#### Chaos, complexity, and random matrices

HJE
Chaos, complexity, and random matrices
Jordan Cotler 0 1 2 4 5
Nicholas Hunter-Jones 0 1 2 5
Junyu Liu 0 1 2 5
Beni Yoshida 0 1 2 3 5
0 Pasadena , California 91125 , U.S.A
1 California Institute of Technology , USA
2 Stanford , California 94305 , U.S.A
3 Perimeter Institute for Theoretical Physics
4 Stanford Institute for Theoretical Physics, Stanford University
5 Waterloo , Ontario N2L 2Y5 , Canada
Chaos and complexity entail an entropic and computational obstruction to describing a system, and thus are intrinsically di cult to characterize. In this paper, we consider time evolution by Gaussian Unitary Ensemble (GUE) Hamiltonians and analytically compute out-of-time-ordered correlation functions (OTOCs) and frame potentials to quantify scrambling, Haar-randomness, and circuit complexity. While our random matrix analysis gives a qualitatively correct prediction of the late-time behavior of chaotic nd unphysical behavior at early times including an O(1) scrambling time and the apparent breakdown of spatial and temporal locality. The salient feature of GUE Hamiltonians which gives us computational traction is the Haar-invariance of the ensemble, meaning that the ensemble-averaged dynamics look the same in any basis. Motivated by this property of the GUE, we introduce k-invariance as a precise de nition of what it means for the dynamics of a quantum system to be described by random matrix theory. We envision that the dynamical onset of approximate k-invariance will be a useful tool for capturing the transition from early-time chaos, as seen by OTOCs, to late-time chaos, as seen by random matrix theory.
AdS-CFT Correspondence; Black Holes; Matrix Models; Random Systems
1 Introduction 2
Form factors and random matrices
2.1
2.2
Random matrix theory
Spectral form factors
2.2.1
2.2.2
2-point spectral form factor at in nite temperature
2-point spectral form factor at nite temperature
2.3
4-point spectral form factor at in nite temperature
3.1
3.2
3.3
4.1
4.2
4.3
4.4
4.5
3
Out-of-time-order correlation functions
Spectral form factor from OTOCs
OTOCs in random matrix theory
Scrambling in random matrices
4
Frame potentials and random matrices
Overview of QI machinery
Frame potentials for the GUE
Higher k frame potentials
Frame potentials at nite temperature
Time scales from GUE form factors
5
6
7
Complexity and random matrices
Characterization of Haar-invariance
Discussion
A Scrambling and 2-designs
A.1 Scrambling
A.2 Unitary designs
A.3 Approximate 2-designs
B Information scrambling in black holes
C Spectral correlators and higher frame potentials
C.1 Expressions for spectral correlators
C.2 Expressions for higher frame potentials
C.3 Expressions for Weingarten
D Additional numerics
D.1 Form factors and numerics
D.2 Frame potentials and numerics
D.3 Minimal realizations and time averaging
{ i {
Introduction
Quantum chaos is a general feature of strongly-interacting systems and has recently
provided new insight into both strongly-coupled many-body systems and the quantum nature
of black holes. Even though a precise de nition of quantum chaos is not at hand,
understanding how chaotic dynamics process quantum information has proven valuable. For
instance, Hayden and Preskill [1] considered a simple model of random unitary evolution
to show that black holes rapidly process and scramble information. The suggestion that
black holes are the fastest scramblers in nature [2, 3] has led to a new probe of chaos
in quantum systems, namely the 4-point out-of-time-order correlation function (OTOC).
Starting with the work of Shenker and Stanford [4, 5], it was shown [6] that black holes are
maximally chaotic in the sense that a bound on the early time behavior of the OTOC is
saturated. Seperately, Kitaev proposed a soluble model of strongly-interacting Majorana
fermions [7, 8], which reproduces many features of gravity and black holes, including the
saturation of the chaos bound [9, 10]. The Sachdev-Ye-Kitaev model (SYK) has since been
used as a testing ground for questions about black hole information loss and scrambling.
In recent work, [11] found evidence that the late time behavior of the SYK model
can be described by random matrix theory, emphasizing a dynamical perspective on more
standard notions of quantum chaos. Random matrix theory (RMT) has its roots in nuclear
physics [12, 13] as a statistical approach to understand the spectra of heavy atomic nuclei,
famously reproducing the distribution of nearest neighbor eigenvalue spacings of nuclear
resonances. Random matrix theory's early success was later followed by its adoption in
a number of sub elds, including large N quantum
eld theory, string theory, transport
in disordered quantum systems, and quantum chaos. Indeed, random matrix eigenvalue
statistics have been proposed as a de ning characteristic of quantum chaos, and it is thought
that a generic classically chaotic system, when quantized, has the spectral statistics of a
random matrix ensemble consistent with its symmetries [14].
Current thinking holds that both spectral statistics and the behavior of the OTOC
serve as central diagnostics of chaos, although the precise relation between the two is
unclear. OTOCs have recently been studied using techniques from quantum information
theory, and it was found that their decay as a function of time quanti es scrambling [15] and
randomness [16]. The goal of this paper is to connect various concepts as a step towards a
quantum information-theoretic de nition of quantum chaos that incorporates scrambling,
chaotic correlation functions, complexity, approximate randomness, and random matrix
universality.
As alluded to above, an important rst step to bridge early-time chaos and late-time
dynamics is to understand the relation between the OTOC and the spectral statistics. We
derive an explicit analytical formula relating certain averages of OTOCs and spectral form
factors which holds for arbitrary quantum mechanical systems. A simple corollary is that
spectral form factors can be approximated by OTOCs de ned with respect to random
(typically non-local) operators, highlighting the fact that spectral statistics are good probes of
macroscopic thermodynamic properties, but may miss important microscopic physics such
as early-time chaos. We also compute correlation functions for an ensemble of
Hamiltoni{ 1 {
ans given by the Gaussian Unitary Ensemble (GUE), and nd that 4-point OTOCs decay
faster than 2-point correlators contrary to
ndings for local quantum Hamiltonians [6].
Due to the basis independence of the GUE, averaged correlation functions do not depend
on sizes of operators, and thus can be expressed solely in terms of spectral form factors.
Furthermore, we nd that correlators for GUE Hamiltonians do not even depend on the
time-ordering of operators. These results imply that the GUE ignores not only spatial but
also temporal locality.
Another important question is to understand the approach to entropic (as well as
quantum complexity) equilibrium via pseudorandomization at late times in strongly
coupled systems. We consider the ensemble of unitaries generated by xed GUE Hamiltonians,
HJEP1(207)48
namely
Et
GUE = e iHt; for H 2 GUE ;
(1.1)
and study its approach to Haar-randomness by computing frame potentials which quantify
the ensemble's ability to reproduce Haar moments. We
nd that the ensemble forms an
approximate k-design at an intermediate time scale, but then deviates from a k-design at
late times. These results highlight that the k-design property fails to capture late time
behavior of correlation functions. An interesting application of unitary k-designs is that
Haar-randomness is a probe of quantum complexity. We apply techniques from [16] to
lower bound the quantum circuit complexity of time evolution by GUE Hamiltonians and
nd a quadratic growth in time.
In order to make precise claims about the behavior of OTOCs and frame potentials for
GUE Hamiltonians, we need explicit expressions for certain spectral quantities.
Accordingly, we compute the 2-point and 4-point spectral form factors for the GUE at in nite
temperature, as well as the 2-point form factor at nite temperature. We then use these
expressions to discuss time scales for the frame potentials. We also analytically compute
the late-time value of the k-th frame potential for arbitrary k.
Under time evolution by strongly-coupled systems, correlations are spread throughout
(i.e., Et
the system and the locality of operators as well as time-ordering appear to be lost from
the viewpoint of correlation functions, as implied by the late-time universality of random
matrix theory. Also motivated by the k-design property's failure to capture late-time chaos
GUE fails to be Haar-random at late times), we propose a new property called
kinvariance, which may provide a better probe of chaos at both early and late times. The
property of k-invariance characterizes the degree to which an ensemble is Haar-invariant,
meaning that the ensemble is invariant under a change of basis.
When the dynamics
becomes approximately Haar-invariant, correlation functions can be captured solely in
terms of spectral functions, which signi es the onset of an e ective random matrix theory
description.
We thus provide an information theoretically precise de nition of what it
means for a system's dynamics to be described by random matrix theory. Speci cally, we
say that an ensemble of Hamiltonian time evolutions Et is described by random matrix
theory at times greater than or equal to t with respect to 2k-point OTOCs when Et is
approximately k-invariant with respect to its symmetry class, for example the symmetry
class of either the unitary, orthogonal, or symplectic groups.
{ 2 {
The paper is organized as follows: in section 2, we provide a brief overview of random
matrix theory and explicitly compute the spectral form factors for the GUE at in nite and
nite temperature. In section 3, we compute correlation functions for the GUE, including
the OTOC, and demonstrate that they can be expressed in terms of spectral correlators as
well. In section 4, we compute frame potentials for the GUE, and extract the timescales
when it becomes an approximate k-design both at
nite and in nite temperatures. We
show that the frame potentials can be also expressed as products of sums of spectral
correlators. In section 5, we discuss complexity bounds and complexity growth for the
GUE. In section 6, we discuss Haar-invariance as a diagnostic of delocalization of spatial
degrees of freedom and random matrix universality at late times.
We conclude with a
holes, more details of our random matrix calculations, and numerics.
2
Form factors and random matrices
For a long time, the spectral statistics of a random matrix were seen as a de ning feature
of quantum chaos. More recently, it has been proposed that the late time behavior of
certain strongly coupled theories with large numbers of degrees of freedom also exhibit a
dynamical form of random matrix universality at late times [11]. The central object of
study in this recent work is the 2-point spectral form factor,1 which is de ned in terms of
the analytically continued partition function
and where h i denotes the average over an ensemble of Hamiltonians. In SYK as well as
standard RMT ensembles, the 2-point spectral form factor decays from its initial value and
then climbs linearly back up to a
oor value at late times. The early time decay of the
form factor is called the slope, the small value at intermediate times is called the dip, the
steady linear rise is called the ramp, and the late time oor is called the plateau. In gure 1
we observe these features in SYK with N = 26 Majoranas, which has GUE statistics at
late times.2 Furthermore, it was found that in SYK, time scales and many features of the
slope, dip, ramp and plateau agree with predictions from RMT.
In this section, we brie y review random matrix theory. Further, we study the
2point spectral form factor for the GUE at both in nite and nite temperature, compute its
1One motivation for studying this object is a simple version of the information loss problem in
AdS/CFT [17], where the apparent exponential decay of 2-point correlation functions in bulk e ective
eld theory contradicts the
nite late-time value of e O(S) implied by the discreteness of the spectrum. As
2 or 6 corresponds to GUE statistics [23]. Furthermore, the spectral density of SYK
and its relation to random matrices has also been discussed in [24].
{ 3 {
R2
1
0.100
S
l
o
p
e
Dip
100
p
m
a
R
Plateau
105
t
analytic form, and extract its dip and plateau times and values.3 In addition, we compute
the 4-point form factor and extract relevant time scales and values.
We
nd that the
late-time rise in the 4-point form factor is quadratic in t, in contrast to the linear rise
in the 2-point form factor. The expressions derived in this section will give us analytic
control over the correlation functions and frame potentials discussed in later sections. For
a detailed treatment of the random matrix ensembles, we refer the reader to [25{27].
