Recent developments in the holographic description of quantum chaos

We review recent developments encompassing the description of quantum chaos in holography. We discuss the characterization of quantum chaos based on the late time vanishing of out-of-time-order correlators and explain how this is realized in the dual gravitational description. We also review the connections of chaos with the spreading of quantum entanglement and diffusion phenomena.


Introduction
The characterization of quantum chaos is fairly complicated. Possible approaches range from semi-classical methods to random matrix theory: in the first case one studies the semi-classical limit of a system whose classical dynamics is chaotic; in the latter approach the characterization of quantum chaos is made by comparing the spectrum of energies of the system in question to the spectrum of random matrices [1]. Despite the insights provided by the above-mentioned approaches, a complete and more satisfactory understanding of quantum chaos remains elusive.
Surprisingly, new insights into quantum chaos have come from black holes physics! In the context of so-called gauge gravity duality [2][3][4], black holes in asymptotically AdS spaces are dual to strongly coupled many-body quantum systems. It was recently shown that the chaotic nature of many-body quantum systems can be diagnosed with certain out-of-time-order correlation (OTOC) functions which, in the gravitational description, are related to the collision of shock waves close to the black hole horizon [5][6][7][8][9]. In addition to being useful for diagnosing chaos in holographic systems and providing a deeper understanding for the inner-working mechanisms of gauge-gravity duality, OTOCs have also proved useful in characterizing chaos in more general non-holographic systems, including some simple models like the kicked-rotor [10], the stadium billiard [11], and the Dicke model [12].
In this paper, we review the recent developments in the holographic description of quantum chaos. We discuss the characterization of quantum chaos based on the late time vanishing of OTOCs and explain how this is realized in the dual gravitational description. We also review the connections of chaos with the spreading of quantum entanglement and diffusion phenomena. We focus on the case of d´dimensional gravitational systems with d ě 3, which excludes the case of gravity in AdS 2 and SYK-like models [13][14][15][16] 1 . Also, due the lack of the author expertise, we did not cover the recent developments in the direct field theory calculations of OTOCs. This includes calculations for CFTs [20], weakly coupled systems [21,22], random unitary models [23][24][25] and spin chains [26][27][28][29][30].

A bird eye's view on classical chaos
In this section we briefly review some basic aspects of classical chaos. For definiteness we consider the case of a classical thermal system with phase space denoted as X " pq, pq, where q and p are multi-dimensional vectors denoting the coordinates and momenta of the phase space. We can quantify whether the system is chaotic or not by measuring the stability of a trajectory in phase space under small changes of the initial condition. Let us consider a reference trajectory in phase space, Xptq, with some initial condition Xp0q " X 0 . A small change in the initial condition X 0 Ñ X 0`δ X 0 leads to a new trajectory Xptq Ñ Xptq`δXptq. This is illustrated in figure 1. For a chaotic system, the distance between the new trajectory and the reference one increases exponentially with time |δXptq| " |δX 0 |e λt or where λ is the so-called Lyapunov exponent. This should be contrasted with the behaviour of non-chaotic systems, in which δXptq remains bounded or increase algebraically [31].
The exponential increase depends on the orientation of δX 0 and this leads to a spectrum of Lyapunov exponents, tλ 1 , λ 2 , ..., λ K u, where K is the dimensionality of the p q δqp0q δqptq Figure 1: Variation of a trajectory in the phase space under small modifications of the initial condition. For a chaotic system the distance between two initially nearby trajectories increases exponentially with time, i.e., |δqptq| " |δqp0q|e λt . phase space. A useful parameter characterizing the trajectory instability is which is called the maximum Lyapunov exponent. When the above limits exist and λ max ą 0 the trajectory shows sensitive to initial conditions and the system is said to be chaotic [31].
The chaotic behavior can be either a consequence of a complicated Hamiltonian or simply due to the contact with a thermal heat bath. This is because chaos is a common property of thermal systems. To later to make contact with black holes physics we consider the case of a classical thermal system with inverse temperature β. If F pXq is some function of the phase space coordinates we define its classical expectation value as where HpXq is the system's Hamiltonian.
Classical thermal systems have two exponential behaviors that have analogues in terms of black holes physics: the Lyapunov behavior, characterizing the sensitive dependence on initial conditions; and the Ruelle behavior, characterizing the approach to thermal equilibrium [32,33].
To quantify the sensitivity to initial conditions in a thermal system we need to consider thermal expectation values. Note that (1) can have either signs. To avoid cancellations in a thermal expectation values we consider the square of this derivative The expected behavior of this quantity is the following [34] F ptq " where c k are constants and λ k are the Lyapunov exponents. At later times the behavior is controlled by the maximum Lyapunov exponent F " e 2λmaxt .
The approach to thermal equilibrium or, in other words, how fast the system forgets its initial condition, can be quantified by two-point functions of the form whose expected behavior is [34] Gptq " where b j are constants and µ j are complex parameters called Ruelle resonances. The late time behavior is controlled by the smallest Ruelle resonance G " e´µ min t .

Some aspects of quantum chaos
In this section, we review some aspects of quantum chaos. For a long time, the characterization of quantum chaos was made by comparing the spectrum of energies of the system in question to the spectrum of random matrices or using semiclassical methods [1]. Here we follow a different approach, which was first proposed by Larkin and Ovchinnikov [35] in the context of semi-classical systems, and it was recently developed by Shenker and Stanford [6][7][8] and by Kitaev [9].
