# Quantum speed limits on operator flows and correlation functions

Nicoletta Carabba  Department of Physics and Materials Science, University of
Luxembourg, L-1511 Luxembourg, G. D. Luxembourg  Niklas Hörnedal  Department
of Physics and Materials Science, University of Luxembourg, L-1511 Luxembourg,
G. D. Luxembourg  Fysikum, Stockholms Universitet, 106 91 Stockholm, Sweden
Adolfo del Campo  Department of Physics and Materials Science, University of
Luxembourg, L-1511 Luxembourg, G. D. Luxembourg  Donostia International
Physics Center, E-20018 San Sebastián, Spain

###### Abstract

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.

##  I 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
Mandelstam and Tamm [1945], Margolus and Levitin [1998] was restricted to
unitary evolution between two pure quantum states, by now they have been
generalized to the case of arbitrary mixed states Uhlmann [1992], Campaioli et
al. [2018], driven Hamiltonians Anandan and Aharonov [1990], Uhlmann [1992],
Deffner and Lutz [2013a], Okuyama and Ohzeki [2018a], open quantum dynamics
Taddei et al. [2013], del Campo et al. [2013], Deffner and Lutz [2013b],
Campaioli et al. [2019], continuous quantum measurements García-Pintos and del
Campo [2019], and classical processes Shanahan et al. [2018], Okuyama and
Ohzeki [2018b], Shiraishi et al. [2018]. Their applications are thus manifold
and range over various branches of physics Deffner and Campbell [2017]. Their
use is prominent in quantum technologies, including quantum computation Lloyd
[2000, 2002] and quantum metrology Giovannetti et al. [2011], Beau and del
Campo [2017]. QSL are known to limit the performance of quantum control
algorithms Caneva et al. [2009], Hegerfeldt [2013], and play a key role in
shortcuts to adiabaticity by counterdiabatic driving both in isolated Funo et
al. [2017], Campbell and Deffner [2017] and open quantum systems Alipour et
al. [2020], Funo et al. [2021]. They have also been applied to many-body
physics, in which care is needed given the orthogonality catastrophe Bukov et
al. [2019], Suzuki and Takahashi [2020], del Campo [2021], Hamazaki [2021],
Gong and Hamazaki [2022]. 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 Jing et al.
[2016], Marvian et al. [2016], and more generally, quantum resources Campaioli
et al. [2022].

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) Gingrich et al. [2016],
Hasegawa [2021] has been explored in Nicholson et al. [2020], Vo et al.
[2020], identifying QSL-inspired constraints on non-equilibrium fluctuations.
Recently, a closely related work García-Pintos et al. [2022], following the
initial spirit of Mandelstam and Tamm [1945], 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 Hamazaki [2021], Gong and Hamazaki [2022].

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 and continuous
renormalization group flows, such as the one proposed by Wegner Wegner [2001].
Preceding efforts in this direction have focused on the preparation of unitary
Poggi [2019] and non-unitary Uzdin [2013] 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 Uzdin and Kosloff [2016].
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 von Keyserlingk
et al. [2018], Khemani et al. [2018], Nahum et al. [2018], Gopalakrishnan et
al. [2018], Rakovszky et al. [2018], 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
Susskind [2016], that is also subject to QSLs Brown et al. [2016a, b], Chapman
et al. [2018], Molina-Vilaplana and del Campo [2018]. In this context, a QSL-
like uncertainty bound has recently been introduced Hörnedal et al. [2022] on
the growth of a particular notion of operator complexity, known as Krylov
complexity Parker et al. [2019], Barbón et al. [2019], Rabinovici et al.
[2021], Caputa et al. [2021].

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 Kubo [1957] 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) Mandelstam and Tamm [1945] type of QSL
and a second one in which a Margolus-Levitin (ML) Margolus and Levitin [1998]
type of QSL yields a more accurate description. We illustrate this crossover,
analogous to that recently observed experimentally for quantum state evolution
Ness et al. [2021], 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 relation
between the dynamical susceptibility and the quantum Fisher information Hauke
et al. [2016], we upper bound this quantity with an easily computable
correlation function.

##  II Quantum speed limits for operators

The paradigmatic example of operator flow in quantum mechanics is the unitary
flow $\dot{O}_{t}=\frac{i}{\hbar}[H,O_{t}],$ describing the time evolution of
an initial operator $O_{0}$ generated by the Liouvillian
$\mathbb{L}[\cdot]=\frac{i}{\hbar}[H,\cdot]$. The unitary character of the
flow is manifest in the formal solution $O_{t}=U_{t}^{\dagger}O_{0}U_{t}$,
where $U_{t}=\exp(-itH/\hbar)$ 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, as it
allows a direct analogy between operator flow and quantum state evolution.
Therefore, let us represent a general bounded operator
$A=\sum_{i,j}A_{ij}\ket{i}\bra{j}$, where $\\{\ket{i}\\}$ can be any basis of
the Hilbert space, as a normalized vector

$\ket{A}=\frac{1}{\|A\|}\sum_{i,j}A_{ij}\ket{i}\otimes\ket{j},$ (1)

where $\|A\|=\sqrt{\langle A,A\rangle}$ is the Hilbert-Schmidt norm associated
with the Hilbert-Schmidt inner product
$\braket{A,B}=\operatorname{Tr}[A^{\dagger}B]$. In Liouville space, the latter
is proportional to the standard scalar product over $\mathbb{C}^{d^{2}}$,
being $\langle A,B\rangle=\|A\|\|B\|\braket{A}{B}$. In the isomorphic real
vector space $\mathbb{R}^{2d^{2}}$ the real part
$\operatorname{Re}\braket{A,B}$ defines the standard Euclidean inner product.

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
$\partial_{t}\ket{O_{t}}=\mathbb{L}\ket{O_{t}}$, where the Liouvillian, which,
with a slight abuse of notation, we shall continue to indicate as
$\mathbb{L}$, takes the form

$\mathbb{L}=\frac{i}{\hbar}(H\otimes\mathbb{1}-\mathbb{1}\otimes H^{T}).$ (2)

A particularly useful object to quantify the displacement of the operator
along the flow is the operator overlap

$\braket{O_{0}}{O_{t}}=\frac{\operatorname{Tr}(O^{\dagger}_{0}O_{t})}{\|O\|^{2}},$
(3)

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 finite-temperature
autocorrelations functions as well. Note that $\braket{O_{0}}{O_{t}}$ is
closely related to the notion of operator fidelity introduced in the context
of Loschmidt echoes and quantum phase transitions Wang et al. [2009].