2.1
Random matrix theory
the normalization GUE(L; 0; 1=p
The Gaussian Unitary Ensemble GUE(L; ; ) is an ensemble of L
L random Hermitian
matrices, where the o -diagonal components are independent complex Gaussian random
variables N ( ; )
C with mean
and variance 2, and the diagonal components are
independent real Gaussian random variables N ( ; )R. It is common in the math literature to
work with GUE(L; 0; 1) which has zero mean and unit variance, but we will instead use
L) so that the eigenvalues do not scale with the system
size.4 The probability density function of the ensemble has a Gaussian form
P (H) / e L2 TrH2 ;
(2.2)
up to a normalizing factor. As the GUE is invariant under unitary conjugation H ! U HU y,
the integration measure dH = d(U HU y) is likewise invariant. The probability measure
P (H) dH on the ensemble integrates to unity.
3We consider the GUE since it corresponds to the least restrictive symmetry class of Hamiltonians. The
4The reason for using the normalization GUE(L; 0; 1=p
generalization of our analysis to the GOE or GSE is left for future work.
the standard normalization GUE(L; 0; 1), the energy spectrum ranges from
normalization GUE(L; 0; 1=p
that by applying a local operator, one may change the energy of the system by O( L). With the physical
L), the energies lie within the range
2 to 2, and local operators act with
O(1) energy. See [28] for discussions on normalizing q-local Hamiltonians.
L) instead of GUE(L; 0; 1) is as follows: with
p
2 L to 2
p
pL. This implies
{ 4 {
dH = C j ( )j2 Y d idU ;
i
( ) = Y( i
i>j
j ) :
where dU is the Haar measure on the unitary group U(L) and
( ) is the Vandermonde
determinant
The joint probability distribution of eigenvalues is
P ( 1; : : : ; L) = Ce L2 Pi i2 j ( )j2 ;
and is symmetric under permutations of its variables. For simplicity, we de ne a measure
which absorbs the Gaussian weights, eigenvalue determinant, and constant factors. We
integrate over the GUE in the eigenvalue basis as
hO( )iGUE
Z
D
O( )
where
D
= C
Z
i
Y d ij ( )j2e L2 Pi i2 = 1 :
The probability density of eigenvalues ( ), where
d
( ) = 1 ;
Instead of integrating over dH directly, it is convenient to change variables to
eigenvalues and diagonalizing unitaries. Up to a normalizing constant C de ned in eq. (C.1) in
appendix C, the measure becomes
(2.3)
(2.4)
(2.5)
(2.6)
(2.7)
(2.8)
(2.9)
(2.10)
(2.11)
(2.12)
can be expressed in terms of a disconnected piece and a squared sine kernel as [25]
(
2
)( 1; 2) =
L2
L(L
1)
( 1) (
2
)
L2
L(L
1)
can be written in terms of the joint eigenvalue probability density by integrating over all
but one argument
Z
( ) =
d 1 : : : d L 1P ( 1; : : : ; L 1; ) :
The spectral n-point correlation function, i.e. the joint probability distribution of n
eigenvalues, (n) is de ned as
(n)( 1; : : : ; n)
With these de nitions at hand, we quote a few central results. In the large L limit,
the density of states for the Gaussian ensembles gives Wigner's famous semicircle law,
where the semicircle diameter is xed by our chosen eigenvalue normalization. Also in the
large L limit, the spectral 2-point function
( ) =
1 p
2
4
2
as
L ! 1 ;
(n)( 1; 2) =
d 3 : : : d LP ( 1; : : : ; L) ;
The 2-point spectral form factor for a single Hamiltonian H is given in terms of the
analytically continued partition function Z( ; t) = Tr (e
H iHt) as
R2H ( ; t)
Z( ; t)Z ( ; t) = Tr (e
H iHt)Tr (e
H+iHt) :
Similarly, the spectral form factor averaged over the GUE is denoted by
HJEP1(207)48
(2.13)
(2.14)
(2.15)
Z
i
which is the Fourier transform of the spectral 2-point function. At in nite temperature
= 0, the Fourier transform of the density of states is just Z(t) = Tr (e iHt), the trace of
unitary time evolution. Using the semicircle law, we take the average of Z(t) at large L
hZ(t)iGUE =
Z
D
X e i it = L
Z 2
2
d
( )e i t =
LJ1(2t)
t
;
where J1(t) is a Bessel function of the rst kind. The function J1(2t)=t is one at t = 0 and
oscillates around zero with decreasing amplitude that goes as
1=t3=2, decaying at late
times. At in nite temperature, the 2-point spectral form factor for the GUE is
R2(t) = Z(t)Z (t) GUE =
Z
dH Tr e iHt Tr eiHt =
D
X ei( i j)t :
(2.16)
More generally, we will also be interested in computing 2k-point spectral form factors
R2k(t) =
D Z(t)Z (t) kE
GUE
=
Z
D
X
i0s;j0s
ei( i1 +:::+ ik j1 ::: jk )t ;
(2.17)
the Fourier transform of the spectral 2k-point function (2k).5 Although the form factors
can be written exactly at
nite L, our analysis will focus on analytic expressions that
capture the large L behavior.6
Note that in [11], 2-point form factors were normalized via dividing by Z( )2. At
in nite temperature, this simply amounts to dividing by L2, but at
nite temperature
the situation is more subtle. As we will comment on later, the correct object to study
is the quenched form factor hZ( ; t)Z ( ; t)=Z( )2i, but since we only have analytic
control over the numerator and denominator averaged separately, we instead work with the
unnormalized form factor R2 as de ned above.
5In the random matrix literature, the 2-point form factor is often de ned as the Fourier transform of
the connected piece of the spectral 2-point correlation function, where the connected piece of the spectral
2k-point function is often referred to as the 2k-level cluster function. Our de nition for the 2k-point spectral
form factor R2k includes both connected and disconnected pieces.
6In addition to relating the form factor to the delty of certain states, [29] also studies the 2-point spectral
form factor for the GUE, computing an analytic form at nite L and discussing the dip and plateau.
{ 6 {
Z
2.2.1
Here we calculate the 2-point form factor at
= 0. Working at large L, we can evaluate
R2 by rst pulling out the contribution from coincident eigenvalues R2(t) =
eq. (2.12). Using eq. (2.15), we integrate the rst term, a product of 1-point functions,
and nd
In order to integrate the sine kernel, we make the change of variables:
which allows us to rewrite the integral
L
2
Having decoupled the variables, in order to integrate over u1 and u2, we must employ a
short distance cuto . We develop a certain approximation method which we refer to as the
`box approximation,' and explain its justi cation in appendix C. Speci cally, we integrate
u1 from 0 to u2, and integrate u2 from
=2 to =2,
L
2
Z
du1du2
L u
2
1
sin2(Lu1) eiu1t = L
(
1
0 ;
2tL ; for t < 2L
Note that in the random matrix theory literature, a common treatment [30] is to
approximate the short-distance behavior of (
2
)( 1; 2) by adding a delta function for coincident
points 1 =
2 and inserting a 1-point function into the sine kernel. For R2 this gives the
same result as the approximation above, but this short-distance approximation does not
generalize to higher k-point form factors, as discussed in appendix C. The 2-point form
factor we compute is7
where we de ne the functions
As was discussed in [11], we can extract the dip and plateau times and values from R2.
From the ramp function r2, we observe that the plateau time is given by
7We emphasize that this function relied on an approximation and while it captures certain desired
behavior, it should not be viewed as exact. In appendix D we provide numerical checks and discuss an
improvement of the ramp function r2(t).
tp = 2L
{ 7 {
0.100
0.010
0.001
10-4
10-5
10-6
L=10
L=102
L=103
L=104
for various values of L and normalized by the initial value L2. We observe the linear ramp and
scaling of the dip and plateau with L.
where after the plateau time, the height of the function R2 is the constant L. This value
can also be derived by taking the in nite time average of R2.
The other important time scale is the dip time td, which we can estimate using the
asymptotic form of the Bessel function at large t, which gives
r1(t)
1 cos(2t
3 =4)
p t
;
t
t
d
p
L ;
oscillating at times
O(1) with decaying envelope
O(1), we will be interested in the dip time as seen by the envelope, especially because
the oscillatory behavior disappears at
nite temperature (see
gure 3). Solving for the
t 3=2. While the rst dip time is
minimum of the envelope of R2, we nd
(2.26)
(2.27)
1:18p
nd the dip value R2(td)
L in gure 2.
p
up to order one factors. The true minimum of the envelope and ramp is (6= )1=4pL
L, but in light of the approximations we made, and the fact that the precise ramp
behavior is somewhat ambiguous, we simply quote the dip time as td
L. At td, we
p
L. We plot the 2-point form factor for di erent dimensions
The oscillations in the early time slope behavior of the form factor simply arise from the
oscillatory behavior of the Bessel function, i.e. the zeros of r1(t)2.
2.2.2
2-point spectral form factor at nite temperature
Recall that spectral 2-point function at nite temperature is de ned as
As described in appendix C, we insert the spectral 2-point function (
2
) and, using the
short-distance kernel, nd R2(t; ) in terms of the above functions:
First we comment on the validity of the approximations used in the nite temperature
case. The rst and third terms of eq. (2.28), dominating at early and late times respectively,
are computed from the 1-point function. Therefore, the expression captures the early time,
slope, and plateau behaviors. The dip and ramp behavior, encoded in the r2 term, are
more subtle. The expression correctly captures the slope of the ramp, but deviates from
the true ramp at large . We will discuss this more in appendix C, but here only discuss
quantities around the dip for small , where eq. (2.28) is a good approximation.
The ramp function r2, which is the same as at in nite temperature, gives the
plateau time
tp = 2L :
For convenience we de ne the function h1( )
initial value and plateau value are thus given by
J1(2i )=i , which is real-valued in .8 The
R2(0) = (h1( ))2L2 ;
R2(tp) = h1(
2
)L :
To nd the dip time, we make use of the asymptotic expansion of the Bessel function as
L2r1(t + i )r1( t + i )
L2
2 t3 cosh(4 )
sin(4t)
L2
t
3 cosh2(
2
) :
Finding the minimum of the expression gives the dip time
p
td = h2( ) L
where h2( )
1 +
+ O( 4) ;
2
2
and evaluating R2 at the dip gives
R2(td)
up to order one factors. While we could write down full expressions for the dip time h2
and dip value h3 in terms of the Bessel function, we only trust eq. (2.28) in this regime for
small , and thus report the functions perturbatively.
The 2-point form factor is plotted in
gure 3 for various values of L and . While
where S(
2
) is the thermal Renyi-2 entropy.
increasing the dimension L lowers the dip and plateau values and delays the dip and
plateau times, decreasing temperature raises the dip and plateau values and delays the dip
times. We also note that lowering the temperature smooths out oscillations from the Bessel
function.9 After normalizing R2( ; t) by its initial value, the late-time value is ' 2 S(
2
)
8For instance, to emphasize its real-valuedness, we could equivalently write h1( ) as a regularized
hypergeometric function h1( )
0Fe1(2; 2).
9While the oscillatory behavior still persists at nite temperature, the width of the dips become very
sharp as we increase
and thus the oscillations are not observed when plotted. Furthermore, if we average
over a small time window, the oscillations are also smoothed out.
{ 9 {
(2.29)
(2.30)
(2.31)
(2.32)
(2.33)
HJEP1(207)48
0.100
0.010
R2 β
1
plotted at di erent values of L, and on the right plotted at di erent temperatures, normalized by
the initial value. We see that the dip and plateau both scale with
and L and that lowering the
temperature smooths out the oscillations in R2.
