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Abstract
An uncertainty relation between energy and time having a simple physical meaning is rigorously deduced from the principles of quantum mechanics. Some examples of its application are discussed.
... Subsequently, using Eq. (25) in Eq. (24), WSPC Proceedings -9.61in x 6.69in ws-procs961x669 page 8 8 the survival probability of the initial state |ν 1 ⟩ at any time t can be expressed as a function of radial distance r from the black hole as ...
... In the relativistic limit, using Eq. (25) in Eq. (38), Fig. 5(a) shows the survival probability P s as a function of the small range of radial distance r from the black hole for a low (dashed line) and a high (solid line) value of specific angular momentum a. We observe that near the black hole, the peak of survival probability P s fluctuates more frequently between 1 and 0 for a high a (solid line) compared to a low a (dashed line). ...
... The quantum speed limit (QSL) concept originated from the uncertainty relationship between conjugate variables in quantum mechanics 25,26 . It represents a fundamental constraint set by quantum mechanics on the rate of evolution for any quantum system undergoing a specific dynamical process. ...
We investigate the quantum speed limit (QSL) during the time evolution of neutrino-antineutrino system under the influence of the gravitational field of a spinning primordial black hole (PBH). We derive an analytical expression for the four-vector gravitational potential in the underlying Hermitian Dirac Hamiltonian using the Boyer-Lindquist (BL) coordinates. This gravitational potential leads to an axial vector term in the Dirac equation in curved spacetime, contributing to the effective mass matrix of the neutrino-antineutrino systems. Our findings indicate that the gravitational field, expressed in BL coordinates, significantly influences the transition probabilities in two-flavor oscillations of the neutrino-antineutrino system. We then apply the expression for transition probabilities between states to analyze the Bures angle, which quantifies the closeness between the initial and final states of the time-evolved flavor state. We use this concept to probe the QSL for the time evolution of the initial flavor neutrino state.
... The first QSL bound on the evolution time was discovered by Mandelstam and Tamm [15] using the Heisenberg-Robertson uncertainty relation. It depends on the shortest distance between the initial and the final state and the variance of the driving Hamiltonian. ...
... The concept of quantum speed limit has been widely studied for closed quantum system dynamics [15][16][17] as well as for open quantum system dynamics [9,[73][74][75][76][77][78][79]. It is essential to mention that a method to measure speed of quantum system in interferometry was proposed in [62] and recently, an experiment was reported where the quantum speed limits were tested in a multi-level quantum system by tracking the motion of a single atom in an optical trap using fast matter-wave interferometry [80]. ...
... This is the famous Mandelstam-Tamm bound [15] which suggests that T QSL is the minimum time required for the quantum system to evolve from |Ψ(0)⟩ to |Ψ(T)⟩ by unitary evolution. ...
The quantum speed limit describes how quickly a quantum system can evolve in time from an initial state to a final state under a given dynamics. Here, we derive a generalised quantum speed limit (GQSL) for arbitrary time-continuous evolution using the geometrical approach of quantum mechanics. The GQSL is applicable for quantum systems undergoing unitary, non-unitary, completely positive, non-completely positive and relativistic quantum dynamics. This reduces to the well known standard quantum speed limit (QSL), i.e., the Mandelstam-Tamm bound when the quantum system undergoes unitary time evolution. Using our formalism, we then obtain a quantum speed limit for non-Hermitian quantum systems. To illustrate our findings, we have estimated the quantum speed limit for a time-independent non-Hermitian system as well as for a time-dependent non-Hermitian system namely the Bethe-Lamb Hamiltonian for general two-level system.
... While this relationship is well known for non-commuting observables, e.g., position and momentum, the time-energy uncertainty relation has been controversial over decades, resulting in several attempts to address this issue [1,2]. In their seminal work, Mandelstam and Tamm (MT) [3] reinterpreted this question by introducing the concept of quantum speed limit (QSL), which is a threshold imposed by quantum mechanics to the minimum evolution time between two orthogonal states. QSLs have been addressed for either closed and open quantum systems [4][5][6][7][8][9][10][11][12][13], and find applications ranging from quantum many-body systems [14][15][16][17][18][19][20][21], to quantum thermodynamics [22][23][24][25]. ...
... Thus, to evaluate the geometric QSL of a given Riemannian metrics, in the experiments, we simply prepare the state ρ 0 by the application of a single (π/4) rotation on the y-axis and let the system evolve according to Eq. (3). We control the relaxation rates 1/T 1,H and 1/T 2,C by adding the paramagnetic salt iron(III) acetylacetonate (Fe(acac) 3 ) to the solution, with the relaxation rates growing linearly with the concentration of Fe(acac) 3 [44]. In Appendix A, we present technical details about the experiment and sample preparation. ...
... Thus, to evaluate the geometric QSL of a given Riemannian metrics, in the experiments, we simply prepare the state ρ 0 by the application of a single (π/4) rotation on the y-axis and let the system evolve according to Eq. (3). We control the relaxation rates 1/T 1,H and 1/T 2,C by adding the paramagnetic salt iron(III) acetylacetonate (Fe(acac) 3 ) to the solution, with the relaxation rates growing linearly with the concentration of Fe(acac) 3 [44]. In Appendix A, we present technical details about the experiment and sample preparation. ...
The quantum speed limit sets a lower bound on the evolution time for quantum systems undergoing general physical processes. Here, using an ensemble of chloroform molecules, we study the speed of evolution of a qubit subject to decoherence. In this solution, the carbon nuclear spin encodes the two-level system, while the hydrogen spin plays the role of an environment for the latter. By adding a paramagnetic salt, we control the system-reservoir interaction as the hydrogen spin relaxation rates change, and we probe the speed of qubit evolution. We address geometric QSLs based on two distinguishability measures of quantum states, quantum Fisher information (QFI) and Wigner-Yanase skew information (WY) metrics. For high concentrations of the salt, the system undergoes a Markovian dynamics, and the tighter QSL is set by the WY metric. For low concentrations, we observe crossovers between QSLs related to the QFI and WY metrics, while the system exhibits non-Markovian dynamics. The QSLs are sensitive to even small fluctuations in spin magnetization, from low to high concentrations. Our results find applications in quantum computing and optimal control.
... Time-energy uncertainty relations have long been used to estimate characteristic time scales in physical processes, including lifetimes in quantum decay, tunneling times, and the duration of a quantum jump, among others [1][2][3][4]. In the quantum domain, Mandelstam and Tamm put the time-energy uncertainty relation on firm ground in their 1945 work [5]. They provided its rigorous derivation by combining the Heisenberg equation of motion and the Robertson uncertainty relation. ...
