Precision of estimation and entropy as witnesses of non-Markovianity in the presence of random classical noises

We investigate in detail the quantum Fisher information (QFI) behavior by examining a single qubit model in the presence of random classical noises in both Markovian and non-Markovian regimes. In particular, we precisely study the effects of noise switching rate ξ$\xi$ and qubit–environment coupling strength ν$\nu$ on the precision of estimation, when the qubit is subjected to random telegraph noise with a Lorentzian spectrum or colored noise with a spectrum of the form 1/f^α$1/f^{\alpha }$. In the Markovian regime, a monotone decay of the QFI with time is found, whereas for non-Markovian noise sudden death and revivals may occur. Despite these oscillations of the QFI in non-Markovian regime, we find that non-Markovian parameter γ=ξ/ν$\gamma =\xi /\nu$ is not the principal parameter controlling the collapse and revival of the QFI. In fact, in both Markovian and non-Markovian regimes, parameters ξ$\xi$ and ν$\nu$independently determine how the QFI varies. We also find that the QFI in the case of colored environments decreases when the number of fluctuators realizing the noise increases, and therefore the parameter estimation becomes more inaccurate. Furthermore, by analyzing the von Neumann entropy of the system density matrix, we illustratively unveil a fundamental relationship between the dynamics of this quantity and non-Markovian behavior in the presence of random classical noises. We also show that this result may lead to a better non-Markovianity interpretation, based on quantum memory effects. Moreover, we demonstrate the connection between the precision of parameter estimation and rising the non-Markovianity in our model where the environment is modeled as classical.     

Enhancing the Precision of Parameter Estimation in Band Gap

Recently, the dynamics of quantum Fisher information(QFI) in various environment are investigated and many kinds of schemes to overcome the drawback of decoherence are designed. Here we propose the pseudomode method to enhance the phase parameter precision of optimal quantum estimation of a qubit coupled to a non-Markovian structured environment. We find that the QFI can be enhanced in the weak-coupling regime with non-perfect band gap and can be trapped permanently with a large value in the perfect band gap. The effects of qubit-pseudomode detuning and the spectrum of reservoir are discussed, a reasonable physical explanation is given, too.     

Dynamical Change of Quantum Fisher Information of Cavity-Reservoir Systems

We study the quantum Fisher information(QFI) dynamics of the phase parameter in the enlarged cavityreservoir 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.     

Improving Quantum Metrology via Purifying and Distilling Noisy Probe States

In quantum metrology, the precision of unknown parameter estimation is studied in the quantum regime, and the choice of the probe state plays an important role in determining the precision of the parameter to be estimated. The quality of quantum metrology will be reduced in the presence of quantum noise during the memory time of probe states after preparation. Meanwhile the noisy probe state can be manipulated by different protocols such as single‐qubit purification, entanglement purification, and entanglement distillation etc. In this paper, the effects of these manipulations on the usefulness, that is, quantum Fisher information (QFI), of the noisy probe state in quantum metrology are studied. The results show that joint operations in single‐qubit purification and entanglement purification processes play positive roles in enhancing the QFI of the probe states, and local measurements in entanglement purification and entanglement distillation processes play both positive and negative roles in enhancing the QFI of the probe states. In this sense, single‐qubit purification will always be helpful in parameter estimation by using single qubits as probe, and entanglement purification process maybe more suitable for improving the estimation precision when entangled‐state probe is adopted.     

Fisher information due to a phase noisy laser under non-Markovian environment

More recently, K. Berrada [Annals of Physics 340 (2014) 60-69] [1] studied the geometric phase of a two-level atom system driven by a phase noise laser under non-Markovian dynamics in terms of different parameters involved in the whole system, and collapse and revival phenomena were found for large class of states. In this paper, using this noise effect, we study the quantum fisher information (QFI) for a two-level atom system driven by a phase noise laser under non-Markovian dynamics. A new quantity, called QFI flow is used to characterize the damping effect and unveil a fundamental connection between non-Markovian behavior and dynamics of system–environment correlations under phase noise laser. It is shown that QFI flow has disappeared suddenly followed by a sudden birth depending on the kind of the environment damping. QFI flow provides an indicator to characterize the dissipative quantum system’s decoherence by analyzing the behavior of the dynamical non-Markovian coefficients.     

