# Work Statistics, Loschmidt Echo and Information Scrambling in Chaotic
Quantum Systems

Aurélia Chenu<EMAIL_ADDRESS>http://dipc.ehu.es/chenu Donostia International
Physics Center, E-20018 San Sebastián, Spain  IKERBASQUE, Basque Foundation
for Science, E-48013 Bilbao, Spain  Theory Division, Los Alamos National
Laboratory, MS-B213, Los Alamos, NM 87545, USA  Massachusetts Institute of
Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA  Javier Molina-
Vilaplana  Technical University of Cartagena, UPCT, 30202, Cartagena, Spain
Adolfo del Campo  Donostia International Physics Center, E-20018 San
Sebastián, Spain  IKERBASQUE, Basque Foundation for Science, E-48013 Bilbao,
Spain  Theory Division, Los Alamos National Laboratory, MS-B213, Los Alamos,
NM 87545, USA  Department of Physics, University of Massachusetts, Boston, MA
02125, USA

###### Abstract

Characterizing the work statistics of driven complex quantum systems is
generally challenging because of the exponential growth with the system size
of the number of transitions involved between different energy levels. We
consider the quantum work distribution associated with the driving of chaotic
quantum systems described by random matrix Hamiltonians and characterize
exactly the work statistics associated with a sudden quench for arbitrary
temperature and system size. Knowledge of the work statistics yields the
Loschmidt echo dynamics of an entangled state between two copies of the system
of interest, the thermofield double state. This echo dynamics is dictated by
the spectral form factor. We discuss its relation to frame potentials and its
use to assess information scrambling.

Quantum thermodynamics has become a research field in bloom, building on the
dialogue between technological progress and foundations of physics [1, 2]. In
the presence of thermal and quantum fluctuations, familiar concepts from
traditional thermodynamics at the macroscale, such as heat and work, become
stochastic variables. Their analysis has been a fertile ground of inquiry,
leading to the discovery of fluctuation theorems [3, 4, 5, 6, 7] and time-work
uncertainty relations [8, 9, 10].

In the quantum domain, even for isolated systems governed by unitary dynamics,
work is not considered to be an observable. The definition of work requires
two projective measurements, one at the beginning and another at the end of
the physical process under consideration [6, 11]. Measuring the work
statistics done by or on a driven system requires analyzing the transitions
between its energy levels at the beginning and at the end of the process. In
simple quantum systems, this is an amenable task and pioneering experiments
have measured the quantum work statistics, e.g., in coupled qubits in a NMR
setting [12], a driven quantum oscillator describing the dynamics of a trapped
ion [13], and ultracold atoms [14].

For complex quantum systems, determining the work probability density function
associated with a given process is intrinsically challenging, due to the
exponential growth in the number of transitions with the dimension of the
Hilbert space. Progress in this direction has been achieved avoiding their
explicit computation. To this end, it has proved useful to recast the
characteristic function of the work distribution function as a Loschmidt echo
amplitude [15]. This interpretation has facilitated the study of quantum work
statistics in driven many-body spin systems [15, 16, 17] and quantum fluids
[18].

Among complex systems, those exhibiting quantum chaos constitute a
paradigmatic reference example. Following pioneering work by Wigner and Dyson,
quantum chaotic systems are often defined as systems whose spectral properties
are well described by random-matrix Hamiltonian ensembles [19, 20]. Random-
matrix Hamiltonians can actually characterize a variety of physical systems
ranging from atomic nuclei to spin chains. Quantum chaos plays as well a
prominent role in black hole physics, as it is widely believed that quantum
correlations in these systems are scrambled at a maximum rate saturating an
analogue of the Lyapunov exponent [21]. It should be noted that, very
recently, preliminary steps have been taken to study work statistics in
quantum chaotic systems [22, 23, 24].

In addition, black hole physics has motivated recent progress to characterize
information scrambling, i.e., the spreading of initially localized quantum
information across different degrees of freedom in many-body systems.
Approaches to this end include the study of the decay dynamics of certain
entangled states (the thermofield double state, associated with the
purification of a thermal density matrix) [25, 26, 27], related analytically-
continued partition functions [28, 29], out-of-time-order correlators [21] and
frame potentials [30], some of which have been related to work statistics [24,
31, 32].

In this work, we characterize the work statistics when a quantum chaotic
system is driven between two different chaotic Hamiltonians. We show that the
characteristic function of the work probability distribution of a thermal
state is equivalent to the Loschmidt echo of an entangled state. Using this
relation, we show that the work statistics can be used to elucidate the
dynamics responsible for information scrambling in chaotic quantum systems.
The connection between work statistics and information scrambling is further
discussed by relating the Loschmidt echo to the spectral form factor and frame
potentials.

##  1 Quantum work statistics in chaotic quantum systems

_Work probability distribution function.—_ Let us consider a driven, isolated
system described by the instantaneous Hamiltonian at time $s$

$\displaystyle\hat{H}_{s}=\sum_{n}E_{n}^{s}|n_{s}\rangle\langle n_{s}|\,,$ (1)

where $E_{n}^{s}$ denotes the real energy eigenvalue for the eigenstate
$|n_{s}\rangle$. We focus on the time-evolution of a system initially prepared
in a mixed state $\hat{\rho}$ at time $s=0$ and driven to a final Hamiltonian
$\hat{H}_{\tau}$ at time $s=\tau$. The corresponding time-evolution operator
is

$\displaystyle\hat{U}(\tau)={\cal
T}\exp\left[-i\int_{0}^{\tau}ds\hat{H}_{s}\right]\,,$ (2)

where the time-ordering operator ${\cal T}$ is required whenever the
instantaneous Hamiltonians at two different times of evolution
$0\leq(s,s^{\prime})\leq\tau$ do not commute. The associated work probability
distribution function (work pdf, hereafter) is given by [6, 11]

$p_{\tau}(W):=\sum_{n,m}p^{0}_{n}\;p^{\tau}_{m|n}\delta\left[W-\left(E_{m}^{\tau}-E_{n}^{0}\right)\right]\,.$
(3)

Here, a first projective energy measurement leads to the probability
$p^{0}_{n}=\langle n_{0}|\hat{\rho}|n_{0}\rangle$ to find the initial state
$\hat{\rho}$ in the $n$-th eigenmode of the initial Hamiltonian $\hat{H}_{0}$.
Following the evolution associated with $\hat{U}(\tau)$, a second projective
energy measurement is used to determine the transition probability from
$|n_{0}\rangle$ to $|m_{\tau}\rangle$ given by $p^{\tau}_{m|n}=|\langle
m_{\tau}|\hat{U}(\tau)|n_{0}\rangle|^{2}$. The work associated with such a
transition is set by the energy difference between the associated eigenvalues,
$W=E_{m}^{\tau}-E_{n}^{0}$. The definition of the work pdf in (3) encompasses
all possible transitions with the corresponding probabilities. As already
advanced in the introduction, the need for two projective-energy measurements
prevents the introduction of a Hermitian work operator [11]. However, it is
possible to describe work in terms of a positive-operator value measure (POVM)
[33, 34].

_Work in chaotic quantum systems.—_ Quantum chaotic systems are described by
random matrix Hamiltonians that have been extensively studied following the
seminal works by Wigner and Dyson [19, 20]. We here focus on the quantum work
statistics associated with the driving of a system between two chaotic
Hamiltonians, each independently sampled from the Gaussian unitary ensemble
(GUE), i.e., described by random $N\times N$ Hermitian matrices invariant
under conjugation by unitaries. Here, $N$ is the dimension of the Hilbert
space that can be related to the size of a $N_{p}$ particle system (for
qubits, $N=2^{N_{p}}$).

We shall consider the initial state to be described by the canonical density
matrix of a thermal state at inverse temperature $\beta=(k_{B}T)^{-1}$,

$\displaystyle\hat{\rho}_{\rm th}=\frac{e^{-\beta\hat{H}_{0}}}{Z(\beta)}\,,$
(4)

where $Z(\beta)={\rm Tr}\left(e^{-\beta\hat{H}_{0}}\right)=\sum_{n}e^{-\beta
E_{n}^{0}}$ is the partition function of the initial Hamiltonian
$\hat{H}_{0}$. This density matrix, being diagonal in the energy eigenbasis of
the initial Hamiltonian, is left unchanged by the first projective energy
measurement. The occupation probability of the $n$-initial eigenmode is thus
given by the Boltzmann factor, $p_{n}^{0}=\exp(-\beta E_{n}^{0})/Z(\beta)$
where $E_{n}^{0}$ is the $n$-th eigenstate of the initial Hamiltonian
$\hat{H}_{0}$, that is sampled from GUE. The Hamiltonian at the end of the
evolution $\hat{H}_{\tau}$ is also sampled from the GUE. As a result, we are
led to consider the average work pdf $\langle p_{\tau}(W)\rangle$ that
includes a double GUE average over $\hat{H}_{0}$ and $\hat{H}_{\tau}$,

$\langle
p_{\tau}(W)\rangle=\left\langle\\!\\!\left\langle\sum_{n,m}p^{0}_{n}\;p^{\tau}_{m|n}\delta\left[W\\!-\left(E_{m}^{\tau}-E_{n}^{0}\right)\right]\right\rangle\\!\\!\right\rangle.$
(5)

For its study, we shall use the average of an arbitrary operator $\hat{O}$
over the probability density function $\rho(\hat{H})$ from which realizations
of the chaotic Hamiltonian $\hat{H}$ are sampled in GUE. This average can be
written as [19, 23]

$\displaystyle\langle\hat{O}\rangle$ $\displaystyle\equiv$ $\displaystyle\int
d\hat{H}\rho(\hat{H})\,\hat{O}=\int DE\,\hat{O}(E)\,,$ (6)

where, in terms of the eigenvalues $E_{n}$ of $\hat{H}$, the GUE measure $DE$
has been defined to include the eigenvalue Vandermonde determinant
$\Delta(E)=\prod_{k>j}(E_{k}-E_{j})\,$ and all the Gaussian terms in the pdf.
Therefore, the normalization condition gives the scaling of eigenvalues with
the system size and reads $\int
DE=C_{N}\,\int\prod_{k}dE_{k}|\Delta(E)|^{2}e^{-\sum_{j}E_{j}^{2}}=1$, the
constant $C_{N}$ being given in Ref. [19, 20].

