Quantum speed limits on operator flows and correlation functions

Quantum speed limits (QSLs) identify fundamental time scales of physical processes by providing lower bounds on the rate of change of a quantum state or the expectation value of an observable. We introduce a generalization of QSL for unitary operator flows, which are ubiquitous in physics and relevant for applications in both the quantum and classical domains. We derive two types of QSLs and assess the existence of a crossover between them, that we illustrate with a qubit and a random matrix Hamiltonian, as canonical examples. We further apply our results to the time evolution of autocorrelation functions, obtaining computable constraints on the linear dynamical response of quantum systems out of equilibrium and the quantum Fisher information governing the precision in quantum parameter estimation.


Introduction
Unraveling the fundamental time scale of a physical process is crucial in many theoretical and experimental scenarios. Quantum speed limits (QSLs) constitute a set of fundamental results in quantum physics that bound the minimum time for a physical process to happen. Although their initial formulation [1,2] was restricted to unitary evolution between two pure quantum states, by now they have been generalized to the case of arbitrary mixed states [3,4], driven Hamiltonians [5,3,6,7], open quantum dynamics [8,9,10,11], continuous quantum measurements [12], and classical processes [13,14,15]. Their applications are thus manifold and range over various branches of physics [16]. Their use is promi-nent in quantum technologies, including quantum computation [17,18] and quantum metrology [19,20]. QSL are known to limit the performance of quantum control algorithms [21,22], and play a key role in shortcuts to adiabaticity by counterdiabatic driving both in isolated [23,24] and open quantum systems [25,26]. They have also been applied to many-body physics, in which care is needed given the orthogonality catastrophe [27,28,29,30,31]. In addition, some formulations of QSLs have transcended the notion of quantum state distinguishability and focused on the rate of change of other quantities such as quantum coherence [32,33,34], and more generally, quantum resources [35].
Bounds to the pace of evolution naturally have important implications in nonequilibrium thermodynamics. In this context, the connection between QSLs and thermodynamic uncertainty relations (TURs) [36,37] has been explored in [38,39], identifying QSL-inspired constraints on non-equilibrium fluctuations. Recently, a closely related work [40], following the initial spirit of [1], established a generalized QSL on the evolution of observables, in particular, of their expectation values, under arbitrary dynamics. The notion of the speed of an observable expectation value was also considered in [30,31,41].
A natural question then arises: is it possible to formulate QSLs directly on the operator flow, rather than on the time-dependent expectation value of an observable? An operator flow describes the evolution of an operator under some given dynamics. In the unitary case, the evolution of the operator results from its conjugation by a unitary. Unitary operator flows are ubiquitous in physics, ranging from the quantum evolution of an observable in the Heisenberg picture to the Lax pair flow in classical integrable systems [42] and continuous renormal-ization group flows, such as the one proposed by Wegner [43]. Preceding efforts in this direction have focused on the preparation of unitary [44] and non-unitary [45] quantum operations in the context of quantum control. Moreover, a Liouville space formulation of the QSL on the evolution of density matrices was introduced in [46]. However, a framework to describe general operator flows, for general operators, is still lacking. Constraining the speed of these flows would naturally prove useful also in characterizing operator growth [47,48,49,50,51], which has attracted increasing attention as a powerful approach to describe the buildup of complexity in quantum systems, in parallel to the earlier notion of quantum state complexity [52], that is also subject to QSLs [53,54,55,56]. In this context, a QSL-like uncertainty bound has recently been introduced [57] on the growth of a particular notion of operator complexity, known as Krylov complexity [58,59,60,61].
