The dynamical behaviors of many-body systems after switching of even a single, weak interacting impurity qubit are interesting. The phenomenon of Anderson’s orthogonality catastrophe (OC) [1] is then witnessed, when the initial many-body wave function loses its essential overlap with the perturbed one. It captures the main feature of many-body systems and reveals the sensitivity of them to local perturbations. The study of OC has aroused considerable interest within various areas of physics, such as quantum spin systems [2–6], quantum phase transitions (QPTs) [7–9] and nonequilibrium dynamics [10, 11]. Recently, the behavior of OC has been linked to the quantum speed limit (QSL), which originates from the uncertainty theory [12]. It demonstrates that the OC follows as a consequence of the vanishing QSL time in many-body systems [13]. In addition, the OC is also related to the decoherence of the impurity, characterizing the dynamical evolution of the impurity [14]. Thus, it is natural to relate the evolution speed of the impurity qubit to the OC since the two quantities are associated with the fidelity.

In this paper, we consider a qubit-spin-bath system consisting of a central impurity qubit coupled to an isotropic Lipkin-Meshkov-Glick (LMG) model [15]. The system was first proposed to describe how the QPT of the bath influences the quantum coherence of the central qubit [16]. Tian et al analyzed the influences of the environment parameters to the non-Markovian effect of the system [17]. The non-Markovian feature is the intrinsic mechanism for the quantum acceleration phenomenon of the central qubit in LMG bath [18]. The LMG model, as a clean system, has also been used to study the time crystal effect. It shows that the model being infinite range and having an extensive amount of symmetry-breaking eigenstates is crucial to exhibit the time-crystal behavior [19, 20]. Within the considered system, we propose an alternative method of characterizing the OC dynamics through the speed of quantum evolution, which relies on a determined metric. Thus, we exploit the actual speed of quantum evolution in terms of the quantum Fisher information metric [21] to investigate its relationships to the OC and the QSL, respectively.

The rest of this paper is organized as follows. In section 2, we briefly review the qubit-spin-bath system and present the relationship between the OC dynamics and the quantum average speed. In section 3, we measure the QSL in an explicit form, and show the numerical comparison and analysis between the QSL and the actual speed quantum evolution speed. Section 4 is the conclusion.

The LMG model was first introduced to describe a set of N spin-1/2 mutually interacting in atomic nuclei, and this model is currently studied in various fields of physics ranging from magnetic molecules [22] to Bose–Einstein condensates [23, 24], etc. For its unique phase transition feature, the LMG model has been widely used to investigate quantum critical phenomena [25–27]. The total Hamiltonian of a central qubit interacting with an isotropic LMG spin model is written aswith where H_S and H_B represent the Hamiltonians of the central qubit and the isotropic LMG bath, respectively. H_SB denotes the interacting Hamiltonian between the central qubit and the environment. λ refers to the coupling strength between the spins in the bath, while ${\lambda }^{{\prime} }$ denotes the coupling strength between the central qubit and the bath, fixed as ${\lambda }^{{\prime} }=\lambda /\sqrt{N}$. N is the particle number.

In terms of the collective spin operators ${J}_{N}^{\alpha }=1/2{{\rm{\Sigma }}}_{i=1}^{N}{\sigma }_{i}^{\alpha }$ with $\alpha =\left\{x,y,z\right\}$, the total Hamiltonian can be rewritten aswhere ${J}_{N}^{\pm }={J}_{N}^{x}\pm {\rm{i}}{J}_{N}^{y}$ and s_± = s^x ± is^y correspond, respectively, to the ladder operators of the N-spin bath and the central qubit. The LMG bath undergoes a QPT at the critical point λ = 1, i.e. the ground state of the bath on two sides of the critical point has different symmetry properties. The bath is in a symmetry broken phase for 0 < λ < 1 and in a symmetric phase for λ > 1.

In an invariant subspace H_M of H spanned by the ordered basis vectors {∣N/2, M⟩ ⨂ ∣0⟩, ∣N/2, M + 1⟩ ⨂ ∣1⟩}, the time evolution operator $U\left(t\right)=\exp \left(-{\rm{i}}{Ht}\right)$ can be expressed as a quasi-diagonal matrix with the diagonal blocksand in another invariant subspace H_M−1 spanned by {∣N/2, M − 1⟩ ⨂ ∣0⟩, ∣N/2, M⟩ ⨂ ∣1⟩}, the time evolution operator iswhere the matrix elements of $U\left(t\right)$ and ${U}^{{\prime} }\left(t\right)$ are relevant to the environment parameters. We assume that the bath is initially in the ground state, and the initial state of the central qubit is in a pure superposition state $\rho (0)=c| 0\rangle \langle 0| +d| 0\rangle \langle 1| +{d}^{* }| 1\rangle \langle 0| +\left(1-c\right)| 1\rangle \langle 1| $. Since the ground state of the bath is different for two phases, the reduced state of the central qubit in the symmetry broken phase can be expressed asand in the symmetric phase, we have

