formulations, especially those that are exact up to second-order but nonequivalent in high-order system-bath coupling contributions. Discrimination is made in favor of an unconventional formulation that in a sense combines the merits of both the conventional time-local and memory-kernel prescriptions, where the latter is least favorite in terms of the applicability range of parameters for system-bath coupling, non-Markovian, and temperature. Also highlighted is the importance of correlated driving and dissipation effects, not only on the dynamics under strong external field driving, but also in the calculation of field-free correlation and response functions.
In this paper we investigate the averaged dynamics of quantum systems under the influence of classical stochastic fields. This influence is modeled using a stochastic Hamiltonian evolution for the system-density matrix. From the averaged dynamics, a general characterization of the short-time decoherence behavior is obtained. General applicable short-time perturbation expansions for the input-output fidelity and its generalizations for mixed states are developed. The master equations of the systems that can be worked out without a perturbative expansion, i.e., any system subject to dispersive noise and the quantum harmonic oscillator subject to amplitude noise, are extensively analyzed. In both cases all non-Markovian features are worked out in an exact way. We apply these cases to the study of heating of trapped ions.