Chaos and entanglement spreading in a non-commutative gauge theory

Holographic theories with classical gravity duals are maximally chaotic: they saturate a set of bounds on the spread of quantum information. In this paper we question whether non-locality can affect such bounds. Specifically, we consider the gravity dual of a prototypical theory with non-local interactions, namely, N=4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N}=4 $$\end{document} non-commutative super Yang Mills. We construct shock waves geometries that correspond to perturbations of the thermofield double state with definite momentum and study several chaos related properties of the theory, including the butterfly velocity, the entanglement velocity, the scrambling time and the maximal Lyapunov exponent. The latter two are unaffected by the non-commutative parameter θ, however, both the butterfly and entanglement velocities increase with the strength of the non-commutativity. This implies that non-local interactions can enhance the effective light-cone for the transfer of quantum information, eluding previously conjectured bounds encountered in the context of local quantum field theory. We comment on a possible limitation on the retrieval of quantum information imposed by non-locality.


Probes of quantum chaos
Recent studies of many-body quantum chaos have shed light into the inner-working mechanisms of the gauge/gravity duality [1][2][3]. For example, the characteristic velocity of the butterfly effect is known to play an important role in determining the bulk causal structure [4], while the saturation of the maximal Lyapunov exponent might be a necessary condition for the existence of a gravity dual [5,6]. There are also some interesting proposals connecting chaos and hydrodynamics [7][8][9][10][11][12][13], and chaos and the spread of quantum entanglement [14,15].
One way to diagnose chaos in quantum many-body systems is to consider the influence of an early perturbation V on the later measurement of some other operator W . Such an effect is encoded in the quantity [16] where · · · = Z −1 tr[e βH · · · ] denotes the thermal expectation value at temperature T = β −1 . For chaotic systems the expected behavior is the following [17,18] where N 2 is the number of degrees of freedom of the system. Here, we have assumed V and W to be few-body Hermitian operators normalized such that V V = W W = 1. The exponential growth of C(t) is characterized by the Lyapunov exponent λ L and takes place at intermediate time scales bounded by the dissipation time t d and the scrambling time t * . The dissipation time characterizes the exponential decay of two-point correlators, e.g., V (0)V (t) ∼ e −t/t d , while the scrambling time t * ∼ λ −1 L log N 2 is defined as the time at which C(t) becomes of order O(1) [19,20]. The behavior of C(t) can be understood in terms of the expansion of V in the space of degrees of freedom. Under time evolution, the operator V gets scrambled with an increasing number of degrees of freedom and this causes C(t) to grow. Eventually, V gets scrambled with all degrees of freedom available in the system and, as consequence, C(t) saturates to a constant O(1) value.
In holographic theories, the dissipation time is controlled by the black hole quasinormal modes, so one generally expects t d ∼ β for low dimension operators. On the other hand, the scrambling time for black holes is found to be t * ∼ β log N 2 . For general quantum systems with such a large hierarchy between these two time scales, t d < < t * , the Lyapunov exponent was shown to have a sharp upper bound [5] λ L ≤ 2π β . (1.2) Interestingly, this bound is saturated by black holes in Einstein gravity, leading to the speculation that any large N system that saturates this bound will necessarily have an