Optimal quantum estimation of the coupling constant of Jaynes-Cummings interaction

We address the estimation of the coupling constant of the Jaynes-Cummings Hamiltonian for a coupled qubit-oscillator system. We evaluate the quantum Fisher Information (QFI) for the system undergone the Jaynes-Cummings evolution, considering that the probe initial state is prepared in a Fock state for the oscillator and in a generic pure state for the qubit; we obtain that the QFI is exactly equal to the number of excitations present in the probe state. We then focus on the two subsystems, namely the qubit and the oscillator alone, deriving the two QFIs of the two reduced states, and comparing them with the previous result. Next we focus on possible measurements on the system, and we find out that if population measurement on the qubit and Fock number measurement on the oscillator are performed together, the Cramer-Rao bound is saturated, that is the corresponding Fisher Information (FI) is always equal to the QFI. We compare also the performances of these energy measurements performed alone, that is when one of the two subsystem is ignored. We show that, when the qubit is prepared in either the ground or the excited state, the local measurements are still optimal. Finally we investigate the case when the harmonic oscillator is prepared in a thermal state and observe how, particularly for small values of the coupling constant, the QFI increases with the average number of thermal photons of the initial state.     

Geometric phase,entanglement, and quantum Fisher information near the saturation point

Considering a collection of two-level atoms in the presence of a saturating monochromatic, steady-state field, we investigate the geometric phase (GP) of an arbitrary medium’s atom. We find that it is possible to detect the saturation of the atomic response by the GP computation. This is an interesting result, because we can predict the collective behaviour of the atomic system—i.e., the saturation of the optical response of the medium- by investigating the GP of a single medium’s atom, described as a qubit. Moreover, we find that the plot of the atomic GP in terms of the detuning Δ$\Delta$ is very similar to the absorption spectrum of the medium. In addition, it is shown that when the intensity of the driving laser field tends to saturation intensity, the qubit approaches maximum correlation with its environment described by the driving field and other qubits in the atomic system. Furthermore, we find that the behaviour of the entanglement is very analogous to that of the GP and the absorption coefficient. Besides, we adopt the atom to estimate the decoherence parameter by using the quantum Fisher information (QFI), an important measure of the information content of quantum states. Interestingly, we find that when the atomic system approaches its saturation point, the QFI decays with increasing the laser intensity, and therefore the parameter estimation becomes more inaccurate.     

Quantum Fisher information of the GHZ state due to classical phase noise lasers under non-Markovian environment

The dynamics of N$N$-qubit GHZ state quantum Fisher information (QFI) under phase noise lasers (PNLs) driving is investigated in terms of non-Markovian master equation. We first investigate the non-Markovian dynamics of the QFI of N$N$-qubit GHZ state and show that when the ratio of the PNL rate and the system–environment coupling strength is very small, the oscillations of the QFIs decay slower which corresponds to the non-Markovian region; yet when it becomes large, the QFIs monotonously decay which corresponds to the Markovian region. When the atom number N$N$ increases, QFIs in both regions decay faster. We further find that the QFI flow disappears suddenly followed by a sudden birth depending on the ratio of the PNL rate and the system–environment coupling strength and the atom number N$N$, which unveil a fundamental connection between the non-Markovian behaviors and the parameters of system–environment couplings. We discuss two optimal positive operator-valued measures (POVMs) for two different strategies of our model and find the condition of the optimal measurement. At last, we consider the QFI of two atoms with qubit–qubit interaction under random telegraph noises (RTNs).     

Quantum Discord Dynamics of Three Qubits in Non-Markovian Environments

We investigate the dynamics of palrwise quantum discord （QD） for a mixed three-qubit W-type state in three independent non-Markovian reservoirs at zero temperature, each of which is modeled by a leaky cavity with Lorentzian spectral density. The influence of the environment＇s amount of non-Markovianity, the detuning between the qubit frequency and the cavity centre frequency, and the purity of the initial state on the QD dynamics are analyzed in detail. It is found that in the non-Maxkovian regime the system-reservoir interactions induce QD revivals and oscillations no matter whether the detuning is zero or not. Moreover, QD can be preserved for a long time if the non-Markovian condition and the detuning condition are satisfied simultaneously.     