2.3
4-point spectral form factor at in nite temperature
We can also compute the 4-point form factor at in nite temperature, de ned as
R4(t)
Z(t)Z(t)Z (t)Z (t) GUE =
ei( i+ j k `)t :
(2.34)
Z
D
X
i;j;k;`
As we explain in appendix C, we compute R4 by replacing (4) by a determinant of sine
kernels and carefully integrating each term using the box approximation. The result is
R4(t) = L4r14(t)+2L2r22(t) 4L2r2(t) 7Lr2(2t)+4Lr2(3t)+4Lr2(t)+2L2
L ;
(2.35)
given in terms of the functions r1(t) and r2(t) de ned above. The initial value of R4 is L4.
Given the dependence on the ramp function, the plateau time is still tp = 2L. The plateau
value 2L2
L matches the in nite time average of eq. (2.34). The dip time is found again
by considering the leading behavior of R4 and expanding the Bessel functions
R4
L
t
4
+ (t
2)
L4
t6 2 + 2 (t
t
2) :
Solving for the minimum, we nd the dip time
t
2
t
d
p
L ;
where at the dip time R4(td)
L. We plot the R4(t) for various values of L in gure 4.
(2.36)
(2.37)
0.01
10-4
10-6
10-8
10-10
L=8
L=40
L=200
L=1000
GUE R4 at β = 0
values of L and normalized by their initial values. We observe the scaling of the dip and plateau,
and the quadratic rise
t2.
Let us summarize the time scales and values for the form factors considered above:
form factor time scale
time
R2(t)
The -dependent functions were de ned above.
With an understanding of the rst few form factors, we brie y describe the expected
after the dip levels o at the plateau time 2L, with plateau value
kLk.
behavior for 2k-point form factors R2k(t) (with k
J12k(2t)=t2k, reaching the dip at time td
p
L where R2k(td)
L). Initially, R2k decays from L2k as
Lk=2. The
tk growth
Given that we employed some approximation to compute the form factors, we perform
numerical checks for the expressions above in appendix D. At both in nite and
nite
temperature, we correctly capture the time scales, early time decay, dip behavior, and the
late-time plateau, but nd slight deviations from the analytic prediction for the ramp. We
discuss this and possible improvements to the ramp function in appendix D.
Later we will study frame potentials which diagnose whether an ensemble forms a
k-design. We will nd that the frame potentials for the ensemble of unitaries generated
by the GUE can be written in terms of the spectral form factors discussed here, thereby
allowing us to extract important time scales pertaining to k-designs.
3.1
Spectral form factor from OTOCs
Although quantum chaos has traditionally focused on spectral statistics, recent
developments from black hole physics and quantum information theory suggest an alternative way
of characterizing quantum chaos via OTOCs [1, 4, 6, 15]. In this subsection, we bridge
the two notions by relating the average of 2k-point OTOCs to spectral form factors. We
work at in nite temperature (
= 0), but note that by distributing operator insertions
around the thermal circle, the generalization to nite temperature is straightforward. The
results in this subsection are not speci c to GUE and are applicable to any quantum
mechanical system.
Consider some Hamiltonian H acting on an L = 2n-dimensional Hilbert space, i.e.
consisting of n qubits.
We start by considering the 2-point autocorrelation function
hA(0)Ay(t)i, time evolved by H. We are interested in the averaged 2-point function:
HJEP1(207)48
dAhA(0)Ay(t)i
dA Tr(Ae iHtAyeiHt)
where R dA represents an integral with respect to a unitary operator A over the Haar
measure on U(2n). We note that since the 2-point Haar integral concerns only the rst moment
of the Haar ensemble, we can instead average over the ensemble of Pauli operators10
Z
Z
10This is because the Pauli operators form a 1-design.
where Aj are Pauli operators and L2 = 4n is the number of total Pauli operators for a
system of n qubits. To derive the spectral form factor, we will need the rst moment of
the Haar ensemble
Z
dA AjkAy`m =
1 j `
L m k ;
Z
or equivalently
dA AOAy =
Tr(O)I:
1
L
Applying eq. (3.3) to eq. (3.1), we obtain
Z
dAhA(0)Ay(t)i = jTr(e iHt) 2
j = R2H (t)
L2 ;
L2
where R2Hk(t)
jTr(e iHt)j2k is the same as R2k(t) from before, but written for a
single Hamiltonian H instead of averaged over the GUE. Thus, the 2-point form factor is
proportional to the averaged 2-point function.
This formula naturally generalizes to 2k-point OTOCs and 2k-point form factors.
Consider 2k-point OTOCs with some particular ordering of operators
hA1(0)B1(t)
Ak(0)Bk(t)i
where
A1B1
AkBk = I:
(3.5)
(3.1)
(3.2)
(3.3)
(3.4)
Operators which do not multiply to the identity have zero expectation value at t = 0, and
the value stays small as we time-evolve. We are interested in the average of such 2k-point
OTOCs. By using eq. (3.3) 2k
1 times, we obtain
B1yAy1. Thus, higher-point spectral form factors can be also computed
from OTOCs. In fact, by changing the way we take an average, we can access various
types of form factors. For instance, let us consider OTOCs hA1(0)B1(t)
The fact that the expression on the right-hand side is asymmetric is because the operator
These expressions not only provides a direct link between spectral statistics and
physical observables, but also give a practical way of computing the spectral form factor. If
one wishes to compute or experimentally measure the 2-point form factor R2(t), one just
needs to pick a random unitary operator A and study the behavior of the 2-point correlator
hA(0)Ay(t)i. In order to obtain the exact value of R2(t), we should measure hA(0)A(t)i
for all possible Pauli operators and take their average. Yet, it is possible to obtain a pretty
good estimate of R2(t) from hA(0)A(t)i with only a few instances of unitary operator A.
Consider the variance of hA(0)A(t)i,
Z
hA(0)Ay(t)ia2vg
dAjhA(0)Ay(t)ij2
Z
dAhA(0)Ay(t)i :
2
If the variance is small, then the estimation by a single A would su ce to obtain a good
estimate of R2(t). Computing this, we obtain
(3.8)
(3.9)
(3.10)
This implies that the estimation error is suppressed by 1=L. By choosing a Haar unitary
operator A (or 2-design operator, such as a random Cli ord operator), one can obtain a
good estimate of R2(t).
A check in a non-local spin system.
To verify eq. (3.4) and the claim that the variance
of the 2-point functions is small, consider a random non-local (RNL) spin system with the
Hamiltonian given as the sum over all 2-body operators with random Gaussian couplings
Jij
[31]:
HRNL =
Jij
Si Sj ;
11BY learned eq. (3.7) from Daniel Roberts.
hA(0)Ay(t)ia2vg
O
1
L2
:
X
i;j; ;
for n = 5 sites and averaged over 500 samples. The thick blue line is R2=L2 and surrounding bands
of lines are all 1024 Pauli 2-point functions of di erent weight.
where i; j sum over the number of sites and ; sum over the Pauli operators at a given
site. Such Hamiltonians have a particularly useful property where locally rotating the spins
of HRNL with couplings Jij
creates another Hamiltonian HR0NL with di erent couplings
Ji0j . More precisely, if we consider an ensemble of such 2-local Hamiltonians;
ERNL = fHRNL; for Jij
2 Gaussiang
the ensemble is invariant under conjugation by any 1-local Cli ord operator
ERNL = V ERNLV y ;
V 2 1-body Cli ord:
Here a Cli ord operator refers to unitary operators which transform a Pauli operator to a
Pauli operator. For this reason, the 2-point correlation function hA(0)Ay(t)iERNL depends
only of the weight of Pauli operator A:
hA(0)Ay(t)iERNL = cm ;
where A is an m-body Pauli operator
(3.13)
and where h iERNL denotes the ensemble (disorder) average. Thus, this system is desirable
for studying the weight dependence of 2-point correlation functions.
As mentioned above, we can write the average over 2-point correlation functions as
the average over all Paulis as
Z
time evolving with HRNL. Numerically, for a single instance of HRNL, we nd that the
average over all 2-point functions of Pauli operators gives R2 as expected. In gure 5, for
(3.11)
(3.12)
(3.14)
n = 5 sites and averaged over 500 random instances of HRNL to suppress uctuations, we
plot R2 along side all 2-point functions of Pauli operators. We observe that correlation
functions depend only on the weight of A, with the higher weight Pauli operators clustered
around R2. The arrangement of the 2-point functions for Paulis of di erent weight depends
on the number of sites n. But for n = 5, the even and odd weight Paulis are respectively
below and above R2 at later times and weight 2 and 3 Paulis are the closest to R2. We
will comment on the size dependence of correlators in section 6.
The conclusion is that we can choose a few random Paulis, and by computing 2-point
functions, quickly approximate R2. We also checked that by increasing the number of
spins, the variance becomes small and 2-point functions become closer to R2.
HJEP1(207)48
Operator averages and locality.
Let us pause for a moment and discuss the meaning
of considering the operator average from the perspective of spatial locality in quantum
mechanical systems. In deriving the above exact formulae relating the spectrum and
correlators, we considered the average of OTOCs over all the possible Pauli operators. For a
system of n qubits, a typical Pauli operator has support on ' 3n=4 qubits because there
are four one-body Pauli operators, I; X; Y; Z. It is essential to recognize that the average of
correlation functions is dominated by correlations of non-local operators with big supports
covering the whole system. Thus, the spectral statistics have a tendency to ignore the
spatial locality of operators in correlation functions.12
In fact, the spectral statistics ignore not only spatial locality but also temporal locality
of operators. Namely, similar formulas can be derived for correlation functions with various
ordering of time. For instance, consider the following 4-point correlation function:
hA(0)B(t)C(2t)D(t)i
where the C operator acts at time 2t instead of 0 such that the correlator is not
out-oftime-ordered. Computing the average of the correlator with ABCD = I, we obtain
(3.15)
(3.16)
Z
dAdBdChA(0)B(t)C(2t)D(t)i = R4(t)
L4
which is exactly the same result as the average of 4-point OTOCs in eq. (3.6). Indeed,
time-ordering is washed away since GUE Hamiltonians cause a system to rapidly delocalize,
thus destroying all local temporal correlations.
In strongly coupled systems with local Hamiltonians, correlation functions behave
rather di erently depending on the time ordering of operators, as long as the time gaps
involved are small or comparable to the scrambling time [4, 5, 9, 33]. This observation hints
that the spectral statistics are good probes of correlations at long time scales, but may miss
some important physical signatures at shorter time scales, such as the exponential growth
of OTOCs with some Lyapunov exponent.
12Signatures of the locality of an individual Hamiltonian may be seen in properties of its spectrum, as
argued in [32].
3.2
Next, we turn our attention to correlators averaged over random matrices, analytically
computing the 2-point correlation functions and 4-point OTOCs for the GUE. We begin
with the 2-point correlation functions for the GUE
Z
hA(0)B(t)iGUE
dHhA(0)B(t)i
where
B(t) = e iHtB(0)eiHt ;
(3.17)
where R dH represents an integral over Hamiltonians H drawn from the GUE. Since the
GUE measure dH is invariant under unitary conjugation dH = d(U HU y) for all U , we can
express the GUE average as
hA(0)B(t)iGUE =
Z Z
dHdU AU e iHtU yBU eiHtU y
(3.18)
by inserting U; U y where dU is the Haar measure. Haar integrating, we obtain
hA(0)B(t)iGUE = hAihBi + R2(t)
L2
1
1 hhABii ;
hhABii
hABi
hAihBi
(3.19)
where hhABii represents the connected correlator. If A; B are non-identity Pauli operators,
for any non-identity Pauli operator A. It is worth emphasizing the similarity between
eq. (3.21) and eq. (3.4). Recall that eq. (3.4) was derived by taking an average over all
Pauli operators A and is valid for any quantum mechanical system while eq. (3.21) was
derived without any additional assumption on the locality of Pauli operator A. Namely,
the key ingredient in deriving eq. (3.21) was the Haar-invariance of the GUE measure dH.