In this manuscript, we introduce two notions of QSL for operator flows. Although we focus on the unitary time evolution in the Heisenberg picture, our results can be applied to a wider class of unitary flows, including some examples of inhomogeneous flows. Moreover, they do not require the flowing operator to be Hermitian. In addition, we introduce QSLs for the two-point autocorrelation functions, which play a key role in many-body physics, determine the linear response [62] and transport properties of a quantum system, and characterize the operator growth in Krylov space. The exact computation of correlation functions is generally a challenging task, requiring solving the dynamics, while our bounds are easy to compute provided that the Hamiltonian and the initial operator are known. Our results show the presence of a universal crossover between an initial regime in which the dynamics follows a Mandelstam-Tamm (MT) [1] type of QSL and a second one in which a Margolus-Levitin (ML) [2] type of QSL yields a more accurate description. We illustrate this crossover, analogous to that recently observed experimentally for quantum state evolution [63], in the case of a two-level system and of a random matrix Hamiltonian. Furthermore, our results provide easily computable constraints on the non-equilibrium dynamical response of arbitrary systems to an external perturbation within linear response theory. Finally, exploiting the re-lation between the dynamical susceptibility and the quantum Fisher information [64], we upper bound this quantity with an easily computable correlation function.

Quantum speed limits for operators
The paradigmatic example of operator flow in quantum mechanics is the is the unitary time-evolution operator associated with a time-independent Hamiltonian H. It is convenient to visualize this flow upon vectorizing operators in Liouville space [46], as it allows a direct analogy between operator flow and quantum state evolution. Therefore, let us represent a general bounded operator A = i,j A ij |i j|, where {|i } can be any basis of the Hilbert space, as a normalized vector where A = A, A is the Hilbert-Schmidt norm associated with the Hilbert-Schmidt inner product A, B = Tr[A † B]. In Liouville space, the latter is proportional to the standard scalar product over C d 2 , being A, B = A B A|B . In the isomorphic real vector space R 2d 2 , the real part Re A, B defines the standard Euclidean inner product. Let us stress that, despite the formal similarity, vectorized operators and physical states are fundamentally different. A quantum state |ψ , being a ray in the projective Hilbert space, is defined up to a global phase, i.e., exp(iθ)|ψ with θ ∈ R is equivalent to |ψ . By contrast, vectorized operators differing by a global phase are not equivalent. We further remark that when the operator is Hermitian, its evolution in Liouville space is equivalent to the quantum evolution of the corresponding physical observable in the Heisenberg picture.
Let us now consider the Heisenberg evolution of an operator O 0 generated by a time-independent Hamiltonian H. For the moment, we do not restrict ourselves to observables, meaning that O 0 can also be non-Hermitian. In Liouville space, we can rewrite the Heisenberg equation as ∂ t |O t = iL |O t , where the Liouvillian, which, with a slight abuse of notation, we shall continue to indicate as L, takes the form A particularly useful object to quantify the displacement of the operator along the flow is the operator overlap which can be complex in general and becomes real when Hermitian operators are considered.
A key observation is that the operator overlap is proportional to the infinite temperature autocorrelation function: we shall see how this feature will allow us to extend our result to finitetemperature autocorrelations functions as well.
Note that O 0 |O t is closely related to the notion of operator fidelity introduced in the context of Loschmidt echoes and quantum phase transitions [65]. Similarly to the fidelity of quantum states, the operator overlap (3) is potentially affected by the orthogonality catastrophe in many-body systems [27,28,29,30,31], which would render it asymptotically small for large system sizes. Nevertheless, as shown in the random matrix example below, we can provide non-trivial bounds on the correlation functions. To measure how "far" the operator O t has flowed, we define the operator angle L t between the vector |O t and the inital one |O 0 : This quantity defines a distance over the uni- Being the vectorized operator normalized as in Eq. (1), the unitary flow in Liouville space lies on the unit sphere. Since Re O 0 |O t is the Euclidean inner product, Eq. (4) reduces to the angle between the corresponding real vectors in R 2d 2 and is, therefore, a distance. We also note that L t resembles the notion of Bures angle = arccos | ψ 0 |ψ t |, but it does not reduce to it when the initial operator is chosen to be the projector over the pure state |ψ 0 , O 0 = |ψ 0 ψ 0 |. Moreover, let us stress that arccos | O 0 |O t | would not be a good notion of distance between operators, as it would vanish every time they differ only by a phase. While physical states are defined up to an irrelevant global phase, the same is not true for operators, as global phases between