As we have seen, the measurement of work singles out the energy eigenbasis as
a preferred basis. In the following, we shall be able to express all the
ensemble averages using the density of states
$\rho(E)=\sum_{n}\,\delta(E-E_{n})$. When the Hilbert space dimension $N$ is
large, the averaged $\langle\rho(E)\rangle$ is well described by Wigner’s
semicircle law [19]. A common approach in random matrix theory to account for
a finite Hilbert space dimension $N$ relies on perturbative expansions in
$1/N$. However, in the GUE, the exact $\langle\rho(E)\rangle$ for any finite
$N$ is simply read [19],

$\displaystyle\langle\rho(E)\rangle=\sum_{j=0}^{N-1}\varphi_{j}(E)^{2}\,,$ (7)

where the
$\varphi_{j}(E)={(2^{j}j!\sqrt{\pi})^{-1/2}}e^{-\frac{E^{2}}{2}}\mathcal{H}_{j}(E)$
are given in terms of the Hermite polynomials $\mathcal{H}_{j}$. Knowledge of
the density of states will allow us to characterize the work statistics in
systems of arbitrary size. To this end, we shall also need the connected two-
level correlation function [19],

$\displaystyle\left\langle\rho_{c}^{(2)}(E,E^{\prime})\right\rangle$
$\displaystyle=$
$\displaystyle\langle\rho(E)\rho(E^{\prime})\rangle-\langle\rho(E)\rangle\langle\rho(E^{\prime})\rangle$
(8) $\displaystyle=$
$\displaystyle-\left(\sum_{j=0}^{N-1}\varphi_{j}(E)\varphi_{j}(E^{\prime})\right)^{2}\,,$

that accounts for the presence of correlations between different energy
levels.

Thus, the results we obtain in this manuscript are nonperturbative in $N$ and
account exactly for the finiteness of the Hilbert space dimension, which can
be varied from small to arbitrarily large values. This is of particular
importance to describe the quantum work statistics of finite quantum chaotic
systems as well as to explore the connections between work fluctuations and
information scrambling.

###  1.1 Characteristic function and work distribution

The characteristic function of the work pdf is defined as its Fourier
transform

$\displaystyle\chi(t,\tau)=\int_{-\infty}^{\infty}dWp_{\tau}(W)\,e^{iWt}\,.$
(9)

Using the definition of $p_{\tau}(W)$, the explicit form of the characteristic
function can be written as

$\displaystyle\chi(t,\tau)=\sum_{n}p_{n}^{0}\langle
n_{0}|e^{it\hat{H}_{\tau}^{\rm eff}}e^{-it\hat{H}_{0}}|n_{0}\rangle\,,$ (10)

which suggests an interpretation of the variable $t$ as a second time of
evolution – different from the physical time $s$ – with a quench from the
initial Hamiltonian to the effective Hamiltonian [15, 22, 35, 24]

$\displaystyle\hat{H}_{\tau}^{\rm
eff}=\hat{U}^{\dagger}(\tau)\hat{H}_{\tau}\hat{U}(\tau)\,,$ (11)

defined in terms of the time-evolution operator $\hat{U}(\tau)$ in Eq. (2).

In what follows, we aim at deriving an exact analytical solution of the
characteristic function valid at finite $N$. We shall focus on the average of
this quantity for Hamiltonians sampled from the GUE. Specifically, for an
initial thermal state $\hat{\rho}_{\rm th}$, the characteristic function of
the work distribution associated with a sudden quench protocol between two GUE
Hamiltonians can be written as

$\left\langle\chi(t,\tau)\right\rangle=\left\langle\\!\\!\left\langle\frac{1}{Z(\beta)}{\rm
Tr}\left(e^{-\sigma_{\tau}\hat{H}_{\tau}}\,e^{-\sigma_{0}\hat{H}_{0}}\right)\right\rangle\\!\\!\right\rangle\,,$
(12)

with $\sigma_{0}=\beta+it,\,\sigma_{\tau}=-it$, and where the brackets denote
a double average over the initial and final GUE ensembles.

Using the spectral resolution of the initial and final Hamiltonians, we find

$\displaystyle\left\langle\chi(t,\tau)\right\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\langle Z(\beta)\rangle}\times$
$\displaystyle\left\langle\\!\\!\left\langle\sum_{n,m}|\langle
m_{\tau}|n_{0}\rangle|^{2}e^{-\sigma_{\tau}E_{m}^{\tau}}e^{-\sigma_{0}E_{n}^{0}}\right\rangle\\!\\!\right\rangle\,.$

Note that in going from (12) to (1.1), the average of the fraction has been
approximated as the ratio of the numerator and denominator averages. This kind
of approximation is known as annealing in the context of random matrix theory.
The resulting ‘annealed’ expression of the characteristic function is required
for an analytical treatment, and we verify numerically that it is a good
approximation. Introducing the density of states
$\rho(E)=\sum_{n}\,\delta(E-E_{n})$ and using the average of the transition
probability over the GUE $\left\langle p^{\tau}_{m|n}\right\rangle=1/N$ [23],
we find

$\displaystyle\left\langle\chi(t,\tau)\right\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{N\,\langle Z(\beta)\rangle}\iint
dE_{0}\,dE_{\tau}\,\langle\rho(E_{0})\rangle\,\langle\rho(E_{\tau})\rangle\,$
(14) $\displaystyle\times e^{-\sigma_{\tau}E_{\tau}}\,e^{-\sigma_{0}E_{0}}\,.$

A compact form can be derived by defining the Laplace transform of the density
of states averaged over a GUE,

$\displaystyle\mathcal{I}(\sigma)=\int dE\,\langle\rho(E)\rangle\,e^{-\sigma
E}\,.$ (15)

It is worth pointing out that the related real Fourier transform of the
density of states is used as a tool to measure statistical level properties of
systems with a complex energy spectrum [36, 37, 38, 39].

The explicit evaluation of (15) for finite $N$ requires the expression for the
GUE density of states (7), and leads to

$\displaystyle\mathcal{I}(\sigma)=e^{\frac{\sigma^{2}}{4}}L_{N-1}^{1}\left(-\tfrac{\sigma^{2}}{2}\right)=\langle
Z(\sigma)\rangle\,,$ (16)

where $L_{n}^{\alpha}(x)=\sum_{i=0}^{n}\binom{n+\alpha}{n-i}(-x)^{i}/i!$
denotes the generalized Laguerre polynomial. Physically, $\mathcal{I}(\sigma)$
corresponds to the average of the analytically continued partition function
[27]. The GUE averaged partition function is readily obtained as,

$\displaystyle\langle
Z(\beta)\rangle=\mathcal{I}(\beta)\xrightarrow[\beta\rightarrow
0]{}L^{1}_{N-1}(0)=N\,,$ (17)

where we verify that in the infinite-temperature limit the partition function
measures the Hilbert space dimension $N$. Thus, we obtain the following
compact and explicit expression for the characteristic function

$\displaystyle\left\langle\chi(t,\tau)\right\rangle=\frac{1}{N}\frac{\langle
Z(\sigma_{0})\rangle\langle Z(\sigma_{\tau})\rangle}{\langle Z(\beta)\rangle}$
(18) $\displaystyle=\frac{1}{N}e^{\frac{1}{2}(\beta^{2}-t^{2})}e^{i\frac{\beta
t}{2}}\frac{L^{1}_{N-1}(-\frac{(\beta+it)^{2}}{2})L^{1}_{N-1}(\frac{t^{2}}{2})}{L^{1}_{N-1}(-\frac{\beta^{2}}{2})}\,.$

Note that the characteristics function in the limit of large $N$ has also been
studied in [40].

In the infinite temperature limit, it simplifies to
$\left\langle\chi(t,\tau)\right\rangle=e^{-t^{2}/2}[L^{1}_{N-1}(t^{2}/2)]^{2}/N^{2}$
and can be written in the form of a Gaussian multiplied by a polynomial of
even powers up to $2(N-1)$. A log-log representation exhibits a number of dips
that corresponds to the positive roots. The number of dips is thus $(N-1)$,
before a final monotonic decay. Figure 1 presents the analytical results for
different temperatures and Hilbert space dimensions. It illustrates the
importance of the closed-form expression (18) over the numerical simulations
that exhibit poor convergence properties.

Figure 1: Characteristic function of the work pdf for a sudden quench between
two GUE Hamiltonians. $\langle\chi(t,\tau)\rangle$ from the analytical
solution (solid line, Eq. 18), and as obtained numerically averaging over
50000 realizations of the GUE at infinite temperature (red) and $\beta=1$
(blue). The numerical solutions converge very slowly and tend to 0 at large
time. The dimension of the Hilbert space is (a) $N=5$ and (b) $N=20$.