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Einstein gravity dual, at least in the near horizon region [5,6]. Such a claim triggered an enormous interest in the community, and lead to many works attempting to use the saturation of the bound as a criterion to discriminate between CFTs with potential Einstein gravity duals [21][22][23][24][25][26][27]. However, it was recently proved that this criterion by itself is insufficient (albeit necessary) to guarantee a dual description with gravitational degrees of freedom [28]. A further diagnose of quantum chaos comes from considering the response of the system to arbitrary local perturbations. This effect can be studied by upgrading the commutator in (1.1) to (1.3) Calculations for holographic systems [29,30] and the SYK chain suggested that for chaotic systems, the exponential growth regime in (1.1) generalizes to: The butterfly velocity v B characterizes the rate of expansion of the operator V in space. This quantity defines an emergent light cone, defined by t − t * = | x|/v B . Within the cone, i.e. for t − t * > | x|/v B , one has that C(t, x) ∼ O(1), whereas outside the cone, for t − t * < | x|/v B , one has C(t, x) ≈ 0. Interestingly, in [31] it was argued that v B acts as a low-energy Lieb-Robinson velocity, which sets a bound for the rate of transfer of quantum information. In [15] it was proved that for asymptotically AdS black holes in two-derivative (Einstein) gravity, satisfying null energy condition (NEC), the butterfly velocity is bounded by , (1.5) where v Sch B is the value of the butterfly velocity for a (d + 1)-dimensional AdS-Schwarzschild black brane. It is tempting to conjecture that (1.5) might be a bound for any (local) QFT, in the same sense as the bound for the Lyapunov exponent (1.2). However, (1.5) was shown to fail for higher derivative gravities [29], as well as for anisotropic theories in Einstein gravity [32,33], which is reminiscent of the well-known violation of the shear viscosity to entropy density ratio [34][35][36][37][38][39]. In such cases, however, v B is still bounded from above and never reaches the speed of light c = 1, provided that the theory respects causality. 1 Naively, one would expect the speed of light to define a region of causal influence in a relativistic system. However, as clarified in [4], when we only have access to a subset of the Hilbert space, the propagation velocity of causal influence is generically smaller than the speed of light. So, we usually have v B < 1 because the butterfly velocity characterizes the velocity of causal influence in a subset of the Hilbert space defined by the thermal ensemble, i.e. the states with a fixed energy density. Indeed, the authors of [4] showed that, for any JHEP11(2018)072 asymptotically AdS geometry in two-derivative gravity, the butterfly velocity is bounded by the speed of light, i.e. v B ≤ 1 , (1.6) as it should for a theory with a (Lorentz invariant) UV fixed point. A natural question one can ask is whether non-local interactions can lead to a violation of either the Lyapunov exponent bound (1.2) or the butterfly velocity bound (1.6). Since non-local interactions break Lorentz invariance, a priori one does not expect c to play a role. Furthermore, non-local theories with holographic duals, have bulk metric that are in fact non-asymptotically AdS, so the bound derived in [4] does not apply. Known examples of non-local holographic theories are, for instance, i) the near horizon limit of a stack of D3-branes with a constant Neveu-Schwarz B µν [43,44], dual to non-commutative N = 4 super Yang Mills, ii) the near horizon limit of a stack of D3-branes with global R-symmetry charges [45], dual to a dipole deformation of N = 4 super Yang Mills and iii) the theory dual to the near horizon limit of a stack of NS5-branes [46], the so-called little string theory.
In this paper we will explore the aforementioned question in a prototypical theory with non-local interactions, namely, N = 4 non-commutative super Yang Mills. 2 Holography has already been useful to explore several dynamical effects of the non-locality inherent to non-commutative theories, with some surprising findings. For instance, in [47] it was shown from a quasinormal mode analysis that non-commutative gauge theories display a parametrically shorter dissipation time for light probes, i.e. t d (θ) < < t d (0) for T √ θ > > 1 and k √ θ > > 1, where θ measures the strength of the non-commutativity. Heavy probes were further analyzed in [48,49] showing a qualitative reduction in the viscosity felt by the probe. Lastly, the holographic complexity was recently studied in [50] which, remarkably, was shown to violate the so-called Lloyd's bound at late times.