The resemblance of eq. (3.21) and eq. (3.4) implies that the GUE is suited for studying
physical properties of chaotic Hamiltonians at macroscopic scales such as thermodynamic
quantities.
Next, we compute the 4-point OTOCs for the GUE
hA(0)B(t)C(0)D(t)iGUE :
Inserting U; U y, we must compute the fourth Haar moment
hA(0)B(t)C(0)D(t)iGUE =
Z Z
dHdU AU e iHtU yBU eiHtU yCU e iHtU yDU eiHtU y :
(3.20)
(3.21)
(3.22)
(3.23)
We can avoid dealing directly with the (4!)2 terms generated by integrating here and
focus on the leading behavior. Assuming that A; B; C; D are non-identity Pauli operators,
we obtain
hA(0)B(t)C(0)D(t)iGUE ' hABCDi RL4(4t) :
Thus, OTOCs are almost zero unless ABCD = I.13;14 A similar analysis allows us to
obtain the following result for 2k-point OTOCs:
(3.24)
(3.25)
hA1(0)B1(t) : : : Ak(0)Bk(t)iGUE ' hA1B1 : : : AkBki L2k :
R2k(t)
The above equation is nonzero when A1B1 : : : AkBk = I. Again, note the similarity between
eq. (3.25) and eq. (3.6). Recall that in order to derive eq. (3.6), we took an average over
OTOCs with A1B1 : : : AkBk = I. This analysis also supports our observation that the
GUE tends to capture global-scale physics very well.
Similar calculations can be carried out for correlation functions with arbitrary
timeordering. For m-point correlators, at the leading order, we have
hA1(t1)A2(t2) : : : Am(tm)iGUE ' hA1 : : : Ami Lm Tr(e it12H )Tr(e it23H ) : : : Tr(e itm1H )
1
where tij = tj
ti. Namely, we have:
hA(0)B(t)C(2t)D(t)iGUE ' hABCDi RL4(4t) :
So, for the GUE, hA(0)B(t)C(2t)D(t)iGUE ' hA(0)B(t)C(0)D(t)iGUE. This implies that
the GUE does not care if operators in the correlator are out-of-time-ordered or not, ignoring
both spatial and temporal locality.
Careful readers may have noticed that the only property we used in the above
derivations is the unitary invariance of the GUE ensemble. If one is interested in computing
correlation functions for an ensemble of Hamiltonians which are invariant under
conjugation by unitary operators, then correlation functions can be expressed in terms of spectral
form factors. Such techniques have been recently used to study thermalization in
manybody systems, see [35] for instance. We discuss this point further in section 6.
3.3
Scrambling in random matrices
Finally, we discuss thermalization and scrambling phenomena in random matrices by
studying the time scales for correlation functions to decay.
We begin with 2-point correlators and thermalization. In a black hole (or any thermal
system), quantum information appears to be lost from the viewpoint of local observers.
This apparent loss of quantum information is called thermalization, and is often associated
13In fact, one can prove that the GUE averaged OTOCs are exactly zero if ABCD is non-identity Pauli
operator for all times.
14For analysis related to eq. (3.24) in the context of SYK, see [34].
with the decay of 2-point correlation functions hA(0)B(t)i where A and B are some local
operators acting on subsystems HA and HB which local observers have access to. In the
context of black hole physics, HA and HB correspond to infalling and outgoing Hawking
radiation and such 2-point correlation functions can be computed from the standard
analysis of Hawking and Unruh [36, 37]. 2-point correlation functions of the form hA(0)B(t)i
have an interpretation as how much information about initial perturbations on HA can be
detected from local measurements on HB at time t. A precise and quantitative relation
between quantum information (mutual information) and 2-point correlation functions is
derived in appendix B. The upshot is that the smallness of hA(0)B(t)i implies the
information theoretic impossibility of reconstructing from Hawking radiation (de ned on HB)
an unknown quantum state (supported on HA) that has fallen into a black hole.
Is the GUE a good model for describing thermalization? For the GUE, we found
hA(0)B(t)i ' R2(t)=L2 for non-identity Pauli operators with AB = I. Since the early
time behavior of R2(t) factorizes and is given by
the time scale for the decay of 2-point correlation functions, denoted by t2, is O(1). This
is consistent with our intuition from thermalization in strongly coupled systems where
t2 '
. As such, quantum information appears to be lost in O(1) time for local observers
in systems governed by GUE Hamiltonians.
Next, let us consider 4-point OTOCs and scrambling. To recap the relation between
OTOCs and scrambling in the context of black hole physics, consider a scenario where Alice
has thrown an unknown quantum state into a black hole and Bob attempts to reconstruct
Alice's quantum state by collecting the Hawking radiation. Hayden and Preskill added an
interesting twist to this classic setting of black hole information problem by assuming that
the black hole has already emitted half of its contents and Bob has collected and stored
early radiation in some quantum memory he possesses. The surprising result by Hayden
and Preskill is that, if time evolution U = e iHt is approximated by a Haar random unitary
operator, then Bob is able to reconstruct Alice's quantum state by collecting only a few
Hawking quanta [1]. This mysterious phenomenon, where a black hole re ects a quantum
information like a mirror, relies on scrambling of quantum information where Alice's input
quantum information is delocalized over the whole system [15]. The de nition of scrambling
can be made precise and quantitative by using quantum information theoretic quantities
as brie y reviewed in appendix A and appendix B.
The scrambling of quantum information can be probed by the decay of 4-point OTOCs
of the form hA(0)B(t)Ay(0)By(t)i where A; B are some local unitary operators. An
intuition is that an initially local operator B(0) grows into some non-local operator under time
evolution via conjugation by e iHt, and OTOCs measure how non-locally B(t) has spread.
For this reason, the time scale t4 when OTOCs start decaying is called the scrambling time.
Having reviewed the concepts of scrambling and OTOCs, let us study scrambling in
random matrices. For the GUE, we found hA(0)B(t)C(0)D(t)i ' R4(t)=L4 for non-identity
Pauli operators with ABCD = I. Since one can approximate R4 as R4(t) ' R2(t)2 at
HJEP1(207)48
of operators and the emergence of k-invariance would be to compare connected pieces of
OTOCs with local and non-local operators and observe their eventual convergence. Of
particular interest is to nd the 2-invariance time when all the 4-point OTOCs, regardless
of sizes of operators, start to behave in a similar manner. This time scale must be at
least the scrambling time since OTOCs with local operators start to decay only around the
scrambling time while OTOCs with non-local operators decay immediately. Relatedly, we
would like to draw attention to an upcoming work [63] which studies the onset of random
matrix behavior at early times.
In this paper, we computed correlation functions averaged over an ensemble of
Hamiltonians. Chaotic systems described by disordered ensembles tend to have small variance
in their correlators, and their averaged correlation functions are close to those computed
for a simple instance of the ensemble. Even in regimes where replica symmetries are
broken, performing time bin averaging reproduces the averaged behaviors very well. We nd
in appendix D.3 that the time bin-averaged frame potential in the large L limit for two
samples agrees with averaging over the whole ensemble.
We conclude by mentioning a far reaching goal, but one that provides the conceptual
pillars for these ideas, namely understanding black holes as quantum systems.
While
black holes are thermodynamic systems whose microscopic details remain elusive, questions
about information loss can be precisely framed by late-time values of correlation functions
within AdS/CFT [17], where unitary evolution can be discussed in terms of the boundary
CFT. Ultimately, we would like to use random matrix theory to characterize chaos and
complexity in local quantum systems and identify late-time behaviors which are universal
for gravitational systems. An interesting future question is to see if gravitational systems
are described by random matrices in the sense of k-invariance and pinpoint some late-time
behavior which results from gravitational universality.
Acknowledgments
We thank Yoni Bentov, Fernando Branda~o, Cli ord Cheung, Patrick Hayden, Alexei
Kitaev, John Preskill, Daniel Ranard, Daniel Roberts, Lukas Schimmer, and Steve Shenker
for valuable comments and insights. JC is supported by the Fannie and John Hertz
Foundation and the Stanford Graduate Fellowship program. JC and NHJ would like to thank
the Perimeter Institute for their hospitality during the completion of part of this work.
BY and NHJ acknowledge support from the Simons Foundation through the \It from
Qubit" collaboration. NHJ is supported the Institute for Quantum Information and
Matter (IQIM), an NSF Physics Frontiers Center (NSF Grant PHY-1125565) with support
from the Gordon and Betty Moore Foundation (GBMF-2644). JL is supported in part by
the U.S. Department of Energy, O
ce of Science, O
ce of High Energy Physics, under
Award Number DE-SC0011632. Research at Perimeter Institute is supported by the
Government of Canada through Industry Canada and by the Province of Ontario through the
Ministry of Research and Innovation.
Scrambling and 2-designs
Recently there has been growing interest in scrambling and unitary designs from the high
energy and quantum information communities. Here we provide a short summary of
different ways of quantifying them for in nite temperature cases.
A.1
Scrambling
We begin with scrambling. Consider a system of qubits and non-overlapping local
(O(1)body) Pauli operators V; W and compute OTOC = hV W (t)V W (t)i where W (t) = U W U y.
The initial value of OTOC at t = 0 is 1. Scrambling is a phenomenon where the OTOC
V W (t)V W (t)i = O( )
for all pairs of local operators V; W
(A.1)
It is often the case that OTOCs with local operators are the slowest to decay. This can be
seen from our analysis on 4-point spectral form factors. So, by the scrambling time, OTOCs
with non-local operators are already O( ) or smaller. The scrambling time is lower bounded
by O(log(n)) in the case of 0-dimensional O(1)-local systems due to a Lieb-Robinson-like
argument [3].
Scrambling has caught signi cant attention from the quantum gravity community since
it is closely related to the Hayden-Preskill thought experiment on black hole information
problems [1]. Assume that V; W act on qubits on some local regions A; D respectively, and
de ne their complements by B = Ac; C = Dc. Imagine that A is an unknown quantum state
j i thrown into a \black hole" B, and the whole system evolves by some time-evolution
operator U = e iHt. At time t, we collect the \Hawking radiation" D and attempt to
reconstruct (an unknown) j i from measurement on D. Such a thought experiment was
considered by Page who argued that, if a black hole's dynamics U is approximated by a
random unitary operator, then reconstructing j i is not possible unless we collect more
than n=2 qubits of the Hawking radiation [64]. As we shall show in appendix B, the
impossibility of reconstruction of A from D is re ected in the smallness of the 2-point
correlation functions:
jhV W (t)ij = O( )
for local V; W
!
no reconstruction of A from D.
(A.2)
The famous calculations by Hawking and Unruh imply that these two-point correlators are
thermal, and quickly become small.
Hayden and Preskill considered a situation where a black hole B has already emitted
half of its contents, and we have collected its early radiation and stored it in some secure
quantum memory M . The quantum memory M is maximally entangled with B, and the
question is whether we can reconstruct j i by having access to M . It has been shown that
scrambling, as de ned above, implies that we can reconstruct j i with some good average
delity by collecting the Hawking radiation on D at time t:
h
V W (t)V W (t)i = O( )
!
reconstruction of A from D and M .
(A.3)
Therefore, scrambling implies the possibility of recovering local quantum information via
local measurements on the Hawking radiation. A random unitary operator U typically
gives very small OTOCs which enables reconstruction of A in the Hayden-Preskill thought
experiment.
Reconstruction problems in the Hayden-Preskill setting are closely related to the
problem of decoupling. A crucial di erence between scrambling and decoupling is that
decoupling typically considers A; D to be some nite fraction of the whole system and concerns
the reconstruction of unknown many-body quantum states supported on a big region A.