For simplicity, let us consider the case of a one-dimensional system, with phase space variables pq, pq. Classically, we know that Bqptq{Bqp0q grows exponentially with time for a chaotic system. The quantum version of this quantity can be obtained by noting that Bqptq Bqp0q " tqptq, pp0qu P.B. , where tqptq, pp0qu P.B. denotes the Poisson bracket between the coordinate qptq and the momentum pp0q. The quantum version of Bqptq{Bqp0q can then be obtained by promoting the Poisson bracket to a commutator where nowqptq andpp0q are Heisenberg operators.
We will be interested in thermal systems, so we would like to calculate the expectation value of rqptq,pp0qs in a thermal state. However, this commutator might have either sign in a thermal expectation value and this might lead to cancellations. To overcome this problem, we consider the expectation value of the square of this commutator where β is the system's inverse temperature and the overall sign is introduced to make Cptq positive. More generally, one might replaceqptq andpp0q by two generic hermitian operators V and W and quantify chaos with the double commutator This quantity measures how much an early perturbation V affects the later measurement of W . As chaos means sensitive dependence on initial conditions, we expect Cptq to be 'small' in non-chaotic system, and 'large' if the dynamics is chaotic. In the following, we give a precise meaning for the adjectives 'small' and 'large'.
For some class of systems, the quantum behavior of Cptq has a lot of similarities with the classical behavior of xpBqptq{Bqp0qqy β . However, the analogy between the classical and quantum quantities is not perfect because there is not always a good notion of a small perturbation in the quantum case (remember that classical chaos is characterized by the fact that a small perturbation in the past has important consequences in the future). If we start with some reference state and then perturb it, we easily produce a state that is orthogonal to the original state, even when we change just a few quantum numbers. Because of that, it seems unnatural to quantify the perturbation as small. Fortunately, there are some quantum systems in which the notion of a small perturbation makes perfect sense. An example is provided by systems with a large number of degrees of freedom. In this case, a perturbation involving just a few degrees of freedom is naturally a small perturbation.
For some class of chaotic systems, which include holographic systems, Cptq is expected to behave as 2 where N dof is the number of degrees of freedom of the system. Here, we have assumed V and W to be unitary and hermitian operators, so that V V " W W " 1.  If we perturb the system with an operator that involves only a few degrees of freedom, the information about this operator will spread among the other degrees of freedom of the system. After a scrambling time, the information will be scrambled among all the degrees of freedom and the operator will have a large commutator with almost any other operator.
To understand how the above behavior relates to chaos, we write the double commutator as " 2´2 xW ptqV p0qW ptqV p0qy β , where we made the assumption that W and V are hermitian and unitary operators.
Note that all the relevant information about Cptq is contained in the OTOC OTOptq " xW ptqV p0qW ptqV p0qy .
The fact that Cptq approaches 2 at later times implies that the OTOptq should vanish in that limit. To understand why this is related to chaos we think of OTOptq as an inner-product of two states OTOptq " xψ 2 |ψ 1 y , where |ψ 1 y " W p´tqV p0q|βy , |ψ 2 y " V p0qW p´tq|βy (16) where |βy is some thermal state and we replace t Ñ´t to make easier the comparison with black holes physics.
If rV p0q, W ptqs « 0 for any value of t, the two states are approximately the same, and xψ 1 |ψ 2 y « 1, implying Cptq « 0. That means the system displays no chaosthe early measurement of V has no effect on the later measurement of W . If, on the other hand, rV p0q, W ptqs ‰ 0, the states |ψ 1 y and |ψ 2 y will have a small superposition xψ 1 |ψ 2 y « 0, implying Cptq « 2. That means that V has a large effect on the later measurement of W .
In figure 3 we construct the states |ψ 1 y and |ψ 2 y and explain why xψ 1 |ψ 2 y « 0 for large t means chaos. Let us start by constructing the state |ψ 1 y " W p´tqV p0q|βy " e´i Ht W p0qe iHt V p0q|βy. The unperturbed thermal state is represented by a horizontal line. We initially consider the state V p0q|βy, which is the thermal state perturbed by V . If we evolve the system backward in time (applying the operator e iHt ) for some time which is larger than the dissipation time, the system will thermalize and it will no longer display the perturbation V . After that, we apply the operator W , which should be thought of as a small perturbation, and then we evolve the system forwards in time (applying the operator e´i Ht ). The final result of this set of operations depends on the nature of the system. If the system is chaotic, the perturbation W will have a large effect after a scrambling time, and the perturbation that was present at t " 0 will no longer re-materialize. This is illustrated in figure 3. In contrast, for a non-chaotic system, the perturbation W will have little effect on the system at later times, and the perturbation V will (at least partially) re-materialize at t " 0.
We now construct the state |ψ 2 y " V p0qW p´tq|βy " V p0qe´i Ht W p0qe iHt |βy. This is illustrated in figure 4. We start with the thermal state |βy and then we evolve this state backwards in time e iHt |βy. After that we apply the operator W and then we evolve the system forwards in time, obtaining the state e´i Ht W p0qe iHt |βy. Finally, we apply the operator V , obtaining the state V p0qW p´tq|βy. Note that, by construction, this state displays the perturbation V at t " 0, while the state W p´tqV p0q|βy does not. As a consequence, the two states are expected to have a small superposition xW p´tqV p0q|W p´tqV p0qy β « 0. This should be contrasted to the case where the system is not chaotic. In this case the perturbation V re-materializes at t " 0, and the states |ψ 1 y and |ψ 2 y have a large superposition, i.e., xW p´tqV p0q|W p´tqV p0qy β « 1.