From the characteristic function in Eq. (18), the average work probability
distribution can be obtained by inverting the Fourier transform (9). The work
pdf is found to be given by the auto-convolution of the average density of
states $\langle\rho(E)\rangle$ with an exponential weight,

$\langle
p(W)\rangle=\frac{1}{N\mathcal{I}(\beta)}\int\limits_{-\infty}^{\infty}dE\langle\rho(E)\rangle\langle\rho(E{+}W)\rangle
e^{{-}\beta E}\,.$ (19)

We present the average work pdf in Figure 2, and verify that results obtained
from the numerical integration of the above equation coincide with those
obtained from direct sampling over many realizations of the GUE. In the
infinite temperature limit, the even parity of the density of states
$\langle\rho(E)\rangle=\langle\rho(-E)\rangle$ carries over the work pdf,
$\langle p(W)\rangle=\langle p(-W)\rangle$. As a result, any odd moment of the
distribution $\langle W^{n}\rangle=\int dWp(W)W^{n}$, including the mean work
($n=1$), vanishes identically in this limit. Away from it, the work pdf is
described by a bell-shaped function whose mean shifts from zero to higher
positive values as the temperature of the initial canonical state is
decreased.

Figure 2: Work statistics in a chaotic quantum system driven by a sudden
quench. Work pdf $\langle p(W)\rangle$ obtained from the analytical expression
Eq. (19) (circles), and the numerical solution averaging over 50000
realizations of the GUE (solid lines), at infinite temperature (red) and
$\beta=1$ (blue). The vertical dashed lines mark the corresponding average
work given in Eq. (22). We verify the symmetric form at infinite temperature,
yielding an average work $\langle W\rangle=0$. The dimension of the Hilbert
space is (a) $N=5$ and (b) $N=20$.

###  1.2 Mean work in a random quench

The first moment of the work pdf, the mean work, is of both practical and
fundamental interest. It controls the performance of quantum thermodynamic
devices such as heat engines and refrigerators. In addition, it can be used to
assess irreversibility and inner friction of quantum thermodynamic processes
[41, 42].

The mean of the work pdf is directly given by the difference between the
initial and final mean energies,

$\displaystyle\langle W\rangle$ $\displaystyle=$
$\displaystyle\langle\hat{H}_{\tau}\rangle-\langle\hat{H}_{0}\rangle$
$\displaystyle=$ $\displaystyle\left\langle\\!\\!\left\langle{\rm
Tr}\left[(\hat{H}_{\tau}-\hat{H}_{0})\frac{e^{-\beta\hat{H}_{0}}}{{\rm
Tr}\left(e^{-\beta\hat{H}_{0}}\right)}\right]\right\rangle\\!\\!\right\rangle\,,$

which involves a double GUE average over $\hat{H}_{0}$ and $\hat{H}_{\tau}$.
The mean final energy vanishes identically, i.e.,

$\displaystyle\langle\hat{H}_{\tau}\rangle$ $\displaystyle=$
$\displaystyle\left\langle\\!\\!\left\langle{\rm
Tr}\left[\hat{H}_{\tau}\frac{e^{-\beta\hat{H}_{0}}}{{\rm
Tr}\left(e^{-\beta\hat{H}_{0}}\right)}\right]\right\rangle\\!\\!\right\rangle=0\,,$
(21)

because of the spectral symmetry of the final ensemble. The mean work is thus
set exclusively by the initial mean energy and reads

$\displaystyle\langle W\rangle=-\,\langle\hat{H}_{0}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\langle Z(\beta)\rangle}\frac{d\langle
Z(\beta)\rangle}{d\beta}$ (22) $\displaystyle=$
$\displaystyle\frac{\beta}{2}+\beta\,\frac{L_{N-2}^{2}(-\beta^{2}/2)}{L_{N-1}^{1}(-\beta^{2}/2)}\,.$

For high temperature ($\beta\ll 1$), the exact asymptotic behavior is thus

$\displaystyle\langle W\rangle\xrightarrow[\beta\rightarrow
0]{}\frac{N}{2}\beta-\frac{N^{2}-1}{24}\beta^{3}+\mathcal{O}(\beta^{5})\,,$
(23)

verifying the fact that the mean work vanishes exactly in the infinite
temperature limit and increases as the temperature is decreased, as seen in
Figure 2. The mean work is lower-bounded by the low-temperature limit,

$\displaystyle\langle
W\rangle\xrightarrow[\beta\rightarrow\infty]{}\frac{\beta}{2}\,,$ (24)

as in this regime the second term in the r.h.s. of Eq. (22) vanishes. The
behavior of the mean work for chaotic systems as a function of the inverse
temperature is presented in Figure 3 for different dimensions of the Hilbert
space and illustrates the accuracy of the high temperature asymptotes.

Comparing the analytical results with the numerical simulations, we find
discrepancies between the ‘annealed’ and exact expression at low temperatures.
This can be understood by the shape of the distribution of the GUE
Hamiltonians that, as the temperature lowers, has an increasing non-zero tail
that eventually requires more sampling—in contrast with the large $N$ limit
where the semi-circle distribution ensures a finite sampling domain.
Physically, the work, set by (the negative of) the ensemble initial energy,
should be bounded at low temperature. This is clearly not the case in the
analytical asymptote in Eq. (24), which shows the limitations of the annealed
approximation made in Eq. (1.1) in the low temperature regime, needed for the
pursued analytical treatment.

Figure 3:  Temperature dependence of mean work. For chaotic systems of various
Hilbert space dimension $N$, the mean work $\langle W\rangle$ associated with
a sudden quench is shown as function of the inverse temperature $\beta$. Both
the annealed expression in Eq. (22) (solid lines) and numerical results (dots)
are plotted. Comparison with the numerical results without annealing (dotted
lines) shows the importance of the latter, as discussed in the text. The
dashed grey line shows the asymptote $\beta/2$ at low temperatures; the dashed
black lines correspond to the asymptotes at high temperatures, as given in Eq.
(23). The star symbols indicate the conditions for which the detailed
characteristic function and work pdf are given in Figs. 1 and 2.

###  1.3 Variance of the work pdf in a random quench

The stochastic nature of work in the presence of quantum thermal fluctuations
becomes manifest when considering fluctuations away from the mean $\langle
W\rangle$. We focus on the variance that naturally characterizes the width of
the work pdf. As we shall see, it also governs the decay of the Loschmidt echo
at early times of evolution [24] and constitutes the leading contribution to
the irreversible entropy in linear response theory [43].

An explicit computation shows that the second moment of the work pdf
simplifies to

$\displaystyle\langle W^{2}\rangle$ $\displaystyle=$
$\displaystyle\left\langle\int_{-\infty}^{\infty}dWp(W)\,W^{2}\right\rangle$
$\displaystyle=$ $\displaystyle\left\langle\\!\\!\left\langle{\rm
Tr}\\!\left((\hat{H}_{\tau}^{2}+\hat{H}_{0}^{2})\frac{e^{-\beta\hat{H}_{0}}}{{\rm
Tr}(e^{-\beta\hat{H}_{0}})}\right)\\!\right\rangle\\!\\!\right\rangle\,,$

as a result of the aforementioned symmetry in the final ensemble, that yields
$\langle\\!\langle\hat{H}_{\tau}\hat{H}_{0}\rangle\\!\rangle=0$. The GUE-
averaged expectation value of the second moment for the initial Hamiltonian
can be derived from the partition function in Eq. (17),

$\left\langle{\rm
Tr}\left(\hat{H}_{0}^{2}e^{-\beta\hat{H}_{0}}\right)\right\rangle=\frac{d^{2}\langle
Z(\beta)\rangle}{d\beta^{2}}\,.$ (26)

By contrast, the second moment of the final Hamiltonian requires averaging
over both the initial and final state ensembles, specifically

$\displaystyle\left\langle\\!\\!\left\langle{\rm
Tr}\left(\hat{H}_{\tau}^{2}\,e^{-\beta\hat{H}_{0}}\right)\right\rangle\\!\\!\right\rangle$
$\displaystyle=\left\langle\\!\\!\left\langle\sum_{m,n}|\langle
m_{\tau}|n_{0}\rangle|^{2}\\!\left(E^{\tau}_{m}\right)^{2}\\!e^{-\beta
E^{0}_{n}}\right\rangle\\!\\!\right\rangle$ $\displaystyle=\langle
p_{m|n}^{\tau}\rangle\left\langle\sum_{m}\left(E^{\tau}_{m}\right)^{2}\right\rangle\left\langle\sum_{n}e^{-\beta
E^{0}_{n}}\right\rangle$
$\displaystyle=\frac{1}{N}\left.\frac{d^{2}\mathcal{I}(\beta)}{d\beta^{2}}\right|_{\beta=0}\langle
Z(\beta)\rangle$ $\displaystyle=\frac{N}{2}\langle Z(\beta)\rangle\,.$ (27)

Equations (1.3)-(1.3), along with the definitions (16) and (17), give the
exact, non-perturbative expression for the work fluctuations as

$\displaystyle\langle W^{2}\rangle$ $\displaystyle=$
$\displaystyle\frac{N}{2}+\frac{1}{\langle Z(\beta)\rangle}\frac{d^{2}\langle
Z(\beta)\rangle}{d\beta^{2}}$ (28) $\displaystyle=$
$\displaystyle\frac{N{+}1}{2}+\frac{\beta^{2}}{4}+(1{+}\beta^{2})\frac{L_{N-2}^{2}\left(-\frac{\beta^{2}}{2}\right)}{L_{N-1}^{1}\left(-\frac{\beta^{2}}{2}\right)}$
$\displaystyle+\beta^{2}\frac{L_{N-3}^{3}\left(-\frac{\beta^{2}}{2}\right)}{L_{N-1}^{1}\left(-\frac{\beta^{2}}{2}\right)}\,.$