observable are physical. Said differently, the fact that Liouville space is a Hilbert space but not a projective Hilbert space favors the use of the distance (4) over the conventional Bures angle. Now, let {|i } be the energy eigenbasis, such that H |i = E i |i , and let the initial operator be O 0 = ij O ij |i j| in such basis. The Liouvillian is diagonal in the vectorized basis |i ⊗|j and has the energy gaps ∆ ij ≡ E i −E j as diagonal entries. The operator overlap takes the form Using trigonometric inequalities, we can derive two QSLs in terms of the operator overlap, which, as we will argue below, can be regarded as the generalization of the where, for an arbitrary operator A, we define |A| = √ A † A and the brackets in the right-hand side stand for the expectation value over the vectorized operator: · = O 0 | · |O 0 . Further, its time derivative Re O 0 |Ȯ t can be upper bounded by making use of the inequality −x 2 ≤ x sin x ≤ x 2 , valid ∀x: The derivative of the operator overlap is related to that of the operator angle L t by which, after time integration, when combined with Eq. (7), yields Therefore, from Eqs. (6) and (9) we obtain the two following QSLs, formulated in terms either of the operator overlap or the operator angle: These results identify |L| and L 2 as upper bounds on the speeds of the operator flow. We stress that the quantities |L| and L 2 are time-independent under unitary dynamics when the evolution is generated by a time-independent Hamiltonian. Moreover, we note that if O 0 is Hermitian, the operator overlap is real at any time and L = 0 by parity so that L 2 = (∆L) 2 is the variance of the Liouvillian. The analogy with ML and MT bounds, at least for flows of observables, is already evident. Both for state and operator evolution, the relevant time scale is given in terms of the mean and the variance of the generator of evolution, for ML and MT bounds, respectively. In the case of states, the generator is the Hamiltonian H, while in Liouville space the dynamics of operator flows is generated by the Liouvillian L. We observe that, in the case of operator evolution, the ML bound (10) is given in terms of |L| rather than the mean of the generator L , as this one vanishes for Hermitian operators. Let us further note that, by using other trigonometric inequalities, one can derive analogous QSLs that are proportional to Eqs. (10)-(11) through a numerical constant smaller than one, thus yielding a weaker result, see App. A. It is instructive to identify which operators maximize the upper bounds |L| and L 2 on the speed of the flow for a given Hamiltonian, as these will undergo the fastest operator growth. By looking at Eqs. (7) and (6), and noting that (|O ij |/ O ) 2 ≤ 1 defines a proper probability distribution over the energy states pairs, it is clear that the operators O max flowing at the maximal speed are the ones whose non-zero elements are only between energy eigenstates with the maximum gap |∆ max | = E max − E 0 , where E max and E 0 are the highest and the lowest energy eigenvalues, respectively. This maximal speed for operators is the analog of the one identified by the dual ML bound, recently introduced for state evolution [67]. If these levels are non-degenerate, the fastest operator will be of the form O max = µ |E max E 0 | + ν |E 0 E max | for some complex constant µ and ν (µ = ν * for observables). This is analogous to the well-known result for states [68].
Remarkably, the above QSLs identify a universal crossover between two different time regimes. At early times t ≤ τ c , being τ c = 2α |L| / L 2 the crossover time, the decay of the operator overlap is governed by a quadratic MT bound while for times t ≥ τ c the linear ML bound becomes tighter. A similar crossover was experimentally observed for the state fidelity of a single atom in an optical trap [63]. Furthermore, let us note that the QSLs (10) and (11) can be recast in terms of the Hamiltonian H, which generates the time evolution in the Hilbert space, rather than the Liouvillian, i.e., the generator of evolution in Liouville space. This reformulation will prove advantageous in expressing the relevant timescales as thermal expectation values. Indeed, L 2 is proportional to the norm of the operator velocity which, as emphasized, is time-independent under unitary dynamics. Regarding the ML QSL, by using that |∆ jk | ≤ E j + E k − 2E 0 in terms of the ground state energy E 0 , we find Therefore, the ML QSL can be recast as which is generally weaker than the original Liouvillian bound (10). We shall make use of these results in the next section, to bound the rate of change of autocorrelation functions. Finally, if we choose O 0 = |ψ 0 ψ 0 | and let it evolve backward in time, thus recovering the corresponding forward time evolution of the state |ψ 0 in the Schrödinger picture, we find that the bounds (10) and (11) become proportional to the standard ML and MT QSLs in the case of orthogonal state evolution, thus justifying the given interpretation, see App. B. We note that the proportionality constant is smaller than one, meaning that our bounds are not violated. However, they cannot be tight for state evolution. Furthermore, the result (80) can be extended to driven dynamics, under the assumption that the energy eigenvectors are stationary, see App. C.
