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arXiv:1311.5951 · Journal of Physics A Mathematical and Theoretical

Dynamics of the quantum Fisher information in a spin-boson model

Xiang Hao, Ning-Hua Tong, Shiqun Zhu

Published Aug 8, 2013• 14 citations
arXivHTMLPDF

Abstract

The quantum Fisher information characterizes the phase sensitivity of qubits in the spin-boson model with a finite bandwidth spectrum. In contrast with Markovian reservoirs, the quantum Fisher information flows from the environments to qubits after a number of times if the bath parameter s is larger than a critical value which is related to temperature. The sudden-change behavior will happen during the evolution of the quantum Fisher information of the maximal entanglement state in the non-Markovian environments. The sudden-change times can be varied with the change of the bath parameter s . For a very large number of entangled qubits, the sudden-change behavior of the maximal quantum Fisher information can be used to characterize the existence of the entanglement. The metrology strategy based on the quantum correlated state leads to a lower phase uncertainty when compared with the uncorrelated product state.

DOI
10.1088/1751-8113/46/35/355302

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The quantum Fisher information characterizes the phase sensitivity of qubits in the spin-boson model with a finite bandwidth spectrum. In contrast with Markovian reservoirs, the quantum Fisher information flows from the environments to qubits after a number of times if the bath parameter s is larger than a critical value which is related to temperature. The sudden-change behavior will happen during the evolution of the quantum Fisher information of the maximal entanglement state in the non-Markovian environments. The sudden-change times can be varied with the change of the bath parameter s . For a very large number of entangled qubits, the sudden-change behavior of the maximal quantum Fisher information can be used to characterize the existence of the entanglement. The metrology strategy based on the quantum correlated state leads to a lower phase uncertainty when compared with the uncorrelated product state.
The quantum Fisher information, the quantum analog of the classical Fisher information, is a central quantity in quantum metrology and quantum sensing due to its connection to parameter estimation and fidelity susceptibility. Using numerically exact methods applied to a paradigmatic open quantum system, the spin-boson model, we calculate both static and dynamical quantum Fisher information matrix elements with respect to spin-bath couplings and magnetic field strengths. As the spin-bath interaction increases, we first show that the coupling-coupling matrix elements relative to the ground state of the Hamiltonian are linked to the entanglement growth and signal the Berezinskii-Kosterlitz-Thouless quantum phase transition through their nonmonotonic behavior. We also point out that the static quantum Fisher information exhibits a nonperturbative behavior in the zero-coupling limit, which we justify with an analytic argument. Furthermore, we demonstrate that the time-dependent matrix elements can reveal non-Markovian effects as well as the transition from the coherent to incoherent regime at the Toulouse point, remaining robust under pure dephasing noise. Nonmonotonic signatures of the quantum Fisher information matrix reflect changes in quantum resources such as entanglement and coherence, quantify non-Markovian behavior, and enable criticality-enhanced quantum sensing, thereby shedding light on key features of open quantum systems.
Abstract We analytically investigate the dynamics of quantum Fisher information of two qubits in independent environments for both uncorrelated and entangled input states. Especially, we observe that, in the non-Markovian regime and resonant case, quantum Fisher information of uncorrelated input state vanishes only at discrete time point whereas that of entangled input state can disappear during finite time interval. It shows that quantum Fisher information, which determines the parameter estimation precision, is strongly affected by the environment memory effects, and the advantage of entanglement-enhance metrology no longer exists even for a very short time. We also note that quantum Fisher information for two kinds of input states can be preserved under appropriate qubit-reservoir detuning.
We investigate the dynamical quantum Fisher information of a two-level system coupled to a bosonic environment, focusing on the estimation of the qubit gap. We combine analytical calculations with numerically controlled matrix-product-state simulations. In the exactly solvable pure-dephasing Ohmic regime at zero temperature, the long-time quantum Fisher information displays a coupling-dependent algebraic behavior, leading to a dynamical crossover: it grows without bound at weak coupling, approaches a finite asymptotic value at the crossover coupling, and vanishes at strong coupling. At finite temperature, thermal fluctuations suppress the long-time growth and generate a finite-time maximum, whose dependence on the dephasing coupling retains a clear signa- ture of the zero-temperature crossover. We show that this crossover is absent at zero temperature in non-Ohmic baths: the long-time quantum Fisher information vanishes in the sub-Ohmic and diverges in the super-Ohmic regimes. At non-zero temperature instead, the crossover appears for a super-Ohmic quadratic bath and the crossover coupling becomes temperature-dependent. Moreover, in the zero-temperature Ohmic regime, we introduce an additional amplitude-damping system-bath coupling that induces energy relaxation. This relaxation channel replaces the unbounded long-time growth with a finite asymptotic quantum Fisher information associated with the reduced interacting ground state, while a signature of the pure-dephasing crossover persists in the early-time dynamics. These results establish a direct connection between the low-frequency structure of the bath and the asymptotic metrological behavior of dynamical gap sensing, and show how thermal fluctuations and energy relaxation regularize the ideal pure-dephasing dynamical crossover of the quantum Fisher information.
Abstract We study the quantum Fisher information (QFI) dynamics of the phase parameter in the enlarged cavity-reservoir systems at zero temperature under two situations of large N limit and non-Markovian environment, respectively. We find an important relation that the total quantities of QFI of the cavity and reservoir are equal to unit during the dynamical evolution. The lost QFI of the cavity transfers to its corresponding reservoir with the same quantities simultaneously. Moreover, we also find that the detuning parameter and non-Markovian effect are two significant factors to affect the preservation of QFI.
The problem of quantum metrology in the context of a particular non-Markovian quantum evolution is explored. We study the dynamics of the quantum Fisher information (QFI) of a composite quantum probe coupled to a Lorentzian environment, for a full variety of different classes of parameters. We are able to find the best metrological state, which is not maximally entangled but is the one which evolves the most rapidly. This is shown by demonstrating a connection between QFI and different quantum speed limits. At the same time, by optimizing a control field acting on the probes, we show how the total information flow is actively manipulated by the control so as to enhance the parameter estimation at a given final evolution time. Finally, under this controlled scenario, a sharp interplay between the dynamics of QFI, non-Markovianity, and entanglement is revealed within different control schemes.
We explore the nonunital non-Markovian dynamics of a qubit immersed in a spin bath. The nonunital environmental effects on the precisions of quantum parameter estimation are investigated. The time-dependent transfer matrix and inhomogeneity vector are obtained for the description of the open dynamical process. The evolution of the coherent vectors for a qubit is studied so as to geometrically demonstrate the nonunital non-Markovian effects. The revivals of angles for the fidelity between a maximal mixed state and an arbitrary trajectory state are presented in the nonunital non-Markovian dynamics. The degree for the nonunitality is controllable with the changes of the local magnetic field for a qubit and spin bath temperature. It is found out that the increase of quantum Fisher information for composite states is connected with the nonunital non-Markovian effects, which is helpful for the improvement of quantum metrology.