Knowledge of the first two moments of the distribution, (22) and (28), readily
yields the work variance

$\displaystyle\sigma_{W}^{2}$ $\displaystyle=$ $\displaystyle\langle
W^{2}\rangle-\langle W\rangle^{2}$ $\displaystyle=$
$\displaystyle\frac{N+1}{2}+\frac{L_{N-2}^{2}\left(-\frac{\beta^{2}}{2}\right)}{L_{N-1}^{1}\left(-\frac{\beta^{2}}{2}\right)}$
$\displaystyle+\beta^{2}\left[\frac{L_{N-3}^{3}\left(-\frac{\beta^{2}}{2}\right)}{L_{N-1}^{1}\left(-\frac{\beta^{2}}{2}\right)}-\left(\frac{L_{N-2}^{2}\left(-\frac{\beta^{2}}{2}\right)}{L_{N-1}^{1}\left(-\frac{\beta^{2}}{2}\right)}\right)^{2}\right]\,.$

At infinite temperature, the work variance saturates at the Hilbert space
dimension $N$ following the asymptotic behavior

$\displaystyle\sigma_{W}^{2}\xrightarrow[\beta\rightarrow
0]{}N-\frac{N^{2}-1}{8}\beta^{2}+\mathcal{O}(\beta^{4})\,.$ (30)

In the low temperature regime, the work variance is still finite and saturates
at the value of $(N+1)/2$ beyond $\beta\gtrapprox 1.5$. Both limits are
presented in Figure 4, where the monotonic decay of $\sigma_{W}^{2}$ is shown
as function of $\beta$ for different values of the Hilbert space dimension.
Here, the analytical and numerical results for the annealed approximation
match the exact simulation, as the error observed on the average energy (see
Figure 3) is exactly compensated by that on the energy second moment also used
to obtain the variance.

Figure 4:  Temperature dependence of work fluctuations. Variance of the work
$\sigma_{W}^{2}$ given in Eq. (1.3) for different system sizes as function of
the inverse temperature $\beta$. The fluctuations monotonically decay from the
infinite-temperature value $N$ following the dashed-line asymptote
corresponding to Eq. (30), saturating at $(N+1)/2$ in the low-temperature
regime. The analytical results (solid lines) are in excellent agreement with
the numerical simulations for both the annealed (circles) and the exact
(dotted lines) expressions.

##  2 Loschmidt echo in Chaotic Systems and information scrambling

The study of the work statistics can be recast into a dynamical problem, since
the characteristic function of the work pdf can be related to a Loschmidt
echo. A Loschmidt echo quantifies the degree to which a given dynamics can be
reversed [44, 45]. The connection between work statistics and Loschmidt echoes
was first established for a system initialized in an energy eigenstate and
driven by a sudden quench [15]. Following works extended this relation to
arbitrary evolutions by interpreting the characteristic function as a
Loschmidt echo amplitude where the backward evolution is generated by an
effective Hamiltonian [22, 35]. More recently, we have shown that such a
connection holds as well for an arbitrary initial state, whether it is pure or
not [24]. In turn, the ability to recast work statistics in terms of a
dynamical problem (a Loschmidt echo) for any arbitrary initial states results
is an important connection between work statistics and information scrambling.

To appreciate this, we first consider a general initial state $\hat{\rho}$,
which may include coherences in the initial energy eigenbasis. Following the
first energy measurement associated with the projectors
$\hat{P}^{0}_{n}=|n_{0}\rangle\langle n_{0}|$, the post-measurement state
becomes
$\sum_{n}\hat{P}^{0}_{n}\hat{\rho}\hat{P}^{0}_{n}=\sum_{n}p^{0}_{n}\,|n_{0}\rangle\langle
n_{0}|$. This mixed state can be purified by embedding it in an extended
Hilbert space, e.g., $\mathcal{H}_{L}\otimes\mathcal{H}_{R}$, with
$\mathcal{H}_{L}=\mathcal{H}_{R}=\mathcal{H}$. The purified state is then
given by an entangled state between two copies of the system

$\displaystyle|\Psi_{0}\rangle=\sum_{n}\sqrt{p^{0}_{n}}\,\,|n_{0}\rangle_{L}\otimes|n_{0}\rangle_{R}\,.$
(31)

The characteristic function associated with the evolution $\hat{U}(\tau)$ in
Eq. (2) can then be expressed as the survival amplitude
$\langle\Psi_{0}|\Psi_{t}\rangle$ of the purified state (31) when one of the
copies of the system evolves under the sudden quench

$\hat{H}(t)=\hat{H}_{0}\,\Theta(-t)-\hat{H}_{\tau}^{\rm eff}\,\Theta(t)\,,$
(32)

this is, a sudden quench between the initial Hamiltonian and the effective
Hamiltonian $-\hat{H}_{\tau}^{\rm eff}$ defined in Eq. (11), $\Theta(t)$
denoting the Heaviside function. Specifically, the equivalence becomes
explicit by composition of the unitary evolution operators, and reads [24]

$\displaystyle\chi(t,\tau)$ $\displaystyle\equiv$
$\displaystyle\langle\Psi_{0}|\Psi_{t}\rangle$ (33) $\displaystyle=$
$\displaystyle\langle\Psi_{0}|e^{+it\hat{H}_{\tau}^{\rm
eff}}e^{-it\hat{H}_{0}}\otimes\mathbb{1}_{R}|\Psi_{0}\rangle\,,$

where we emphasize that the quench evolution acts exclusively on one of the
copies, e.g., the left one. Thus, the characteristic function in Eq. (33) also
encodes the dynamics of the Loschmidt echo amplitude of the pure state
$|\Psi_{0}\rangle$ in which the left copy evolves first forward in time under
the Hamiltonian $\hat{H}_{0}$ and then backward under the effective
Hamiltonian $\hat{H}_{\tau}^{\rm eff}$, leaving the right copy unchanged.

We stress here that the connection
$\chi(t,\tau)\equiv\langle\Psi_{0}|\Psi_{t}\rangle$ is valid for any driving
protocol and for any purified state $|\Psi_{0}\rangle$, including those
arising from mixed states at finite temperature [24]. One may thus consider
the initial state to be the canonical thermal state, as in Eq. (4). In this
particular case, the corresponding purified state is the so-called thermofield
double state [46],

$\displaystyle|{\Phi}_{0}\rangle=\frac{1}{\sqrt{Z(\beta)}}\\!\sum_{n}e^{-\frac{\beta}{2}\hat{H}_{0}\,\otimes\,\mathbb{1}_{R}}\,|n_{0}\rangle_{L}\otimes|n_{0}\rangle_{R}\,.$

Its time-evolution is then described by

$|\Phi_{t}\rangle=e^{-i\int_{0}^{t}dt^{\prime}\hat{H}(t^{\prime})\,\otimes\,\mathbb{1}_{R}}\,|\Phi_{0}\rangle\,,$
(35)

with $\hat{H}(t^{\prime})$ given in Eq. (32). Its survival probability
amplitude is equivalent to the characteristic function of the corresponding
work statistics, namely

$\displaystyle\chi(t,\tau)$ $\displaystyle=$
$\displaystyle\langle\Phi_{0}\,|\,\Phi_{t}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{Z(\beta)}\,{\rm Tr}\left(e^{+it\hat{H}_{\tau}^{\rm
eff}}e^{-(\beta+it)\hat{H}_{0}}\right)\,.$

In this interpretation, the thermofield double state (2) is not invariant
under time-evolution and the equivalence established above gives the Loschmidt
echo $\mathcal{L}(t)$ as the fidelity between the initial thermofield double
state $|\Phi_{0}\rangle$ and its time-evolution, also known as the survival
probability:

$\displaystyle\mathcal{L}(t)=|\langle\Phi_{0}\,|\,\Phi_{t}\rangle|^{2}\,=\left|\int_{-\infty}^{\infty}dWp_{\tau}(W)\,e^{iWt}\right|^{2}\,.$

Said differently, the Loschmidt echo of the thermofield double state can be
obtained from the Fourier transform of the work pdf. This observation leads to
a direct connection between quantum thermodynamics – the analysis of work
statistics – and information scrambling, as characterized by $\mathcal{L}(t)$.

Information scrambling was first discussed in black hole physics [47, 48, 49,
50]. In particular, within the AdS/CFT correspondence, an eternal black hole,
or so-called Einstein-Rosen bridge, is dual to the thermofield double state,
and described by a state with a highly localized entanglement distribution
between the degrees of freedom of the two non-interacting copies of a CFT
[51]. The notion of information scrambling refers to the redistribution of any
quantum information, initially localized in a subsystem, over the entire
system under unitary time evolution. Thus, in the AdS/CFT framework, it is
natural to study the spreading of information in black holes via the survival
probability of a thermofield double state, that is, via the fidelity between
the initial state and its unitary time evolution, as in Eq. (2) [25, 26, 28,
29, 27]. The time-evolved state $|\Phi_{t}\rangle$ in Eq. (35) can be
understood, in the gravitational picture, as an Einstein-Rosen bridge which
interior has grown [52, 53]. This notion of information scrambling has proved
useful in a more general setting, in which initially localized information in
a subsystem spreads and gets scrambled in the degrees of freedom of a many-
body system. In the following, we characterize this dynamics in chaotic
quantum systems represented by the Gaussian unitary ensemble.