Since we quantify the reconstruction via
delity for many-body quantum states, the
requirement tends to be more stringent. The relation between scrambling and decoupling is
discussed in [65] in the context of local random circuits.
A.2
Unitary designs
E and the Haar ensemble are
E
( ) =
Next let us discuss unitary 2-designs. Consider an ensemble of time evolution operators
Uj with probability distributions pj ; E = fUj ; pj g with Pj pj = 1. The 2-fold channels of
Z
Haar
Uj ( )Ujy
U y
j
Haar( ) =
dU U
U ( )U y
U y:
(A.4)
If
E
( ) =
Haar( ) for all , then we say E is 2-design. One can check if E is 2-design
or not by looking at OTOCs. Consider the OTOC h
V W (t)V W (t)i for arbitrary Pauli
operators V; W
which are not necessarily local operators. We will be interested in the
ensemble averages of OTOCs:
h
V W (t)V W (t)iE
X pj hV Uj W U yV Uj W Ujyi:
j
If hV W (t)V W (t)iE = hV W (t)V W (t)iHaar for all pairs of Pauli operators V; W , then the
ensemble forms a unitary 2-design [16].
A typical unitary operator from a 2-design achieves scrambling because
jhV W (t)V W (t)ijHaar ' L
h
V W (t)V W (t)iHaar ' L2
1
for any (possibly non-local) Pauli operators V; W . The rst equation implies that the
OTOC value for a single instance from the ensemble is typically 1=L in absolute value
while the second equation implies that the OTOC, after ensemble averaging, is 1=L2.
Since OTOCs are small, a typical 2-design unitary operator U implies scrambling, but the
converse is not always true. Recall that scrambling only requires OTOC = O( ). There is
thus a big separation in the smallness of the OTOC, and the scrambling time may be much
shorter than the 2-design time. Also, scrambling requires OTOC = O( ) only for local
operators while a 2-design unitary makes the OTOC small for all pairs of Pauli operators.
The lower bound for the exact 2-design time is O(log(n)), but no known protocol achieves
this time scale.
One important distinction between scrambling and the 2-design time is how small the
OTOCs becomes. The phenomena of scrambling concerns the deviation of OTOC values
1
j
(A.5)
(A.6)
from the maximal value 1. The concept of a 2-design concerns the deviation of OTOC
values from the minimal value O(1=L). The former is related to early-time chaos and the
latter is related to late-time chaos.
A.3
Approximate 2-designs
Finally, let us brie y discuss the notion of approximate 2-design. When two quantum
operations
E and
Haar are close to each other, we say that E is an approximate 2-design.
In order to be quantitative, however, we need to pick appropriate norms with which two
quantum operations can be compared. The 2-norm distance can be de ned in a simple
2-norm =
tr(SSy)
S =
U y
j
dU U
U
U y
U y:
Z
Haar
If S = 0, then
in the 2-norm if ptr(SSy)
.
E and
Haar would be the same. We say that E is a -approximate 2-design
Frame potentials are closely related to the 2-norm distance because tr(SSy) = FE
0. In [16], a relation between the frame potential and OTOCs has been derived
dAdBdCdDjhAB(t)CD(t)iE j2 = FLE6 :
(
2
)
In practice, the main contribution to the left-hand side comes from OTOCs of the form
hAB(t)AB(t)iE . For simplicity of discussion, let us assume that hAB(t)CD(t)iE = 0 when
C 6= A or D 6= B (where A; B; C; D are non-identity Pauli operators). Then, a simple
analysis leads to
jhAB(t)AB(t)iE j2 '
2
for typical non-identity Pauli operators A; B. Thus, being a -approximate 2-design in the
2-norm implies that OTOCs are typically small. However, this does not necessarily imply
scrambling because OTOCs with local operators are often the slowest to decay. In order
to guarantee scrambling, we would need a L -approximate design in the 2-norm (under an
assumption on hAB(t)CD(t)iE = 0 for C 6= A or D 6= B). For this reason, an alternative
distance measure called the diamond norm is often used in quantum information literature.
See [66] for relations between di erent norms.
B
Information scrambling in black holes
In this appendix, we discuss behaviors of 2-point correlators and 4-point OTOCs from the
viewpoint of information scrambling in black holes. We begin by deriving a formula which
relates two-point autocorrelation functions and mutual information. We will be interested
in the following quantity
way via
FHaar
q
Z
j
Z
hOAOD(t)iavg
2
1
L2AL2D OA2PA OD2PD
X
X
jhOAOD(t)ij2
(A.7)
(A.8)
(A.9)
(B.1)
HJEP1(207)48
where hOAOD(t)i = L1 Tr(OAU ODU y) and U is the time-evolution operator of the system,
and PA and PD are sets of Pauli operators on A and D. There are L2A and L2D Pauli
operators.
The relation between apparent information loss and two-point correlators can be
understood by using the state representation jU i of a unitary operator U . Given a unitary
operator U acting on an n-qubit Hilbert space H, one can view U as a pure quantum state
jU i de ned on a 2n-qubit Hilbert space H
jU i
U
IjEPRi;
1
2n
X
jEPRi = p2n j=1
j i
jji:
(B.2)
jji where U = Pi;j Ui;j jiihjj. One easily see
that the quantum state jU i is uniquely determined by a unitary operator U . The state
representation allows us to view jU iABCD as a four-partite quantum state:
where B = Ac and D = Cc in the original system of qubits. Given the state representation
jU i of a unitary operator, we can derive the following formula
graphical representation is
where I(
2
)(A; D) is the Renyi-2 mutual information between A and D for j i, de ned by
To derive the formula, let
AD be the reduced density matrix of jU i on AD. Its
jU i = p2n
1
AD =
1
L
(B.3)
(B.4)
(B.5)
The averaged 2-point correlator is given by
hOAOD(t)iavg
2 =
1
SWAP, we obtain
where dotted lines represent averaging over Pauli operators. By using L1 P
O2P
O
Oy =
jhOAOD(t)iavej2 =
Tr( 2AD)
LALD
1
L2AL2D 2I(
2
)(A;D):
Let us further ponder this formula. For strongly interacting systems, it is typically the
case that
hOAOD(t)i ' 0
if Tr(OAOD) = 0:
So, the following relation for the autocorrelation functions holds approximately:
X
OA2PA
jhOAOA(t)ij2 ' 2I(
2
)(A;D)
where we took A and D to be the same subset of qubits.
The above formula has an interpretation as information retrieval from the early
Hawking radiation. Consider scenarios where Alice throws a quantum state j i into a black hole
and Bob attempts to reconstruct it from the Hawking radiation. In accordance with such
thought experiments, let A be qubits for Alice's quantum state, B be the black hole, C
be the remaining black hole and D be the Hawking radiation. Then, the averaged 2-point
correlation functions have an operational interpretation as Bob's strategy to retrieve Alice's
quantum state. Let us assume that the initial state of the black hole is unknown to Bob
and model it by a maximally mixed state B = LIBB . Alice prepares an EPR pair jEPRiAR
on her qubits and her register qubits. Notice the di erence from the Hayden-Preskill setup
where Bob had access to some reference system B0 which is maximally entangled with the
black hole B. In this decoding problem, we do not grant such access to Bob. He just
collects the Hawking radiation D and tries to reconstruct Alice's quantum state.
The most obvious strategy is to apply the inverse U y. However, Bob does not have an
access to qubits on C. So, he applies UCy D
IR to C
DR where C = LICC . Graphically,
(B.6)
(B.7)
(B.8)
(B.9)
HJEP1(207)48
this corresponds to
The success of decoding is equivalent to distillation of an EPR pair between A and R. So,
we compute the EPR delity. Namely, letting
be a projector onto an EPR pair between
A and R, we have
j i = p
L
LALBLC
F = h j j i =
1
L2
which leads to
F = Tr( 2BC ) = Tr( 2AD) = LALDjhOAOD(t)iavgj2:
Therefore, the decay of 2-point correlation functions indeed implies that Bob cannot
reconstruct Alice's quantum state.
information:
Finally, let us summarize the known relations between correlation functions and mutual
2 I(
2
)(A;BD) = hOAOD(t)OAOD(t)iavg
2I(
2
)(A;D) = jhOAOD(t)iavgj2 L2AL2D:
Note that the rst formula proves that the decay of OTOCs leads to large I(
2
)(A; BD)
which implies the possibility of Bob decoding Alice's quantum state by accessing both the
(B.10)
(B.11)
(B.12)
(B.13)
(B.14)
early radiation B and the new Hawking radiation D. These two formulae allow us to
formally show that a black hole can be viewed as a quantum error-correcting code. Let
A; D be degrees of freedom corresponding to incoming and outgoing Hawking radiation,
and B; C be degrees of freedom corresponding to other exotic high energy modes at the
stretched horizon. Since a black hole is thermal, we know that jhOAOD(t)iavgj decays at
t
O( ). Also, due to the shockwave calculation by Shenker and Stanford [4], we know
that hOAOD(t)OAOD(t)iavg decays at t
O( log N ). These results imply that after the
scrambling time:
I(
2
)(A; D) ' 0
I(
2
)(A; C) ' 0:
(B.15)
The implication is that quantum information injected from A gets delocalized and
nonlocally is hidden between C and D. The error-correction property can be seen by
I(
2
)(A; BD) ' 2a
I(
2
)(A; BC) ' 2a
I(
2
)(A; CD) ' 2a
(B.16)
where a is the number of qubits on A. Namely, if we see the black hole as a quantum
code which encodes A into BCD, then the code can tolerate erasure of any single region
B; C; D. In other words, accessing any two of B; C; D is enough to reconstruct Alice's
quantum state. Thus, black hole dynamics, represented as a four-partite state jU iABCD,
can be interpreted as a three-party secret sharing quantum code.
C
Spectral correlators and higher frame potentials
For GUE(L; 0; 1=p
normalizing factors, is
In this appendix we will present formulas for form factors from random matrix theory.
L), L
L matrices with o -diagonal complex entries and real diagonal
entries chosen with variance 2 = 1=L, the joint probability of eigenvalues for GUE, with
P ( 1; : : : ; L) =
LL2=2
(
2
)L=2 QpL=1 p!
e L2 Pi i2 Y( i
j )
2
i<j
and the joint probability distribution of n eigenvalues (i.e., the n-point spectral correlation
function), de ned as
Z
(n)( 1; : : : ; n) =
We can compactly express (n)( 1; : : : ; n) in terms of a kernel K [25, 26] as
(n)( 1; : : : ; n) =
(L
det K( i; j ) in;j=1
In the large L limit, the kernel K is approximately
K( i; j )
>
>
>
8 L sin(L( i
L( i
>>> L q
>
: 2
4
2
i
j ))
j )
for i 6= j
for i = j
(C.1)
(C.2)
(C.3)
(C.4)
where the i 6= j case is called the sine kernel, and the i = j case is simply the Wigner
semicircle. In the large L limit, the basic approach for computing spectral form factors will
be expanding the determinant in eq. (C.3) using the kernel in eq. (C.4), and computing
the Fourier transform of the resulting sums of product of kernels. Thus we will have sums
of integrals of the form [25]
Z
m
i=1
Y d iK( 1; 2)K( 2; 3) : : : K( m 1
; m)K( m; 1) ei Pim=1 ki i
d ei Pim=1 ki i
k1
2L
2L
km 1
2L
dk g(k)g k +
g k +
: : : g k +
(C.5)
(C.6)
(C.7)
= L :
(C.8)
g(k)
L Z
Z
Z
Z
=2
k2
2L
where we de ne the Fourier transform of the sine kernel
dr e2 ikr sin( r)
10 ffoorr jjkkjj <> 1212 :
The delta function singularity from the R d ePim=1 iki integral in eq. (C.5) is an artifact
of our expansion around in nite L, namely that L sin(L( i
j)) is not regulated in the
( i + j ) direction. The most direct method to soften this divergence is to impose a cuto
L( i
d ei Pim=1 ki i
L Z =2
d ei Pim=1 ki i
which is xed by the normalization condition
L Z =2
=2
d eiPim=1 ki i
2L
dk g(k)g k +
g k +
: : : g k +
km 1
2L
k1;:::;km=0
and late times greater than O( L).
p
While the `box approximation' of applying the cuto allows us to compute higher-point
spectral correlators in the large L limit, it does lead to errors relative to an exact answer
whose closed form is not tractable.25
Thus we must be careful to keep track of these
errors and compare with numerics. However, we nd that at in nite temperature, the box
approximation of the spectral form factors is analytically controlled at early times like O(1)
To understand the errors of the box approximation, we rst consider various cases
normalization. When Pi ki 6= 0, the
heuristically: when we have Pi ki = 0, the
integral in eq. (C.5) is directly
xed by
integral in eq. (C.5) dephases and so decays when
is also small and the value is still close to the Pi ki = 0 value.
j Pi kij is large, and thus the induced error is unimportant at long times. At small, O(1)
values of the jkij's (assuming that m is O(1)), the error induced by the box approximation
For instance, carefully keeping track of factors of L tells us that in R4, for early times
like O(1) the error is suppressed by O(1=L) relative to largest order terms, while for late
times after O(pL) the error is suppressed by O(1=pL) relative to the largest order terms.