... is some real-valued function of t. 4 Given that S is non-zero, the corresponding curve of must then be situated on an effective Bloch sphere spanned by the operators X, Y and S. 5 More explicitly, we have that A = ∥V ∥ cos θ(t)X + ∥V ∥ sin θ(t)Y + S will trace out a curve following a circle centered at S (see figure 1). Since this circle is not centered at the origin, it will not be a great circle on the sphere S ∥A∥ . ...
... A consequence of this is that the length of this curve must be strictly larger than the geodesic distance on S ∥A∥ . This length is precisely the numerator 4 In fact, if we choose X and Y so that X = V , then θ is the angle between V and Vt and is given by 5 We emphasize that the operators S, X and Y are orthogonal with respect to Re⟨·, ·⟩ and not necessarily the Hilbert-Schmidt inner product. Figure 2: As ∥S∥ grows larger, the center of the circle that t follows will be closer to the poles of the sphere. ...
Quantum speed limits (QSLs) provide lower bounds on the minimum time required for a process to unfold by using a distance between quantum states and identifying the speed of evolution or an upper bound to it. We introduce a generalization of QSL to characterize the evolution of a general operator when conjugated by a unitary. The resulting operator QSL (OQSL) admits a geometric interpretation, is shown to be tight, and holds for operator flows induced by arbitrary unitaries, i.e., with time- or parameter-dependent generators. The derived OQSL is applied to the Wegner flow equations in Hamiltonian renormalization group theory and the operator growth quantified by the Krylov complexity.
... The Quantum Speed Limits (QSLs) are inequalities that evaluate the minimum time required to transition from one quantum state to another. They were initially formulated for isolated quantum systems [1][2][3][4][5][6] as generalizations of the time-energy uncertainty relations, and afterwards they have been extended to open quantum systems [7][8][9][10][11][12][13][14][15]. They provide universal constraints on the rates of state changes across various dynamics, regardless of the specific characteristics of the system. ...
... First, we transform the von Neumann entropy flux and heat flux using Eqs. (1) and (14) as follows [10]: ...
In the context of quantum speed limits, it has been shown that the minimum time required to cause a desired state conversion via the open quantum dynamics can be estimated using the entropy production. However, the established entropy-based bounds tend to be loose, making it difficult to accurately estimate the minimum time for evolution. In this research, we have combined the knowledge of the entropy-based speed limits with that of the resource theory of asymmetry (RTA) and provided much stricter inequalities. Our results show that the limitation on the change rate of states and expectation values can be divided into two parts: quantum coherence for energy (i.e., asymmetry) contributed by the system and the heat bath and the classical entropy-increasing effect from the bath. As a result, our inequalities demonstrate that the difference in the speed of evolution between classical and quantum open systems, i.e., the quantum enhancement in speed, is determined by the quantum Fisher information, which measures quantum fluctuations of energy and serves as a standard resource measure in the resource theory of asymmetry. We further show that a similar relation holds for the rate of change of expectation values of physical quantities.
... Here, we are interested in a speed limit for the time-evolved state rather than the operator. In the standard formulation, one considers as a distance the Fubini-Study angle Θ FS (t) = arccos |⟨ψ(0)|ψ(t)⟩| [27]. The minimum time for sweeping Θ FS (t) is then lower bounded by the time-average energy variance [28][29][30]. ...
... The spread complexity is written as K(t) = d−1 k=0 sin 2 Θ k (t). By using similar techniques to those in the derivation of the celebrated Mandelstam-Tamm time-energy uncertainty relation [27,31], we can derive upper bounds to Θ n (t). The time derivative of Θ n (t) leads to ...
Krylov subspace methods in quantum dynamics identify the minimal subspace in which a process unfolds. To date, their use is restricted to time evolutions governed by time-independent generators. We introduce a generalization valid for driven quantum systems governed by a time-dependent Hamiltonian that maps the evolution to a diffusion problem in a one-dimensional lattice with nearest-neighbor hopping probabilities that are inhomogeneous and time-dependent. This representation is used to establish a novel class of fundamental limits to the quantum speed of evolution and operator growth. We also discuss generalizations of the algorithm, adapted to discretized time evolutions and periodic Hamiltonians, with applications to many-body systems.
... Hence, it cannot be represented by a self-adjoint operator (conjugated to the Hamiltonian) as argued by Pauli due to the bounded nature of the continuous energy spectrum [4]. The time-energy uncertainty relation was initially deduced in the work by Mandelstam and Tamm [5], and subsequently discussed by various authors [6][7][8][9][10][11][12][13][14]. ...
... A rigorous derivation of the uncertainty relation between time and energy was first provided by [5]. In this work, we will follow a different approach, as presented by [15], where a new canonical variable, called tempus T , conjugate to energy E is defined through a canonical transformation. ...
The Generalized Uncertainty Principle (GUP) modifies the Heisenberg Uncertainty Principle (HUP) between position and momentum by introducing a nonzero minimum uncertainty in position. In a previous study, we demonstrated the emergence of GUP from non-extensive entropies, particularly for dependent only on the probability. In this new research, we derive a generalized energy-time uncertainty relation from these entropies. Consequently, we observe that the dispersion relation undergoes modification due to the non-extensivity introduced by the entropies and . These modifications play a significant role at the Planck scale, but are negligible in the classical regime of large distances and low energies. Moreover, the modified uncertainty relation results in a maximum uncertainty in energy, attributed to the negative deformation parameter associated with entropy . Conversely, the deformation parameter linked with leads to the emergence of a minimum time interval. In the Planck regime, the minimum time interval is on the order of the Planck time, while the maximum uncertainty in energy reaches the Planck energy. These findings imply that quantum gravity effects can be connected with non-extensive statistics.
... We find that the rate of synchronization is determined by a competition between the irreversible entropy production caused by damping, which slows synchronization, and the strength of the anti-Hermitian coupling, which speeds up synchronization. The resulting upper bound on the synchronization rate has terms of the form of the Mandelstam-Tamm inequality [63], where speed scales with the uncertainty of the energy, except in this case even the uncertainties of the Hermitian and anti-Hermitian parts of the Hamiltonian are crucial. As an example, we consider a dissipatively coupled photonic dimer and find that the quantum system synchronizes in a parameter regime wherein it is impossible for the classical model to synchronize, thereby displaying a quantum advantage. ...