###  2.1 Loschmidt Echo in GUE

The Loschmidt echo, defined in Eq. (2), averaged over the GUE follows from the
characteristic function in Eq. (12) as

$\displaystyle\langle\mathcal{L}(t)\rangle$ $\displaystyle=$
$\displaystyle\Big{\langle}\\!\\!\Big{\langle}\frac{1}{{\rm
Tr}\left(e^{-\beta\hat{H}_{0}}\right)^{2}}{\rm
Tr}\left(e^{-\sigma_{\tau}\hat{H}_{\tau}^{\rm
eff}}e^{-\sigma_{0}\hat{H}_{0}}\right)\quad$ (38) $\displaystyle\times{\rm
Tr}\left(e^{-\sigma_{\tau}^{*}\hat{H}_{\tau}^{\rm
eff}}e^{-\sigma_{0}^{*}\hat{H}_{0}}\right)\Big{\rangle}\\!\\!\Big{\rangle}\,.$

Using the property of the GUE invariance under conjugation by a unitary, the
double average reduces to two independent GUE averages, one over $\hat{H}_{0}$
and one over $\hat{H}_{\tau}$. The ‘annealed’ version of the Loschmidt echo is
thus

$\displaystyle\langle\mathcal{L}(t)\rangle=\frac{1}{\langle
Z(\beta)^{2}\rangle}\left\langle\\!\\!\left\langle\left|{\rm
Tr}\left(e^{-\sigma_{\tau}\hat{H}_{\tau}}e^{-\sigma_{0}\hat{H}_{0}}\right)\right|^{2}\right\rangle\\!\\!\right\rangle\,.$

Its explicit form can be conveniently written in terms of the partition
function and the GUE spectral form factor defined as

$\displaystyle g(\beta,t)\equiv\left\langle Z(\beta+it)Z(\beta-
it)\right\rangle\,.$ (40)

As shown in Appendix A, it is found that

$\displaystyle\langle\mathcal{L}(t)\rangle=\frac{1}{\langle
Z(\beta)^{2}\rangle}\,\frac{1}{N^{2}{-}1}\Big{(}g(0,t)g(\beta,t)$ (41)
$\displaystyle+N\langle Z(2\beta)\rangle-\frac{1}{N}\langle Z(2\beta)\rangle
g(0,t)-\frac{1}{N}g(\beta,t)N\Big{)}\,.$

The dependence of the Loschmidt echo on the spectral form factor illustrates
the importance of correlations between energy eigenvalues. In the infinite-
temperature regime ($\beta=0$), this result simplifies to

$\displaystyle\langle\mathcal{L}(t)\rangle=\frac{1}{N^{2}}\,\frac{1}{N^{2}{-}1}\Big{(}g(0,t)^{2}{+}N^{2}{-}2g(0,t)\Big{)}\,.$

The Loschmidt echo can thus be fully characterized from the GUE spectral form
factor (40), for which we give an explicit formulation below.

###  2.2 Loschmidt echo and the spectral form factor

Figure 5: Decay of the Loschmidt echo in GUE. The time-evolution of the
Loschmidt echo is shown for (a) $\beta=0$ and (b) $\beta=1$ and various system
sizes $N$. Three different stages are clearly distinguished in the decay
dynamics. The short-time behavior is governed by a Gaussian decay dictated by
the work variance according to (46), see the dashed lines with $e^{-t^{2}N}$
for $\beta=0$. The ensuing dynamics is characterized by a broad dip, and a
slow increase towards the plateau at which the Loschmidt echo saturates at
long times. The long-time asymptotic value predicted by Eq. (48) is marked by
the horizontal dashed lines. The figure also shows the difference between the
‘annealed’ (solid lines) and exact (dotted lines) Loschmidt echo, which are
indistinguishable in the infinite temperature limit.

We have shown that the averaged value of $\mathcal{L}(t)$ can be expressed in
terms of the spectral form factor, as seen in Eq. (41). The latter admits an
exact analytical expression. Indeed, the form factor can be decomposed into
three contributions as [27]

$\displaystyle g(\beta,t)$ $\displaystyle=$ $\displaystyle\langle
Z(\beta+it)Z(\beta-it)\rangle$ $\displaystyle=$ $\displaystyle\langle
Z(2\beta)\rangle+|\langle Z(\beta+it)\rangle|^{2}+g_{c}(\beta,t)\,,$

where $g_{c}(\beta,t)$ is the double complex Fourier transform of the
connected contribution,

$\displaystyle g_{c}(\beta,t)$ $\displaystyle=$ $\displaystyle\int
dEdE^{\prime}\left\langle\rho_{c}^{(2)}(E,E^{\prime})\right\rangle$ (44)
$\displaystyle\times e^{-(\beta+it)E-(\beta-it)E^{\prime}}\,.$

The analytically-continued partition function averaged over the GUE is known
from [27] and given in Eq. (16). Similarly, the connected two-level
correlation function
$\left\langle\rho_{c}^{(2)}(E,E^{\prime})\right\rangle=\langle\rho(E)\rho(E^{\prime})\rangle-\langle\rho(E)\rangle\langle\rho(E^{\prime})\rangle$
is given by Eq. (8). Its complex Fourier transform, Eq. (44), is evaluated in
Appendix B following [27] and reads

$\displaystyle g_{c}(\beta,t)$ $\displaystyle=$
$\displaystyle-e^{\frac{\sigma_{0}^{2}+\sigma_{0}^{*2}}{4}}\sum_{n,m=0}^{N-1}c_{m,n}\left(\frac{|\sigma_{0}|^{2}}{2}\right)^{|n-m|}$
(45)
$\displaystyle\times\left|L_{\min(m,n)}^{|n-m|}\left(-\frac{\sigma_{0}^{2}}{2}\right)\right|^{2}\,,$

where $c_{m,n}=\min(m,n)!/\max(m,n)!$. We verify that at infinite temperature
and initial time ($\sigma_{0}=0$), the only non-zero terms in the sum bring a
Kronecker delta $\delta_{n,m}$, while the Laguerre polynomials reduce to
binomial coefficients, yielding $g_{c}(0,0)=-N$ and $\langle
Z(0)^{2}\rangle=g(0,0)=N^{2}$, as expected.

###  2.3 Loschmidt echo dynamics

With the exact analytical expressions at hand, we proceed to characterize the
dynamics of the Loschmidt echo. The Loschmidt echo is shown in Figure 5 in two
different panels corresponding to infinite temperature ($\beta=0$) and a
finite value ($\beta=1$). In both cases, the decay dynamics is shown for
different values of the Hilbert space dimension, and one can distinguish a
sequence of three stages. The Loschmidt echo first decays monotonically until
reaching a minimum value. The second stage is characterized by a broad dip in
which the Loschmidt echo stabilizes around $1/N^{2}$ and gradually starts to
grow. Subsequently, a saturation of the Loschmidt echo occurs around an
approximately constant value.

The short time behavior of the Loschmidt echo was first given in our recent
work [24] and is governed by work fluctuations – the variance of the work pdf
– according to

$\langle\mathcal{L}(t)\rangle=\big{\langle}e^{-t^{2}\sigma_{W}^{2}+\mathcal{O}(t^{4})}\big{\rangle}\geq
e^{-t^{2}\langle\sigma_{W}^{2}\rangle+\mathcal{O}(t^{4})}\,,$ (46)

where we have used Jensen’s inequality. For $\beta\leavevmode\nobreak\
=\leavevmode\nobreak\ 0$, the work variance given in Eq. (1.3) simplifies to
$N$, leading to a Gaussian decay that is simply given by $e^{-t^{2}N}$, as
verified in Figure 5.

As for the long-time asymptotics, it is best analyzed in terms of the spectral
form factor. The latter can be written, using Eq. (2.2), as

$\displaystyle g(\beta,t)$ $\displaystyle=$ $\displaystyle\langle
Z(2\beta)\rangle$ $\displaystyle+\left\langle\sum_{n}\sum_{n^{\prime}\neq
n}e^{-\beta(E_{n}+E_{n^{\prime}})}e^{-it(E_{n}-E_{n^{\prime}})}\right\rangle\,.$

At long times, the complex exponential exhibits rapid oscillations and only
the first term survives under a short time average that enters in the feature
of a plateau. Taking the limit $g(\beta,t)\to\langle Z(2\beta)\rangle$, Eq.
(41) gives the asymptote of the Loschmidt echo at long times as

$\langle\mathcal{L}(t)\rangle\xrightarrow[t\to\infty]{}\frac{\langle
Z(2\beta)\rangle}{\langle Z(\beta)^{2}\rangle}\frac{2}{(N+1)}$ (48)

for the ‘annealed’ expression, Eq. (2.1). We numerically validate the
approximation splitting the numerator and denominator averages and compare
with the exact expression, defined in Eq. (38), which is found slightly below
this asymptote with a limit given by
$\left\langle\frac{Z(2\beta)}{Z(\beta)^{2}}\right\rangle\frac{2}{(N+1)}$. At
infinite temperature, $\beta=0$, the exact and annealed asymptotes are
identical and equal to $2/(N(N+1))$.

###  2.4 Loschmidt Echo, frame potentials and scrambling

As discussed in Section 2, quantum work statistics can be used to assess
information scrambling, as the generating function of the work pdf is
equivalent to the Loschmidt echo amplitude of a thermofield double state.
Scrambling amounts to the effective randomization of the state of a quantum
system subject to unitary time evolution. Depending on the dynamics,
scrambling produces different degrees of randomness. There is a wide variety
of approaches to pursue its study, including the use of survival
probabilities, analytically-continued partition functions, and out-of-time-
order correlators, among other examples.

In black hole physics, the common reference is the Haar/Page information
scrambling, which consists in drawing Haar unitaries from the Haar ensemble.
This yields a complete randomization of a quantum state, making it impossible
to recover information about the initial state by carrying out only local
measurements on the system. In this context, it is relevant to analyze the
extent to which a concrete ensemble of unitaries can match Haar scrambling.