25For instance, the Fourier transform of the semicircle distribution decays as t 3=2, whereas the Fourier
transform of a box decays as t 1.
In this discussion, particularly for Pi ki = 0, we assumed simple sine kernel correlations
and found r2 to be a pure linear function. However, a more delicate treatment shows some
other transition time scale at early times, which likely complicates the functional form of
r2 and gives a di erent slope for the ramp. We brie y address this issue for our numerics
in appendix D.
Since the dephasing of the integral at large j Pi kij is suppressed at nite temperature,
to better capture long-time nite temperature eigenvalue correlations we use a modi ed
kernel Ke which is valid in the short distance limit j a
bj
O(1=L) [55, 67],
Ke ( i; j ) =
sin L ( i
j ) (1)(( i + j )=2)
( i
j )
(C.9)
which naturally provides a cuto in the ( i + j ) direction. However, this approximation
assumes the continued domination of the regulated integral in the short distance limit,
which may not be true for large . However, for small
the modi ed kernel is reliable. In
the generic case, one should consider the full expression of Hermite polynomials as the sine
kernel, and correctly take the limit. A complicated formula has been derived in [55, 67]
from a saddle point approximation.
C.1
Expressions for spectral correlators
Using the analysis above, it is straightforward to compute form spectral correlation
functions for the GUE. It is convenient to de ne
2L
for t < 2L
sin( t=2)
t=2
:
(C.10)
as mentioned earlier. The in nite temperature form factors which appear in the calculation
of the rst and second frame potentials are
L
X ei( i j)t ;
i;j=1
L
X
i;j;k;`=1
ei( i+ j k `)t ;
R4;1(t) =
R4;2(t) =
Z
Z
D
D
L
X
i;j;k=1
L
i;j=1
ei( i+ j 2 k)t ;
X e2i( i j)t :
(C.11)
As R4;2 is simply R2(2t), we only need to compute the rst three spectral correlation
functions. We will also investigate the nite temperature version of R2, which we de ned as
Z
D
R2 at in nite temperature.
We start by computing R2 at in nite temperature: R2(t) = L +
Z
d 1 d 2 K( 1; 1)K( 2; 2)
K2( 1; 2) ei( 1 2)t :
L
i;j=1
(C.12)
(C.13)
Evaluating the rst term in the integral, we nd
The second term can be evaluated using eq. (C.5), and we nd
Z
Z
d 1K( 1; 1)ei 1t
d 2K( 2; 2)e i 2t = L2r12(t) :
Z
d 1d 2K2( 1; 2)ei( 1 2)t = Lr2(t) :
The nal result is
R2 at nite temperature.
at
nite temperature we will use the short-distance-limit kernel Ke . Firstly, for i = j,
R2(t) = L + L2r12(t)
Lr2(t):
As explained above, to better capture long-time correlations
L
Z
D e 2 1 = Lr1(2i ) :
1)
Z
Z
L(L
D ei( 1 2)t ( 1+ 2)
d 1d 2 Ke ( 1; 1)Ke ( 2; 2)
Ke 2( 1; 2) ei( 1 2)t ( 1+ 2)
= L2r1(t + i )r1( t + i )
Putting everything together, we obtain
R2 = Lr1(2i ) + L2r1(t + i )r1( t + i )
R4 at in nite temperature.
We now compute R4(t), again by separately considering
coincident eigenvalues, using the determinant of kernels, and Fourier transforming to nd
R4(t) = L4r14(t)
2L3r12(t)r2(t)r3(2t)
4L3r12(t)r2(t) + 2L3r1(2t)r12(t) + 4L3r12(t)
+ 2L2r22(t) + L2r22(t)r32(2t) + 8L2r1(t)r2(t)r3(t)
2L2r1(2t)r2(t)r3(2t)
4L2r1(t)r2(2t)r3(t) + L2r12(2t)
4L2r12(t)
4L2r2(t) + 2L2
7Lr2(2t) + 4Lr2(3t) + 4Lr2(t)
L :
We can simplify this formula at early times of O(1) and late times greater than O( L) by
dropping subdominant terms and nd
R4
L4r14(t)+2L2r22(t) 4L2r2(t)+2L2 7Lr2(2t)+4Lr2(3t)+4Lr2(t) L ;
(C.21)
where the 2L2r22 term gives a quadratic rise at late times, akin to the ramp in R2.
(C.14)
(C.15)
(C.16)
(C.17)
(C.18)
(C.19)
(C.20)
p
HJEP1(207)48
8L2 L
4
+ L
4
+ 16L L
2
4L L
2
.
8R2
L2
2R24;1 +
L4
64R2R4
L6
16R2R4;1 +
L5
8R2R4 ;
L4
R4;1 at in nite temperature.
R4;1(t) = L3r1(2t)r12(t)
L2r1(2t)r2(t)r3(2t)
2L2r1(t)r2(2t)r3(t)
+ L2r12(2t) + 2L2r12(t) + 2Lr2(3t)
Lr2(2t)
2Lr2(t) + L :
Just as above, we can approximate R4;1 at early and late times by
R4;1
L3r1(2t)r12(t) + 2Lr2(3t)
Lr2(2t)
2Lr2(t) + L :
(C.22)
(C.23)
C.2
Expressions for higher frame potentials k = 2 frame potential.
We computed the second frame potential for the GUE to be
L
4
8L2 + 6 R42 + 4L2 L
2
9 R4 + 4 L
6
9L4 + 4L2 + 24 R2
2
HJEP1(207)48
11L2 + 18 R2 + 2 L
4
7L2 + 12 R4;1
2
4L2 L
8L2 + 6 R4;2
2
8L2 + 6 R2R4
4L L
8 L2 + 6 R2R4;2 + 2 L2 + 6 R4R4;2
4 R4;1R4;2 + 2L4 L
4
12L2 + 27
1)L2(L + 1)(L + 2)(L + 3) :
with form factors as de ned in eq. (C.11). Let us try and extract the interesting behavior
encoded in the expression. We know the maximal value of the spectral n-point functions
de ned above at early times, R2
L2, R4
L4, R4;1
L3, and R4;2
L2. From the
expression for the frame potential above, we keep the terms that are not suppressed in
1=L, i.e. can contribute at least at zeroth order:
FGUE
36R22 +
L4
4LR222 +
4LR44 +
6R24
L8
2 2
8LR624 + RL44 + R4;2
L4
16R4R4;1
L7
4R4R4;1 +
L5
2R4R4;2
L6
14R24;1
L6
L5
4R4;1R4;2
L, and R4;1; R4;2 ! L, we have
(
2
)
FGUE =
10L2 + 22L
20
L2 + 5L + 6
which gives
R4 ! 2L2
with the Haar value appearing at the beginning. At early times, the leading order behavior
is FGUE
R42=L4. From our calculation of the n-point form factors, we know that at the
dip time all form factor terms above are suppressed in L, meaning the frame potential goes
like the Haar value. Knowing the late time value of the 2-point and 4-point form factors,
the terms above that will contribute at late times are
Late :
FGUE
2
2 + RL44 +
4LR222 ;
(C.24)
10 in the large L limit. In the strict t !
1 limit, where R2 ! L,
and
(
2
)
FGUE
10 for L
1 :
(C.25)
As the left-hand side expression is valid for any L at late times, in doing the numerics and
taking the sample size to be large, this is the value for L we should converge to.