To measure how “far” the operator $O_{t}$ has flowed, we define the operator
angle $\mathcal{L}_{t}$ between the vector $\ket{O_{t}}$ and the inital one
$\ket{O_{0}}$:

$\mathcal{L}_{t}\equiv\arccos{\operatorname{Re}\braket{O_{0}}{O_{t}}}.$ (4)

This quantity defines a distance over the unitary flow
$O_{t}=U_{t}^{\dagger}O_{0}U_{t}$. Being the vectorized operator normalized as
in Eq. (1), the unitary flow in Liouville space lies on the unit sphere. Since
$\operatorname{Re}\braket{O_{0}}{O_{t}}$ is the Euclidean inner product, Eq.
(4) reduces to the angle between the corresponding real vectors in
$\mathbb{R}^{2d^{2}}$ and is therefore a distance. We also note that
$\mathcal{L}_{t}$ resembles the notion of Bures angle
$\ell=\arccos{|\braket{\psi_{0}}{\psi_{t}}|}$, but it does not reduce to it
when the initial operator is chosen to be the projector over the pure state
$\ket{\psi_{0}}$, $O_{0}=\ket{\psi_{0}}\bra{\psi_{0}}$. Moreover, let us
stress that $\arccos{|\braket{O_{0}}{O_{t}}|}$ 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 $\\{\ket{i}\\}$ be the energy eigenbasis, such that
$H\ket{i}=E_{i}\ket{i}$, and let the initial operator be
$O_{0}=\sum_{ij}O_{ij}\ket{i}\bra{j}$ in such basis. The Liouvillian is
diagonal in the vectorized basis $\ket{i}\otimes\ket{j}$ and has the energy
gaps $\Delta_{ij}\equiv E_{i}-E_{j}$ as diagonal entries. The operator overlap
takes the form

$\braket{O_{0}}{O_{t}}=\frac{1}{\|O\|^{2}}\sum_{j,k}e^{i\Delta_{jk}t/\hbar}|O_{jk}|^{2}.$
(5)

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 Margolus-Levitin (ML) Margolus and Levitin [1998] and
Mandelstam-Tamm (MT) Mandelstam and Tamm [1945] QSLs for Schrödinger
evolution. Indeed, by using that $\cos x\geq 1-\alpha|x|$, where the parameter
$\alpha\approx 0.724$ is chosen such that $1-\alpha x$ is tangent to $\cos x$
for $x>0$ Andersson [2019], we can bound the real part of the overlap from
below,

$\operatorname{Re}\braket{O_{0}}{O_{t}}\geq 1-\frac{\alpha
t}{\hbar\|O\|^{2}}\sum_{j,k}|\Delta_{jk}||O_{jk}|^{2}=1-\alpha\braket{}{\mathbb{L}}{}t,$
(6)

where the brackets in the right-hand side stand for the expectation value over
the vectorized operator: $\braket{\cdot}=\braket{O_{0}}{\cdot}{O_{0}}$.
Further, its time derivative $\operatorname{Re}\braket{O_{0}}{\dot{O}_{t}}$
can be upper bounded by making use of the inequality $-x^{2}\leq x\sin x\leq
x^{2}$, valid $\forall x$:

$|\operatorname{Re}\braket{O_{0}}{\dot{O}_{t}}|\leq\frac{t}{\hbar^{2}\|O\|^{2}}\sum_{j,k}\Delta_{jk}^{2}|O_{jk}|^{2}=\braket{\mathbb{L}^{2}}t.$
(7)

The derivative of the operator overlap is related to that of the operator
angle $\mathcal{L}_{t}$ by

$\operatorname{Re}\braket{O_{0}}{\dot{O}_{t}}=-\sin{(\mathcal{L}_{t})}\,\dot{\mathcal{L}_{t}},$
(8)

which, after time integration, when combined with Eq. (7), yields

$1-\cos{\mathcal{L}_{t}}\leq\frac{\braket{\mathbb{L}^{2}}}{2}t^{2}.$ (9)

Therefore, from Eqs. (6) and (9) we obtain the two following QSLs, formulated
in terms either of the operator overlap or the operator angle:

$\displaystyle t$
$\displaystyle\geq\frac{1-\operatorname{Re}\braket{O_{0}}{O_{t}}}{\alpha\braket{}{\mathbb{L}}{}}=\frac{1-\cos{\mathcal{L}_{t}}}{\alpha\braket{}{\mathbb{L}}{}},$
(10) $\displaystyle t$
$\displaystyle\geq\sqrt{\frac{2(1-\operatorname{Re}\braket{O_{0}}{O_{t}})}{\braket{\mathbb{L}^{2}}}}=\sqrt{\frac{2(1-\cos{\mathcal{L}_{t})}}{\braket{\mathbb{L}^{2}}}}.$
(11)

These results identify $\braket{}{\mathbb{L}}{}$ and
$\sqrt{\braket{\mathbb{L}^{2}}}$ as upper bounds on the speeds of the operator
flow. We stress that the quantities $\braket{}{\mathbb{L}}{}$ and
$\braket{\mathbb{L}^{2}}$ 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
$\braket{\mathbb{L}}=0$ by parity, so that
$\braket{\mathbb{L}^{2}}=(\Delta\mathbb{L})^{2}$ 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 $\mathbb{L}$. We
observe that, in the case of operator evolution, the ML bound (10) is given in
terms of $\braket{}{\mathbb{L}}{}$ rather than the mean of the generator
$\braket{\mathbb{L}}$, 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
$\braket{}{\mathbb{L}}{}$ and $\sqrt{\braket{\mathbb{L}^{2}}}$ 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
$(|O_{ij}|/\|O\|)^{2}\leq 1$ defines a proper probability distribution over
the energy states pairs, it is clear that the operators $O_{\text{max}}$
flowing at the maximal speed are the ones whose non-zero elements are only
between energy eigenstates with the maximum gap
$|\Delta_{\text{max}}|=E_{\text{max}}-E_{0}$, where $E_{\text{max}}$ and
$E_{0}$ are the highest and the lowest energy eigenvalues, respectively. This
maximal speed is the analogous for operators of the one identified by the dual
ML bound, recently introduced for state evolution Ness et al. [2022]. If these
levels are non-degenerate, the fastest operator will be of the form
$O_{\text{max}}=\mu\ket{E_{\text{max}}}\bra{E_{0}}+\nu\ket{E_{0}}\bra{E_{\text{max}}}$
for some complex constant $\mu$ and $\nu$ ($\mu=\nu^{*}$ for observables).
This is analogous to the well-known result for states Levitin and Toffoli
[2009].