Chaos and entanglement spreading
One way to diagnose chaos in holographic theories is by studying the disruption of mutual information between subregions of the two boundaries in a maximally extended black brane geometry. This approach is particularly interesting because it makes a clear connection between chaos and spreading of entanglement, which is another topic that will be relevant for our discussion.
Holographically, a maximally extended (two-sided) black brane geometry is dual to a thermofield double state (TFD) of two identical copies of the theory, which we call QFT L and QFT R , respectively [51]. At t = 0, the TFD state is given by This state displays a very atypical left-right pattern of entanglement at t = 0 and the chaotic nature of the boundary theories is manifested by the fact that small perturbations JHEP11(2018)072 added to the system in the asymptotic past destroy these delicate correlations [17]. This phenomenon is known as the butterfly effect. An efficient way to diagnose this pattern of entanglement and how it is destroyed by small perturbations is to consider the mutual information I(A, B) between subsystems A ⊂ QFT L and B ⊂ QFT R , defined as where S A is the entanglement entropy of the subsystem A, and so on. Importantly, this quantity is always positive and provides an upper bound for correlations between operators O L and O R defined on A and B, respectively [52] (1.9) Let us consider a small perturbation by acting with an operator W at some time t 0 in the past. From the point of view of the gravitational theory the state W |TFD is represented by an excitation near the boundary of the space time, which then falls into the black hole. This excitation gets blue shifted as it fall into the black hole and generates a shock wave geometry, in which the wormhole becomes larger. If this perturbation is early enough, the operator will scramble the Hilbert space and the state W |TFD will have a zero mutual information between A and B at t = 0, signaling the destruction of the left-right correlations. In this setup, then, the disruption of mutual information sets a bound on the two-sided correlators of the form which are related by analytic continuation to the one-sided out-of-time-order correlators that appear in the chaos commutator (1.1) [30]. Therefore, the disruption of mutual information effectively provides a concrete realization of the butterfly effect in holographic theories [17]. This setup has been studied and extended in various directions in [53][54][55][56][57][58][59][60][61]. The disruption of the two-sided mutual information in the TFD state takes place at time scales of the order of the scrambling time t * ∼ β log N 2 , and is controlled by the so-called entanglement velocity v E . Upon inspection one finds that the non-trivial part of the computation comes from the last term in (1.8) which, for large enough subsystems and times in the range t d < < t < < t * , is found to vary linearly with the shock wave time, where s th is the thermal entropy and A Σ is the area of Σ = ∂(A ∪ B). This behavior can be explained in terms of the so-called 'entanglement tsunami' that appears in the study of entanglement entropy following a quantum quench, both in field theory [62] and holographic calculations [63][64][65][66][67]. In [66,67], the authors conjectured the entanglement velocity should be bounded by

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where v Sch E is the entanglement velocity for a (d+1)-dimensional Schwarzschild black brane. Later in [15], this bound was proven to be valid for quite generic holographic theories in Einstein gravity satisfying the NEC. However, once again, the bound was shown to be violated once the assumption of isotropy is relaxed [33]. In this case, though, v E is still bounded and never exceeds the speed of light.
More generally, [14] conjectured that in any quantum system v E ≤ v B . So, if the bound (1.6) holds true, then, the entanglement velocity must also be bounded v E ≤ 1 . (1.13) The authors of [68] proved this using the positivity of mutual information, while [69] used inequalities of relative entropy. 3 However, both [68,69] assumed that the theory is Lorentz invariant. If the theory is Lorentz invariant the entanglement entropy depends not on the particular Cauchy slice, but on the causal development of the subregion. This means that one can split the Hilbert space in various ways (basically we can pick any space-like slice) and the entanglement entropy of boosted regions is trivially related. Various consequences follows from it, such as the entropic proof of the c-theorem [72,73], monotonicity properties of various entanglement related quantities, e.g. [74][75][76], and so on. Since we are studying a system that does not have Lorentz invariance, these proofs do not apply and, in particular, we do not expect the speed of light to play a role. See for instance [77] for a discussion of entanglement entropy on generic time slices for theories that are not Lorentz invariant.

Plan of the paper
The paper is organized as follows. In section 2 we give a brief overview of the background material needed to set up the problem. We introduce and explain the basic properties of the N = 4 non-commutative super Yang Mills theory and its gravity dual, and then we discuss some subtleties in the definition of the gauge invariant observables of interest, namely, correlation functions and entanglement entropies. In section 3 we explain how to construct shock wave solutions with definite momentum for a very general two-sided black hole geometry, including geometries which are not asymptotically AdS. Then, we show how to extract from the shock wave profiles several chaotic quantities of interest: the maximal Lyapunov exponent, the scrambling time and the butterfly velocity. We specialize our formulas to the gravity dual of N = 4 non-commutative super Yang Mills. In section 4 we compute the two-sided mutual information for strip-like regions both in the unperturbed geometry and in the prepense of homogeneous shock waves. We also discuss the role of the spread of entanglement in the disruption of the two-sided mutual information in the shock wave geometries. In section 5 we discuss an alternative derivation of the butterfly velocity, based on entanglement wedge subregion duality, and we show that the final result agrees with the shock wave calculations. Finally, we close in section 6 with a discussion of our results and open questions. We relegate some technical details to the appendices.