In this construction we assumed the operators V p0q and W p´tq to be separated by a scrambling time, i.e., |t| ą t˚. This is important because, at earlier times, the two operators, which in general involve different degrees of freedom of the system, generically commute. The operators manage to have a non-zero commutator at later times because of the phenomenon of operation growth that we will describe in the next section.
V p0q|βy t " 0 t ă 0 Figure 3: Construction of the state W p´tqV p0q|βy. For a chaotic system the perturbation V fails to re-materialize at t " 0. In a non-chaotic system we expect the perturbation V to re-materialize at t " 0.

Operator growth and scrambling
The operators V and W act generically at different parts of the physical system, yet they can have a non-zero commutator at later times. This is possible because in chaotic systems the time evolution of an operator makes it more and more complicated, involving and increasing number of degrees of freedom. As a result, an operator that initially involves just a few degrees of freedom becomes delocalized over a region that grows with time. The growth of the operator W ptq is maybe more evident from the point of view of the Baker-Campbell-Hausdorff (BCH) formula, in terms of which we can write From the above formula it is clear that, at each order in t there is a more complicated contribution to W ptq. In chaotic systems the operator becomes more and more delocalized as the time evolves, and it eventually becomes delocalized over the entire system. The time scale at which this occurs is the so-called scrambling time t˚. After the scrambling time the operator W ptq manages to have a non-zero and large commutator with almost any other operator, even operators involving only a few degrees of freedom.
This can be clearly illustrated in the case of a spin-chain. Let us follow [7] and consider an Ising-like model with Hamiltonian where X i , Y i and Z i denote Pauli matrices acting on the ith site of the spin chain. The above system is integrable if we take g " 1 and h " 0, but it is strongly chaotic if we choose g "´1.05 and h " 0.5.
To illustrate the concept of scrambling, we consider the time evolution of the operator Z 1 . Using the BCH formula we can write Ignoring multiplicative constants and signs we can write the above terms (schematically) as As the the time evolves higher order terms become important in the series (19), and the operator Z 1 ptq becomes more and more complicated, involving terms in an increasing number of sites. For large enough t the operator will involve all the sites of the spin chain and it will manage to have a non-zero commutator with a Pauli operator in any other site of the system. In this situation the information about Z 1 is essentially scramble among all the degrees of freedom of the system. As discussed before, this occurs after a scrambling time. Above this time the double commutator Cptq saturates to a constant value. This should be contrasted to what happens for an integrable system. In this case the operator grows, but it also decreases at later times. In the chaotic case, the operator remains large at later times [7].

Probing chaos with local operators
In quantum field theories we can upgrade (11) to the case where the operators are separated in space Cpt, xq " x´rV p0, 0q, W pt, xqs 2 y β .
Strictly speaking, the above expression is generically divergent, but it can be regularized by adding imaginary times to the time arguments of the operators V and W . For a large class of spin-chains, higher dimensional SYK-models and CFTs the above commutator is roughly given by where v B is the so-called butterfly velocity 4 . This velocity describes the growth of the operator W in physical space and it acts as a low-energy Lieb-Robinson velocity [37], which sets a bound for the rate of transfer of quantum information. From the above formula, we can see that there is an additional delay in scrambling due to the physical separation between the operators. The butterfly velocity defines an effective light-cone for the commutator (21). Inside the cone, for t´t˚ě |x|{v B , we have Cpt, xq " Op1q, whereas for outside the cone, for t´t˚ă |x|{v B , the commutator is small, Cpt, xq " 1{N dof ăă 1. Outside the light-cone the Lorentz invariance implies a zero commutator. The light-cone and the butterfly effect cone are illustrated in figure  5.

Chaos & Holography
In this section, we review how the chaotic properties of holographic theories can be described in terms of black holes physics. Black holes behave as thermal systems and thermal systems generically display chaos. This implies that black holes are somehow chaotic. This statement has a precise realization in the context of the gauge/gravity duality. According to this duality, some strongly coupled non-gravitational systems are dual to higher dimensional gravitational systems. In the most known and studied example of this duality the N " 4 super Yang-Mills (SYM) theory living in R 3,1 is dual to type IIB supergravity in AdS 5ˆS 5 . More generically, a d´dimensional non-gravitational theory living in R d´1,1 is dual to a gravity theory living in a higherdimensional space of the form AdS d`1ˆM , where M is generically a compact manifold. The non-gravitational theory can be thought as living in the boundary of AdS d`1 and because of that is usually called the boundary theory. The gravitational theory is also called the bulk theory.
There is a dictionary relating physical quantities in the boundary and bulk description [3,4]. An example is provided by the operators of the boundary theory, which are related to bulk fields. The boundary theory at finite temperature can be described by introducing a black hole in the bulk. The thermalization properties of the boundary theory have a nice visualization in terms of black holes physics. By applying a local operator in the boundary theory we produce some perturbation that describes a small deviation from the thermal equilibrium. The information about V pxq is initially contained around the point x, but it gets delocalized over a region that increases with time until it completely melts into the thermal bath. In the bulk theory, the application of the operator V pxq produces a particle (field excitation) close to the boundary of the space, which then falls into the black hole. The return to the thermal equilibrium in the boundary theory corresponds to the absorption of the bulk particle by the black hole. Figure 6 illustrates the bulk description of thermalization. The black hole's interior is shown in gray. The boundary operator V is dual to the bulk field φ. From the point of view of the boundary theory the perturbation produced by V is initially localized around the point x, but it gets delocalized over a region that increases with time. In the bulk description this is described by a particle (field excitation) that is initially close to the boundary and then falls into the black hole.