Remarkably, the above QSLs identify a universal crossover between two
different time regimes. At early times $t\leq\tau_{\textrm{c}}$, being
$\tau_{\textrm{c}}=2\alpha\braket{}{\mathbb{L}}{}/\braket{\mathbb{L}^{2}}$ the
crossover time, the decay of the operator overlap is governed by a quadratic
MT bound

$\operatorname{Re}\braket{O_{0}}{O_{t}}\geq
1-\frac{\braket{\mathbb{L}^{2}}}{2}t^{2},$ (12)

while for times $t\geq\tau_{\textrm{c}}$ the linear ML bound

$\operatorname{Re}\braket{O_{0}}{O_{t}}\geq 1-\alpha\braket{}{\mathbb{L}}{}t$
(13)

becomes tighter. A similar crossover was experimentally observed for the state
fidelity of a single atom in an optical trap Ness et al. [2021].

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, $\braket{\mathbb{L}^{2}}$ is
proportional to the norm of the operator velocity

$\braket{\mathbb{L}^{2}}=\frac{1}{\hbar}\frac{\|[H,O_{t}]\|}{\|O\|}=\frac{\|\partial_{t}O\|}{\|O\|},$
(14)

which, as emphasized, is time-independent under unitary dynamics. Regarding
the ML QSL, by using that $|\Delta_{jk}|\leq E_{j}+E_{k}-2E_{0}$ in terms of
the ground state energy $E_{0}$, we find

$\braket{}{\mathbb{L}}{}\leq\frac{1}{\hbar}\frac{\operatorname{Tr}(O_{0}^{\dagger}\\{H-E_{0},O_{0}\\})}{\|O\|^{2}}.$
(15)

Therefore, the ML QSL can be recast as

$t\geq\hbar\frac{\|O\|^{2}}{\alpha}\frac{1-\operatorname{Re}\braket{O_{0}}{O_{t}}}{\operatorname{Tr}(O_{0}^{\dagger}\\{H-E_{0},O_{0}\\})},$
(16)

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 $O_{0}=\ket{\psi_{0}}\bra{\psi_{0}}$ and let it evolve
backward in time, thus recovering the corresponding forward time evolution of
the state $\ket{\psi_{0}}$ 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 (78) 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.

##  III 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 Kubo [1957], are the
two-point time-correlation functions, whose explicit form is generally
unknown. In particular, the so-called autocorrelation function

$C_{O}(t)=\operatorname{Tr}(O^{\dagger}_{t}O_{0}\rho)$ (17)

determines the operator growth of $O_{0}$ in Krylov space and therefore
accounts for the build-up of the corresponding notion of operator complexity
Parker et al. [2019], Barbón et al. [2019], Rabinovici et al. [2021], Caputa
et al. [2021], Dymarsky and Smolkin [2021].

Our QSLs on operators provide easily computable lower bounds to these
quantities. In the following, we take $O_{0}$ to be Hermitian, $\rho$ to be a
stationary state, $[\rho,H]=0$, and the Hamiltonian $H$ to be time
independent. In practical applications, $\rho$ is often chosen to be the Gibbs
state $e^{-\beta H}/Z$ at inverse temperature $\beta$, with
$Z=\operatorname{Tr}e^{-\beta H}$, 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

$C_{O}(t)=\braket{\tilde{O}_{t},\tilde{O}_{0}}$ (18)

between a non-Hermitian operator $\tilde{O}_{0}$ and its time-evolved operator
$\tilde{O}_{t}$, defined as follows:

$\tilde{O}_{0}\equiv
O_{0}\sqrt{\rho},\quad\tilde{O}_{t}=U^{\dagger}\tilde{O}_{0}U=O_{t}\sqrt{\rho}.$
(19)

By making the commutator explicit and using that $[\rho,H]=0$, we obtain
$\|[H,\tilde{O}_{0}]\|^{2}=\hbar^{2}\braket{\dot{O}^{2}_{t}}$, where the
brackets stand for the expectation value with respect to $\rho$. We note that
this characteristic velocity of the operator flow is time independent and thus
does not require computing $O_{t}$. Being $\|\tilde{O}_{0}\|^{2}=C_{O}(0)$ the
norm of $\tilde{O}_{t}$, the autocorrelation function becomes
$C_{O}(t)=C_{O}(0)\braket{\tilde{O}_{t}}{\tilde{O}_{0}}$. Substituting these
expression into Eqs. (11) and (14), we can recast our MT QSL as a lower bound
on the symmetrized autocorrelation function

$\operatorname{Re}C_{O}(t)=\frac{1}{2}\braket{\\{O_{t},O_{0}\\}}\geq
C_{O}(0)-\frac{1}{2}\braket{\dot{O}^{2}_{t}}|_{t=0}t^{2},$ (20)

which, as also occurs for the decay of the state fidelity, corresponds to a
short-time Taylor expansion up to second order. This early timescale was also
identified in Alhambra et al. [2020] 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 instead a linear decay,

$\operatorname{Re}C_{O}(t)\geq
C_{O}(0)-\frac{\alpha}{\hbar}\braket{O_{0}\\{H-E_{0},O_{0}\\}}t,$ (21)

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 time

$\tau_{\textrm{c}}=\frac{2\alpha}{\hbar}\frac{\braket{O_{0}\\{H-E_{0},O_{0}\\}}}{\braket{\dot{O}^{2}_{t}}|_{t=0}},$
(22)