... This term unsurprisingly implies that stronger non-Hermitian coupling leads to faster synchronization. The second term is reminiscent of the Mandelstam-Tamm quantum speed limit [63], and involves the second moments of both the Hermitian and anti-Hermitian parts of the Hamiltonian; however there is a penalty that scales with the square of their commutator. This term arises from the uncertainty relation [113], and can be explained by the fact that synchronization is most effective when there is a large correlation between the observables corresponding to the Hermitian and anti-Hermitian parts of the Hamiltonian. ...
Quantum synchronization is crucial for understanding complex dynamics and holds potential applications in quantum computing and communication. Therefore, assessing the thermodynamic resources required for finite-time synchronization in continuous-variable systems is a critical challenge. In the present work, we find these resources to be extensive for large systems. We also bound the speed of quantum and classical synchronization in coupled damped oscillators with non-Hermitian anti-PT-symmetric interactions, and show that the speed of synchronization is limited by the interaction strength relative to the damping. Compared to the classical limit, we find that quantum synchronization is slowed by the noncommutativity of the Hermitian and anti-Hermitian terms. Our general results could be tested experimentally, and we suggest an implementation in photonic systems.
... [28,29] The QSL refers to the minimum time required for a quantum system to evolve from one state to another. Two well-known formulae are the Mandelstam-Tamm (MT) [30] and Margolus-Levitin (ML) [31] relations, which establish a lower bound on the evolution time in closed quantum systems under unitary evolution with a time-independent Hamiltonian. ...
... [28,29] The QSL refers to the minimum time required for a quantum system to evolve from one state to another. Two well-known formulae are the Mandelstam-Tamm (MT) [30] and Margolus-Levitin (ML) [31] relations, which establish a lower bound on the evolution time in closed quantum systems under unitary evolution with a time-independent Hamiltonian. [32] Subsequent developments in the field have expanded these formulae to address a broader range of quantum complexities, including time-dependent Hamiltonian, [33][34][35] open quantum systems, [36][37][38][39][40] and many-body systems, [41][42][43] etc. ...
Adiabatic time-optimal quantum controls are extensively used in quantum technologies to break the constraints imposed by short coherence times. However, practically it is crucial to consider the trade-off between the quantum evolution speed and instantaneous energy cost of process because of the constraints in the available control Hamiltonian. Here, we experimentally show that using a transmon qubit that, even in the presence of vanishing energy gaps, it is possible to reach a highly time-optimal adiabatic quantum driving at low energy cost in the whole evolution process. This validates the recently derived general solution of the quantum Zermelo navigation problem, paving the way for energy-efficient quantum control which is usually overlooked in conventional speed-up schemes, including the well-known counter-diabatic driving. By designing the control Hamiltonian based on the quantum speed limit bound quantified by the changing rate of phase in the interaction picture, we reveal the relationship between the quantum speed limit and instantaneous energy cost. Consequently, we demonstrate fast and high-fidelity quantum adiabatic processes by employing energy-efficient driving strengths, indicating a promising strategy for expanding the applications of time-optimal quantum controls in superconducting quantum circuits.
... For a comprehensive review on the subject, one can refer to [1], while a recent discussion on the non-uniqueness of TEUR formulation appears in [2]. In what follows, we shall basically resort to the most general Mandelstam-Tamm version of TEUR, which is based on the only assumption that the evolution of Heisenberg operators in quantum theory is ruled by the Hamiltonian H [3]. Using natural units ℏ = c = G = 1, TEUR in this framework takes the form ...
... H → 3 He +ν e ,3 He +ν e → 3 H ,(9) ...
The time–energy uncertainty relation (TEUR) plays a fundamental role in quantum mechanics, as it allows the grasping of peculiar aspects of a variety of phenomena based on very general principles and symmetries of the theory. Using the Mandelstam–Tamm method, TEUR has recently been derived for neutrino oscillations by connecting the uncertainty in neutrino energy with the
characteristic timescale of oscillations. Interestingly, the suggested interpretation of neutrinos as unstable-like particles has proved to naturally emerge in this context. Further aspects were later discussed in semiclassical gravity theory, by computing corrections to the neutrino energy uncertainty in a generic stationary curved spacetime, and in quantum field theory, where the clock observable turns out to be identified with the non-conserved flavor charge operator. In the present work, we give an overview on the above achievements. In particular, we analyze the implications of TEUR and explore the impact of gravitational and non-relativistic effects on the standard condition for neutrino oscillations.
... Variance in energy determines the precision in estimating both the time [38] and temperature [39] parameters. Both moments, when combined, provide a tight bound on the characteristic time scale of a quantum system [40][41][42]. ...
... We argued for using this method for estimating moments of energy, which have a wide range of applications while being difficult to measure directly in many-body systems. For instance, using this method, one can bound the characteristic timescale through the Mandelstam-Tamm and Margolus-Levitin bounds [40][41][42], to estimate the amount of extractable work from an unknown source of states [37], or to estimate temperature. Moreover, the latest can be used to benchmark the cooling function of quantum annealers [94][95][96][97] and adiabatic quantum computers [98]. ...
We present a method to estimate the probabilities of outcomes of a quantum observable, its mean value, and higher moments by measuring any other observable. This method is general and can be applied to any quantum system. In the case of estimating the mean energy of an isolated system, the estimate can be further improved by measuring the other observable at different times. Intuitively, this method uses interplay and correlations between the measured observable, the estimated observable, and the state of the system. We provide two bounds: one that is looser but analytically computable and one that is tighter but requires solving a nonconvex optimization problem. The method can be used to estimate expectation values and related quantities such as temperature and work in setups where performing measurements in a highly entangled basis is difficult, finding use in state-of-the-art quantum simulators. As a demonstration, we show that in Heisenberg and Ising models of ten sites in the localized phase, performing two-qubit measurements excludes 97.5% and 96.7% of the possible range of energies, respectively, when estimating the ground-state energy.
... We find that the rate of synchronization is determined by a competition between the irreversible entropy production caused by damping, which slows synchronization, and the strength of the anti-Hermitian coupling, which speeds up synchronization. The resulting upper bound on the synchronization rate has terms of the form of the Mandelstam-Tamm inequality [58], where speed scales with the uncertainty of the energy, except in this case even the uncertainties of the Hermitian and anti-Hermitian parts of the Hamiltonian are crucial. As an example, we consider a dissipatively coupled photonic dimer. ...