The general results derived in this section can be applied in a variety of theoretical settings. In particular, as shown below, the QSLs on operator flows bound the decay of autocorrelation functions, thus providing constraints to the dynamical susceptibility in linear response theory and the quantum Fisher information in quantum metrology.

QSL on autocorrelation functions
Solving the dynamics of an arbitrary many-body quantum system is generally a demanding task. In many-body physics, the central objects that characterize the dynamics, determining for example the linear response [62], are the two-point time-correlation functions, whose explicit form is generally unknown. In particular, the so-called autocorrelation function determines the operator growth of O 0 in Krylov space and therefore accounts for the build-up of the corresponding notion of operator complexity [58,59,60,61,69]. Our QSLs on operators provide easily computable lower bounds to these quantities. In the following, we take O 0 to be Hermitian, ρ to be a stationary state, [ρ, H] = 0, and the Hamiltonian H to be time independent. In practical applications, ρ is often chosen to be the Gibbs state e −βH /Z at inverse temperature β, with Z = Tr e −βH , but this assumption is not necessary for the derivation of the results below. Now, let us note that this autocorrelation function can be rewritten as the Hilbert-Schmidt inner product between a non-Hermitian operatorÕ 0 and its time-evolved operatorÕ t , defined as follows: By making the commutator explicit and using that [ρ, H] = 0, we obtain [H,Õ 0 ] 2 = 2 Ȯ 2 t 0 , where the brackets · 0 stand for the expectation value with respect to ρ. We note that the characteristic velocity Ȯ 2 t 0 of the operator flow is time independent and thus does not require solving for O t . Being Õ 0 2 = C O (0) the norm ofÕ t , the autocorrelation function becomes Substituting this expression into Eqs. (11) and (14), we can recast our MT QSL as a lower bound on the symmetrized autocorrelation function which, as also occurs for the decay of the state fidelity, corresponds to a short-time Taylor expansion up to the second order. This early timescale was also identified in [70] and found to be proportional to the equilibration timescale at late times. While the MT QSL accounts for the short-time quadratic decay of the autocorrelation function, the ML QSL (16) yields a linear decay instead, which identifies a new timescale. Again, as for the evolution of the operator overlap, we observe a crossover between the MT and the ML regimes, occurring at the time which is illustrated explicitly in Figs. 1 and 3 for a two-level system and a random matrix Hamiltonian, respectively. Furthermore, as the operator dynamics is governed by the full autocorrelation function (17), one wishes to have a bound also on the antisymmetrized, imaginary autocorrelation function This quantity determines the linear response of the operator when the Hamiltonian is perturbed with an external, time-dependent driving [62]. Let us then discuss an ML-type upper bound on the imaginary part of the autocorrelation function. This can be achieved by noting that the imaginary part of the operator overlap (5) is upper bounded by the averaged Liouvillian |L| , as one can see by using that −|x| ≤ sin x ≤ |x| ∀x. Now, being This result will be used below to derive analogous bounds on the linear response under an external perturbation and the thermal quantum Fisher information associated with an arbitrary observable O.

The two-level system
Before considering further applications of our results, let us analyze a simple model for which we can compute both the bounds (20)- (22) and the actual autocorrelation function C O (t), thus illustrating explicitly the existence of the aforementioned crossover. Let us consider a two-level Hamiltonian where − → σ = (σ x , σ y , σ z ) is the Pauli matrix vector and − → r = (a, b, c). Let us choose O 0 = σ x as the initial operator and ρ = e −βH /Z as the state of the system. Then, using natural units = 1, the symmetric autocorrelation function is given by where r ≡ | − → r | = √ a 2 + b 2 + c 2 , the temperature dependence being contained only in the antisymmetric part Moreover, the MT and ML time scales have the following expressions respectively, and (29) These can be inserted into Eqs. (20) and (21) to compute lower bounds on Re C O (t) and Im C O (t), as shown in Fig. 1. The symmetric, real part is the only nonvanishing term at the initial time and undergoes a decay in parallel with the onset of the antisymmetric, imaginary contribution. This decay is initially captured by the MT QSL, as expected from Taylor expansion, while after the crossover time τ c , the ML QSL becomes tighter.