Frame potentials have been argued to act in this context as natural probes of
the randomness created by a concrete time evolution, by quantifying an
ensemble ability to reproduce moments of the Haar random unitary ensemble
[30]. The $k$-th frame potential with respect to an ensemble $\mathcal{E}$ is
defined as

$\displaystyle\mathcal{F}^{(k)}_{\mathcal{E}}=\int_{\hat{A},\hat{B}\,\in\mathcal{E}}D\hat{A}\,D\hat{B}\left|\,{\rm
Tr}\,\hat{A}^{\dagger}\,\hat{B}\,\right|^{2k}\,.$ (49)

For any ensemble, $\mathcal{F}^{(k)}_{\mathcal{E}}\geq\mathcal{F}^{(k)}_{\rm
Haar}$, where the Haar ensemble frame potential is $\mathcal{F}^{(k)}_{\rm
Haar}=k!$.

In our case, the relevant ensemble consists of the unitary time evolutions at
a fixed time $t$ with Hamiltonians drawn from the GUE and reads
$\mathcal{E}_{t}^{\rm GUE}=\big{\\{}e^{-i\hat{H}t},\leavevmode\nobreak\ {\rm
for}\leavevmode\nobreak\ \hat{H}\in{\rm GUE}\big{\\}}\,.$ Remarkably, we note
that the $k=1$ frame potential with respect to the GUE is given by

$\displaystyle\mathcal{F}^{(1)}_{\rm
GUE}(t)=\left\langle\\!\\!\left\langle\left|\,{\rm
Tr}\,\left(e^{it\hat{H}_{\tau}}e^{-it\hat{H}_{0}}\right)\,\right|^{2}\right\rangle\\!\\!\right\rangle\,.$
(50)

This is directly related to the expression of the GUE-averaged Loschmidt echo
in Eq. (2.1) at infinite temperature as

$\displaystyle\langle\mathcal{L}(t)\rangle=\frac{1}{N^{2}}\,\mathcal{F}^{(1)}_{\rm
GUE}(t)\quad{\rm for}\quad\beta=0\,.$ (51)

This correspondence thus shows that whenever $\mathcal{L}(t)\approx
1/N^{2}\,$, the frame potential $\mathcal{F}^{(1)}_{\rm GUE}(t)\approx 1$,
indicating that the GUE time evolution achieves Haar randomness. At very early
times $g(0,0)=N^{2}$, and the dominant term $g(0,t)^{2}$ yields
$\mathcal{L}(t)\approx 1$ and $\mathcal{F}^{(1)}_{\rm GUE}(t)\approx N^{2}$,
showing that the GUE time evolution is far from achieving Haar randomness. At
very late times, $g(0,t\to\infty)\approx N$, which implies that
$\mathcal{L}(t)\approx 2/N^{2}$ and $\mathcal{F}^{(1)}_{\rm GUE}(t)\approx 2$.
Between these two limits, there is a timescale for which the spectral form
factor is about $\sqrt{N}$, yielding $\mathcal{L}(t)\approx 1/N^{2}$ as far as
$\mathcal{F}^{(1)}_{\rm GUE}(t)\approx N^{2}/(N^{2}-1)\approx 1$, and
signaling a GUE-mediated Haar scrambling. The relation between
$\langle\mathcal{L}(t)\rangle$ and $\mathcal{F}^{(1)}_{\rm GUE}(t)$ can be
extended to other correlation functions, as discussed in Appendix C.

However, the relation shown in Eq. (51) is restricted to the infinite
temperature limit, since the first frame potential for GUE Hamiltonians [30]

$\mathcal{F}_{{\rm
GUE}}^{(1)}(\beta,t)=\frac{1}{N^{2}{-}1}\Big{(}g(\beta/2,t)^{2}+N^{2}-2g(\beta/2,t)\Big{)}\,,$
(52)

cannot be directly related to the expression for $\mathcal{L}(t)$ in Eq. (41)
at finite temperature $\beta>0$.

##  3 Summary and Discussion

The work associated with driving a quantum mechanical system can be determined
via two-projective energy measurements and is characterized by a probability
distribution, reflecting the stochastic nature of this quantity. Analyzing the
work statistics of complex systems is generally challenging due to the number
of transitions generated during the driving between the energy levels of the
initial and final Hamiltonians of the system. Indeed, the latter blows up
exponentially as the size of the system is increased.

In this manuscript, we have reported the exact work statistics of quantum
chaotic systems of arbitrary size that are described by random matrix
Hamiltonians. In particular, we have characterized the mean work and work
fluctuations resulting from driving a canonical thermal state by a sudden
quench. We have further shown that the generating function of the work pdf can
be interpreted as the Loschmidt echo of the purified thermal state, the
thermofield double state. The dynamics of this Loschmidt echo accounts for the
reversibility of quenches between chaotic Hamiltonians. This dynamics also
describes the scrambling of information of the thermofield double state, which
acquires a natural interpretation in AdS/CFT being dual to an eternal black
hole. The connection between work statistics and information scrambling has
been established by mapping the work pdf to the Loschmidt echo, that can be
expressed in terms of the spectral form factor is related to the frame
potential.

Our work establishes an unexpected and firm connection between different
fields, including quantum thermodynamics, the emergence of irreversibility,
and information scrambling. This connection is amenable to experimental tests
in a variety of quantum platforms. A prominent example is offered by nuclear
magnetic resonance (NMR) experiments, in which measurements of work statistics
[12], Loschmidt echo dynamics in chaotic systems [54] and information
scrambling [55] have been reported. Alternative setups include trapped ions
[13, 14, 56] and superconducting qubits [57], where digital quantum simulation
techniques [58, 59, 60] can be used to explore quantum chaos and systems
described by AdS/CFT that exhibit scrambling [61]. Chaotic systems could also
be implemented in controllable many-body systems by applying random pulses to
create an evolution with a Haar-uniform distribution [62].

Furthermore, our results should find broad applications in quantum
thermodynamics of many-body chaotic systems. Quantum many-body systems can be
used as a working medium in thermodynamic devices including heat engines and
refrigerators [63, 64, 65] and it is natural to explore the role of quantum
chaos in this context.

The authors are indebted to Zhenyu Xu for numerous discussions that helped to
improve the manuscript. Further, it is a pleasure to thank Luis Pedro García-
Pintos for comments on the manuscript. We acknowledge funding support from the
U.S. Department of Energy, the Spanish Ministerio de Economía y Competitividad
(project FIS2015-69512-R), Programa de Excelencia de la Fundación Séneca
Región de Murcia (project 19882/GERM/15), the John Templeton Foundation, and
UMass Boston (project P20150000029279). A.C. and A.d.C. acknowledge the
hospitality of the Simons Center for Geometry and Physics during completion of
this work.

##  Appendix A Loschmidt echo averaged over the GUE

Starting with the definition of the GUE-averaged Loschmidt echo in Eq. (2.1),
we provide here the details to derive its explicit analytical form, as given
in Eq. (41).

The unitary conjugation invariance of the GUE measure allows us to add an
integration over the unitary group. So denoting by $dV,\,dW$ the Haar measures
on the unitary group $\mathcal{U}(N)$, we follow [30] and write

$\displaystyle\langle\mathcal{L}(t)\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\langle
Z(\beta)^{2}\rangle}\int\\!\\!d\hat{H}_{0}\,d\hat{H}_{\tau}\,\rho(\hat{H}_{0})\,\rho(\hat{H}_{\tau})\left|{\rm
Tr}\,\left(e^{-\sigma_{\tau}\hat{H}_{\tau}}e^{-\sigma_{0}\hat{H}_{0}}\right)\right|^{2}$
(53) $\displaystyle=$ $\displaystyle\frac{1}{\langle
Z(\beta)^{2}\rangle}\,\int
d\hat{H}_{0}\,d\hat{H}_{\tau}\,\rho(\hat{H}_{0})\rho(\hat{H}_{\tau})\int_{\mathcal{U}(N)}dVdW\Big{|}\,{\rm
Tr}\big{(}V^{\dagger}\Lambda_{\tau}^{\dagger}VW^{\dagger}{\Lambda}_{0}W\big{)}\,\Big{|}^{2}\,,$

where ${\Lambda}_{0}\equiv We^{-\sigma_{0}\hat{H}_{0}}W^{\dagger}$ and
$\Lambda_{\tau}\equiv Ve^{-\sigma_{\tau}\hat{H}_{\tau}}V^{\dagger}$ are the
matrix exponentials of the GUE initial and final Hamiltonians in the diagonal
basis. The first integrals can be expressed in the eigenvalue basis, yielding

$\displaystyle\langle\mathcal{L}(t)\rangle=\frac{1}{\langle
Z(\beta)^{2}\rangle}\,\int D\lambda_{\tau}\,D\lambda_{0}\int_{\rm
Haar}dV\,{\rm
Tr}\big{(}V^{\dagger}\Lambda_{\tau}^{\dagger}V{\Lambda}_{0}\big{)}\,{\rm
Tr}\big{(}{\Lambda}_{0}^{\dagger}V^{\dagger}\Lambda_{\tau}V\big{)}\,,$

where we have used the GUE measure defined in Eq. (6). Here, the Haar
integrals have been simplified to a single integral using the left and right
invariance of the Haar measure. The latter can be explicitly solved by using
an exact formula for integrals of monomials of Haar random unitaries [30, 66,
67]. Namely, following [30], we first explicitly write the traces as