Gravity dual of non-commutative SYM
Non-commutative quantum field theory has been an important theoretical arena and a topic of great research interest in the past few decades. The basic postulate of non-commutativity is that space-time coordinates do not commute. Instead, they satisfy the following commutation relation where θ µν is a real and antisymmetric rank-2 tensor. The algebra of functions in a noncommutative theory can be viewed as an algebra of ordinary functions with the product deformed to the so-called Moyal product, Non-commutative theories arise naturally in string theory, as the worldvolume theory of D-branes with non-zero NS-NS B-field, provided that one takes a special limit to decouple the open and closed string sectors [78][79][80][81]. In the context of gauge/gravity duality, this implies that the dynamics of certain strongly-coupled, large N , non-commutative field theories can be described in terms of a classical gravity dual. The first example of this kind of dualities was presented in [43,44], which studied a specific decoupling limit of a stack of D3-branes with non-zero B 23 . The decoupling limit consists of scaling the string tension to infinity, and the closed string metric to zero, while keeping the B-field fixed. This limit provided a gravity dual for finite temperature SU(N ) non-commutative super Yang Mills theory at large N and large 't Hooft coupling λ, with non-commutative parameter non-zero only in the (x 2 , x 3 )-plane, i.e., [x 2 , x 3 ] ∼ iθ. The gravity dual of this theory is type IIB supergravity, with where R 4 = 4πĝN α 2 ,ĝ denotes the string coupling and α is the string tension. The 't Hooft coupling is related to the curvature of the background and the string tension through the standard relation √ λ = R 2 /α . Notice that for future convenience, we have given the metric above in the Einstein frame. 4 Moreover,

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is the standard blackening factor, with r H = πT , while is a function that encodes the effects of the non-commutativity. 5 The parameter a is related to the non-commutative parameter θ through a = λ 1/4 √ θ. This parameter can be thought of as a "renormalized" non-commutative length scale at strong coupling, since this is the parameter that will enter in every holographic computation.
As usual in holography, the radial direction r is mapped into an energy scale in the field theory, in such a way that r → ∞ and r → r H correspond to the UV and IR limits, respectively. The directions x µ ≡ (t, x) are parallel to the boundary and are directly identified with the field theory directions. Finally, the five-sphere coordinates are associated with the global SU(4) internal symmetry group, but they will play no role in our discussion.
For r − r H < < a −1 , the background (2.3) goes over to the AdS 5 -Schwarzschild × S 5 solution, which is dual to a thermal state of the standard SU(N ) super Yang Mills theory. Indeed, it can be shown that all the thermodynamic quantities derived from (2.3) are the same as the ones obtained from the AdS 5 -Schwarzschild solution [43,44]. This observation just reflects the fact that the non-commutative boundary theory goes over to ordinary super Yang Mills at length scales much greater than λ 1/4 √ θ. On the other hand, for r > > a −1 the background (2.3) exhibits significant differences with respect to AdS 5 × S 5 and, in particular, is no longer asymptotically AdS. From the boundary perspective, this just means that the effect of the non-commutativity becomes pronounced for length scales of order or smaller than λ 1/4 √ θ.

Gauge invariant operators in non-commutative theories
In non-commutative gauge theories, the non-commutativity of the spacetime mixes with the gauge transformations and, therefore, there are no gauge invariant operators in position space. However, one can construct gauge invariant operators in momentum space,Õ(k µ ), by smearing the gauge covariant operators O(x µ ) transforming in the adjoint representation of the gauge group over an open Wilson line W (x, C) according to [82][83][84] where denotes the Moyal product. A few comments are in order: • For k √ θ < < 1, the length of the Wilson line W goes to zero and (2.6) reduces to the standard operators in commutative field theory, (2.7) • For k √ θ > > 1, the length of the Wilson line W becomes large as dictated by the noncommutativity. In this limit, the operator is dominated by the Wilson line regardless