The purpose of this research is to investigate a connection between the OC and the actual speed of quantum evolution. Hence, we start by considering the overlap between the perturbed and unperturbed many-body states, which describes the OC. The original work of Anderson’s OC effect focused on stationary states [1], however, the many-body state after a local perturbation is generally time-dependent. The dynamical OC is given by a time-dependent overlap, read aswhere H_f denotes the perturbed Hamiltonian. H_i is the unperturbed Hamiltonian whose eigenstate is the initial state ∣ψ⟩ with the corresponding eigenenergy ϵ_i. To be precise, the initial state is assumed to be ∣ψ⟩ = ∣ψ₀⟩ and the time-evolved state is $| {{\rm{\Psi }}}_{t}\rangle ={{\rm{e}}}^{-{\rm{i}}{H}_{f}t}| {\psi }_{0}\rangle $, the dynamical overlap is thus reduced toThe time-dependent fidelity or survival probability, as a signature of the OC, is referred to as the square modulus of the overlapwhich measures the susceptibility of dynamics to a local perturbation. The decay dynamics of the fidelity are widely used in quantum information theory.

The speed of quantum evolution can be quantified by various measures [28–30]. In this work, we look to the actual speed of quantum evolution derived by Cianciaruso et al due to the quantity relative to the time-dependent fidelity [21]. Theoretical studies have proposed using the non-Markovian effect, and the associated information backflow from the environment, to enhance the actual evolution speed [31]. Recent effort has observed that as much as the classical correlation of the channel increases, the speed of quantum evolution decreases [32]. Based on the unified form of the Riemannian metric g, the squared infinitesimal distance between the neighboring states ρ and ρ + dρ is ${\left({\rm{d}}{s}\right)}^{2}\,={g}_{\rho }\left({\rm{d}}\rho ,{\rm{d}}\rho \right)$. The speed of quantum evolution $\rho \left(t\right)={{\rm{\Lambda }}}_{t}\left[\rho \left(0\right)\right]$ at time t is given as followswhere $g\left(t\right)={g}_{\rho \left(t\right)}\left(\dot{\rho }\left(t\right),\dot{\rho }\left(t\right)\right)$. As proven by the Morozova–Chencov–Petz theorem, the exact solution of $g\left(t\right)$ depends on a chosen metric. We exploit the Bures-Uhlmann metric, known as the quantum Fisher information metric, to quantify the actual speed of quantum evolution. Since the second derivative of the Uhlmann fidelity with respect to time is proportional to the quantum Fisher information, the following useful relation holds as well [33]whereis the Uhlmann fidelity. For the sake of simplicity, we impose the initial state is pure, $\rho \left(0\right)=| {\psi }_{0}\rangle \langle {\psi }_{0}| $, and the fidelity is $F\left(\rho \left(0\right),\rho \left(t\right)\right)=\langle {\psi }_{0}| \rho \left(t\right)| {\psi }_{0}\rangle $, which equals the survival probability between the states (12).

Next, we turn our attention to calculating the exact solution of the OC and quantum evolution speed. Regarding the qubit-spin-bath system of interest, the central qubit is initialized as the excited state $\rho \left(0\right)=| 1\rangle \langle 1| $, then we obtain the measure of dynamical speed in an explicit form in the symmetry broken phase by the second derivative of the fidelityunder the same initial condition, the dynamical speed in the symmetric phase isTo relate the OC and the actual speed of quantum evolution, we focus our analysis on the minimal fidelity ${F}_{\min }$ and the corresponding average speedwith ${t}_{\min }$ being the time when the minimum value of fidelity occurs. When ${F}_{\min }$ is shown to vanish, the overall system evolves to almost orthogonal states, leading to an OC effect. According to the properties of the oscillatory function, in the symmetry broken phase, the minimum value of fidelity is reduced to

Equation (19) indicates that ${F}_{\min }$ keeps a stable value for fixed particle number regardless of the coupling strength. In the symmetric phase, we write the minimal fidelity as

The variations of the minimal fidelity ${F}_{\min }$ and the corresponding average speed ${\overline{v}}_{\min }$ with respect to the bath size N for different λ are plotted in figure 1. In the region 0 < λ < 1, ${F}_{\min }$ never reaches zero and its behavior is insensitive to the bath size. Therefore, the OC is not witnessed in this phase, which is in agreement with the findings in [13]. In contrast, for λ > 1, ${F}_{\min }$ is shown to disappear as the coupling strength is sufficiently large but finite, which indicates that the evolved state approaches a fully orthogonal state. The quantum averaged speed also exhibits a plateau independent of the bath size in figure 1(b) for the symmetry broken phase and then enlarges as we increase the coupling strength λ. The qualitatively different behaviors of the two quantities demonstrate that the increment of the average speed may accelerate the occurrence of the OC.