$\displaystyle\langle\mathcal{L}(t)\rangle=\frac{1}{\langle
Z(\beta)^{2}\rangle}\,\int D\lambda_{\tau}\,D\lambda_{0}\,\int_{\rm
Haar}dV\Big{(}V^{\dagger}_{k_{1},k^{\prime}_{1}}\Lambda^{\dagger}_{\tau;k^{\prime}_{1},l_{1}}V_{l_{1},l^{\prime}_{1}}\Lambda_{0;l^{\prime}_{1},k_{1}}V_{k_{2},k^{\prime}_{2}}\Lambda^{\dagger}_{0;k^{\prime}_{2},l_{2}}V^{\dagger}_{l_{2},l^{\prime}_{2}}\Lambda_{\tau;l^{\prime}_{2},k_{2}}\Big{)}\,,$
(54)

and evaluate the Haar integral using the second moment [66, 30]

$\displaystyle\int_{\rm
Haar}dV\,V_{l_{1},l^{\prime}_{1}}V_{k_{2},k^{\prime}_{2}}V^{\dagger}_{k_{1},k^{\prime}_{1}}V^{\dagger}_{l_{2},l^{\prime}_{2}}$
$\displaystyle=$ $\displaystyle{\rm
Wg}(1^{2},N)\delta_{l_{1},k^{\prime}_{1}}\delta_{l^{\prime}_{1},k_{1}}\delta_{k_{2},l^{\prime}_{2}}\delta_{k^{\prime}_{2},l_{2}}+{\rm
Wg}(1^{2},N)\delta_{l_{1},l^{\prime}_{2}}\delta_{l^{\prime}_{1},l_{2}}\delta_{k_{2},k^{\prime}_{1}}\delta_{k^{\prime}_{2},k_{1}}$
(55) $\displaystyle+$ $\displaystyle{\rm
Wg}(2,N)\delta_{l_{1},k^{\prime}_{1}}\delta_{l^{\prime}_{1},l_{2}}\delta_{k_{2},l^{\prime}_{2}}\delta_{k^{\prime}_{2},k_{1}}+{\rm
Wg}(2,N)\delta_{l_{1},l^{\prime}_{2}}\delta_{l^{\prime}_{1},k_{1}}\delta_{k_{2},k^{\prime}_{1}}\delta_{k^{\prime}_{2},l_{2}}\,,$

where we have used the Weingarten functions [68] given by Collins [66] as
${\rm Wg}(1^{2},N)=1/(N^{2}-1)$ and ${\rm Wg}(2,N)=-1/(N(N^{2}-1))$.

This simplifies Eq. (54) to

$\displaystyle\langle\mathcal{L}(t)\rangle=\frac{1}{\langle
Z(\beta)^{2}\rangle}\,\int
D\lambda_{\tau}\,D\lambda_{0}\,\frac{1}{N^{2}-1}\bigg{[}$ $\displaystyle{\rm
Tr}(\Lambda_{\tau}^{\dagger})\,{\rm Tr}(\Lambda_{\tau})\,{\rm
Tr}({\Lambda}_{0}^{\dagger})\,{\rm Tr}({\Lambda}_{0})+{\rm
Tr}({\Lambda}_{0}^{\dagger}{\Lambda}_{0}){\rm
Tr}({\Lambda}_{\tau}^{\dagger}{\Lambda}_{\tau})$
$\displaystyle-\frac{1}{N}{\rm Tr}({\Lambda}_{0}^{\dagger}{\Lambda}_{0})\,{\rm
Tr}(\Lambda_{\tau}^{\dagger})\,{\rm Tr}(\Lambda_{\tau})-\frac{1}{N}\,{\rm
Tr}({\Lambda}_{0}^{\dagger})\,{\rm Tr}({\Lambda}_{0}){\rm
Tr}({\Lambda}_{\tau}^{\dagger}{\Lambda}_{\tau})\bigg{]}\,.$

By noting that the integrals in the eigenstate basis can be expressed in terms
of the partition function,

$\displaystyle\int D\lambda_{i}\,{\rm
Tr}\left({\Lambda_{i}}^{\dagger}{\Lambda_{i}}\right)$ $\displaystyle=$
$\displaystyle\Big{\langle}Z\big{(}\sigma^{R}_{i}\big{)}\Big{\rangle}\,,$ (57)
$\displaystyle\int D\lambda_{i}\,{\rm Tr}({\Lambda_{i}}^{\dagger})\,{\rm
Tr}({\Lambda_{i}})$ $\displaystyle=$
$\displaystyle\Big{\langle}Z(\sigma_{i})Z(\sigma_{i}^{*})\Big{\rangle}\,,$
(58)

we obtain the following expression

$\displaystyle\langle\mathcal{L}(t)\rangle=\frac{1}{\langle
Z(\beta)^{2}\rangle}\,\frac{1}{N^{2}{-}1}\bigg{[}$
$\displaystyle\Big{\langle}\\!Z(\sigma_{\tau})Z(\sigma_{\tau}^{*})\Big{\rangle}\,\Big{\langle}\\!Z(\sigma_{0})Z(\sigma_{0}^{*})\Big{\rangle}+\Big{\langle}Z\big{(}2\sigma_{0}^{R}\big{)}\Big{\rangle}\,\Big{\langle}Z\big{(}2\sigma_{\tau}^{R}\big{)}\Big{\rangle}$
$\displaystyle-\frac{1}{N}\Big{\langle}Z\big{(}2\sigma_{0}^{R}\big{)}\Big{\rangle}\Big{\langle}Z(\sigma_{\tau})Z(\sigma_{\tau}^{*})\Big{\rangle}-\frac{1}{N}\Big{\langle}Z(\sigma_{0})Z(\sigma_{0}^{*})\Big{\rangle}\Big{\langle}Z\big{(}2\sigma_{\tau}^{R}\big{)}\Big{\rangle}\bigg{]}\,,$

where $\sigma^{R}$ denotes the real part ${\rm Re}(\sigma)$. This expression
can be conveniently written in terms of the spectral form factor as shown in
Eq. (41) of the main text.

_Alternative derivation.—_ The average of the Loschmidt echo over the GUE can
be obtained via an alternative route. As any Hermitian Hamiltonian $\hat{H}$
can be diagonalized by a unitary similarity transformation, it is possible to
express the GUE average in Eq. (6) as an average over the eigenvalues
$\boldsymbol{E}$ and $N(N-1)$ angles $\boldsymbol{\theta}$ associated with the
unitary [23],

$\displaystyle\langle\hat{O}\rangle$ $\displaystyle=$ $\displaystyle\int
d\hat{H}\,\rho(\hat{H})\,\hat{O}=\int
d\boldsymbol{E}d\boldsymbol{\theta}\rho_{\boldsymbol{E}}(\boldsymbol{E})\rho_{\boldsymbol{\theta}}(\boldsymbol{\theta})\,\hat{O}(\boldsymbol{E},\boldsymbol{\theta})\,,$
(60)

where $\rho_{\boldsymbol{E}}(\boldsymbol{E})$ and
$\rho_{\boldsymbol{\theta}}(\boldsymbol{\theta})$ are the corresponding
distributions given by

$\displaystyle\rho_{\boldsymbol{E}}(\boldsymbol{E})$ $\displaystyle=$
$\displaystyle\frac{2^{N(N-1)/2}}{\pi^{N/2}\prod_{n=1}^{N}n!}\prod_{k>j}(E_{k}-E_{j})^{2}e^{-\sum_{k}E_{k}^{2}}\,,$
$\displaystyle\rho_{\boldsymbol{\theta}}(\boldsymbol{\theta})$
$\displaystyle=$ $\displaystyle\frac{1}{(2\pi)^{N(N-1)}}\,.$ (61)

Since the energy eigenvalues $\boldsymbol{E}$ and the angles
$\boldsymbol{\theta}$ are statistically independent from each other, the
double average in Eq. (2.1) factorizes as

$\displaystyle\langle\mathcal{L}(t)\rangle=\frac{1}{\langle
Z(\beta)^{2}\rangle}\,\sum_{n,n^{\prime}}\sum_{m,m^{\prime}}\left\langle
p_{m|n}^{\tau}p_{m^{\prime}|n^{\prime}}^{\tau}\right\rangle_{\boldsymbol{\theta}^{0},\boldsymbol{\theta}^{\tau}}\left\langle
e^{-\sigma_{\tau}E_{m}^{\tau}-\sigma_{\tau}^{*}E_{m^{\prime}}^{\tau}}\right\rangle_{\boldsymbol{E}^{\tau}}\\!\left\langle
e^{-\sigma_{0}E_{n}^{0}-\sigma_{0}^{*}E_{n^{\prime}}^{0}}\right\rangle_{\boldsymbol{E}^{0}}\,,$
(62)

where we have made use of the transition probabilities, as in Eq. (1.1). While
the average of the transition probabilities generates a uniform distribution,
known as $\left\langle
p_{m|n}^{\tau}\right\rangle_{\boldsymbol{\theta}^{0},\boldsymbol{\theta}^{\tau}}=1/N$
[23], the average of two transition probabilities is quite more involved and
has not yet been reported. The difficulty in evaluating this expression comes
from the fact that the second transition is not independent from the first
one, creating a complex combinational problem. Inspired by the approach in the
previous derivation, we here find its explicit form using integration over the
unitary group $\mathcal{U}(N)$.