The approach to thermal equilibrium is controlled by the black hole's quasi-normal modes (QNMs). In holographic theories, the quasi-normal modes control the decay of two-point functions of the boundary theory where the dissipation time t d is related to the lowest quasi-normal mode (Impωq " t´1 d ).
From the point of view of the bulk theory the QNMs describes how fast a perturbed black hole returns to equilibrium. Clearly, the black hole's quasi-normal modes correspond to the classical Ruelle resonances. In holographic theories the dissipation time is roughly give by t d " β.
Another important exponential behavior of black holes is provided by the blueshift suffered by the in-falling quanta, or, equivalently, the red shift suffered by the quanta escaping from the black hole. The blueshift suffered by the in-falling quanta is determined by the black hole's temperature. If the quanta asymptotic energy is E 0 , this energy increases exponentially with time where β is the Hawking's inverse temperature. Later we will see that this exponential increase in the energy of the in-falling quanta gives rise to the Lyapunov behavior of Cpt, xq of holographic theories.

Holographic setup
The TFD state & Two-sided black holes In the study of chaos is convenient to consider a thermofield double state made out of two identical copies of the boundary theory where L and R label the states of the two copies, which we call QFT L and QFT R , respectively. The two boundary theories do not interact and only know about each other through their entanglement. This state is dual to an eternal (two-sided) black hole, with two asymptotic boundaries, where the boundary theories live [38]. This is a wormhole geometry, with an Einstein-Rosen bridge connecting the two sides of the geometry. The wormhole is not traversable, which is consistent with the fact that the two boundary theories do not interact.
For definiteness we assume a metric of the form where the boundary is located at r " 8, where the above metric is assumed to asymptote AdS d`1 . We take the horizon as located at r " r H , where G tt vanish and G rr has a first order pole. For future purposes, let β be the Hawking's inverse temperature, and S BH be the Bekenstein-Hawking entropy.
In the study of shock waves is more convenient to work with Kruskal-Szekeres coordinates, since these coordinate cover smoothly the globally extended spacetime. We first define the tortoise coordinate dr˚" and then we introduce the Kruskal-Szekeres coordinates U, V as U "`e In terms of these coordinates the metric reads where In these coordinates the horizon is located at U " 0 or at V " 0. The left and right boundaries are located at U V "´1 and the past and future singularities at U V " 1.
The Penrose diagram for this metric is shown in figure 7. AdS. This geometry is dual to a thermofield double state constructed out of two copies of the boundary theory.
The global extended spacetime can also be described in terms of complexified coordinates [39]. In this case one defines the complexified Schwarzschild time where t L and t E are the Lorentzian and Euclidean times, and then one describes the time in each of the four patches (left and right exterior regions, and the future and past interior regions) as having a constant imaginary part t E " 0 pright exterior regionq t E "´β{4 pfuture interior regionq (32) t E "´β{2 pleft exterior regionq t E "`β{4 ppast interior regionq The Euclidean time has a period of β. The Lorentzian time increases upward (downward) in the right (left) exterior region, and to the right (left) in the future (past) interior.
Note that, with the complexified time, one can obtain an operator acting on the left boundary theory by adding (or subtracting) iβ{2 to the time of an operator acting on the right boundary theory.

Perturbations of the TFD state & Shock wave geometries
We now turn to the description of states of the form W ptq|TFDy where W is a thermal scale operator that acts on the right boundary theory. This state can be describe by a 'particle' (field excitation) in the bulk that comes out of the past horizon, reaches the right boundary at time t, and then falls into the future horizon, as illustrated in figure 8.
W ptq W ptq|T F Dy " Figure 8: Bulk description of the state W ptq|TFDy. In blue is shown the trajectory of a 'particle' that comes out of the past horizon, reaches the boundary at time t producing the perturbation W , and then falls into the future horizon. For now on, we will refer to this bulk excitation as the W-particle.
If |t| is not too large, the state W ptq|TFDy will represent just a small perturbation of the TFD state and the corresponding description in the bulk will be just an eternal two-sided black hole geometry slightly perturbed by the presence of a probe particle. This is no longer the case if |t| is large. In this case, there is a non-trivial modification of the geometry. A very early perturbation, for example, is described in the bulk in terms of a particle that falls towards the future horizon for a very long time and gets highly blueshifted in the process. If the particle's energy is E 0 in the asymptotic past, this energy will be exponentially larger from the point of view of the t " 0 slice of the geometry, i.e., E " E 0 e 2π β t . Therefore, for large enough |t|, the particle's energy will be very large and one needs to include the corresponding back-reaction.
The back-reaction of a very early (or very late) perturbation is actually very simple -it corresponds to a shock wave geometry [40,41]. To understand that, we first need to notice that, under boundary time evolution, the stress-energy of a generic perturbation W gets compressed in the V´direction, and stretched in the U´direction. For large enough |t| we can approximate the stress tensor of the W-particle as where P U " β´1e 2π β t is the momentum of the W-particle in the U´direction and ap xq is some generic function that specifies the location of the perturbation in the spatial directions of the right boundary. Note that T V V is completely localized at V " 0 and homogeneous along the U´direction. Besides that, even if the W-particle is massive, the exponential blue-shift will make it follow an almost null trajectory, as shown in figure 9.  The shock wave geometry produced by the W-particle is described by the metric that is completely specified by the shock wave transverse profile hpt, xq. This geometry can be seen as two pieces of a eternal black hole glued together along V " 0 with a shift of magnitude hpt, xq in the U´direction. We find it useful to represent this geometry with the same Penrose diagram of the unperturbed geometry, but with the prescription that any trajectory crossing the shock wave gets shifted in the U´direction as U Ñ U`hpt, xq. See figure 9.