... This term unsurprisingly implies that stronger non-Hermitian coupling leads to faster synchronization. The second term is reminiscent of the Mandelstam-Tamm quantum speed limit [58], and involves the second moments of both the Hermitian and anti-Hermitian parts of the Hamiltonian, however there is a penalty that scales with the square of their commutator. This term arises from the uncertainty relation [96], and can be explained by the fact that synchronization is most effective when there is a large correlation between the observables corresponding to the Hermitian and anti-Hermitian parts of the Hamiltonian. ...
Quantum synchronization is crucial for understanding complex dynamics and holds potential applications in quantum computing and communication. Therefore, assessing the thermodynamic resources required for finite-time synchronization in continuous-variable systems is a critical challenge. In the present work, we find these resources to be extensive for large systems. We also bound the speed of quantum and classical synchronization in coupled damped oscillators with non-Hermitian anti-PT-symmetric interactions, and show that the speed of synchronization is limited by the interaction strength relative to the damping. Compared to the classical limit, we find that quantum synchronization is slowed by the non-commutativity of the Hermitian and anti-Hermitian terms. Our general results could be tested experimentally and we suggest an implementation in photonic systems.
... The initial discovery of QSL stems from the uncertainty relationship between conjugate variables in quantum mechanics [17,18]. The QSL time formulation depends on factors such as the shortest distance (or geodesic distance) between the initial and final states and the variance (or fluctuation) of the driving Hamiltonian [19]. ...
The quantum speed limits (QSLs) determine the minimal amount of time required for a quantum system to evolve from an initial to a final state. We investigate QSLs for the unitary evolution of the neutrino-antineutrino system in the presence of a gravitational field. It is known that the transition probabilities between neutrino and antineutrino in the framework of one and two flavors depend on the strength of the gravitational field. The behavior of the QSL time in the two-flavor system indicates fast flavor transitions as the gravitational field strength increases. Subsequently, we observe quick suppression of entanglement by exploring the speed limit for entanglement entropy of two-flavor oscillations in the neutrino-antineutrino system in the proximity of a spinning primordial black hole.
... If we consider two vector states such that one evolves into the other in a given time T , that is, for |φy, |ψy we have |ψy " exp`´i HT ℏ˘| φy for some Hamiltonian H, we could then ask, what is the smallest value of T ?, that is, what is the shortest time to transform |φy into the vector state |ψy? This is a problem that is known as quantum speed limits or simply QSL [17,18]. ...
We present a characterization of the Hamiltonians that generate optimal-speed unitary time evolution and the associated dynamical trajectory, where the initial states are either pure states or quasi-pure quantum states. We construct the manifold of pure states as an orbit under the conjugation action of the Lie group \SU(n) on the manifold of one-dimensional orthogonal projectors, obtaining an isometry with the flag manifold \SU(n)/\textnormal{S}(\textnormal{U}(1)\times \textnormal{U}(n-1 )). From this construction, we show that Hamiltonians generating optimal-speed time evolution are fully characterized by equigeodesic vectors of \SU(n)/\textnormal{S}(\textnormal{U}(1)\times \textnormal{U}(n-1)). We later extend that result to quasi-pure quantum states.
... Introduction-The laws of quantum mechanics impose a fundamental bound on the time scale for the evolution of a quantum system. Mandelstam and Tamm were the first to realise this in the context of unitary evolution [1]. They proved that this time scale is the minimum time required for the system to evolve from an initial state to some final state and gave a clear interpretation of the time-energy uncertainty relation. ...
Given the initial and final states of a quantum system, the speed of transportation of state vector in the projective Hilbert space governs the quantum speed limit. Here, we ask the question what happens to the quantum speed limit under continuous measurement process. We model the continuous measurement process by a non-Hermitian Hamiltonian which keeps the evolution of the system Schr{\"o}dinger-like even under the process of measurement. Using this specific measurement model, we prove that under continuous measurement, the speed of transportation of a quantum system tends to zero. Interestingly, we also find that for small time scale, there is an enhancement of quantum speed even if the measurement strength is finite. Our findings can have applications in quantum computing and quantum control where dynamics is governed by both unitary and measurement processes.
... The uncertainty relations are widely regarded as a fundamental aspect of modern physics, representing the uncertainty relationship between conjugate variables in a quantum mechanical system [1,2]. Mandelstam and Tamm (MT) demonstrated that the energy, time uncertainty relation can be understood as imposing limits on the evolution rate of a quantum system [3]. Another bound on the quantum evolution time was subsequently obtained by Margolus and Levitin (ML) in terms of the initial mean energy of the quantum system [4]. ...
The quantum speed limits (QSLs) set fundamental lower bounds on the time required for a quantum system to evolve from a given initial state to a final state. In this work, we investigate CP violation and the mass hierarchy problem of neutrino oscillations in matter using the QSL time as a key analytical tool. We examine the QSL time for the unitary evolution of two- and three-flavor neutrino states, both in vacuum and in the presence of matter. Two-flavor neutrino oscillations are used as a precursor to their three-flavor counterparts. We further compute the QSL time for neutrino state evolution and entanglement in terms of neutrino survival and oscillation probabilities, which are experimentally measurable quantities in neutrino experiments. A difference in the QSL time between the normal and inverted mass hierarchy scenarios, for neutrino state evolution as well as for entanglement, under the effect of a CP violation phase is observed. Our results are illustrated using energy-varying sets of accelerator neutrino sources from experiments such as T2K, NOvA, and DUNE. Notably, three-flavor neutrino oscillations in constant matter density exhibit faster state evolution across all these neutrino experiments in the normal mass hierarchy scenario. Additionally, we observe fast entanglement growth in DUNE assuming a normal mass hierarchy.
... Sen et al. [32] explained microscopic phenomena that cannot be differentiated discontinuously over time in an environment involving uncertainty using a quantum-mechanicsbased SDE. Oh and Lee [33] proposed an optimization study that more efficiently generated routes in ACO by predicting the probability-based drift of pheromones through quantum mechanics, while Mandelstam and Tamm [34] conducted research that considered the uncertainty relationship between energy and discontinuous time in non-relativistic quantum mechanics. Xiao et al. [35] applied a combination of complex evidence theory and quantum mechanics to address uncertainty in decision-making problems and predicted the effects of interference on decision-making behavior. ...