Finally, let us note that, for the parameters chosen (a = 10, c = b = 1 and β = 10), the two ML QSLs, formulated in terms either of the Hamiltonian (16) or the Liouvillian (10), are equivalent, as shown in Fig. 2. Indeed, in the latter case, the velocity is given by so that the difference between the two ML veloc- ities is However, if one increases the temperature sufficiently the Liouvillian bound gives a tighter result.

Random matrix example
As already stressed, the above scenario consisting of an early quadratic decay governed by MT (20), a crossover, and a subsequent time window in which ML (21) becomes tighter, proves to be a general feature of autocorrelations in isolated quantum systems. We further illustrate this scenario in the following generic setting. First, we sample the Hamiltonian H from the Gaussian Orthogonal Ensemble (GOE), with standard deviation σ = 1 and dimension d = 200; we subsequently diagonalize H and thus fix as a reference basis the energy eigenbasis. We then construct the initial operator, using this basis, by sampling it from the same GOE as the Hamiltonian. Alternatively, we could sample the operator and the Hamiltonian using the same computational basis. The comparison between the autocorrelation function and the speed limits is shown in Fig. 3. While the tangent character of the ML QSL to the real curve after the crossover is a feature specific to the qubit case (see Fig. 1), we observe that the divergence of the bound does not keep increasing when considering an increasing Hilbert space dimension. ML-type bounds were recently found to lose tightness for higher dimensions than

Dynamical susceptibilities
Thermal correlation functions determine the nonequilibrium response of an observable at the first order in the perturbation [62,71]. Therefore, the results illustrated in the previous section allow us to bound the linear response and, in particular, the dynamical susceptibility, which is the quantity that characterizes the response of a system to an external perturbation.
Let us then consider the situation in which a time-independent Hamiltonian H 0 is perturbed by an external, time-dependent driving, where the perturbation operator V does not depend explicitly on time, λ is a real positive constant that quantifies the strength of the perturbation and the driving force f (t), which can be taken to be |f | ≤ 1, is assumed to vanish for t ≤ 0.
Let the initial state of the system be the thermal Gibbs state ρ 0 , relative to the unperturbed Hamiltonian H 0 , at inverse temperature β. We are interested in determining the linear response of an observable A, namely the shift of its expectation value at time t from the initial equilibrium value, Here, the operators are evaluated in the interaction picture (35) To the first order in λ, the linear response is given by the celebrated Kubo formula [62] (36) Thus, the non-equilibrium response, at the linear order, is determined by an equilibrium correlation function. As already stressed above, computing correlation functions at all times is generally a challenging task, equivalent to solving the dynamics. Therefore, having universal, modelindependent bounds on these quantities is extremely useful. We next present several ways in which one can bound the right-hand side of Eq. (36).