The precise form of hpt, xq can be determined by solving the V V´component of Einstein's equation. For a local perturbation, i.e., ap xq " δ d´1 p xq, the solution reads where, for simplicity, G ij has been assumed to be diagonal and isotropic.
Interestingly, the shock wave profile contains information about the parameters characterizing the chaotic behavior of the boundary theory. Indeed, the double commutator has a region of exponential growth at which Cpt, xq " hpt, xq. From this identification, we can write hpt, xq " e 2π β´t´t˚´| x| v B¯ ( 37) where (the leading order contribution) to the scrambling time scales logarithmically with the Bekenstein-Hawking entropy while the Lyapunov exponent is proportional to the Hawking's temperature.
The butterfly velocity is determined from the near-horizon geometry 5 v 2 B "

Bulk picture for the behavior of OTOCs
In this section we present the bulk perspective for the vanishing of OTOCs at later times. In order to do that, we write the OTOC as a superposition of two states OTOptq " xTFD|W p´tqV p0qW p´tqV p0q|TFDy " xψ out |ψ in y , where the 'in' and 'out' states are given by |ψ in y " W p´tqV p0q|TFDy , |ψ out y " V : p0qW : p´tq|TFDy The interpretation of a vanishing OTOC in terms of the bulk theory is actually very simple. Let us go step by step and construct first the state V p0q|βy. This state is described by a particle that comes out of the past horizon, reaches the boundary at t " 0, and then falls back into the future horizon. See the left panel of figure 10.
Now the 'in' state can be obtained as |ψ in y " W p´tqV p0q|TFDy " e´i Ht W p0q e iHt V p0q|TFDy .
This amounts to evolve the state V p0q|TFDy backwards in time, apply the operator W , and then evolve the system forwards in time. The corresponding description in the bulk is shown in the right panel of figure 10. From this picture we can see that the perturbation W produces a shock wave that causes a shift in the trajectory of the V-particle, which no longer reaches the boundary at time t " 0, but rather with some time delay. The physical interpretation is that a small perturbation in the asymptotic past (represented by W ) is amplified over time and destroys the initial configuration (represented by the state V p0q|TFDy).
W p´tq V is produced with some time delay V p0q V p0q|TFDy W p´tqV p0q|TFDy h Figure 10: Left panel: bulk description of the state V p0q|TFDy. The V-particle comes out of the past horizon, reaches the boundary at time t " 0 producing the perturbation V , and then falls into the future horizon. Right panel: bulk description of the 'in' state |ψ in y " W p´tqV p0q|TFDy. The W-particle (shown in blue) produces a shock wave along V " 0. The trajectory of the V-particle (shown in red) suffers a shift and the perturbation V is produced at the boundary with some time delay.
V p0qW p´tq|TFDy W p´tq V p0q h Figure 11: Bulk description of the 'out' state |ψ out y " V p0qW p´tq|TFDy. The W-particle produces the shock wave geometry. The trajectory of the V-particle is such that, after suffering the shift U Ñ U`hpt, xq, it reaches the boundary at time t " 0, producing the perturbation V .
The bulk description of the 'out' state can be obtained in the same way. As this state displays the perturbation V at t " 0, the V-particle should be produced in the asymptotic past in such a way that, after its trajectory gets shifted as U Ñ U`h, it reaches the boundary at the time t " 0 producing the perturbation V .
Comparing the bulk description of the state |ψ in y (shown in the right panel of figure 10) with the description of the state |ψ out y (shown in figure 11) we can see that these states are indistinguishable when hpt, xq is zero, but they become more and more different for large values of hpt, xq. As a consequence, the overlap Cptq " xψ out |ψ in y is equal to one when h " 0, but it decreases to zero as we increase the value of h.
The exponential behavior of hpt, xq implies that an early enough perturbation can produce a very large shift in the V-particle's trajectory, causing it to be captured by the black hole, and preventing the materialization of the V perturbation at the boundary. See figure 12. This should be compared with the physical picture given in figure 3.
W p´tq hpt, xq V fails to materialize Figure 12: Bulk description of the state W p´tqV p0q|TFDy for the case where |t| t˚. The V-particle's trajectory undergoes a shift, and it is captured by the black hole. The perturbation V never forms, and the corresponding state have no superposition with the 'out' state V p0qW p´tq|TFDy, resulting in a vanishing OTOC.
The physical picture of the process described in figure 12 is quite simple. The state V p0q|TFDy can be represented by a black hole geometry in which a particle (the Vparticle) escapes from the black holes and reaches the boundary at time t " 0. The state W p´tqV p0q|TFDy is obtained by perturbing the state V p0q|TFDy in the asymptotic past. This corresponds to add a W-particle to the system in the asymptotic past. This particle gets highly blue shifted as it falls towards the black hole. The black hole captures the W-particle and becomes bigger. The V-particle fails to escape from the bigger black hole, and never reaches the boundary to produce the V perturbation. This physical picture is illustrated in figure 13.  Figure 13: Left panel: bulk description of the state V p0q|TFDy. The V-particle (whose trajectory is shown in red) escapes from the black hole (shown in gray), reaches the boundary at time t " 0, and then falls back towards the horizon. Right panel: bulk description of the state W p´tqV p0q|TFDy for the case where |t| t˚. The W-particle (whose trajectory is shown in blue) gets blue shifted and increases the black hole size as it falls into it. The V-particle fails to scape from the larger black hole, and the perturbation V never forms at the boundary.