k = 3 frame potential. The full expression for the third frame potential of the GUE is
+144R2R4;1L9 36R4R4;1L9 36R4;1R4;2L9+11574R22L8 369R42L8+R62L8 828R42;1L8+9R22R24;2L8 18R2R42;2L8 441R42;2L8+6R62;1L8
+4R62;2L8+12R62;3L8+4R62;4L
8 29772R2L8+3276R2R4L8 1728R4L8+36R2R6L8 18R4R6L8 12R6L8 36R22R4;2L8+18R4R4;2L
8
+1800R4;2L8 36R4;1R6;1L8 24R6;4L8 37158L8 6192R2R4;1L7+1332R4R4;1L7+36R6R4;1L7+108R2R4;1R4;2L7+1548R4;1R4;2L
7
144R2R6;1L7+108R4R6;1L7 12R6R6;1L7 36R2R4;2R6;1L7+36R4;2R6;1L7+72R4;1R6;2L7 24R6;1R6;2L7+144R2R6;3L7 72R2R4;2R6;3L
7
+72R4;2R6;3L7 24R6;2R6;3L7 48R6;3R6;4L7 39978R22L6+3726R42L6 41R62L6+11610R42;1L6 297R22R24;2L6+594R2R42;2L6+6750R42;2L6
HJEP1(207)48
204R62;1L6 156R62;2L6 348R62;3L6 148R62;4L6+169812R2L6 42768R2R4L6+24732R4L6 1512R2R6L6+738R4R6L6+528R6L
6
+1512R22R4;2L6 432R2R4;2L6 162R2R4R4;2L6 486R4R4;2L6+18R2R6R4;2L6 18R6R4;2L6 27972R4;2L6+1224R4;1R6;1L6+144R2R6;2L
6
144R4R6;2L6+16R6R6;2L6+72R2R4;2R6;2L6 72R4;2R6;2L6 48R6;2L6 360R4;1R6;3L6+120R6;1R6;3L6 144R2R6;4L6+72R2R4;2R6;4L
6
72R4;2R6;4L6+32R6;2R6;4L6+1032R6;4L6+89040L6+72576R2R4;1L5 11232R4R4;1L5 1188R6R4;1L5 3132R2R4;1R4;2L5 18792R4;1R4;2L
5
+5040R2R6;1L5 3564R4R6;1L5+396R6R6;1L5+1044R2R4;2R6;1L5 1044R4;2R6;1L5 2232R4;1R6;2L5+744R6;1R6;2L5 5040R2R6;3L
5
52128R22L4+458R62L4 55692R42;1L4+2430R22R24;2L4 4860R2R42;2L
4 35190R42;2L4+1794R62;1L4+1660R62;2L4+2388R62;3L4+1440R62;4L4
274320R2L4+146412R2R4L4+17172R2R6L4 8244R4R6L4 6276R6L4 15876R22R4;2L4+18144R2R4;2L4+3078R2R4R4;2L4+324R4R4;2L
4
342R2R6R4;2L4+342R6R4;2L4+141408R4;2L4 10764R4;1R6;1L4 4608R2R6;2L4+3672R4R6;2L4 408R6R6;2L4 1368R2R4;2R6;2L
4
+1368R4;2R6;2L4+1968R6;2L4+7200R4;1R6;3L4 2400R6;1R6;3L4+3312R2R6;4L4 288R4R6;4L4+32R6R6;4L4 1368R2R4;2R6;4L
4
+1368R4;2R6;4L4 752R6;2R6;4L4 11568R6;4L4 96000L4 199728R2R4;1L3 4392R4R4;1L3+9144R6R4;1L3+26352R2R4;1R4;2L
3
+51552R4;1R4;2L3 37296R2R6;1L3+27432R4R6;1L3 3048R6R6;1L3 8784R2R4;2R6;1L3+8784R4;2R6;1L3+17928R4;1R6;2L3 5976R6;1R6;2L
3
720R4;1R6;4L3+240R6;1R6;4L3 11952R6;3R6;4L3+141840R22L2 49284R42L2 1258R62L2+111852R42;1L2+1098R22R24;2L2 2196R2R42;2L2
+53712R42;2L2 3756R62;1L2 3188R62;2L2+108R62;3L2 2736R62;4L2+288000R2L2+5472R2R4L2 47376R2R6L2+22644R4R6L2+14400R6L
2
16488R4R6;2L2+1832R6R6;2L2+4176R2R4;2R6;2L2 4176R4;2R6;2L2 19200R6;2L2 45720R4;1R6;3L2+15240R6;1R6;3L2+8352R2R6;4L
2
8352R4R6;4L2+928R6R6;4L2+4176R2R4;2R6;4L2 4176R4;2R6;4L2+5520R6;2R6;4L2+19200R6;4L2+133200R2R4;1L+53208R4R4;1L
12312R6R4;1L 62208R2R4;1R4;2L+4608R4;1R4;2L+32400R2R6;1L 36936R4R6;1L+4104R6R6;1L+20736R2R4;2R6;1L 20736R4;2R6;1L
33048R4;1R6;2L+11016R6;1R6;2L 32400R2R6;3L 25272R4R6;3L+2808R6R6;3L+41472R2R4;2R6;3L 41472R4;2R6;3L+16632R6;2R6;3L
16848R4;1R6;4L+5616R6;1R6;4L+22032R6;3R6;4L 216000R22 2160R42+240R62 105840R42;1 12960R22R24;2+25920R2R4;2 34560R42;2
2
2160R62;1 2160R62;2 19440R62;3 960R62;4+43200R2R4+14400R2R6 4320R4R6+172800R2R4;2+25920R2R4R4;2 69120R4R4;2
2880R2R6R4;2+2880R6R4;2+12960R4;1R6;1+14400R2R6;2+4320R4R6;2 480R6R6;2 11520R2R4;2R6;2+11520R4;2R6;2+90720R4;1R6;3
30240R6;1R6;3 28800R2R6;4 2880R6R6;4+25920R4R6;4 11520R2R4;2R6;4+11520R4;2R6;4 6720R6;2R6;4
. (L 5)(L 4)(L 3)(L 2)(L 1)L2(L+1)(L+2)(L+3)(L+4)(L+5) :
The expression is best appreciated from a distance.
Lastly, we give the de nition of the unitary Weingarten function, which appeared in the
integration of Haar random unitaries in eq. (4.9). The 2k-th moment of the Haar ensemble
appeared in the k-th frame potential. For the n-th moment, the Weingarten function is a
function of an element
of the permutation group Sn and presented as de ned in [46],
Wg( ) =
1
where we sum over integer partitions of n (recall that the conjugacy classes of Sn are
HJEP1(207)48
labeled by integer partitions of n).
is an irreducible character of Sn labeled by
(as
each irrep of Sn can be associated to an integer partition) and e is the identity element.
s (1) = s (1; : : : ; 1) is the Schur polynomial evaluated on L arguments and indexed by
the partition . For instance, the Weingarten functions needed to compute the rst frame
potential were
1
L2
1
Wg(f1; 1g) =
and
Wg(f2g) =
1
L(L2
1)
:
(C.26)
(C.27)
D
Additional numerics
and frame potentials.
D.1
Form factors and numerics
We conclude with a few numerical checks on the formulae we derived for the form factors
As we mentioned in section 2.2 and discussed in appendix C.1, in order to derive expressions
for the form factors for the GUE we had to make approximations which should be compared
to numerics for the GUE.
We brie y remind the reader that at in nite temperature, we derived the expression
R2(t) = L2r12(t)
Lr2(t) + L :
(D.1)
Numerical checks of this expression are shown in gure 8. We see that the approximations
employed work well at
= 0, reproducing the early time oscillations, dip, plateau, and
ramp features. But there is some discrepancy in the ramp behavior which merits discussion.
As we take L ! 1, the di erence between the predicted ramp and numerical ramp is not
suppressed. In
gure 8, we see that the relative error between the numerics and analytic
prediction does not decrease as we increase L, indicating that this di erence in the ramp
prediction is not an artifact of
nite L numerics. On a log-log plot, this shift from the
numerics suggests that we capture the correct linear behavior, but with a slightly di erent
slope for the ramp.
The r2(t) = 1
t=2L function which controls the slope behavior comes from the
Fourier transform of the square of the sine kernel. Recall that in our approximation, we
integrated over the entire semicircle. A phenomenological observation is that the modi ed
ramp function de ned by r~2(t)
1
2t= L, where we change the slope to 2= , does a
0.100
0.010
10-4
10-5
R2/L2
1
0.100
0.010
10-4
10-5
L=20
L=50
L=100
L=200
L=500
L=20
L=50
L=100
L=200
L=500
Numerics
Analytics
Numerics
Δ0.2R52/R2
0.20
0.1
1
100
0.1
1
100 1000 104 t
HJEP1(207)48
various values of L and normalized by L2. The analytic expressions derived in section 2 are in
the lighter shades and the numerics for GUE are in darker shades. Numerics were done 10000
samples from the GUE. On the right we plot the relative error between the numerics and analytic
predictions. We observe good agreement at early and late times, and see deviations around the
ramp.
Numerics for modified R2 at β = 0
100
0.1
1
100 1000 104 t
with the modi ed ramp behavior r~2(t).
much better job of capturing the ramp behavior. Working in the short-distance limit of
the 2-point correlator (
2
)( 1; 2) (as in [30]) and integrating the sine kernel over the entire
semicircle, we obtain r~2 whose behavior we only trust near the dip.
Numerically, we nd that this modi ed slope of 2 =L better captures the r2 function
near the dip, with error that is suppressed as we take L ! 1. The same numerics are
reported in gure 9, but with the modi ed ramp behavior. There is still some discrepancy
near the plateau time when we transition to the constant plateau value, but the ramp
behaviors near the dip are in much better agreement.
We understand the Bessel function contribution to R2(t), which arises from 1-point
functions. The subtlety above is really in the connected piece of the 2-point function
R2(t)conn
R2(t)
L2r12(t) :
(D.2)
1
0.01
10-4
Numerics
connected R2
104
0.01
1
100
(D.4)
plotted for L = 500 with 10000 random samples.
The dashed line is the expression eq. (D.4)
approximating the three regimes of the connected form factor.
Numerically, we see that the connected 2-point form factor for the GUE exhibits three
di erent behaviors: an early time quadratic growth, an intermediate linear growth, and
then a late-time constant plateau. The closed form expression we derived in section 2
should be viewed as a coarse approximation before the plateau, approximately capturing
the linear regime. The modi ed ramp function r~2(t) = 1
2t= L appears to capture the
linear behavior near the dip with the correct slope. In [55], a more detailed treatment of
the connected correlator is given at early times. From the integral representation of the
connected 2-point form factor, they nd that
Early :
1 t6 + : : :
to leading order in L (eq. (2.28) in [55]). The three behaviors are compared with numerics
in gure 10.
tured by
In summary, the three regimes of the connected 2-point form factor are roughly
capR2(t)conn = <
8
>
>
>
>:L
for t . 1 ;
for 1 . t . 2L ;
for t & 2L :
The early time quadratic behavior does not play an important role in our analysis of GUE
correlation functions and frame potentials, but is of independent physical interest. This
intruiging early-time behavior of the connected 2-point form factor will be explored in [63].
At
nite temperature we
nd good agreement between the expression R2(t; ) and
numerics at early and late times, but again see a deviation of the dip and ramp behaviors
from the analytic prediction, as shown in
gure 11. Using the modi ed ramp r~2 we
nd
closer agreement at small , but as we increase
the predicted ramp behavior again starts
to deviate from the numerics, indicating that there is a -dependence to the slope that we
do not fully understand. But as we discussed in appendix C.1, we only trust the
shortdistance approximation at nite temperature, and thus R2(t; ), for small . We also
report numerics for the R4 expression in gure 12.
Numerics
1
10-4
R4/L4
10-5
10-7
L=20
L=50
L=100
L=200
L=500
L=20
L=50
L=100
L=200
L=500
Numerics
1
0.100
10-5
10-7
L=20
L=50
L=100
L=200
L=500
L=20
L=50
L=100
L=200
L=500
0.1
1
10
100
1000
0.1
1
10
100
1000
= 0:5, plotted for various values of L and normalized by their initial values. Numerics were
done with a GUE sample size of 10000. The left gure uses the expression for R2(t; ) derived in
section 2.2 and C.1, whereas the right gure uses the modi ed ramp r~2 discussed above.
Numerics for GUE R4 at β = 0
Numerics for modified R4 at β = 0
0.1
1
10
100
1000
0.1
1
10
100
1000
10000 samples, plotted for various values of L and normalized by their initial values. The left gure
uses the R4 expression derived in appendix C.1, and the right gure uses r~2.
D.2
Frame potentials and numerics
As the frame potential depends on the eigenvectors of the elements in the ensemble (and not
just the eigenvalues as per the form factors) and requires a double sum over the ensemble,
numerical simulation of the frame potential is harder than for the form factors. For an
ensemble of L
L matrices, we need to consider sample sizes greater than L2k for the
k-th frame potential, which amounts to summing over many samples for fairly modest
Hilbert space dimension. Instead, for a given L, we can sequentially increase the sample
size and extrapolate to large jEGUEj. In
gure 13 we consider the
rst frame potential
for the GUE at L = 32 and, in the limit of large sample size, nd good agreement with
the analytic expression computed from R2. Alternatively, we can numerically compute
the frame potentials by ignoring the coincident contributions to the double sum in F
i.e. when U = V . For a
nite number of samples, these terms contribute L2=jE j to the
sum, meaning we must look at large ensembles before their contribution does not dominate
entirely. Ignoring these terms, we can time average over a sliding window to compute the
frame potential with only a few samples, as shown in gure 13.
|ε|=400
|ε|=500
|ε|=600
|ε|=800
|ε|=1000
|ε|=1200
|ε|=1600
Prediction from R2
Large |ε| extrapolation
Analytic F(1)
F(1)
1000
100
10
1
Time bin average of F(1) for GUE
F(1) for |ε|=100
time bin avg
0.1
1
10
100
t
0.1
1
10
100
t
sequentially increase the number of samples and extrapolate to large sample size (red line), which
agrees with the both the frame potential computed from R2 numerics as in eq. (4.15) (blue line) and
the analytic expression we derived for F G(1U)E. On the right, we time bin average FGUE as described
(1)
above and, for L = 32 and 100 samples, we nd good agreement with the quantities on the left.