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 anti-symmetrized,
imaginary autocorrelation function $\braket{[O_{t},O_{0}]}_{0}$. This quantity
determines the linear response of the operator when the Hamiltonian is
perturbed with an external, time-dependent driving Kubo [1957]. Let us then
discuss a 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
$\braket{}{\mathbb{L}}{}$,

$|\operatorname{Im}\braket{\tilde{O}_{0}}{\tilde{O}_{t}}|=\frac{1}{2}|\braket{[O_{t},O_{0}]}_{0}|\leq\braket{}{\mathbb{L}}{}t,$
(23)

as one can see by using that $-|x|\leq\sin{x}\leq|x|$ $\forall x$. Now, being
$\operatorname{Im}C_{O}(t)=-C_{O}(0)\operatorname{Im}\braket{\tilde{O}_{0}}{\tilde{O}_{t}}$,
we derive

$|\operatorname{Im}C_{O}(t)|\leq\braket{O^{\dagger}_{0}\\{H-E_{0},O_{0}\\}}\frac{t}{\hbar}.$
(24)

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 $O$.

###  III.1 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.

Figure 1: Comparison of the real (a) and imaginary (b) parts of the
correlation function with the MT and ML QSLs given by Eqs. (20), (21) and (24)
for a two-level system. The parameters are chosen to be $a=10$, $c=b=1$ and
the inverse temperature is $\beta=10$. The identity in the Hamiltonian (25)
plays no role, so we can fix $k=0$ without loss of generality. (a) The initial
decay of the symmetric correlation function undergoes a crossover from a
regime dominated by the MT QSL to a regime in which the ML QSL becomes
tighter. The vertical line corresponds to the crossover time
$\tau_{\textrm{c}}$. (b) The onset of the antisymmetric contribution is
characterized by the ML QSL at short times.

Let us consider a two-level Hamiltonian

$H=k\mathbb{1}+\overrightarrow{r}\cdot\overrightarrow{\sigma},$ (25)

where $\overrightarrow{\sigma}=(\sigma_{x},\sigma_{y},\sigma_{z})$ is the
Pauli matrix vector and $\overrightarrow{r}=(a,b,c)$. Let us choose
$O_{0}=\sigma_{x}$ as the initial operator and $\rho=e^{-\beta H}/Z$ as the
state of the system. Then, using natural units $\hbar=1$, the symmetric
autocorrelation function is given by

$\operatorname{Re}C_{O}(t)=\frac{a^{2}+(b^{2}+c^{2})\cos{2rt}}{r^{2}},$ (26)

where $r\equiv|\overrightarrow{r}|=\sqrt{a^{2}+b^{2}+c^{2}}$, the temperature
dependence being contained only in the antisymmetric part

$\operatorname{Im}C_{O}(t)=\frac{b^{2}+c^{2}}{r^{2}}\tanh{\beta r}\sin{2rt}.$
(27)

Moreover, the MT and ML time scales have the following expressions,
respectively,

$\braket{\dot{O}^{2}_{t}}=4(b^{2}+c^{2})$ (28)

and

$\braket{O_{0}\\{H-E_{0},O_{0}\\}}=\frac{2}{r}\left(\frac{2a^{2}}{1+e^{2\beta
r}}+b^{2}+c^{2}\right).$ (29)

These can be inserted into Eqs. (20) and (21) to compute lower bounds on
$\operatorname{Re}C_{O}(t)$ and $\operatorname{Im}C_{O}(t)$, 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
$\tau_{\textrm{c}}$ the ML QSL becomes tighter.

Finally, let us note that, for the parameters chosen ($a=10$, $c=b=1$ and
$\beta=10$), 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

$C_{O}(0)\braket{\tilde{O}}{\,}{\mathbb{L}|\,|\tilde{O}}=\frac{2}{r}(b^{2}+c^{2}),$
(30)

so that the difference between the two ML velocities is

$\frac{4a^{2}}{r}\frac{\alpha}{1+e^{2\beta r}}\ll 1.$ (31)

However, if one increases the temperature sufficiently the Liouvillian bound
gives a tighter result.

Figure 2: The Liouvillian (10) and Hamiltonian (16) formulations of the ML
bound for a two-level system are shown to be equivalent in the time window of
interest. The parameters are chosen to be $a=10$, $c=b=1$ and the inverse
temperature is $\beta=10$.

###  III.2 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 feature in a generic
setting, by sampling both the Hamiltonian and the initial operator (in the
energy eigenbasis) from the Gaussian Orthogonal Ensemble (GOE) with standard
deviation $\sigma=1$ and dimension $d=200$. 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
loose tightness for higher dimensions than a qubit also in Ness et al. [2022].

Figure 3: Comparison of the real (a) and imaginary (b) parts of the
(normalized) correlation function with the MT and ML QSLs given by Eqs. (20),
(21) and (24), for $H,\,O_{0}\in$ GOE with standard deviation $\sigma=1$ and
dimension $d=200$. The QSLs have been here normalized dividing by $C_{0}(0)$.
(a) The initial decay of the symmetric correlation function undergoes a
crossover from a regime dominated by the MT QSL to a regime in which the ML
QSL becomes tighter. The vertical line corresponds to the crossover time
$\tau_{\textrm{c}}$. (b) The onset of the antisymmetric contribution is
characterized by the ML QSL at short times.

##  IV Dynamical susceptibilities

Thermal correlation functions determine the non-equilibrium response of an
observable at the first order in the perturbation Kubo [1957], Tuckerman
[2010]. 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 with an external, time-dependent driving,

$H(t)=H_{0}+\lambda Vf(t),$ (32)

where the perturbation operator $V$ does not depend explicitly on time,
$\lambda$ is a real positive constant that quantifies the strength of the
perturbation and the driving force $f(t)$, which can be taken to be $|f|\leq
1$, is assumed to vanish for $t\leq 0$. Let the initial state of the system be
the thermal Gibbs state $\rho_{0}$, relative to the unperturbed Hamiltonian
$H_{0}$, at inverse temperature $\beta$. 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,
$\braket{A}_{t}-\braket{A}_{0}$, where

$\braket{A}_{0}=\operatorname{Tr}{\left(A\rho_{0}\right)},\quad\braket{A}_{t}=\operatorname{Tr}{\left[\rho_{I}(t)A_{I}(t)\right]}.$
(33)