The linear response (36) can be rewritten in terms of the so-called dynamical susceptibility where θ(t) is the Heaviside function, as The dynamical susceptibility χ AV (t), which vanishes for a negative argument (i.e., before the external perturbation is applied), expresses the causal linear response of the system and is a real quantity. Its absolute value can be upper bounded using the Heisenberg uncertainty relation | [A, V ] | ≤ 2∆A∆V , which yields the constant bound where ∆ 2 0 A is the variance with respect to ρ 0 . As the latter is time-independent in the interaction picture, we drop the subscript I. Alternatively, one can apply the Bogoliubov inequality [72] This yields a different upper bound with an explicit dependence on the temperature T where we have defined the characteristic temperature Both (39) and (41) upper bound the modulus of the dynamical susceptibility in terms of the equilibrium fluctuations of the perturbation operator V and the observable A in which the response is studied. We see that at low temperature T ≤ T B the Bogoliubov bound (41) is tighter than the Heisenberg bound (39), while the contrary holds at higher temperatures. In certain experimental settings, one may be interested in quantifying the response of the perturbation operator V itself. For example, this is the case of the magnetic susceptibility in magnetic resonance experiments, where both V and the observable of interest A are given by the magnetization [73]. In such case, being A = V , the dynamical susceptibility χ V V (t) becomes proportional to the anti-symmetrized autocorrelation function [V I (t), V I (0)] 0 = 2i Im C V (t), and therefore one can make use of the techniques that we have illustrated above to bound autocorrelations functions utilizing QSLs for operators. The ML QSL (22) yields the following upper bound on the dynamical susceptibility where we have introduced a new time scale , E 0 being the ground state energy of the unperturbed Hamiltonian H 0 . Unlike the previous bounds (39) and (41), Eq. (43) contains an explicit linear dependence on time. This implies that the QSL approach is the most efficient at early enough times. However, as we let the system evolve in time, the QSL bound (43) no longer governs the dynamics, and one needs to consider different approaches. In particular, at lower temperature T ≤ T B , the QSL bound (43) ceases to be the tightest one at the time and for t ≥ τ B the Bogoliubov bound (41) gives the better description. By contrast, at high temperatures T ≥ T B , the crossover is between the QSL bound and the Heisenberg bound (39), and it occurs at the time The bounds (39), (41) and (43) find applications in many experimentally relevant settings [73], of which we give two concrete examples below.

Examples
The situation in which a system is subject to an external perturbation is widespread in physics. A paradigmatic example is that of a system composed of N charged particles and perturbed with a uniform, time-dependent electric field − → E (t) [73]. In this case, the preferred observable to characterize the response of the system is the current flow − → J and the corresponding dynamical susceptibility is the electrical conductivity. The perturbed Hamiltonian takes the form where − → R is the electric dipole moment in the ori- with q n and − → r n being the charge and the position of the n-th particle. The net current vanishes at equilibrium, − → J 0 = 0, and the perturbed expectation value of the i-th spatial component J i at time t ≥ 0 is expressed, at the linear order, as [73] where σ ij ≡ χ J i R j is the electrical conductivity tensor. By using the bounds (39) and (41) we are able to derive the following constraints on σ ij : where, in the first equation, we note that that the net current vanishes at equilibrium, i.e., ∆ 0 J i = where we note that temperature dependence is also contained in the current fluctuations ∆ 0 J i . Another experimental application in which the theory of linear response provides a useful approach is given by magnetic resonance experiments [74]. In this case, the central quantity is the magnetic susceptibility. Consider a paramagnetic system, initially aligned along a constant magnetic field − → B and subsequently perturbed with a weak time-dependent field − → h (t) so that the total Hamiltonian reads The magnetization of the system is perturbed from its initial equilibrium value − → M 0 = χ 0 − → B , where χ 0 is the static susceptibility, and the linear response of its i-th component can be expressed using the Kubo formula is the magnetic susceptibility tensor. The bounds (39) and (41) now yield (56) We note that this bound is consistent with the fact that χ M,ij vanishes whenever the static magnetic field is along the j-th axis, which follows from the vanishing of the averages M i 0 along directions orthogonal to − → B . Moreover, the diagonal magnetic susceptibilities can be upper bounded through the QSL approach (43) to find

Bounds on the Quantum Fisher Information