The precise form of the above OTOC can be obtained by calculating the overlap xψ out |ψ in y using the Eikonal approximation [8], in which the Eikonal phase δ is proportional to the shock wave profile δ " hpt, xq. The OTOC can be written as an integral of the phase e iδ weighted by kinematical factors which are basically Fourier transforms of bulk-to-boundary propagators for the V and W operators.
The result for Rindler AdS 3 reads 6 xV pi 1 qW pt`i 2 qV pi 3 qW pt`i 4 qy xV pi 1 qV pi 3 qyxW pi 2 qW pi 4 qy "¨1 where ∆ V and ∆ W are the scaling dimensions of the operator V and W , respectively, and ij " ipe i i´e i j q. For this system β " 2π and v B " 1. This formula matches the direct CFT calculation 7 obtained in [20]. It can also be derived using the geodesic approximation for two-sided correlators in a shock wave background [5,20].
The above result is valid for small 8 values of G N , or for any value of G N , but for times in the range t d ăă t ăă t˚, where t˚" β 2π log 1 G N . Despite being true in the Rindler AdS 3 case, the proportionality between the double commutator and the shock wave profile has not been demonstrated in more general cases. However, the authors of [8] argued that, in regions of moderate scattering between the V-and W-particle, the identification Cpt, xq " hpt, xq is approximately valid.
At very late times, the behavior of the OTOptq is expected to be controlled by the black hole quasi-normal modes. Indeed, in the case of a compact space, it is possible to show that where R is the diameter of the compact space and ω is the system lowest quasi-normal frequency [8].

Stringy corrections
In this section, we briefly discuss the effects of stringy corrections to the Einstein gravity results for OTOCs. We start by reviewing the Einstein gravity results from the perspective of scattering amplitudes. In the framework of the Eikonal approximation, the phase shift suffered by the V-particle is given by where we used the fact that hpt, xq " G N P U and introduced a Mandelstam-like variable s " 2Ap0qP U P V . In a small-G N expansion the double commutator Cptq and the phase shift δ scale with s in the same way, namely 8 In AdS/CFT the Newton constant is related to the rank of the gauge group of dual CFT as G N " 1{N a , where a is a positive number that depends on the dimensionality of the bulk space time (cf. section 7.2 of [50]). Our classical gravity calculations are only valid in the large-N limit (that suppresses quantum corrections) so it is natural to consider G N as a small parameter.
The string corrections can be incorporated using the standard Veneziano formula for the relativistic scattering amplitude A " s δ. The phase shift can then be schematically written as an infinite sum δ " where each term correspond to the contribution due to the exchange of a spin-J field. In Einstein gravity the dominant contribution comes from the exchange of a spin-2 field, the graviton. In string theory, we have to include an infinite tower of higher spin fields. Naively, it looks like these higher spin contributions will increase the development of chaos. However, the re-summation of the above sum actually leads to a decrease in the development of chaos. The string-corrected phase shift has a milder dependence with s, namely with the effective spin given by [8] J eff " 2´d pd´1q 2 where s is the string length, AdS is the AdS length scale and d is the number of dimensions of the boundary theory. As a result, the string-corrected double commutator grows in time with an effective smaller Lyapunov exponent and this leads to a larger scrambling time 9 t string " t˚ˆ1`d pd´1q 2 s 4 2 AdS˙.
The above discussion implies that for a theory with a finite number of high-spin fields (J ą 2) chaos would develop faster than in Einstein gravity. These theories, however, are known to violate causality [54]. It is then natural to speculate that the Lyapunov exponent obtained in Einstein gravity has the maximal possible value allowed by causality. This is indeed true and this is the topic of the next section. 9 At small scales, the string-corrected shock wave has a gaussian profile, and the concept of butterfly velocity is not meaningful. It was recently shown, however, that at larger scales is possible to define a string-corrected butterfly velocity. The result for N " 4 SYM theory reads [51]

Bounds on chaos
One of the remarkable insights that came from the holographic description of quantum chaos is the fact that there is a bound on chaos -the quantum Lyapunov exponent is bounded from above, while the scrambling time is bounded from below. A distinct feature of holographic systems is that they saturate these two bounds.
Let us follow the historical order and start by discussing the lower bound on the scrambling time. In black holes physics, the scrambling time defines how fast the information that has fallen into a black hole can be recovered from the emitted Hawking radiation 10 . In the context of the Hayden-Preskill thought experiment, the scrambling time is barely compatible with black hole complementarity [52], since a smaller scrambling time would lead to a violation of the non-clonning principle. This led Susskind and Sekino to conjecture that black holes are the fastest scramblers in nature, i.e., they have the smallest possible scrambling time [53]. The lower bound on the scrambling time of a generic many-body quantum system can be written as where Cpβq is some function of the inverse temperature. In the case of black holes this function is simply given by Cpβq " β 2π . The scrambling time defines a stronger notion of thermalization, and should not be confused with the dissipation time. In fact, for black holes, one expects the dissipation time to be given by the black hole quasinormal modes 11 t d " β, while the scrambling time is parametrically larger t˚" β log N dof . This bring us to the second bound on chaos: for systems with such a large hierarchy between the scrambling and the dissipation time is possible to derive an upper bound for the Lyapunov exponent [44] One should emphasize that this bound does not depend on the existence of a holographic dual. It can be derived for generic many-body quantum systems under some very reasonable assumptions.