Here, the operators are evaluated in the interaction picture

$A_{I}(t)=U^{\dagger}_{0}(t)AU_{0}(t),\quad\rho_{I}(t)=U_{V}(t)\rho_{0}U_{V}^{\dagger}(t),$
(34)

with

$U_{0}(t)=e^{-iH_{0}\frac{t}{\hbar}},\quad
U_{V}(t)=\mathcal{T}e^{-i\frac{\lambda}{\hbar}\int_{0}^{t}V_{I}(s)f(s)ds}.$
(35)

To the first order in $\lambda$, the linear response is given by the
celebrated Kubo formula Kubo [1957]

$\braket{A}_{t}-\braket{A}_{0}\simeq-i\frac{\lambda}{\hbar}\int_{0}^{t}\braket{[A_{I}(t-s),V_{I}(0)]}_{0}f(s)ds.$
(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, model-independent 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

$\chi_{AV}(t)=-i\frac{1}{\hbar}\theta(t)\braket{[A_{I}(t),V_{I}(0)]}_{0},$
(37)

where $\theta(t)$ is the Heaviside function, as

$\braket{A}_{t}-\braket{A}_{0}=\lambda\int_{-\infty}^{\infty}\chi_{AV}(t-s)f(s)ds.$
(38)

The dynamical susceptibility $\chi_{AV}(t)$, which vanishes for 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
$|\braket{[A,V]}|\leq 2\Delta A\Delta V$, which yields the constant bound

$|\chi_{AV}(t)|\leq\frac{2}{\hbar}\theta(t)\Delta_{0}A\Delta_{0}V,$ (39)

where $\Delta^{2}_{0}A$ is the variance with respect to $\rho_{0}$. As the
latter is time-independent in the interaction picture, we drop the subscript
$I$. Alternatively, one can apply the Bogoliubov inequality Ueda [2010]

$|\braket{[A,V]}_{0}|^{2}\leq\frac{\braket{A^{2}}_{0}\braket{[V,[H_{0},V]]}_{0}}{k_{B}T}.$
(40)

This yields a different upper bound with an explicit dependence on the
temperature $T$

$|\chi_{AV}(t)|\leq\frac{2\theta(t)}{\hbar}\sqrt{\frac{T}{T_{B}}}\Delta_{0}V\Delta_{0}A,$
(41)

where we have defined the characteristic temperature

$T_{B}\equiv\frac{\braket{A^{2}}_{0}\braket{[V,[H_{0},V]]}_{0}}{4k_{B}(\Delta_{0}A\Delta_{0}V)^{2}}.$
(42)

We see that at low temperature $T\leq T_{B}$ 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
Mazenko [2006]. In such case, being $A=V$, the dynamical susceptibility
$\chi_{VV}(t)$ becomes proportional to the anti-symmetrized autocorrelation
function $\braket{[V_{I}(t),V_{I}(0)]}_{0}=2i\operatorname{Im}C_{V}(t)$, and
therefore one can make use of the techniques that we have illustrated above to
bound autocorrelations functions by means of QSLs for operators. The ML QSL
(22) yields the following upper bound on the dynamical susceptibility

$|\chi_{VV}(t)|\leq\frac{t}{\tau_{QSL}^{3}}2\theta(t)\hbar,$ (43)

where we have introduced a new time scale
$\tau_{QSL}=\hbar\braket{V\\{H_{0}-E_{0},V\\}}^{-1/3}_{0}$. Unlike the
previous bounds (39) and (41), Eq. (43) contains an explicit linear dependance
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 $T\leq T_{B}$, the QSL bound (43) ceases
to be the tightest one at the time

$\tau_{B}=\Big{(}\frac{\Delta_{0}V}{\hbar}\Big{)}^{2}\sqrt{\frac{T}{T_{B}}}\tau^{3}_{QSL}$
(44)

and for $t\geq\tau_{B}$ the Bogoliubov bound (41) gives the better
description. By contrast, at high temperatures $T\geq T_{B}$, the crossover is
between the QSL bound and the Heisenberg bound (39) and it occurs at time

$\tau_{H}=\Big{(}\frac{\Delta_{0}V}{\hbar}\Big{)}^{2}\tau^{3}_{QSL}.$ (45)

The bounds (39),(41) and (43) find applications in many experimentally
relevant settings Mazenko [2006], of which we give two concrete examples
below.

###  IV.1 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 $\overrightarrow{E}(t)$ Mazenko [2006]. In this case, the preferred
observable to characterize the response of the system is the current flow
$\overrightarrow{J}$ and the corresponding dynamical susceptibility is the
electrical conductivity. The perturbed Hamiltonian takes the form

$H(t)=H_{0}-\overrightarrow{R}\cdot\overrightarrow{E}(t),$ (46)

where $\overrightarrow{R}$ is the electric dipole moment in the origin

$\overrightarrow{R}\equiv\sum_{n}^{N}q_{n}\overrightarrow{r}_{n},$ (47)

with $q_{n}$ and $\overrightarrow{r}_{n}$ being the charge and the position of
the $n$-th particle. The net current vanishes at equilibrium,
$\braket{\overrightarrow{J}}_{0}=0$, and the perturbed expectation value of
the $i$-th spatial component $J_{i}$ at time $t\geq 0$ is expressed, at the
linear order, as Mazenko [2006]

$\braket{J_{i}}_{t}=\sum_{j}\int_{-\infty}^{\infty}\sigma_{ij}(t-s)E_{j}(s)\,ds,$
(48)

where $\sigma_{ij}\equiv\chi_{J_{i}R_{j}}$ is the electrical conductivity
tensor. By using the bounds (39) and (41) we are able to derive the following
constraints on $\sigma_{ij}$:

$\displaystyle|\sigma_{ij}(t)|$
$\displaystyle\leq\frac{2}{\hbar}\theta(t)\Delta_{0}J_{i}\Delta_{0}R_{j},$
(49) $\displaystyle|\sigma_{ij}(t)|$
$\displaystyle\leq\frac{\theta(t)}{\hbar}\sqrt{\frac{\braket{J_{i}^{2}}_{0}\braket{[R_{j},[H_{0},R_{j}]]}_{0}}{k_{B}T}}.$
(50)