We next turn our attention to the application of QSL on operator flows in quantum metrology. In this context, the quantum Fisher information F Q associated with a Hermitian operator O quantifies the maximal precision with which we can estimate the phase θ that parameterizes the unitary flow, of a given quantum state ρ, generated by the operator O. In other words, F Q measures the distinguishability of the "initial" state ρ 0 from the one transformed by the unitary flow Let us note the change in the perspective: instead of looking at the time unitary flow that the observable O of interest undergoes under the action of the Hamiltonian that generates the dynamics, we are now considering the unitary flow (in a different parameter θ) of a given quantum state ρ under the action of O, that generates the state transformation. Remarkably, these two approaches are closely related, as the dynamical susceptibility χ OO obtained using the first framework can be related to the quantum Fisher information F Q [64,75]. More precisely, for a thermal state ρ at temperature T , the following result on the quantum Fisher information has been shown by Hauke et. al in [64] whereχ OO (ω, T ) is the Fourier-transformed dynamical susceptibility, defined as with χ OO (t, T ) being defined in Eq. (37) for A = V = O. Now, the upper bound (24) we have derived above on the anti-symmetrized autocorrelation function Im C O , together with the result (59), provides an upper bound on the quantum Fisher information F Q . To this end, let us reverse the order of the integrals in Eq. (59) and perform first the one in ω. Computing the inverse Fourier transform of tanh(ω/2T ) and using the fact that (61) Therefore, using that ∞ 0 dx x csch(πqx) = (8q 2 ) −1 ∀q > 0, from our previous result (24) we derive the following upper bound on the temperature-dependent quantum Fisher information where O is the operator that generates the transformation whose parameter is to be estimated, H is the Hamiltonian of the system, and the expectation value is taken with respect to the corresponding thermal state at temperature T . Let us finally note that, in this framework, the standard bounds (39) and (41) which suggests that a better precision may be achieved at lower temperatures.

Discussion
Conventional QSLs identify the minimum time scale in which a process can unfold by exploiting the notion of quantum state distinguishability. Yet, many applications in theoretical and experimental physics are naturally formulated in terms of operator flows. We have generalized the notion of QSL to this setting, providing bounds to the rate of unitary flows described by the conjugation of an observable by a one-parameter unitary. Making use of Liouville space, we have derived analogs of the MT and ML QSLs, in which the minimum shift of the parameter required to distinguish the evolving operator from the initial one is lower bounded in terms of the mean and variance of the Liouvillian. These bounds generally exhibit a crossover, that we have characterized, and that is analogous to that observed in recent experiments for conventional QSLs.
We have also shown that QSLs for operator flows constrain the time dependence of autocorrelation functions and thus the dynamic susceptibilities introduced in linear response theory to describe transport coefficients. In the context of quantum parameter estimation, we have shown that QSLs for operator flows yield bounds on the quantum Fisher information that restricts the estimation error through the Cramer-Rao bound. This last application makes explicit the fact that the flow under consideration need not be on time, but can describe shifts of an arbitrary parameter through a continuous symmetry. The situation is thus analogous to the generalization of uncertainty relations for quantum states [76].
Our results should find broad applications in nonequilibrium physics and, in particular, quantum technologies, including quantum metrology, quantum thermodynamics, and quantum computation. As we have demonstrated by several examples, our results are also of relevance in condensed matter physics to bound response functions and transport coefficients. We expect further applications of our results in other scenarios where operator flows naturally arise, such as the formulation of integrable systems in terms of Lax pairs and the Wegner renormalization group.

A Deriving QSLs through a trigonometric approach
In this appendix, we show an alternative derivation that results in two weaker QSLs on the operator overlap, proportional to the tighter QSLs (10) and (11) derived in the main text. Making use of the trigonometric inequality valid for x > 0, we obtain Therefore, being sin x ≤ x for x ≥ 0, we find that is, which is proportional to the ML QSL (10) derived in the main text through a constant smaller than one. Next, let us make use of the trigonometric inequality which holds again ∀x. Combining it with the bound (7) on Re O 0 |Ȯ t , we obtain (68) As for the case of ML, this bound is proportional to our MT QSL (11) for operators through a constant smaller than one, thus yielding a weaker result.