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of what operator is attached at the end. Therefore, the correlation functions of these operators are expected to exhibit a universal behavior at large k √ θ. One concrete example of this fact is the universal dissipation time at large momentum found in the quasinormal mode analysis of [47].
• Finally, the fact thatÕ(k) contains a Wilson line whose length W θk depends on k implies that one should not think of these operators as the "same" operator O(x) with different momentum as we usually do in standard field theory. In particular, one should not expect to obtain a local operator by Fourier transforming to position space. Instead, one should think ofÕ(k) as genuinely different operators at different k.
In the holographic context, there is a one-to-one map between gauge invariant operators O(x) in the boundary theory and local bulk fields ϕ(r, x). According to the standard dictionary, the non-normalizable mode of ϕ(r → ∞, x) in a near-boundary expansion corresponds to the source of the dual operator O(x) while the normalizable mode gives its expectation value. The above map is subtle when the boundary gauge theory is noncommutative because, as explained above, there are no gauge invariant local operators in position space. This issue is solved by working in momentum space. More specifically, one can assume that the bulk field ϕ(r, k) is dual to a gauge invariant operatorÕ(k) of the form (2.6) in the sense that in the boundary theory there is a coupling of the form As usual, the source ϕ 0 (k) is determined from the non-normalizable mode of ϕ(r, k) given some appropriate boundary condition in the IR.

Out-of-time-ordered correlators in momentum space
In order to diagnose chaos we need to compute the norm of the commutator C(t, x), defined in (1.4), for two Hermitian gauge-invariant operators W (t, x) and V (0, 0). As expected, such definition is problematic for non-commutative gauge theories, because in these theories there are no gauge invariant operators in position space. Instead, we will quantify chaos in non-commutative theories by defining an equivalent quantity in momentum space, i.e., There is no need to go to frequency space, since the non-commutativity only acts on the spatial coordinates. In the next section we show that C(t, k) has a pole precisely at | k| = iλ L /v B , from which we can extract the Lyapunov exponent λ L and the butterfly velocity v B . The fact that the pole of C(t, k) gives the Lyapunov exponent and the butterfly velocity is implicit in other holographic calculations. See for instance the appendix C of [85].

Entanglement entropy in non-commutative theories
In standard quantum field theory the entanglement entropy associated to a subsystem A can be calculated by the von Neumann formula, reduced density matrix associated to A. In holographic theories S A can be computed in the bulk by the HRRT prescription [86,87] where γ A is an extremal area surface whose boundary coincides with the boundary of the region A, i.e., ∂γ A = ∂A. Entanglement entropy in local theories follows the so-called area law, which means that the leading UV divergence of S A has a coefficient which is proportional to the area of the boundary of the region Σ = ∂A, The area law basically means that the entanglement between A and its complementĀ is dominated by contributions coming from short-ranged interactions between points close to the boundary between A andĀ. In non-commutative theories it is not always possible to precisely define the curve (or surface) delimiting the region A. One possible way to define a subsystem in these theories was proposed in [88]. The prescription is the following: first, one defines a region A for the commutative case as where the surface Φ(x 1 , x 2 , x 3 ) = 0 defines the boundary of the region A. Then, one promotes Φ to an operator, Let |Φ denote the eingenvector ofΦ, with eigenvalue Φ, i.e., Φ|Φ = Φ|Φ . (2.14) The subsystem A can then be uniquely defined as For holographic theories S A can still be computed by using the standard HRRT prescription (2.10), where the boundary of A is given be the classical entangling surface at a particular cut-off scale [88,89]. With this holographic definition, it has been shown that for small enough regions the entanglement entropy follows instead a volume law, while for large regions the standard area law (2.11) is recovered [88,89]. This transition from a volume law to an area law behaviour has also been observed in quantum field theory calculations [90][91][92] and has been understood as a result of the non-locality inherent of noncommutative theories [93,94]. Very recently, the full cutoff dependence was studied in [95] JHEP11(2018)072 which found an exact match with respect to the results previously obtained in the strong coupling regime through holography. Finally, we point out that the time dependence of entanglement entropy for a free scalar field on a non-commutative sphere following a quantum quench was studied in [96]. In this paper it was found that the entanglement velocity v E is generically larger that the commutative counterpart, even exceeding the speed of light in the limit of very strong noncommutativity. As explained in the introduction, this is not an issue for non-commutative theories since Lorentz invariance is explicitly broken and the standard notions of causality do not apply. However, this raises a number of questions. Does this behavior hold in the strong coupling regime? And more importantly, does the conjecture that v E ≤ v B [14] holds for general non-local theories? If so, what are the implications for the transfer of quantum information?
3 Perturbations of the TFD state