Another experimental application in which the theory of linear response
provides a useful approach is given by magnetic resonance experiments Pake
[1962]. In this case, the central quantity is the magnetic susceptibility.
Consider a paramagnetic system, initially aligned along a constant magnetic
field $\overrightarrow{B}$ and subsequently perturbed with a weak time-
dependent field $\overrightarrow{h}(t)$, so that the total Hamiltonian reads

$H(t)=-\overrightarrow{M}\cdot(\overrightarrow{B}+\overrightarrow{h}(t)).$
(51)

The magnetization of the system is perturbed from its initial equilibrium
value $\braket{\overrightarrow{M}}_{0}=\chi_{0}\overrightarrow{B}$, where
$\chi_{0}$ is the static susceptibility, and the linear response of its $i$-th
component can be expressed using the Kubo formula

$\braket{M_{i}}_{t}-\braket{M_{i}}_{0}=\sum_{j}\int_{-\infty}^{\infty}\chi_{M,ij}(t-s)h_{j}(s)\,ds,$
(52)

where $\chi_{M,ij}(t)\equiv\chi_{M_{i}M_{j}}(t)$ is the magnetic
susceptibility tensor. The bounds (39) and (41) now yield

$\displaystyle|\chi_{M,ij}(t)|$
$\displaystyle\leq\frac{2}{\hbar}\theta(t)\Delta_{0}M_{i}\Delta_{0}M_{j},$
(53) $\displaystyle|\chi_{M,ij}(t)|$
$\displaystyle\leq\frac{\theta(t)}{\hbar}\sqrt{\frac{\braket{M_{i}^{2}}_{0}\braket{[M_{j},[H_{0},M_{j}]]}_{0}}{k_{B}T}}.$
(54)

Moreover, the diagonal magnetic susceptibilities can be upper bounded through
the QSL approach (43) to find

$|\chi_{M,ii}(t)|\leq\frac{2}{\hbar^{2}}\theta(t)\braket{M_{i}^{\dagger}\\{H_{0}-E_{0},M_{i}\\}}t.$
(55)

##  V 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 $\theta$ that parameterizes the unitary flow,
of a given quantum state $\rho$, generated by the operator $O$. In other
words, $F_{Q}$ measures the distinguishability of the “initial” state
$\rho_{0}$ from the one transformed by the unitary flow

$\rho_{\theta}=e^{-i\theta O}\rho_{0}e^{i\theta O}.$ (56)

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 $\theta$) of a given quantum state
$\rho$ under the action of $O$, that generates the state transformation.
Remarkably, these two approaches are closely related, as the dynamical
susceptibility $\chi_{OO}$ obtained using the first framework can be related
to the quantum Fisher information $F_{Q}$ Hauke et al. [2016].

More precisely, for a thermal state $\rho$ at temperature $T$, the following
result on the quantum Fisher information has been shown by Hauke et. al in
Hauke et al. [2016]

$F_{Q}(T)=-\frac{4}{\pi}\int_{0}^{\infty}d\omega\tanh\big{(}\frac{\hbar\omega}{2k_{B}T}\big{)}\operatorname{Im}\tilde{\chi}_{OO}(\omega,T),$
(57)

where $\tilde{\chi}_{OO}(\omega,T)$ is the Fourier-transformed dynamical
susceptibility, defined as

$\tilde{\chi}_{OO}(\omega,T)=\int_{0}^{\infty}e^{i\omega
t}\chi_{OO}(t,T)\,dt,$ (58)

with $\chi_{OO}(t,T)$ being defined in Eq. (37) for $A=V=O$.

Now, the upper bound (24) we have derived above on the anti-symmetrized
autocorrelation function $\operatorname{Im}C_{O}$, together with the result
(57), provides an upper bound on the quantum Fisher information $F_{Q}$. To
this end, let us reverse the order of the integrals in Eq. (57) and perform
first the one in $\omega$. Computing the inverse Fourier transform of
$\tanh(\omega/2T)$ and using the fact that
$\chi_{OO}(t,T)=2\theta(t)\operatorname{Im}C_{O}(t,T)$ yields

$F_{Q}(T)=-\frac{16k_{B}T}{\hbar}\int_{0}^{\infty}dt\,\operatorname{csch}\big{(}\pi
k_{B}T\frac{t}{\hbar})\operatorname{Im}C_{O}(t,T).$ (59)

Therefore, using that $\int_{0}^{\infty}dx\,x\operatorname{csch}(\pi
qx)=(8q^{2})^{-1}$ $\forall q>0$, from our previous result (24) we derive the
following upper bound on the temperature-dependent quantum Fisher information

$|F_{Q}(T)|\leq\frac{4}{k_{B}T}\braket{O\\{H-E_{0},O\\}},$ (60)

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)
given by the Heisenberg and Bogoliubov inequalities are divergent, due to the
divergence of $\operatorname{csch}(x)$ for $x\to 0$. Conversely, the time-
linear dependence introduced by the QSL approach guarantees the convergence of
the integral, yielding a finite upper bound on the quantum Fisher information.
Making use of the celebrated Cramer-Rao bound
$(\Delta\theta)^{2}\geq(MF_{Q})^{-1}$, Eq. (60) results in a lower bound on
the variance of the parameter $\theta$ for $M$ independent measurements

$(\Delta\theta)^{2}\geq\frac{k_{B}T}{4}\frac{1}{\braket{O\\{H-E_{0},O\\}}}.$
(61)

The precision can be enhanced by decreasing the temperature and by considering
operators that develop stronger autocorrelations, since these lower bound
(e.g., through Eq.(22)) the thermal correlation function
$\braket{O\\{H-E_{0},O\\}}$ appearing at the denominator.

##  VI 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 Braunstein et al. [1996].

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.

##  VII Aknowledgements

It is a pleasure to acknowledge discussions with Pablo Martinez-Azcona, Norman
Margolus, Jing Yang and Apollonas S. Matsoukas-Roubeas.