The fact that black holes always have a maximum Lyapunov exponent led to the speculation that the saturation of the chaos bound might be a sufficient condition for a system to have an Einstein gravity dual [44,9]. In fact, there have been many attempts to use the saturation of the chaos bound as a criterion to discriminate holographic CFTs from the non-holographic ones [20, 45-49, 55, 56]. It was recently shown, however, that this criterion, though necessary, is insufficient to guarantee a dual description purely in terms of Einstein gravity [57,58].
Since v B defines the speed at which information propagates it is natural to question whether this quantity is also bounded. From the perspective of the boundary theory, meaning that information should not propagate faster than the speed of light. Indeed, the above bound can be derived in the context of Einstein gravity by using Null Energy Condition (NEC) and assuming an asymptotically AdS geometry 12 [59]. This is consistent with the expectation that gravity theories in asymptotically AdS geometries are dual to relativistic theories. In contrast, for geometries which are not asymptotically AdS, the butterfly velocity can surpass the speed of light [59,42], which is consistent with the non-Lorentz invariance of the corresponding boundary theories.
If we further assume isotropy, it is possible to derive a stronger bound for where v Sch B is the value of the butterfly velocity for an AdS-Schwarzschild black brane in d`1 dimensions. This is also the butterfly velocity for a d-dimensional thermal CFT.
The above formula shows that, for thermal CFTs, v B does not depend on the temperature. However, if we deform the CFT, v B acquires a temperature dependence as we move along the corresponding renormalization group (RG) flow. In fact, by considering deformations that break the rotational symmetry it was noticed that the butterfly velocity violates the above bound, but remains bounded from above by its value at the infra-red (IR) fixed point, never surpassing the speed of light [61][62][63]. The above bound can also be violated by higher curvature corrections, but v B remains bounded by the speed of light as long as causality is respected 13 . The violation of the bound given in (58) by anisotropy or higher curvature corrections is reminiscent of the well-known violation of the shear viscosity to entropy density ratio bound [66][67][68][69][70][71].

Chaos and Entanglement Spreading
The thermofield double state displays a very atypical left-right pattern of entanglement that results from non-zero correlations between subsystems of QFT L and QFT R at t " 0. The chaotic nature of the boundary theories is manifest by the fact that small perturbations added to the system in the asymptotic past destroy this delicate correlations [5].
The special pattern of entanglement can be efficiently diagnosed by considering the mutual information IpA, Bq between spatial subsystems A Ă QFT L and B Ă QFT R , defined as where S A is the entanglement entropy of the subsystem A, and so on. The mutual information is always positive and provides an upper bound for correlations between operators O L and O R defined on A and B, respectively [72] IpA, Bq ě The thermofield double state has non-zero mutual information between large 14 subsystems of the left and right boundary, signaling the existence of left-right correlations. These correlations can be destroyed by small perturbations in the asymptotic past, meaning that an initially positive mutual information drops to zero when we add a very early perturbation to the system.
Interestingly, the vanishing of the mutual information can be connected to the vanishing of the OTOCs discussed earlier. If, for simplicity, we assume that O L and O R have zero thermal one point function, then the disruption of the mutual information implies the vanishing of the following four-point function which is related by analytic continuation to the one-sided out-of-time-order correlator introduced earlier 15 .
The disruption of the mutual information has a very simple geometrical realization in the bulk. The entanglement entropies that appear in the definition of IpA, Bq can be holographically calculated using the HRRT prescription [73,74] where γ A is an extremal surface whose boundary coincides with the boundary of the region A. There is an analogous formula for S B . Both γ A and γ B are U-shaped surfaces lying outside of the event horizon, in the left and right side of the geometry, respectively. There are two candidates for the extremal surface that computes S AYB : the surface γ A Yγ B , or the surface γ wormhole that connects the two asymptotic boundaries of the geometry. See figure 14. According to the RT prescription, we should pick the surface with less area. If γ A Y γ B has less area than γ wormhole , then IpA, Bq " 0, because On the other hand, if γ wormhole has less area than γ A Y γ B , i.e., Area(γ wormhole ) ă Area(γ A )+Area(γ B ), then we have a positive mutual information Now, an early perturbation of the thermofield double state gives rise to a shock wave geometry in which the wormhole becomes longer. As a consequence the area of the surface γ wormhole increases, resulting in a smaller mutual information. It is then clear that the mutual information will drop to zero if the wormhole is long enough. The length of the wormhole depends on the strength of the shock wave, which, by its turn, depends on how early is the perturbation producing it. Therefore, an early enough perturbation will produce a very long wormhole in which the mutual information will be zero. The fact that the shock wave geometry produces a longer wormhole (along the t " 0 slice of the geometry) is clearly seen if we represent the shock wave geometry with a tilted Penrose diagram. See, for instance, the figure 3 of [75].
The mutual information IpA, Bq decreases as a function of the time t 0 at which we perturbed the system. For t 0 t˚, the mutual information decreases linearly with behavior controlled by the so-called entanglement velocity v E [62] dIpA, Bq where s th is the thermal entropy density and AreapA Y Bq is the area of A Y B (or the volume of the boundary of this region). The two-sided black hole geometry with a shock wave can be thought of as an additional example of a holographic quench protocol [62], and the time-dependence of entanglement entropy can be understood in terms of the so-called 'entanglement tsunami' picture. See [83] for field theory calculations, and [84][85][86][87][88] for holographic calculations. However, it was recently shown that the entanglement tsunami picture is not very sharp. See [89] for further details.