Eternal black brane geometry
Let us consider a two-sided black brane geometry of the form where i, j = 1, 2, 3. Here (t, x i ) are the coordinates of the boundary theory, while r is the holographic radial coordinate. We take the boundary to be located at r = ∞ and the horizon at r = r H . We assume the following near-horizon expressions for the metric functions The inverse Hawking temperature associated to the above metric is In the study of shock waves is convenient to work in Kruskal coordinates, since these coordinates cover smoothly the two sides of the geometry. We first define the Tortoise coordinate and then we introduce the Kruskal coordinates U, V as follows, In terms of these coordinates the metric reads where

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The region U > 0 and V < 0 (U < 0 and V > 0) covers the left (right) exterior region, while the region U > 0 and V > 0 (U < 0 and V < 0) covers the black hole (white hole) interior region. The horizon is located at U V = 0. The boundary (left or right) is located at U V = −1 and the singularity at U V = 1. We assume that the unperturbed metric is a solution of Einstein's equations 6 where the stress-energy tensor is assumed to be of the form is the most general stress-energy tensor which is consistent with the Ricci tensor of the unperturbed geometry.

Shock wave geometries
In this section we study how the metric (3.1) changes when we add to the system a null pulse of energy located at U = 0 and moving in the V -direction. The motivation to consider such a perturbation is the following. In the context of gauge/gravity duality the two-sided black brane geometry is dual to a thermofield double state of two copies of the boundary theory. This thermofield double state has a very particular pattern of entanglement between the two boundaries theories at t = 0. We want to know how this pattern of entanglement changes when we perturb one of the boundary theories far in the past. We can do that by inserting an operator in one of the boundary theories at some time t 0 in the past and studying the evolution of the system. In the gravitational description, this corresponds to the creation of a perturbation close to the boundary, which then falls into the black brane. From the point of view of the t = 0 frame, the energy of this perturbation increases exponentially with t 0 , while its distance from the past horizon decreases exponentially with t 0 . As a results, an early enough perturbation will follow an almost null trajectory very close to the past horizon, which can then replaced by a null pulse of energy, located at U = 0 and moving in the V -direction. This pulse of energy will give rise to a shock wave geometry.

Shocks with definite momentum
In non-commutative theories is not possible to define local gauge invariant operators in position space. However, as explained in section 2.2, one can define gauge invariant operators that are local in momentum space. This is done by smearing a gauge covariant operator O(x) over a Wilson line. Despite being non-local, this type of perturbation can also give rise to a shock wave geometry, specifically, a shock wave geometry with definite momentum k. The only requirement is that the perturbation is local in time and is applied in the asymptotic past. This is perfectly possible in non-commutative SYM theory, since the non-commutativity affects only two spatial coordinates, i.e. [x 2 , x 3 ] = iθ.

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Based on this observation, we will consider the following form for the stress-energy tensor of the shock wave, This corresponds to a pulse of energy of definite momentum k and amplitude E. The pulse world line divides the bulk into two regions, the causal future of the pulse (region U > 0), and its causal past (region U < 0), but only the metric in the causal future of the pulse gets modified by its presence. The metric in the causal past, on the other hand, is the same as the unperturbed metric. It turns out that the backreaction of this pulse of energy is very simple. It can be described by a shift V → V +α in the V -coordinate [97,98], where α can be determined from Einstein's equations, as we explain below. Given the form of the stress-energy tensor (3.10), α should take the following form, (3.11) We will now use Einstein's equations to determineα(t, k).
We start by replacing V by V + Θ(U )α in the unperturbed metric (3.1). Note that the Heaviside step function Θ(U ) guarantees that only the causal future of the pulse (U > 0) is affected by its presence. The shock wave geometry can then be written as while the stress-energy tensor reads For simplicity, we define the new coordinateŝ 14) in which terms the metric and the stress-energy tensor take the form and respectively. The hats in these expressions indicate that the corresponding quantities are evaluated at (Û ,V ,x i ). Finally, we determine α by requiring (3.15) to satisfy the Einstein's equations