With the development of automation technology, big data is collected during operation processes, and among various machine learning analysis techniques using such data, deep neural network (DNN) has high analysis performance. However, most industrial data has low-variance or near-zero variance data from the refined processes in the collected data itself. This reduces deep learning analysis performance, which is affected by data quality. To overcome this, in this study, the weight learning pattern of an applied DNN is modeled as a stochastic differential equation (SDE) based on quantum mechanics. Through the drift and diffuse terms of quantum mechanics, the patterns of the DNN and data are quickly acquired, and the data with near-zero variance is effectively analyzed simultaneously. To demonstrate the superiority of the proposed framework, DNN analysis was performed using data with near-zero variance issues, and it was proved that the proposed framework is effective in processing near-zero variance data compared with other existing algorithms.
Graphical abstract
... On general grounds, QSL is defined as the minimum time to evolve from one quantum state to another [87,88]. Mandelstam and Tamm proposed a first approach based on the time-energy uncertainty relation, suggesting that the minimum time is related to the standard deviation of energy [89]. Margolus and Levitin later improved the QSL of Mandelstam and Tamm to make it tighter [90]. ...
We introduce a quantum charging distance as the minimal time that it takes to reach one state (charged state) from another state (depleted state) via a unitary evolution, assuming limits on the resources invested into the driving Hamiltonian. For pure states it is equal to the Bures angle, while for mixed states its computation leads to an optimization problem. Thus, we also derive easily computable bounds on this quantity. The charging distance tightens the known bound on the mean charging power of a quantum battery, it quantifies the quantum charging advantage, and it leads to an always achievable quantum speed limit. In contrast with other similar quantities, the charging distance does not depend on the eigenvalues of the density matrix, it depends only on the corresponding eigenspaces. This research formalizes and interprets quantum charging in a geometric way, and provides a measurable quantity that one can optimize to maximize the speed of charging of future quantum batteries.
... For a comprehensive review on the subject, one can refer to [1], while a recent discussion on the non-uniqueness of TEUR formulation appears in [2]. In what follows, we shall basically resort to the most general Mandelstam-Tamm version of TEUR, which is based on the only assumption that the evolution of Heisenberg operators in quantum theory is ruled by the Hamiltonian H [3]. Using natural units = c = G = 1, TEUR in this framework takes the form ...
Time-energy uncertainty relation (TEUR) plays a fundamental role in quantum mechanics, as it allows to grasp peculiar aspects of a variety of phenomena based on very general principles and symmetries of the theory. Using the Mandelstam-Tamm method, TEUR has been recently derived for neutrino oscillations by connecting the uncertainty on neutrino energy with the characteristic timescale of oscillations. Interestingly enough, the suggestive interpretation of neutrinos as unstable-like particles has proved to naturally emerge in this context. Further aspects have been later discussed in semiclassical gravity by computing corrections to the neutrino energy uncertainty in a generic stationary curved spacetime, and in quantum field theory, where the clock observable turns out to be identified with the non-conserved flavor charge operator. In the present work, we give an overview on the above achievements. In particular, we analyze the implications of TEUR and explore the impact of gravitational and non-relativistic effects on the standard condition for neutrino oscillations. Correlations with the quantum-information theoretic analysis of oscillations and possible experimental consequences are qualitatively discussed.
... On general grounds, QSL is defined as the minimum time to evolve from one quantum state to another [79,80]. Mandelstam and Tamm proposed a first approach based on the time-energy uncertainty relation, suggesting that the minimum time is related to the standard deviation of energy [81]. Margolus and Levitin later improved the QSL of Mandelstam and Tamm to make it tighter [82]. ...
We introduce a quantum charging distance as the minimal time that it takes to reach one state (charged state) from another state (depleted state) via a unitary evolution, assuming limits on the resources invested into the charging. We show that for pure states it is equal to the Bures angle, while for mixed states, its computation leads to an optimization problem. Thus, we also derive easily computable bounds on this quantity. The charging distance tightens the known bound on the mean charging power of a quantum battery, it quantifies the quantum charging advantage, and it leads to an always achievable quantum speed limit. In contrast with other similar quantities, the charging distance does not depend on the eigenvalues of the density matrix, it depends only on the corresponding eigenspaces. This research formalizes and interprets quantum charging in a geometric way, and provides a measurable quantity that one can optimize for to maximize the speed of charging of future quantum batteries.
... These are calculated from (Deffner and Lutz, 2013). Originally derived as two different bounds by Mandelstam and Tamm (1945) (⟨∆E⟩), and Margolus and Levitin (1998) (⟨E⟩), they have been recently unified into the bound of Eq. (12) by Levitin and Toffoli (2009). The QSL for the evolution of pure states is tight, and the bound in Eq. (12) is attainable, in the absence of restrictions on the interaction Hamiltonian H 1 (t) of Eq (2). ...
Recent years have witnessed an explosion of interest in quantum devices for the production, storage, and transfer of energy. In this Colloquium, we concentrate on the field of quantum energy storage by reviewing recent theoretical and experimental progress in quantum batteries. We first provide a theoretical background discussing the advantages that quantum batteries offer with respect to their classical analogues. We then review the existing quantum many-body battery models and present a thorough discussion of important issues related to their open nature. We finally conclude by discussing promising experimental implementations, preliminary results available in the literature, and perspectives.
In this paper we introduce and investigate the concept of a perfect quantum protractor , a pure quantum state | ψ ⟩ ∈ H that generates three different orthogonal bases of H under rotations around each of the three perpendicular axes. Such states can be understood as pure states of maximal uncertainty with regards to the three components of the angular momentum operator, as we prove that they maximise various entropic and variance-based measures of such uncertainty. We argue that perfect quantum protractors can only exist for systems with a well-defined total angular momentum j , and we prove that they do not exist for j ∈ { 1 / 2 , 2 , 5 / 2 } , but they do exist for j ∈ { 1 , 3 / 2 , 3 } (with numerical evidence for their existence when j = 7 / 2 ). We also explain that perfect quantum protractors form an optimal resource for a metrological task of estimating the angle of rotation around (or the strength of magnetic field along) one of the three perpendicular axes, when the axis is not a priori known. Finally, we demonstrate this metrological utility by performing an experiment with warm atomic vapours of rubidium-87, where we prepare a perfect quantum protractor for a spin-1 system, let it precess around x , y or z axis, and then employ it to optimally estimate the rotation angle.