Quantum Fisher information of a qubit-qutrit system in Garfinkle–Horowitz–Strominger dilation space–time

We investigate the quantum Fisher information(QFI) of a qubit-qutrit system in the background of Garfinkle–Horowitz–Strominger dilation black hole. After deriving the analytical expression of the QFI, we examine its dynamics with respect to the dilation parameter D and the state parameter γ of the system. Our results show that the QFI for the estimation of γ is a fixed value,which is independent of the parameters D and γ. And the QFI for the estimation of D varies with the parameters D and γ. Additionally, we propose an effective strategy to steer the QFI by introducing weak measurement reversal. We find that the QFI can be remarkably enhanced by adjusting the appropriate reversing measurement strengths. Our findings might provide some useful insights for the study on parameter estimation of hybrid systems in the framework of relativity theory.     

The Precision of Parameter Estimation for Dephasing Model Under Squeezed Reservoir

We study the precision of parameter estimation for dephasing model under squeezed environment. We analytically calculate the dephasing factor γ(t) and obtain the analytic quantum Fisher information (QFI) for the amplitude parameter α and the phase parameter ?. It is shown that the QFI for the amplitude parameter α is invariant in the whole process, while the QFI for the phase parameter ? strongly depends on the reservoir squeezing. It is shown that the QFI can be enhanced for appropriate squeeze parameters r and θ. Finally, we also investigate the effects of temperature on the QFI.     

Improving the teleportation of quantum Fisher information under non-Markovian environment

Quantum teleportation is designed to send an unknown quantum state between two parties. In the perspective of remote quantum metrology, one may be interested in teleporting the information that is encoded by physical parameters synthesized by quantum Fisher information (QFI). However, the teleported QFI is often destroyed by the unavoidable interaction between the system and the environment. Here, we propose two schemes to improve the teleportation of QFI in the non-Markovian environment. One is to control the quantum system through the operations of weak measurement (WM) and corresponding quantum measurement reversal (QMR). The other is to modify the quantum system based on the monitoring result of the environment (i.e., environment-assisted measurement, EAM). It is found that, in the non-Markovian environment, these two schemes can improve the teleportation of QFI. By selecting the appropriate strengths of WM and QMR, the environment noise can be completely eliminated and the initial QFI is perfectly teleported. A comprehensive comparison shows that the second scheme not only has a higher probability of success than the first one, but also has a significant improvement of the teleported QFI.     

噪声对一种三粒子量子探针态的影响

量子度量学是研究量子测量与统计推断的一门学科,主要利用量子手段来提高参数估计的精度,在量子信息处理与测量中起到关键作用.量子参数估计的一般过程包含四个步骤:探针态的制备、参数化过程、对参数化后的输出态进行测量以及根据测量结果估计待测参数.其中探针态的选取对测量精度起着至关重要的作用.然而在实际的量子探针态的制备过程中,初始探针态会受到环境噪声的影响.目前人们已经研究了W态与Greenberger-Horne-Zeilinger(GHZ)态的量子Fisher信息(QFI)在典型噪声通道下的变化行为.由于W态与GHZ态有着不同的纠缠性质,对于W态与GHZ态的叠加态的QFI动力学研究具有重要的实际意义.故此,本文主要研究典型噪声通道对这两种状态的叠加态的QFI动力学行为的影响,得出了QFI随噪声参数的变化行为.结果表明,叠加态中W态组分可明显对抗相位阻尼噪声对探针态的QFI的影响,而其中的GHZ态组分可明显对抗振幅阻尼噪声的影响,从而为在实际环境中选取高精度的参数估计过程提供参考.     

Nonequilibrium Effects on the Precision of Parameter Estimation

The impact of nonequilibrium environment effects on the accuracy of quantum parameter estimation is investigated, and it is found that these effects can significantly affect estimation accuracy. Using an individual estimation strategy reveals that the nonequilibrium effects consistently enhance accuracy, regardless of the coupling strength between the probe and its environment. In contrast, weak memory effects undermine estimation accuracy. When employing a multi-parameter simultaneous estimation strategy, it is observed that the nonequilibrium effects consistently improve the advantages of simultaneous estimation, as analyzed by the ratio of total variances between the two estimation scenarios. However, the memory effects on these advantages depend on the coupling strength between the qubit and the environment. These findings suggest that appropriate parameters of a nonequilibrium environment can increase the quantum Fisher information (QFI), thereby enhancing the accuracy of quantum parameter estimation. These significant results are essential for improving parameter estimation accuracy in quantum systems interacting with nonequilibrium environments.     

