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arXiv:0807.1370 · Physical Review A

Operator fidelity susceptibility, decoherence, and quantum criticality

Xiao-Ming Lu, Zhe Sun, Xiaoguang Wang, Paolo Zanardi A

Published Sep 4, 2008• 44 citations
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Abstract

The extension of the notion of quantum fidelity from the state-space to the operator level can be used to study environment-induced decoherence. The state-dependent operator fidelity susceptibility (OFS), the leading-order term for slightly different operator parameters, is shown to have a nontrivial behavior when the environment is at critical points. Two different contributions to the OFS are identified which have distinct physical origins and temporal dependence. Exact results are obtained for the finite-temperature decoherence caused by a bath described by the Ising model in a transverse field.

DOI
10.1103/physreva.78.032309

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The extension of the notion of quantum fidelity from the state-space to the operator level can be used to study environment-induced decoherence. The state-dependent operator fidelity susceptibility (OFS), the leading-order term for slightly different operator parameters, is shown to have a nontrivial behavior when the environment is at critical points. Two different contributions to the OFS are identified which have distinct physical origins and temporal dependence. Exact results are obtained for the finite-temperature decoherence caused by a bath described by the Ising model in a transverse field.
The operator fidelity is a measure of the information-theoretic distinguishability between perturbed and unperturbed evolutions. The response of this measure to the perturbation may be formulated in terms of the operator fidelity susceptibility (OFS), a quantity which has been used to investigate the parameter spaces of quantum systems in order to discriminate their regular and chaotic regimes. In this work we numerically study the OFS for a pair of nonlinearly coupled two-dimensional harmonic oscillators, a model which is equivalent to that of a hydrogen atom in a uniform external magnetic field. We show how the two terms of the OFS, being linked to the main properties that differentiate regular from chaotic behavior, allow for the detection of this model's transition between the two regimes. In addition, we find that the parameter interval where perturbation theory applies is delimited from above by a local minimum of one of the analyzed terms.
The behavior of the ground-state fidelity susceptibility in the vicinity of a quantum critical point is investigated. We derive scaling relations describing its singular behavior in the quantum critical regime. Unlike in previous studies, these relations are solely expressed in terms of conventional critical exponents. We also describe in detail a quantum Monte Carlo scheme that allows for the evaluation of the fidelity susceptibility for a large class of many-body systems and apply it in the study of the quantum phase transition for the transverse-field Ising model on the square lattice. Finite-size analysis applied to the so-obtained numerical results confirms the validity of our scaling relations. Furthermore, we analyze the properties of a closely related quantity, the ground-state energy's second derivative, which can be numerically evaluated in a particularly efficient way. The usefulness of both quantities as alternative indicators of quantum criticality is examined.
The inner product between the ground-state eigenvectors with proximate interaction parameters, namely, the fidelity, plays a significant role in the quantum dynamics. In this paper, the critical behaviors of the transverse- and longitudinal-field fidelity susceptibilities for the d=2 quantum (transverse-field) Ising model are investigated by means of the numerical-diagonalization method; the former susceptibility has been investigated rather extensively. The critical exponents for these fidelity susceptibilities are estimated as α(F)((t))=0.752(24) and α(F)((h))=1.81(13), respectively. These indices are independent, and suffice for obtaining conventional critical indices such as ν=0.624(12) and γ=1.19(13).
In this dissertation, we address several problems in condensed matter physics, statistical mechanics, and quantum chaos from the perspective of the quantum information-theoretic notion of state distinguishability, as measured though evaluation of the appropriate fidelity measure. ❧ We find that the fidelity susceptibility, a measure for the response of the fidelity to infinitesimal changes of the Hamiltonian parameters, serves as a useful tool for studying the effects of disorder on quantum phase transitions. We study in detail the phase diagram of the well-known one-dimensional random quantum XY chain. Through an analysis of the fidelity susceptibility’s finite-size scaling and statistics for this model, we find that the fidelity approach reflects large variation between average and typical behavior, may detect the presence of Griffiths phases, and indicates the modifications to the phase diagram due to disorder. ❧ For this same model, though without disorder, we evaluate the dynamics following a sudden Hamiltonian quench using the Loschmidt echo, a form of the fidelity that compares the initial state with the time-evolved state. Through a central limit theorem-type argument we find that the long-time statistics of the Loschmidt echo takes one of two universal forms, even at finite temperature, depending on the system’s proximity to the quantum critical lines. ❧ Finally, for a canonical model exhibiting quantum chaos, the hydrogen atom in a uniform external magnetic field, we evaluate the operator fidelity susceptibility (OFS) between unitary operators generating the dynamics. We find that the part of the OFS which corresponds to the variation of the Hamiltonian eigenvectors serves to indicate the transition from the regular perturbative and n-mixing regimes to the quantum chaotic regime.
We study ground-state fidelity defined as the overlap between two ground states of the same quantum system obtained for slightly different values of the parameters of its Hamiltonian. We focus on the thermodynamic regime of the XYXYXY model and the neighborhood of its critical points. We describe extensively fidelity when it is dominated by the universal contribution reflecting the quantum criticality of the phase transition. We show that proximity to the multicritical point leads to anomalous scaling of fidelity. We also discuss fidelity in a regime characterized by pronounced oscillations resulting from the change in either the system size or the parameters of the Hamiltonian. Moreover, we show when fidelity is dominated by non-universal contributions, study fidelity in the extended Ising model, and illustrate how our results provide additional insight into dynamics of quantum phase transitions. Special attention is given to studies of fidelity from the momentum space perspective. All our main results are obtained analytically. They are in excellent agreement with numerics.
In idealized models of a quantum register and its environment, quantum information can be stored indefinitely by encoding it into a decoherence-free subspace (DFS). Nevertheless, perturbations to the idealized register-environment coupling will cause decoherence in any realistic setting. Expanding a measure for state preservation, the dynamical fidelity, in powers of the strength of the perturbations, we prove stability to linear order is a generic property of quantum state evolution. The effect of noise perturbation is quantified by a concise expression for the strength of the quadratic leading order, which we define as the dynamical fidelity susceptibility of DFSs. Under the physical restriction that noise acts on the register kkk-locally, this susceptibility is bounded from above by a polynomial in the system size. These general results are illustrated by two physically relevant examples. Knowledge of the susceptibility can be used to increase coherence times of future quantum computers.