How fast an observable can evolve in time is answered by so-called observable speed limit. Here, we prove a stronger version of the observable speed limit and show that the previously obtained bound is a special case of the new bound. The stronger quantum speed limit for the state also follows from the stronger quantum speed limit for observables (SQSLO). We apply this to prove a stronger bound for the entanglement rate using the notion of capacity of entanglement (the quantum information theoretic counterpart of the heat capacity) and show that it outperforms previous bounds. Furthermore, we apply the SQSLO for the rate of modular Hamiltonian and in the context of interacting qubits in a quantum battery. These illustrative examples reveal that the speed limit for the modular energy and the time required to charge the battery can be exactly predicted using the new bound. This shows that for estimating the charging time of quantum battery SQSLO is actually tight, i.e., it saturates. Our findings can have important applications in quantum thermodynamics, the complexity of operator growth, predicting the time rate of quantum correlation growth, and quantum technology, in general.
Time continues to be an intriguing physical property in the modern era. On the one hand, we have the classical and relativistic notion of time, where space and time have the same hierarchy, essential in describing events in spacetime. On the other hand, in quantum mechanics, time appears as a classical parameter, meaning that it does not have an uncertain relation with its canonical conjugate. In this work we use a recent spacetime-symmetric proposal [Phys. Rev. A 95, 032133 (2017)] that tries to solve the unbalance in nonrelativistic quantum mechanics by extending the usual Hilbert space: the time parameter t and the position operator X̂ in one subspace, and the position parameter x and time operator T in the other subspace. Time as an operator is better suited for describing tunneling processes. We then solve the 1/2-fractional integrodifferential equation for a particle subjected to strong and weak potential limits and obtain an analytical expression for the tunneling time through a rectangular barrier. Using a Gaussian energy distribution, we demonstrate that for wave packets well resolved in time, the expectation value of the operator T is the energy average of the classical time Tclass=∂S/∂E, where S is the classical action, which can be real or imaginary. The imaginary classical time does not contribute to the traveling time. Furthermore, we apply our results to a Gaussian energy distribution and compare them to previous works. This work is a correction of a previous paper [Phys. Rev. A 107, 052220 (2023)].
Recent years have witnessed an explosion of interest in quantum devices for the production, storage, and transfer of energy. This Colloquium concentrates on the field of quantum energy storage by reviewing recent theoretical and experimental progress in quantum batteries. Provided first is a theoretical background discussing the advantages that quantum batteries offer with respect to their classical analogs. The existing quantum many-body battery models are then reviewed and a thorough discussion of important issues related to their “open nature” is presented. The Colloquium concludes with a discussion of promising experimental implementations, preliminary results available in the literature, and perspectives.
Research in quantum information science aims to surpass the scaling limitations of classical information processing. From a physicist's perspective, performance improvement involves a physical speedup in the quantum domain, achieved by dynamically exploiting quantum correlations. In this study, speed limits in interacting quantum systems are derived by comparing the rates of change in actual quantum dynamics with the quasiclassical evolution confined to the manifold of nonentangled separable states. The utility of the resulting bounds on entanglement-assisted speedup is demonstrated on bipartite qubit systems, bipartite qudit systems, as well as a complex multimode systems. Specifically, the proposed speed limits provide a tight bound on quantum speed advantage, including a quantum gain that can scale exponentially with the system's size. Extensions of the results to open systems and measurable witnesses are discussed.
The quantum speed limit (QSL) is a fundamental lower bound on the evolution time for quantum systems, and its tightness has been observed to be dependent on the properties of the physical process. However, experimental studies exploring the QSL in open quantum systems are still missing. Here, we studied geometric quantum speed limits of a qubit subject to decoherence in an ensemble of chloroform molecules in a Nuclear Magnetic Resonance experiment. We controlled the system-reservoir interaction and the spin relaxation rates by adding a paramagnetic salt, allowing the observation of both Markovian and non-Markovian open system dynamics for the qubit. We used two distinguishability measures of quantum states to assess the speed of the qubit evolution: the quantum Fisher information (QFI) and Wigner-Yanase skew information (WY). For non-Markovianity and low salt concentrations, we found crossovers between QSLs related to those metrics. The WY metric sets the tighter QSL for high concentrations and Markovian dynamics. We also show that QSLs are sensitive even to small fluctuations in spin magnetization.
The quantum dynamical activity constitutes a thermodynamic cost in trade-off relations such as the quantum speed limit and the quantum thermodynamic uncertainty relation. However, calculating the quantum dynamical activity has been a challenge. In this paper, we present the exact solution for the quantum dynamical activity by deploying the continuous matrix product state method. Moreover, using the derived exact solution, we determine the upper bound of the dynamical activity, which comprises the standard deviation of the system Hamiltonian and jump operators. We confirm the exact solution and the upper bound by performing numerical simulations.
A quantum measurement involves energy exchanges between the system to be measured and the measuring apparatus. Some of them involve energy losses, for example because energy is dissipated into the environment or is spent in recording the measurement outcome. Moreover, these processes take time. For this reason, these exchanges must be taken into account in the analysis of a quantum measurement engine, and set limits to its efficiency and power. We propose a quantum engine based on a spin 1/2 particle in a magnetic field and study its limitations due to the quantum nature of the evolution. The coupling with the electromagnetic vacuum is taken into account and plays the role of a measurement apparatus. We fully study its dynamics, work, power, and efficiency.
Quantum reference frames have attracted renewed interest recently, as their exploration is relevant and instructive in many areas of quantum theory. Among the different types, position and time reference frames have captivated special attention. Here, we introduce and analyze a nonrelativistic framework in which each system contains an internal clock in addition to its external (spatial) degree of freedom and, hence, can be used as a spatiotemporal quantum reference frame. We present expressions for expectation values and variances of relevant observables in different perspectives, as well as relations between these quantities in different perspectives in scenarios with no interactions. In particular, we show that even in these simple scenarios the relative uncertainty between clocks affects the relative spatial spread of the systems.
We are investigating the quantum geometric speed limit (QSL) for a Hubbard-type fermionic dimer subjected to the effects of decoherence and driving. The influence of decoherence is explored through the utilization of three distinct models that describe weak connections between the dimer and its environment, employing the Davies approximation. We consider two different types of driving: adiabatic driving and white noise stochastic driving. Our findings demonstrate that the properties of the geometric QSL are affected by both the on-site and inter-site Coulomb interactions among the fermions in the dimer. The interplay between Coulomb interactions and the influence or control exerted by external classical or quantum noise results in distinct time regimes for the evolution of the dimer, characterized by specific and potentially desirable QSL properties.