B QSLs in the Schrödinger picture
In this appendix, we reformulate our results on operator flows in the context of the standard time-evolution of quantum states and compare them with the well-known MT [1] and ML [2] bounds. In this way, we provide a further justification for the bounds (11) and (10) to be regarded as generalizations of the MT and ML quantum speed limits to operator flows, respectively. To this end, let us choose the initial operator O 0 to be the projector onto the pure state |ψ 0 , O 0 = |ψ 0 ψ 0 |, and let the expansion of |ψ 0 in the energy eigenbasis be where H |j = E j |j and j |c j | 2 = 1. Then, the vectorization of O 0 is |O 0 = jk c * j c k |j |k , with O 0 = 1. Since |ψ t = U t |ψ 0 , where U t = e −iHt/ , in order to reproduce the forward timeevolution of the quantum state we need to evolve the operator backward in time: Moreover, being the Bures angle between states defined as we have that 1 − O 0 |O −t = sin 2 t . We observe that the operator angle L t defined in Eq. (4) does not reduce to the standard Bures angle between states, being The typical maximal speeds of the flow, given by the ML (10) and MT (11) QSLs respectively, can be written as From the second equation, being also we derive a proportionality relation between the Hamiltonian and Liouvillian variances (over |ψ 0 and |O 0 , respectively): Therefore, the MT bound (11) derived in the main text for operators reduces to the following QSL for states, given in terms of the energy variance: In the case of orthogonal evolution ψ 0 |ψ τ = 0, we have sin τ = 1 and therefore we obtain a bound proportional to (though weaker than) the standard Mandelstam-Tamm QSL [1], thus justifying the interpretation of Eq. (11) as an MT-type of QSL for operators. On the other hand, the mean energy can be rewritten as which, by using that Therefore, we can recast the bound (10) as which represents a generalization of the ML QSL [2] to the case of an arbitrary angle between the initial and final states. A similar generalization was also claimed in reference [6], though their bound is not given in terms of H − E 0 , but rather in terms of the quantity | H |. To our knowledge, the generalization of the ML bound to arbitrary angles between initial and final states was only proven numerically [77,78,16]. Let us note that, by considering again the orthogonalization time t = τ , our QSL becomes where τ M L = π /2( H − E 0 ) is the well known ML QSL [2] for orthogonal evolution.
In conclusion, our bounds (10) and (11) on operator flows turn out to be proportional to the standard MT and ML QSLs for state evolution, justifying the given interpretation. The constant of proportionality is smaller than one, meaning that our bounds are not violated, though they are not tight for quantum state evolution.
Finally, let us perform a numerical test of our ML QSL for states (80), which can be recast as a lower bound on the state overlap In Fig. 4, we test the bound (82) for a single realization of a random matrix Hamiltonian H of dimension d = 50, generated from the Gaussian Orthogonal Ensemble (GOE) with variance σ = 1. The initial state is chosen to be the coherent Gibbs state at inverse temperature β where Z = n e −βEn is the partition function. In this setting, the quantum state overlap can be conveniently rewritten in terms of the analytically-continued partition function as follows [79,80] ψ 0 |ψ t = Z(β + it) Z(β) .
After the initial decay shown in Fig. 4, the fidelity undergoes an oscillatory behavior that is no longer captured by the QSL (82). Deviations from the bound appear far after the characteristic timescale τ = (σ √ 8d) −1 , where σ √ 8d correspond to the width of the eigenvalue distribution (i.e., to the diameter of the Wigner semicircle law), that is to the largest frequency involved in the evolution.

C Margolous-Levitin QSL on operator flows with driven Hamiltonians
Finally, we partially extend our results to the case in which the Hamiltonian generating the unitary We focus on the case in which O 0 is an observable, i.e., a Hermitian operator. Moreover, let us assume that the Hamiltonians at different times commute so that we can drop the time-ordering operator T in front of the exponential. In addition, if we make the strongest assumption that H t remains diagonal in the initial eigenbasis, where only the eigenvalues E j (t) depend on time, then we can write the action of U t over the energy eigenvector |j as Under this assumption, the operator overlap can be expanded as so that, making use of the inequality cos x ≥ 1 − α|x| [66] as in the main text, we derive an ML bound on operator flows in the case of driven dynamics, where f (t) = 1 t t 0 f (u)du is the time average at time t. As a consequence, by choosing O 0 = |ψ 0 ψ 0 |, we obtain a generalization of the ML QSL for state evolution under driven Hamiltonians and arbitrary angles: valid when the energy eigenvectors are stationary. A related result was also claimed in reference [10], but their derivation was later shown to generalize the MT QSL rather than the ML one, as pointed out in [81].