Non-Markovian dynamics of a qubit in a reservoir: different solutions of non-Markovian master equation

We present a non-Markovian master equation for a qubit interacting with a general reservoir, which is derived according to the Nakajima-Zwanzig and the time convolutionless projection operator technique. The non-Markovian solutions and Markovian solution of dynamical decay of a qubit are compared. The results indicate the validity of non-Markovian approach in different coupling regimes and also show that the Markovian master equation may not precisely describe the dynamics of an open quantum system in some situation. The non-Markovian solutions may be effective for many qubits independently interacting with the heated reservoirs.     

Dynamics of quantum Fisher information in a two-level system coupled to multiple bosonic reservoirs

We consider the optimal parameter estimation for a two-level system coupled to multiple bosonic reservoirs. By using quantum Fisher information(QFI), we investigate the effect of the Markovian reservoirs' number N on QFI in both weak and strong coupling regimes for a two-level system surrounded by N zero-temperature reservoirs of field modes initially in the vacua. The results show that the dynamics of QFI non-monotonically decays to zero with revival oscillations at some time in the weak coupling regime depending on the reservoirs' parameters. Furthermore, we also present the relations between the QFI flow, the flows of energy and information, and the sign of the decay rate to gain insight into the physical processes characterizing the dynamics.     

Dephasing Process of a Single Atom Interacting with a Two-Mode Field

We consider the interaction of a qubit system with a two-mode field in the presence of multi-photon transition and phase damping effect. We use the master equation to obtain the density operator when the qubit is initially prepared in its excited state and the field is in a finite-dimensional pair coherent state. The properties of the considered system, such as the population inversion, amount of the mixedness, parameter estimation, and squeezing, are explored for one- and two-photon transitions. The effects of photon addition to the field and phase damping on the evaluation of these quantumness measures are also investigated.     

Preservation of Quantum Fisher Information and Geometric Phase of a Single Qubit System in a Dissipative Reservoir Through the Addition of Qubits

In this paper, we have investigated the preservation of quantum Fisher information (QFI) of a single-qubit system coupled to a common zero temperature reservoir through the addition of noninteracting qubits. The results show that, the QFI is completely protected in both Markovian and non-Markovian regimes by increasing the number of additional qubits. Besides, the phenomena of QFI display monotonic decay or non-monotonic with revival oscillations depending on the number of additional qubits N ??1 in a common dissipative reservoir. If N < N _c (a critical number depending on the reservoirs parameters), the behavior of QFI with monotonic decay occurs. However, if N ≥ N _c , QFI exhibits non-monotonic behavior with revival oscillations. Moreover, we extend this model to investigate the effect of additional qubits and the initial conditions of the system on the geometric phase (GP). It is found that, the robustness of GP against the dissipative reservoir has been demonstrated by increasing gradually the number of additional qubits N ??1. Besides, the GP is sensitive to the initial parameter ??, and possesses symmetric in a range regime [0,2π].     

Modulation of Entanglement for Coupled Superconducting Qubits Under Non-Markovian Environment

The evolution of entanglement decoherence is investigated for a coupled superconducting qubit under non-Markovian environment by utilizing a commensal entanglement degree. The results show that, owing to the memory feedback effect of environment, the entanglement degree of the coupled qubits at the thermal equilibrium always monotonously tends to zero so that entanglement sudden death occurs briefly in the non-Markovian process. Different from the Markovian process, stronger the dissipation is, faster the entanglement sudden death is. We find that, furthermore, the interaction between the qubits results generally in reduction of entanglement degree in the quantum system. With some special initial states or initial phase angles, however, the influence of the interaction between qubits on the system entanglement degree can be avoided.     

Quantum Fisher information width in quantum metrology

In scenarios of quantum metrology, the unitary parametrization process often depends on space directions. How to characterize the sensitivity of parameter estimation to space directions is a natural question. We propose the concept of the quantum Fisher information(QFI) width, which is the difference between the maximum and minimum values of the QFI, to quantitatively study the sensitivity. We find that Fock states, the bosonic coherent states, and the displaced Fock states all have zero widths, indicating that QFI is completely inert over all directions, while the width for the spin state with all spins down or up is equal to the number of particles, so this concept will enable us to choose appropriate directions to make unitary transformation to obtain larger QFI.The QFI width of the displaced quantum states is found to be independent of the magnitude of the displacement for both spin and bosonic systems. We also find some relations between the QFI width and squeezing parameters.