= 5 and L = 500.
On the right: the time average of the rst frame potential for L = 500 computed for two instances. In
both gures, the time average of the minimal number of instances agrees with the ensemble average.
D.3
Minimal realizations and time averaging
Given an ensemble of disordered systems, one can ask whether a quantity averaged over
the ensemble is the same as for a single random instance of the ensemble. It is known
that up until the dip time, the spectral form factor is self-averaging, meaning that single
instance captures the average for large L [68]. However, the spectral form factor is not
selfaveraging at late times. We can try to extract the averaged behavior from a single instance
in regimes dominated by large
uctuations by averaging over a moving time window. In
gure 14, we see that for a single instance of the GUE, the time average of the spectral
form factor at nite
gives the same result as the ensemble average for su ciently large
L. For the frame potential, we can consider two instances, the smallest ensemble for which
the frame potential makes sense. Ignoring the coincident terms in the sum, we see that
the frame potential is also self-averaging at early times and that the time average at late
times agrees with the ensemble average and analytic expression.
This article is distributed under the terms of the Creative Commons
Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
subsystems, JHEP 09 (2007) 120 [arXiv:0708.4025] [INSPIRE].
[arXiv:1306.0622] [INSPIRE].
[arXiv:1412.6087] [INSPIRE].
[arXiv:1503.01409] [INSPIRE].
February 2015.
[3] N. Lashkari, D. Stanford, M. Hastings, T. Osborne and P. Hayden, Towards the Fast
Scrambling Conjecture, JHEP 04 (2013) 022 [arXiv:1111.6580] [INSPIRE].
[4] S.H. Shenker and D. Stanford, Black holes and the butter y e ect, JHEP 03 (2014) 067
[5] S.H. Shenker and D. Stanford, Stringy e ects in scrambling, JHEP 05 (2015) 132
[6] J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP 08 (2016) 106
[7] A. Kitaev, Hidden correlations in the hawking radiation and thermal noise, talks given at
[8] S. Sachdev and J. Ye, Gapless spin
uid ground state in a random, quantum Heisenberg
magnet, Phys. Rev. Lett. 70 (1993) 3339 [cond-mat/9212030] [INSPIRE].
[9] A. Kitaev, A simple model of quantum holography, talks given at The KITP, 7 April 2015
[10] J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94
(2016) 106002 [arXiv:1604.07818] [INSPIRE].
[11] J.S. Cotler et al., Black Holes and Random Matrices, JHEP 05 (2017) 118
[arXiv:1611.04650] [INSPIRE].
Math. 62 (1955) 548.
(1962) 140 [INSPIRE].
[12] E.P. Wigner, Characteristic vectors of bordered matrices with in nite dimensions, Ann.
[13] F.J. Dyson, Statistical theory of the energy levels of complex systems. I, J. Math. Phys. 3
[14] O. Bohigas, M.J. Giannoni and C. Schmit, Characterization of chaotic quantum spectra and
universality of level uctuation laws, Phys. Rev. Lett. 52 (1984) 1 [INSPIRE].
[15] P. Hosur, X.-L. Qi, D.A. Roberts and B. Yoshida, Chaos in quantum channels, JHEP 02
(2016) 004 [arXiv:1511.04021] [INSPIRE].
[16] D.A. Roberts and B. Yoshida, Chaos and complexity by design, JHEP 04 (2017) 121
[arXiv:1610.04903] [INSPIRE].
[hep-th/0106112] [INSPIRE].
[17] J.M. Maldacena, Eternal black holes in anti-de Sitter, JHEP 04 (2003) 021
[18] D.A. Roberts and D. Stanford, Two-dimensional conformal eld theory and the butter y
05 (2016) 109 [arXiv:1603.08925] [INSPIRE].
[20] A.L. Fitzpatrick and J. Kaplan, On the Late-Time Behavior of Virasoro Blocks and a
Classi cation of Semiclassical Saddles, JHEP 04 (2017) 072 [arXiv:1609.07153] [INSPIRE].
[21] E. Dyer and G. Gur-Ari, 2D CFT Partition Functions at Late Times, JHEP 08 (2017) 075
[arXiv:1611.04592] [INSPIRE].
[22] V. Balasubramanian, B. Craps, B. Czech and G. Sarosi, Echoes of chaos from string theory
black holes, JHEP 03 (2017) 154 [arXiv:1612.04334] [INSPIRE].
[23] Y.-Z. You, A.W.W. Ludwig and C. Xu, Sachdev-Ye-Kitaev Model and Thermalization on the
Boundary of Many-Body Localized Fermionic Symmetry Protected Topological States, Phys.
Rev. B 95 (2017) 115150 [arXiv:1602.06964] [INSPIRE].
[24] A.M. Garc a-Garc a and J.J.M. Verbaarschot, Spectral and thermodynamic properties of the
Sachdev-Ye-Kitaev model, Phys. Rev. D 94 (2016) 126010 [arXiv:1610.03816] [INSPIRE].
[25] M. Mehta, Random Matrices, Pure and Applied Mathematics, Elsevier Science (2004).
[26] T. Tao, Topics in Random Matrix Theory, Graduate studies in mathematics, American
Mathematical Society (2012).
[27] T. Guhr, A. Muller-Groeling and H.A. Weidenmuller, Random matrix theories in quantum
physics: Common concepts, Phys. Rept. 299 (1998) 189 [cond-mat/9707301] [INSPIRE].
[28] A.R. Brown and L. Susskind, The Second Law of Quantum Complexity, arXiv:1701.01107
[29] A. del Campo, J. Molina-Vilaplana and J. Sonner, Scrambling the spectral form factor:
unitarity constraints and exact results, Phys. Rev. D 95 (2017) 126008 [arXiv:1702.04350]
[30] E. Brezin and S. Hikami, Spectral form factor in a random matrix theory, Phys. Rev. E 55
[31] L. Erd}os and D. Schroder, Phase Transition in the Density of States of Quantum Spin
Glasses, Math. Phys. Anal. Geom. 17 (2014) 9164 [arXiv:1407.1552].
[32] J.S. Cotler, G.R. Penington and D.H. Ranard, Locality from the Spectrum,
arXiv:1702.06142 [INSPIRE].
JETP 28 (1969) 1200.
[33] A.I. Larkin and Y.N. Ovchinnikov, Quasiclassical Method in the Theory of Superconductivity,
[34] D. Bagrets, A. Altland and A. Kamenev, Power-law out of time order correlation functions
in the SYK model, Nucl. Phys. B 921 (2017) 727 [arXiv:1702.08902] [INSPIRE].
[35] F.G.S.L. Brand~ao, P. Cwiklinski, M. Horodecki, P. Horodecki, J.K. Korbicz and
M. Mozrzymas, Convergence to equilibrium under a random hamiltonian, Phys. Rev. E 86
[36] S.W. Hawking, Particle Creation by Black Holes, Commun. Math. Phys. 43 (1975) 199
[Erratum ibid. 46 (1976) 206] [INSPIRE].
[37] W.G. Unruh, Notes on black hole evaporation, Phys. Rev. D 14 (1976) 870 [INSPIRE].
arXiv:1703.08104 [INSPIRE].
[39] Z.-W. Liu, S. Lloyd, E.Y. Zhu and H. Zhu, Entropic scrambling complexities,
[40] C. Dankert, R. Cleve, J. Emerson and E. Livine, Exact and approximate unitary 2-designs
and their application to delity estimation, Phys. Rev. A 80 (2009) 012304
Approximate Polynomial-Designs, Commun. Math. Phys. 346 (2016) 397
[41] F.G.S.L. Brand~ao, A.W. Harrow and M. Horodecki, Local Random Quantum Circuits are
[42] F. Pastawski, B. Yoshida, D. Harlow and J. Preskill, Holographic quantum error-correcting
codes: Toy models for the bulk/boundary correspondence, JHEP 06 (2015) 149
[arXiv:1503.06237] [INSPIRE].
[43] P. Hayden, S. Nezami, X.-L. Qi, N. Thomas, M. Walter and Z. Yang, Holographic duality
from random tensor networks, JHEP 11 (2016) 009 [arXiv:1601.01694] [INSPIRE].
[44] A.J. Scott, Optimizing quantum process tomography with unitary 2-designs, J. Phys. A 41
[45] B. Collins, Moments and cumulants of polynomial random variables on unitarygroups, the
itzykson-zuber integral, and free probability, Int. Math. Res. Not. 2003 (2003) 953
(1994) 49.
J. Math. Phys. 19 (1978) 999 [INSPIRE].
[46] B. Collins and P. Sniady, Integration with Respect to the Haar Measure on Unitary,
Orthogonal and Symplectic Group, Commun. Math. Phys. 264 (2006) 773
[47] D. Weingarten, Asymptotic Behavior of Group Integrals in the Limit of In nite Rank,
[48] P. Diaconis and M. Shahshahani, On the eigenvalues of random matrices, J. Appl. Prob. 31
[49] S. Bravyi, M.B. Hastings and F. Verstraete, Lieb-Robinson Bounds and the Generation of
Correlations and Topological Quantum Order, Phys. Rev. Lett. 97 (2006) 050401
44 [arXiv:1403.5695] [INSPIRE].
[50] L. Susskind, Computational Complexity and Black Hole Horizons, Fortsch. Phys. 64 (2016)
[51] A.R. Brown, D.A. Roberts, L. Susskind, B. Swingle and Y. Zhao, Complexity, action and
black holes, Phys. Rev. D 93 (2016) 086006 [arXiv:1512.04993] [INSPIRE].
[52] X. Chen, Z.C. Gu and X.G. Wen, Local unitary transformation, long-range quantum
entanglement, wave function renormalization and topological order, Phys. Rev. B 82 (2010)
155138 [arXiv:1004.3835] [INSPIRE].
[53] D. Harlow and P. Hayden, Quantum Computation vs. Firewalls, JHEP 06 (2013) 085
[arXiv:1301.4504] [INSPIRE].
[54] Z.-C. Yang, A. Hamma, S.M. Giampaolo, E.R. Mucciolo and C. Chamon, Entanglement
complexity in quantum many-body dynamics, thermalization, and localization, Phys. Rev. B
96 (2017) 020408 [arXiv:1703.03420].
Mathematical Physics, Springer Singapore (2017).
correlators at late times, arXiv:1705.07597 [INSPIRE].
HJEP1(207)48
Phys. Rev. Lett. 116 (2016) 030401 [arXiv:1508.05339] [INSPIRE].
[2] Y. Sekino and L. Susskind , Fast Scramblers, JHEP 10 ( 2008 ) 065 [arXiv: 0808 .2096] e ect , Phys. Rev. Lett . 115 ( 2015 ) 131603 [arXiv: 1412 .5123] [INSPIRE].
[19] A.L. Fitzpatrick , J. Kaplan , D. Li and J. Wang , On information loss in AdS3/CFT2 , JHEP [38] D.N. Page , Average entropy of a subsystem , Phys. Rev. Lett . 71 ( 1993 ) 1291 [56] Y. Huang , F.G. S.L. Brandao and Y.-L. Zhang , Finite-size scaling of out-of-time- ordered [57] M.V. Berry , Regular and irregular semiclassical wavefunctions , J. Phys. A 10 ( 1977 ) 2083 . [58] M. Srednicki , Chaos and quantum thermalization , Phys. Rev. E 50 ( 1994 ) 888 [59] L. D'Alessio , Y. Kafri , A. Polkovnikov and M. Rigol , From quantum chaos and eigenstate [67] E. Brezin and S. Hikami , Extension of level-spacing universality , Phys. Rev. E 56 ( 1997 ) 264