##  Appendix A Deriving QSLs through a trigonometric approach

In this appendix we show an alternative derivation that results into 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

$\cos{x}\geq 1-\frac{2}{\pi}x-\frac{2}{\pi}\sin{x},$ (62)

valid for $x>0$, we obtain

$\begin{split}&\operatorname{Re}\braket{O_{0}}{O_{t}}=\frac{1}{\|O\|^{2}}\sum_{j>k}\cos(\frac{\Delta_{jk}t}{\hbar})(|O_{jk}|^{2}+|O_{kj}|^{2})\geq\\\
&\frac{1}{\|O\|^{2}}\sum_{j>k}\big{(}1-\frac{2}{\pi}\frac{\Delta_{jk}t}{\hbar}-\frac{2}{\pi}\sin{\frac{\Delta_{jk}t}{\hbar}}\big{)}(|O_{jk}|^{2}+|O_{kj}|^{2}).\end{split}$

Therefore, being $\sin x\leq x$ for $x\geq 0$, we find

$\operatorname{Re}\braket{O_{0}}{O_{t}}\geq
1-\frac{4}{\pi}\braket{}{\mathbb{L}}{}t,$ (63)

that is,

$t\geq\frac{\pi}{4}\frac{1-\operatorname{Re}\braket{O_{0}}{O_{t}}}{\braket{}{\mathbb{L}}{}}=\frac{\pi}{4}\frac{1-\cos{\mathcal{L}_{t}}}{\braket{}{\mathbb{L}}{}},$
(64)

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

$\cos{x}\geq 1-\frac{4}{\pi^{2}}x\sin{x}-\frac{2}{\pi^{2}}x^{2},$ (65)

which holds again $\forall x$. Combining it with the bound (7) on
$\operatorname{Re}\braket{O_{0}}{\dot{O}_{t}}$, we obtain

$\begin{split}&\operatorname{Re}\braket{O_{0}}{O_{t}}\geq\frac{1}{\|O\|^{2}}\sum_{j,k}|O_{jk}|^{2}[1-\frac{4}{\pi^{2}}\frac{\Delta_{jk}t}{\hbar}\sin{\frac{\Delta_{jk}t}{\hbar}}-\\\
&\frac{2}{\pi^{2}}(\frac{\Delta_{jk}t}{\hbar})^{2}]=1+\frac{4}{\pi^{2}}\operatorname{Re}\braket{O_{0}}{\dot{O}_{t}}t-\frac{2}{\pi^{2}}(\Delta\mathbb{L})^{2}t^{2}\geq\\\
&1-\frac{6}{\pi^{2}}(\Delta\mathbb{L})^{2}t^{2},\end{split}$

that is,

$t\geq\frac{\pi}{\sqrt{6}}\frac{\sqrt{1-\operatorname{Re}\braket{O_{0}}{O_{t}}}}{\Delta\mathbb{L}}=\frac{\pi}{\sqrt{6}}\frac{\sqrt{1-\cos{\mathcal{L}_{t}}}}{\Delta\mathbb{L}}.$
(66)

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.

##  Appendix 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 Mandelstam and Tamm [1945] and ML Margolus and Levitin [1998]
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 $\ket{\psi_{0}}$,
$O_{0}=\ket{\psi_{0}}\bra{\psi_{0}}$, and let the expansion of
$\ket{\psi_{0}}$ in the energy eigenbasis be

$\ket{\psi_{0}}=\sum_{j}c_{j}\ket{j},$ (67)

where $H\ket{j}=E_{j}\ket{j}$ and $\sum_{j}|c_{j}|^{2}=1$. Then, the
vectorization of $O_{0}$ is
$\ket{O_{0}}=\sum_{jk}c^{*}_{j}c_{k}\ket{j}\ket{k}$, with $\|O_{0}\|=1$. Since
$\ket{\psi_{t}}=U_{t}\ket{\psi_{0}}$, where $U_{t}=e^{-iHt/\hbar}$, in order
to reproduce the forward time-evolution of the quantum state we need to evolve
the operator backwards in time:

$O_{-t}=U^{\dagger}_{-t}O_{0}U_{-t}=U_{t}\ket{\psi_{0}}\bra{\psi_{0}}U^{\dagger}_{t}=\ket{\psi_{t}}\bra{\psi_{t}}.$
(68)

The operator overlap then reduces to the square modulus of the state overlap,
that is, to the Uhlmann fidelity

$\braket{O_{0}}{O_{-t}}=\operatorname{Tr}{\left(O_{0}^{\dagger}O_{-t}\right)}=|\braket{\psi_{0}}{\psi_{t}}|^{2}.$
(69)

Moreover, being the Bures angle between states defined as

$\ell_{t}=\arccos|\braket{\psi_{0}}{\psi_{t}}|,$ (70)

we have that $1-\braket{O_{0}}{O_{-t}}=\sin^{2}{\ell_{t}}$. We observe that
the operator angle $\mathcal{L}_{t}$ defined in Eq. (4) does not reduce to the
standard Bures angle between states, being
$\mathcal{L}_{-t}=\arccos{|\braket{\psi_{0}}{\psi_{t}}|^{2}}$ when
$O_{0}=\ket{\psi_{0}}\bra{\psi_{0}}$.

The typical maximal speeds of the flow, given by the ML (10) and MT (11) QSLs
respectively, can be written as

$\begin{split}\braket{}{\mathbb{L}}{}&=\frac{1}{\hbar}\sum_{jk}|\Delta_{jk}||c_{j}|^{2}|c_{k}|^{2},\\\
(\Delta\mathbb{L})^{2}&=\frac{1}{\hbar^{2}}\sum_{jk}\Delta^{2}_{jk}|c_{j}|^{2}|c_{k}|^{2}.\end{split}$
(71)

From the second equation, being also

$(\Delta
H)^{2}=\sum_{j}E_{j}^{2}|c_{j}|^{2}-\sum_{jk}E_{j}E_{k}|c_{j}|^{2}|c_{k}|^{2},$
(72)

we derive a proportionality relation between the Hamiltonian and Liouvillian
variances (over $\ket{\psi_{0}}$ and $\ket{O_{0}}$, respectively):

$(\Delta\mathbb{L})^{2}=2\frac{(\Delta H)^{2}}{\hbar^{2}}.$ (73)

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:

$t\geq\frac{\hbar}{\Delta H}\sin{\ell_{t}}.$ (74)

In the case of orthogonal evolution $\braket{\psi_{0}}{\psi_{\tau}}=0$, we
have $\sin{\ell_{\tau}}=1$ and therefore we obtain a bound proportional to
(though weaker than) the standard Mandelstam-Tamm QSL Mandelstam and Tamm
[1945],

$\tau\geq\tau_{MT}=\frac{\pi\hbar}{2\Delta H}>\frac{\hbar}{\Delta H},$ (75)

thus justifying the interpretation of Eq. (11) as a MT-type of QSL for
operators.