In [87,88], the entanglement velocity was conjectured to be bounded as  Figure 14: Illustration of the entangling surfaces in the t " 0 slice of a two-sided black brane geometry. The U-shaped surfaces (γ A and γ B ) are represented by blue curves. The surface stretching through the wormhole is given by the union of the two red surfaces γ wormhole " γ 1 Y γ 2 . In the left panel we represent the unperturbed geometry, in which the two horizons coincide. In the right panel we represent the geometry in the presence of a shock wave added at some time t 0 in the past. In this case, the size of the wormhole is effectively larger, and the two horizons no longer coincide.
where v Sch E is the entanglement velocity for a pd`1q-dimensional Schwarzschild black brane or, equivalently, the value of v E for a d-dimensional thermal CFT. This bound can be derived in the context of Einstein gravity assuming: an asymptotically AdS geometry, isotropy and NEC [60]. Just like in the case of v B , the entanglement velocity in thermal CFTs does not depend on the temperature. But v E acquires a temperature dependence if we deform the CFT and move along the corresponding RG flow [62,63]. In these cases, v E violate the above bound, but it remains bounded by its corresponding value at the IR fixed point, never surpassing the speed of light.
One can also prove that the entanglement velocity is also bounded by the speed of light 16 . This can be done by using the positivity of the mutual information [90], or using inequalities involving the relative entropy [91]. More generally, the authors of [89] conjecture that v E ď v B , which implies the bound v E ă 1 in the cases where v B is bounded. However, both [90,91] assumed that the theory is Lorentz invariant. In the case of non-Lorentz invariant theories (e.g. non-commutative gauge theories) the entanglement velocity can surpass the speed of light. This has been verified both in holography calculations [42] and in field theory calculations [92].
Finally, we mention that other concepts from information theory can also be used to diagnose chaos in holography. It has been shown, for instance, that the relative entropy is also a useful tool to diagnose chaotic behavior [95]. For a connection between chaos and computational complexity, see, for instance [96,97].
A longstanding goal of quantum condensed matter physics is to have a deeper understanding of the so-called 'strange metals'. These are strongly correlated materials that do not have a description in terms of quasiparticles excitations and whose transport properties display a remarkable degree of universality. In [107,108] Sachdev and Damle proposed that such a universal behavior could be explained by a fundamental dissipative timescale that would govern the transport in such systems.
Interestingly, the Lyapunov exponent defines a time scale τ L " 1{λ L , and the upper bound on λ L translates into a lower bound for τ L that precisely coincides with τ P where we reintroduced and the Boltzmann constant in the expression for the bound on the Lyapunov exponent 17 . Holographic systems saturate the above bound, and this explains the universality observed in the transport properties of these systems.
A prototypical example of universality is the linear resistivity of strange metals. In [109], Hartnoll proposed that the linear resistivity could be explained by the existence of a universal lower bound on the diffusion constants related to the collective diffusion of charge and energy where v is some characteristic velocity of the system. As D is inversely proportional to the resistivity, systems saturating the above bound would display linear resistivity behavior 18 . 17 In systems of units where and k B are not equal to one, the bound on the Lyapunov exponent reads λ L ď 2πk B β . 18 See [110] for a recent successful holographic description of linear resistivity at high-temperature.
One should think of (68) as a reformulation of the Kovtun-Son-Starinets (KSS) bound [111] η s which also relies on the idea of a fundamental dissipative timescale τ L " {pk B T q controlling transport in strongly interacting systems. Naively, the observed violations of the KSS bound would seem to indicate the existence of systems in which the bound (67) is violated. The bound (68) saves the idea of a fundamental dissipative timescale by introducing an additional parameter in the game, namely, the characteristic velocity v. The fact that η{s can be made arbitrarily small in some systems corresponds to the fact that the characteristic velocity is highly suppressed in those cases. See [98] for further details.
In [98,99] Blake proposed that, at least for holographic systems with particle-hole symmetry, the characteristic velocity v should be replaced by the butterfly velocity. More precisely where D c is the electric diffusity and C c is a constant that depends on the universality class of theory. This proposal was motivated by the fact that both D c and v B are determined by the dynamics close to the black hole horizon in the aforementioned systems. Despite working well for systems where energy and charge diffuse independently, this proposal was shown to fail in more general cases [112][113][114][115][116]. This is related to the fact that, in more general cases, the diffusion of energy and charge is coupled, and the corresponding transport coefficients are not given only in terms of the geometry close to the black hole horizon. Hence, there is no reason for these coefficients to be related to the butterfly velocity, which is always determined solely by the near-horizon geometry.
There is, however, a universal piece of the diffusivity matrix that can be related to the chaos parameters at infra-red fixed points. This is the thermal diffusion constant [117] D where C T is a universality constant (different from C c ). This proposal was shown to be valid even for systems with spatial anisotropy [118]. The above relation is not well defined when the system's dynamical critical exponent z is equal to one, but it can be extended this case 19 [119].
Finally, we mention that there is an interesting relation between chaos and hydrodynamics that manifest itself in the so-called 'pole-skipping' phenomenon. See [102][103][104] for further details.

Closing remarks
The holographic description of quantum chaos not only has provided new insights into the inner-workings of gauge-gravity duality, but it has also given insights outside the scope of holography: some examples include the characterization of chaos with OTOCs, the definition of a quantum Lyapunov exponent and the existence of a bound for chaos.
The success of this new approach to quantum chaos explains the growing experimental interest that OTOCs have been received. Indeed, several protocols for measuring OTOCs have been proposed, and there are already a few experimental results. See [120] and references therein.
Finally, one of the remarkable features of quantum chaos is level statistics described by random matrices. The fact that this is present in the infrared limit of the SYK model [121][122][123] suggests that it should also be present in quantum black holes 20 , although this has not yet been verified [124].