Let us first note the explicit the expression, dropping the indices on the
eigenvalues for clarity,

$\displaystyle\left\langle
p_{m|n}^{\tau}p_{m^{\prime}|n^{\prime}}^{\tau}\right\rangle_{\boldsymbol{\theta}^{0},\boldsymbol{\theta}^{\tau}}$
$\displaystyle=$ $\displaystyle\Big{\langle}\\!\langle
m|U(\tau)|n\rangle\langle n|U^{\dagger}(\tau)|m\rangle\langle
m^{\prime}|U(\tau)|n^{\prime}\rangle\langle
n^{\prime}|U^{\dagger}(\tau)|m^{\prime}\rangle\\!\Big{\rangle}_{\boldsymbol{\theta}^{0},\boldsymbol{\theta}^{\tau}}$
(63) $\displaystyle=$ $\displaystyle\left\langle
U_{m,n}(\tau)U^{\dagger}_{n,m}(\tau)U_{m^{\prime},n^{\prime}}(\tau)U^{\dagger}_{n^{\prime},m^{\prime}}(\tau)\right\rangle_{\boldsymbol{\theta}^{0},\boldsymbol{\theta}^{\tau}}\,.$

Because the GUE measure is invariant under unitary conjugation, we can
equivalently evaluate this average over angles as an integral over the unitary
group $\mathcal{U}(N)$, i.e.

$\int_{\mathcal{U}(N)}dUU_{m,n}U^{\dagger}_{n,m}U_{m^{\prime},n^{\prime}}U^{\dagger}_{n^{\prime},m^{\prime}}\,.$
(64)

Recognizing here the second moment of the Haar integral, the evaluation of
this expression can be done using the Weingarten functions as detailed above
and yields

$\displaystyle\left\langle
p_{m|n}^{\tau}p_{m^{\prime}|n^{\prime}}^{\tau}\right\rangle_{\boldsymbol{\theta}^{0},\boldsymbol{\theta}^{\tau}}=\frac{1}{N^{2}-1}\Big{(}1+\delta_{nn^{\prime}}\delta_{mm^{\prime}}-\frac{1}{N}\delta_{nn^{\prime}}-\frac{1}{N}\delta_{mm^{\prime}}\Big{)}\,.$
(65)

This expression can be used to compute the exact average Loschmidt echo Eq.
(41), upon substituting it into Eq. (62) and evaluating the energy averages.

##  Appendix B Spectral form factor

We derive here an explicit form of the spectral form factor that provides an
alternative form to the result in [27], with a corrected sign and expression.
Starting with the definition of the connected form factor in (44), we expand
the square value in Eq. (8) and write

$\displaystyle g_{c}(\sigma)$ $\displaystyle=$
$\displaystyle-\sum_{n=0}^{N-1}\sum_{m=0}^{N-1}\int
dEdE^{\prime}\varphi_{n}(E)\varphi_{m}(E)e^{-\sigma
E}\varphi_{m}(E^{\prime})\varphi_{n}(E^{\prime})e^{-\sigma^{*}E^{\prime}}$
(66) $\displaystyle=$
$\displaystyle-\sum_{n,m=0}^{N-1}\frac{\mathcal{I}_{n,m}(\sigma)}{2^{n}n!\sqrt{\pi}}\frac{\mathcal{I}_{m,n}(\sigma^{*})}{2^{m}m!\sqrt{\pi}}\,,$

where the second line has been obtained using the definition of
$\varphi_{n}(E)$ given in the main text and introducing the integral over the
Hermite polynomials

$\mathcal{I}_{n,m}(\sigma)=\int dEe^{-E^{2}-\sigma
E}\mathcal{H}_{n}(E)\mathcal{H}_{m}(E)\,.$ (67)

Note the symmetrical role played by $(n,m)$ in this function. To find an
explicit form, we follow [27] and, for $n\leq m$, use the identity
$\int_{\infty}dEe^{-E^{2}}\mathcal{H}_{n}(E)f(E)=\int_{\infty}dEe^{-E^{2}}D^{n}f(E)$
for $f(E)=e^{-\sigma E}\mathcal{H}_{m}(E)$ to write

$\mathcal{I}_{n,m}(\sigma)=\int dEe^{-E^{2}}D^{n}(e^{-\sigma
E}\mathcal{H}_{m}(E))\,.$ (68)

The differential $D^{n}=(d/dE)^{n}$ can be expanded in a binomial form,
$D^{n}(e^{-\sigma
E}\mathcal{H}_{m}(E))=\sum_{k=0}^{n}\binom{n}{k}D^{k}(e^{-\sigma
E})D^{n-k}(\mathcal{H}_{m}(E))$, and leads, using the derivatives of the
Hermite polynomials
$D^{j}(\mathcal{H}_{m}(E))=2^{j}m!/(m-j)!\mathcal{H}_{m-j}(E)$, to a sum of
single-Hermite integral

$\displaystyle\mathcal{I}_{n,m}(\sigma)=\sum_{k=0}^{n}\binom{n}{k}\frac{m!}{(m-n+k)!}2^{n-k}(-\sigma)^{k}\int
dEe^{-E^{2}-\sigma E}\mathcal{H}_{m-n+k}(E)\,.$ (69)

Using the same identity with $f(E)=e^{-\sigma E}$, the last integral can be
evaluated as [27]

$\displaystyle\int dEe^{-E^{2}-\sigma
E}\mathcal{H}_{m-n+k}(E)=\mathcal{I}_{m-n+k}(\sigma)=\sqrt{\pi}(-\sigma)^{m-n+k}e^{\sigma^{2}/4}\,.$
(70)

We further note the simplification of the sum

$\displaystyle\sum_{k=0}^{n}\binom{n}{k}\frac{x^{k}}{(m-n+k)!}$
$\displaystyle=$
$\displaystyle\frac{n!}{m!}\sum_{k=0}^{n}\binom{m}{n-k}\frac{x^{k}}{k!}=\frac{n!}{m!}L_{n}^{m-n}(-x)$
(71)

in terms of the generalized Laguerre polynomials. After symmetrization and
defining $\tilde{n}\equiv{\rm min}(n,m)$, this gives Eq. (69) as

$\mathcal{I}_{n,m}(\sigma)=\sqrt{\pi}e^{\frac{\sigma^{2}}{4}}(-\sigma)^{|m-n|}\,2^{\tilde{n}}\,\tilde{n}!\,L_{\tilde{n}}^{|m-n|}(-\sigma^{2}/2)\,.$
(72)

Using this expression and the fact that
$\mathcal{I}_{n,m}(\sigma)\mathcal{I}_{m,n}(\sigma^{*})=|\mathcal{I}_{n,m}(\sigma)|^{2}$,
we obtain an analytical expression for the connected form factor, Eq. (66),
given in Eq. (45) of the main text.

##  Appendix C Loschmidt echo and correlation functions

Finally, following [69] we briefly comment on a connection between the GUE-
averaged Loschmidt echo and correlation functions at infinite temperature. We
focus on a particular time-ordered $2k$-point correlation function

${\rm
Tr}\left\\{\hat{A}_{1}\,\hat{B}_{1}(t)\cdots\hat{A}_{k}\,\hat{B}_{k}(t)\,\hat{\rho}\right\\}\,,$
(73)

where $\hat{A}_{1},\cdots,\hat{A}_{k}$ and $\hat{B}_{1},\cdots,\hat{B}_{k}$
are Pauli operators acting on a system of $N_{p}$ qubits and the dimension of
the Hilbert space of the system is $N=2^{N_{p}}$. This can be understood as a
correlator evaluated at infinite temperature ($\beta=0$), i.e, for the
completely mixed state $\hat{\rho}=\mathbb{1}/N$. Here
$\hat{B}_{j}(t)\equiv\hat{U}^{\dagger}(t)\,\hat{B}_{j}\,\hat{U}(t)$, with the
unitary $\hat{U}(t)\in\mathcal{E}_{t}^{\rm GUE}$. The average of these kind of
out-of-time ordered correlation functions for an ensemble of unitary operators
such as $\mathcal{E}_{t}^{GUE}$ can be written as

$\displaystyle\int_{\mathcal{E}_{t}^{GUE}}\,d\hat{U}\,{\rm
Tr}\left\\{\hat{A}_{1}\,\hat{B}_{1}(t)\cdots\hat{A}_{k}\,\hat{B}_{k}(t)\,\hat{\rho}\right\\}\,.$
(74)

These ensemble averaged correlators were shown to be related to the $k-$th
frame potential through [69]

$\displaystyle\frac{N^{4k}}{N^{2(k+1)}}\,\mathcal{F}^{(k)}_{\rm
GUE}=\sum_{\left\\{\hat{A}\right\\}\,\left\\{\hat{B}\right\\}}\left|{\int_{\mathcal{E}_{t}^{GUE}}}d\hat{U}\,{\rm
Tr}\left\\{\hat{A}_{1}\,\hat{B}_{1}(t)\cdots\hat{A}_{k}\,\hat{B}_{k}(t)\,\hat{\rho}\right\\}\right|^{2}\,,$
(75)

where the sum is over the $N^{4k}$ possible Pauli operators
$\hat{A}_{1},\cdots,\hat{A}_{k}\,,\hat{B}_{1},\cdots,\hat{B}_{k}$. Thus,
considering the case for $k=1$ and the correspondence established in Eq. (51),
it is possible to express the GUE-averaged Loschmidt echo in terms of an
averaged 2-point function

$\left\langle\mathcal{L}(t)\right\rangle=\frac{1}{N^{2}}\\!\\!\\!\sum_{\left\\{\hat{A}\right\\}\,\left\\{\hat{B}\right\\}}\left|\int_{\mathcal{E}_{t}^{GUE}}\,d\hat{U}\,{\rm
Tr}\left\\{\hat{A}\,\hat{B}(t)\,\hat{\rho}\right\\}\right|^{2}\,,$ (76)

both measuring through their decrease, the degree of randomness induced by
GUE-time evolution.

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