Chaos, complexity, and random matrices

Journal of High Energy Physics, Nov 2017

Chaos and complexity entail an entropic and computational obstruction to describing a system, and thus are intrinsically difficult 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 systems, we find unphysical behavior at early times including an \( \mathcal{O}(1) \) 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 definition 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.

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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. 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Jordan Cotler, Nicholas Hunter-Jones, Junyu Liu, Beni Yoshida. Chaos, complexity, and random matrices, Journal of High Energy Physics, 2017, 48, DOI: 10.1007/JHEP11(2017)048