As shown in figure 1, the rising of the bath size in the symmetric phase corresponds to the increase in average speed and reduction of fidelity. To illustrate clearly the influence of the bath size N on the OC and the quantum average speed, we plot figure 2. We find a simple relationship, and it is that ${F}_{\min }$ tends to zero in the thermodynamical limit N → ∞ , thus, the system displays the OC effect. For each line with a fixed λ, the increase of the bath size N leads to an increase of ${\overline{v}}_{\min }$ , which indicates that a manifestation of the OC corresponds to a sufficiently large but finite averaged speed. Figure 3 shows intuitively how the minimal fidelity decays with the quantum average speed for different λ or N. We see that the fidelity eventually vanishes induced by the increased average speed. These features combined demonstrate that the actual speed of quantum evolution with a judicious choice of environment parameters can be used to accelerate the emergence of the OC. Our study sheds further light on the interplay between the OC and the quantum average speed of an open quantum system.

The above results show that the dynamical occurrence of OC is attributed to the actual speed of quantum evolution, more specifically, the evolution speed scales with the environment parameters exhibits the OC effect. Previous research concluded that the OC follows as a consequence of the vanishing QSL time in many-body systems [13]. It promotes us to further investigate a relationship between the quantum average speed and the QSL in the open quantum system.

The QSL sets the minimum evolution time with which any quantum system can evolve, and the fundamental expression of QSL depends on the choice of a measure of distinguishability of quantum states. In this work, we exploit the unified expression for the QSL time derived by Deffner and Lutz [34] under a quantum master equation ${\dot{\rho }}_{t}=L\left({\rho }_{t}\right)$, with a positive operator $L\left({\rho }_{t}\right)$. The QSL time is obtained byand accordingly, the QSL ${v}_{\mathrm{QSL}}\left(t\right)$ is defined aswhere ${{\rm{\Lambda }}}_{\tau }^{p}={\tau }^{-1}{\int }_{0}^{\tau }{\parallel L\left({\rho }_{t}\right)\parallel }_{p}{\rm{d}}t$, ${\parallel L\left({\rho }_{t}\right)\parallel }_{p}={\left({\sum }_{i=1}{\sigma }_{i}^{p}\right)}^{1/p}$. p = 1, 2 and ∞ correspond to the trace, Hilbert–Schmidt and operator norms σ_i are the singular values of $L\left({\rho }_{t}\right)$, which equals the eigenvalues of $\sqrt{L{\left({\rho }_{t}\right)}^{\dagger }L\left({\rho }_{t}\right)}$. ${ \mathcal L }\left({\rho }_{0},{\rho }_{\tau }\right)\,=\arccos \sqrt{{Tr}\left({\rho }_{0}{\rho }_{\tau }\right)}$represents the Bures angle between the initial and final states. In particular, equation (21) reveals that the minimum evolution time between two distinguishable states is inversely proportional to the average QSL from time zero to τwith τ being the actual driving time. The theoretical maximal speed of quantum evolution is given by the minimum norm of the generator of the dynamics. As a consequence, we measure the QSL based on the operator norm. The initial state of the central qubit is still chosen as the excited state, and the QSL in the symmetry broken phase isWhen the LMG bath is in the symmetric phase, we obtainNote that in contrast to equation (22), we did not include the denominator $2\cos \left({ \mathcal L }\right)\sin \left({ \mathcal L }\right)$ into the expressions of ${\overline{v}}_{\mathrm{QSL}}$. The reason for this choice is to achieve a tighter bound of the QSL time [18].

We examine the dynamics of the time-average of the QSL and the quantum evolution speed from t = 0 to t = τ under the effects of the coupling strength λ and the LMG bath size N in figure 4. The average QSL ${\overline{v}}_{\mathrm{QSL}}$ behaves in a way similar to the average evolution speed $\overline{v}$, and the two quantities have qualitatively different behaviors on two sides of the critical point λ = 1. In contrast, the effects of λ and N in the symmetry broken phase are negligible compared to the symmetric phase. The different patterns depending on the value of the environment parameters can characterize the QPT of the LMG model.

We have applied the concept of the OC to the qubit-spin-bath system composed of a central qubit interacting with an isotropic LMG spin model to investigate the relationship between the OC and the actual speed of quantum evolution. Within the considered system, the numerical results of the fidelity exhibit how the OC dynamics are affected by the initial environment phase and the OC can only be observed in the symmetric phase of the LMG bath. In particular, we propose an alternative method of characterizing the dynamical occurrence of the OC by the quantum average speed since both are associated with fidelity. We demonstrate that the OC effect manifests itself followed by the vanishing fidelity and the sufficiently large but finite average speed. The actual speed of quantum evolution is as vital as the QSL time in [13] to the OC dynamics in many-body systems. Additionally, we reveal a similar behavior between the actual speed of quantum evolution and the QSL of the central qubit. The two quantities experience a dramatic acceleration around the second-order QPT point, which can be used to characterize the QPT in the LMG model.