We uncover a novel dynamical quantum phase transition, using random matrix theory and its associated notion of planar limit. We study it for the isotropic XY Heisenberg spin chain. For this, we probe its real-time dynamics through the Loschmidt echo. This leads to the study of a random matrix ensemble with a complex weight, whose analysis requires novel technical considerations, that we develop. We obtain three main results: 1) There is a third order phase transition at a rescaled critical time, that we determine. 2) The third order phase transition persists away from the thermodynamic limit. 3) For times below the critical value, the difference between the thermodynamic limit and a finite chain decreases exponentially with the system size. All these results depend in a rich manner on the parity of the number of flipped spins of the quantum state conforming the fidelity.
Trade-off relations place fundamental limits on the operations that physical systems can perform. This Letter presents a trade-off relation that bounds the correlation function, which measures the relationship between a system’s current and future states, in Markov processes. The obtained bound, referred to as the thermodynamic correlation inequality, states that the change in the correlation function has an upper bound comprising the dynamical activity, a thermodynamic measure of the activity of a Markov process. Moreover, by applying the obtained relation to the linear response function, it is demonstrated that the effect of perturbation can be bounded from above by the dynamical activity.
Time continues to be an intriguing physical property in the modern era. On the one hand, we have the classical and relativistic notions of time, where space and time have the same hierarchy, which is essential in describing events in spacetime. On the other hand, in quantum mechanics time appears as a classical parameter, meaning that it does not have an uncertainty relation with its canonical conjugate. In this work, we use a recent spacetime-symmetric proposal [Phys. Rev. A 95, 032133 (2017)] that tries to solve the unbalance in nonrelativistic quantum mechanics by extending the usual Hilbert space, having the time parameter t and the position operator X̂ in one subspace and the position parameter x and time operator T in the other subspace. Time as an operator is better suitable for describing tunneling processes. We then solve the 1/2-fractional integrodifferential equation for a particle subjected to strong and weak potential limits and obtain an analytical expression for the tunneling time through a rectangular barrier. Using a Gaussian energy distribution, we demonstrate that, for wavepackets well resolved in time, the expectation value of the operator T is the energy average of the classical time Tclass=∂S/∂E, where S is the classical action, which can be real or imaginary. For wavepackets not well resolved in time, the contribution of Tclass consistently vanishes, and solely properties of the energy distribution contribute to T. We show that the time of travel for nontunneling particles is purely real. When tunneling is involved, complex arrival times emerge, becoming a signature of tunneling. Furthermore, we apply our results to a constant energy distribution, obtaining pure imaginary times for energies below the barrier while obtaining complex times for particles with a wavepacket spreading energies below and above the barrier, and show a comparison to previous works.
Many-particle quantum systems with intermediate anyonic exchange statistics are supported in one spatial dimension. In this context, the anyon-anyon mapping is recast as a continuous transformation that generates shifts of the statistical parameter κ . We characterize the geometry of quantum states associated with different values of κ , i.e., different quantum statistics. While states in the bosonic and fermionic subspaces are always orthogonal, overlaps between anyonic states are generally finite and exhibit a universal form of the orthogonality catastrophe governed by a fundamental statistical factor, independent of the microscopic Hamiltonian. We characterize this decay using quantum speed limits on the flow of κ , illustrate our results with a model of hard-core anyons, and discuss possible experiments in quantum simulation.
Gravitational redshift can cause the distortion of photon wave packets; thus, the non-Markovian speedup evolution of a photon system must be controlled by considering gravitational redshift. We analyze the non-Markovian speedup dynamics of a photon system in the amplitude-damping channel or pure-dephasing channel under the effect of gravitational redshift, which is modelled as a beam-splitter operation. We show that the gravitational redshift weakens the non-Markovian dynamical behavior in both channels. For the amplitude-damping channel, the gravitational redshift promotes further speedup evolution of a photon system when the initial coherence of the system is nonzero. A stronger gravitational redshift and larger initial coherence make improving the evolution speed of the photon system easier. For the pure-dephasing channel, the non-Markovian dynamical behavior decreases owing to the gravitational redshift; however, the evolution speed of the photon is not influenced by the gravitational redshift.
Detection of very weak forces and precise measurement of time are two of the many applications of quantum metrology to science and technology. To sense an unknown physical parameter, one prepares an initial state of a probe system, allows the probe to evolve as governed by a Hamiltonian H for some time t, and then measures the probe. If H is known, we can estimate t by this method; if t is known, we can estimate classical parameters on which H depends. The accuracy of a quantum sensor can be limited by either intrinsic quantum noise or by noise arising from the interactions of the probe with its environment. In this work, we introduce and study a fundamental trade-off, which relates the amount by which noise reduces the accuracy of a quantum clock to the amount of information about the energy of the clock that leaks to the environment. Specifically, we consider an idealized scenario in which a party Alice prepares an initial pure state of the clock, allows the clock to evolve for a time that is not precisely known, and then transmits the clock through a noisy channel to a party Bob. Meanwhile, the environment (Eve) receives any information about the clock that is lost during transmission. We prove that Bob’s loss of quantum Fisher information about the elapsed time is equal to Eve’s gain of quantum Fisher information about a complementary energy parameter. We also prove a similar, but more general, trade-off that applies when Bob and Eve wish to estimate the values of parameters associated with two noncommuting observables. We derive the necessary and sufficient conditions for the accuracy of the clock to be unaffected by the noise, which form a subset of the Knill-Laflamme error-correction conditions. A state and its local time-evolution direction, if they satisfy these conditions, are said to form a metrological code. We provide a scheme to construct metrological codes in the stabilizer formalism. We show that there are metrological codes that cannot be written as a quantum error-correcting code with similar distance in which the Hamiltonian acts as a logical operator, potentially offering new schemes for constructing states that do not lose any sensitivity upon application of a noisy channel. We discuss applications of the trade-off relation to sensing using a quantum many-body probe subject to erasure or amplitude-damping noise.