On the other hand, the mean energy can be rewritten as

$\braket{H}=\sum_{j}E_{j}|c_{j}|^{2}=\frac{1}{2}\sum_{jk}(E_{j}+E_{k})|c_{j}|^{2}|c_{k}|^{2},$
(76)

which, by using that $|\Delta_{jk}|=|E_{j}-E_{k}|\leq E_{j}+E_{k}-2E_{0}$,
implies

$\braket{}{\mathbb{L}}{}\leq 2\frac{\braket{H}-E_{0}}{\hbar}.$ (77)

Therefore, we can recast the bound (10) as

$t\geq\frac{\hbar}{2\alpha}\frac{\sin^{2}{\ell_{t}}}{\braket{H}-E_{0}},$ (78)

which represents a generalization of the ML QSL Margolus and Levitin [1998] to
the case of arbitrary angle between the initial and final states. A similar
generalization was also claimed in reference Deffner and Lutz [2013a], though
their bound is not given in terms of $\braket{H}-E_{0}$, but rather in terms
of the quantity $|\braket{H}|$. To our knowledge, the generalization of the ML
bound to arbitrary angles between initial and final states was only proven
numerically Giovannetti et al. [2003, 2004], Deffner and Campbell [2017]. Let
us note that, by considering again the orthogonalization time $t=\tau$, our
QSL becomes

$\tau\geq\tau_{ML}>\frac{\hbar}{2\alpha(\braket{H}-E_{0})}\approx
0.44\,\tau_{ML},$ (79)

where $\tau_{ML}={\pi\hbar}/{2(\braket{H}-E_{0})}$ is the well known ML QSL
Margolus and Levitin [1998] 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 (78), which
can be recast as a lower bound on the state overlap

$|\braket{\psi_{0}}{\psi_{t}}|\geq 1-2\alpha(\braket{H}-E_{0})t.$ (80)

In Fig. 4, we test the bound (80) for a single realization of a random matrix
Hamiltonian $H$ of dimension $d=50$, generated from the Gaussian Orthogonal
Ensemble (GOE) with variance $\sigma=1$. The initial state is chosen to be the
coherent Gibbs state at inverse temperature $\beta$

$\ket{\psi_{0}}=\frac{1}{\sqrt{Z}}\sum_{n=1}^{d}e^{-\beta\frac{E_{n}}{2}}\ket{n},$
(81)

where $Z=\sum_{n}e^{-\beta E_{n}}$ is the partition function. In this setting,
the quantum state overlap can be conveniently rewritten in terms of the
analitically-continued partition function as follows del Campo et al. [2017],
Xu et al. [2021]

$\braket{\psi_{0}}{\psi_{t}}=\frac{Z(\beta+it)}{Z(\beta)}.$ (82)

Deviations from the bound appear to be significant after the characteristic
timescale $\tau=(\sigma\sqrt{8d})^{-1}$, where $\sigma\sqrt{8d}$ 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.
After the initial decay shown in Fig. 4, the fidelity undergoes an oscillatory
behavior that is no longer captured by the QSL (80).

Figure 4: The coherent Gibbs state overlap undergoes an initial decay which is
qualitatively captured by the ML QSL for states (80). Deviations from the
bound appear after the characteristic timescale $\tau=(\sigma\sqrt{8d})^{-1}$,
which corresponds to the largest frequency of the spectrum. The parameter
chosen for the simulation are $\sigma=1$, $d=50$ and inverse temperature
$\beta=10$.

##  Appendix 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 flow $O_{t}=U^{\dagger}_{t}O_{0}U_{t}$ is time
dependent, that is

$U_{t}=\mathcal{T}\exp{-\frac{i}{\hbar}\int_{0}^{t}H(s)\,ds}.$ (83)

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 $\mathcal{T}$ in front
of the exponential. In addition, if we make the strongest assumption that
$H_{t}$ remains diagonal in the initial eigenbasis,

$H_{t}=\sum_{j}E_{j}(t)\ket{j}\bra{j},$ (84)

where only the eigenvalues $E_{j}(t)$ depend on time, then we can write the
action of $U_{t}$ over the energy eigenvector $\ket{j}$ as

$U_{t}\ket{j}=e^{-\frac{i}{\hbar}\int_{0}^{t}E_{j}(s)ds}\ket{j}.$ (85)

Under this assumption, the operator overlap can be expanded as

$\braket{O_{0}}{O_{t}}=\frac{1}{\|O\|^{2}}\sum_{j,k}e^{i\frac{\overline{\Delta_{jk}}(t)}{\hbar}t}|O_{jk}|^{2},$
(86)

so that, making use of the inequality $\cos x\geq 1-\alpha|x|$ Andersson
[2019] as in the main text, we derive a ML bound on operator flows in the case
of driven dynamics,

$t\geq\frac{1-\operatorname{Re}\braket{O_{0}}{O_{t}}}{\alpha\overline{\braket{}{\mathbb{L}}{}}}=\frac{1-\cos{\mathcal{L}_{t}}}{\alpha\overline{\braket{}{\mathbb{L}}{}}},$
(87)

where $\overline{f}(t)=\frac{1}{t}\int_{0}^{t}f(u)du$ is the time average at
time $t$. As a consequence, by choosing $O_{0}=\ket{\psi_{0}}\bra{\psi_{0}}$,
we obtain a generalization of the ML QSL for state evolution under driven
Hamiltonians and arbitrary angles:

$t\geq\frac{\hbar}{2\alpha}\frac{\sin^{2}{\ell_{t}}}{\overline{\braket{H}-E_{0}}},$
(88)

valid when the energy eigenvectors are stationary. A related result was also
claimed in reference Deffner and Lutz [2013b], but their derivation was later
shown to generalize the MT QSL rather than the ML one, as pointed out in
Okuyama et al. [2018].

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