We investigate a symmetric logarithmic derivative (SLD) Fisher information for kinetic uncertainty relations (KURs) of open quantum systems described by the GKSL quantum master equation with and without the detailed balance condition. In a quantum kinetic uncertainty relation derived by Vu and Saito [Phys. Rev. Lett. 128, 140602 (2022)], the Fisher information of probability of quantum trajectory with a time-rescaling parameter plays an essential role. This Fisher information is upper bounded by the SLD Fisher information. For a finite time and arbitrary initial state, we derive a concise expression of the SLD Fisher information, which is a double time integral and can be calculated by solving coupled first-order differential equations. We also derive a simple lower bound of the Fisher information of quantum trajectory. We point out that the SLD Fisher information also appears in the speed limit based on the Mandelstam-Tamm relation by Hasegawa [Nat. Commun. 14, 2828 (2023)]. When the jump operators connect eigenstates of the system Hamiltonian, we show that the Bures angle in the interaction picture is upper bounded by the square root of the dynamical activity at short times, which contrasts with the classical counterpart.
We investigate bounds on speed, nonadiabatic entropy production, and the trade-off relation between them for classical stochastic processes with time-independent transition rates. Our results show that the time required to evolve from an initial to a desired target state is bounded from below by the information-theoretical ∞-Rényi divergence between these states, divided by the total rate. Furthermore, we conjecture and provide extensive numerical evidence for an information-theoretical bound on the nonadiabatic entropy production and a dissipation-time trade-off relation that outperforms previous bounds in some cases..
A fundamental problem in quantum engineering is determining the lowest time required to ensure that all possible unitaries can be generated with the tools available, which is one of a number of possible quantum speed limits. We examine this problem from the perspective of quantum control, where the system of interest is described by a drift Hamiltonian and set of control Hamiltonians. Our approach uses a combination of Lie algebra theory, Lie groups, and differential geometry and formulates the problem in terms of geodesics on a differentiable manifold. We provide explicit lower bounds on the quantum speed limit for the case of an arbitrary drift, requiring only that the control Hamiltonians generate a topologically closed subgroup of the full unitary group, and formulate criteria as to when our expression for the speed limit is exact and not merely a lower bound. These analytic results are then tested and confirmed using a numerical optimization scheme. Finally, we extend the analysis to find a lower bound on the quantum speed limit in the common case where the system is described by a drift Hamiltonian and a single control Hamiltonian.
The second law of thermodynamics states that entropy production cannot be negative. Recent developments concerning uncertainty relations in stochastic thermodynamics, such as thermodynamic uncertainty relations and speed limits, have yielded refined second laws that provide lower bounds of entropy production by incorporating information from current statistics or distributions. In contrast, in this study we bound the entropy production from above by terms comprising the dynamical activity and maximum transition-rate ratio. We derive two upper bounds: One applies to steady-state conditions, whereas the other applies to arbitrary time-dependent conditions. We verify these bounds through numerical simulation and identify several potential applications.
We present a numerical method to simulate the dynamics of continuous-variable quantum many-body systems. Our approach is based on custom neural-network many-body quantum states. We focus on dynamics of two-dimensional quantum rotors and simulate large experimentally relevant system sizes by representing a trial state in a continuous basis and using state-of-the-art sampling approaches based on Hamiltonian Monte Carlo. We demonstrate the method can access quantities like the return probability and vorticity oscillations after a quantum quench in two-dimensional systems of up to 64 (8×8) coupled rotors. Our approach can be used for accurate nonequilibrium simulations of continuous systems at previously unexplored system sizes and evolution times, bridging the gap between simulation and experiment.
We study the probability distribution function of the long-time values of observables being time-evolved by Hamiltonians modeling clean and disordered one-dimensional chains of many spin-1/2 particles. In particular, we analyze the return probability and its version for a completely extended initial state, the so-called spectral form factor. We complement our analysis with the spin autocorrelation and connected spin-spin correlation functions, both of interest in experiments with quantum simulators. We show that the distribution function has a universal shape provided the central limit theorem holds. Explicitly, the shape is exponential for the return probability and spectral form factor, meanwhile it is Gaussian for the few-body observables. We also discuss implications over the so-called many-body localization. Remarkably, our approach requires only a single sample of the dynamics and small system sizes, which could be quite advantageous when dealing specially with disordered systems.
We introduce a simple framework for estimating lower bounds on the runtime of a broad class of adiabatic quantum algorithms. The central formula consists of calculating the variance of the final Hamiltonian with respect to the initial state. After examining adiabatic versions of certain keystone circuit-based quantum algorithms, this technique is applied to adiabatic quantum algorithms with undetermined speedup. In particular, we analytically obtain lower bounds on adiabatic algorithms for finding k-clique in random graphs. Additionally, for a particular class of Hamiltonian, it is straightforward to prove the equivalence between our framework and the conventional approach based on spectral gap analysis.
Starting with the quantum brachistochrone problem of the infinitesimal form, we solve the minimal time and corresponding time-dependent Hamiltonian to drive a pure quantum state with limited resources along arbitrary preassigned trajectories. It is also shown that out of all possible trajectories with limited resources, which are physically accessible and which are not. The solution is then generalized to the mixed quantum state cases and applied to trajectories parametrized by single or multiple parameters with a discrete or continuous spectrum. We then compare the solution to that of the counterdiabatic driving and show how the Berry phase is directly involved in both driving processes.
Markovian open quantum systems display complicated relaxation dynamics. The spectral gap of the Liouvillian characterizes the asymptotic decay rate toward the steady state, but it does not necessarily give a correct estimate of the relaxation time because the crossover time to the asymptotic regime may be too long. We here give a rigorous upper bound on the transient decay of autocorrelation functions in the steady state by introducing the symmetrized Liouvillian gap. The standard Liouvillian gap and the symmetrized one are identical in an equilibrium situation but differ from each other in the absence of the detailed balance condition. It is numerically shown that the symmetrized Liouvillian gap always gives a correct upper bound on the decay of the autocorrelation function, but the standard Liouvillian gap does not.
Quantum speed limit is bound on the minimum time a quantum system requires to evolve from an initial state to final state under a given dynamical process. It sheds light on how fast a desired state transformation can take place, which is pertinent for design and control of quantum technologies. In this paper, we derive speed limits on correlations such as entanglement, Bell-CHSH correlation, and quantum mutual information of quantum systems evolving under dynamical processes. Our main result is a speed limit on an entanglement monotone called negativity, which holds for arbitrary-dimensional bipartite quantum systems and processes. Another entanglement monotone which we consider is the concurrence. To illustrate the efficacy of our speed limits, we analytically and numerically compute the speed limits on the negativity, concurrence, and Bell-CHSH correlation for various quantum processes of practical interest. We are able to show that, for practical examples we have considered, some of the speed limits we derived are actually attainable and hence these bounds can be considered to be tight.
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