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with T shock and T matter given by (3.10) and (3.16), respectively. In order to simplify the notation, in the following we will drop the hat over the symbols, but keeping in mind that we are really dealing with the coordinates defined in (3.14). The analysis of the equations of motion simplifies when we rescale α and T shock as α → α and T shock → T shock . With this rescaling we can recover the equations of motion (3.8) for the unperturbed metric by setting = 0 in (3.17). Furthermore, by using (3.8) and analyzing the terms proportional to in (3.17) we find thatα needs to satisfy the equation 7 Going back to the original coordinates t and r of metric (3.1), the equation forα reads where and Assuming G ij to be diagonal, the shock wave profile is then given bỹ (3.22)

Lyapunov exponent and scrambling time
We can extract the chaotic properties of the boundary theory by identifyingα(t, k) with C(t, k) = − [W (t, k), V (0, 0)] 2 . By setting k = 0 we obtaiñ where the constant of proportionality is of order O(1). This case corresponds to a homogeneous shock wave geometry. From this profile we can readily extract the Lyapunov exponent 24) and the scrambling time t * , given in (3.21). Using the expression for the Bekenstein-Hawking entropy S BH = A(r H )/4G N , we can write the leading order contribution to the scrambling time as Note that, since β and S BH are not affected by the non-commutative parameter, both λ L and t * are precisely the same as for the commutative version of the SYM theory. Later in section 4.2, we will use this type of shock waves to study the disruption of two-sided mutual information. From this study we will extract another quantity of interest, the so-called entanglement velocity v E .

Butterfly velocity
At finite k, we expect the size of the Wilson line coupled to the operator W (t, k) to be small for √ θk < < 1 and large for √ θk > > 1. Hence, in the limit of low momentum one can expect to recover an approximate exponential behaviour as in (1.4), for x > > √ θ and t > > t d . More generally, we can extract v B from the leading pole of C(t, k). More specifically, it can be shown thatα(t, k) has a pole precisely at 8 . (3.26) We refer the reader to appendix A for details. From (3.22), then, this leads to In this formula φ is the angle between the x 1 −direction and x. The dependence of v B on φ is due to the anisotropy of the system (G 11 = G 22 ), however, it is clear that in the limit of vanishing non-commutativity (G 11 = G 22 ) the φ-dependence disappears.
For later convenience, we write the explicit formulas for the butterfly velocity along the and for the butterfly velocity along Specializing these formulas to the gravity dual of non-commutative SYM, whose metric is given by and v 2 (3.31) In figure 1 we plot v 2 B,x 1 and v 2 B,x 2 as a function of the non-commutative parameter ar H . Both curves approach the conformal value v 2 B = 2/3 in the IR, and grow monotonically as ar H is increased (although v 2 B,x 1 saturates to a constant value). It is interesting to note that both v 2 B,x 1 and v 2 B,x 2 exceed the speed of light in the regime of strong non-commutativity. As explained in the introduction, this is not an issue for non-commutative theories since Lorentz invariance is explicitly broken and the standard notions of causality do not apply. Nevertheless, this result is remarkable in the context of quantum information theory, since it represents a novel violation of the known bounds on the rate of transfer of information. We will comment more on this result in the conclusions. Finally, it is instructive to discuss the physical meaning of v B in non-commutative theories. In the commutative case, the butterfly velocity describes the spatial growth of an operator around a point x where the operator is inserted. In the case of non-commutative theories, the operator is smeared over a Wilson line. Here v B presumably describes the growth of this operator around the curve C. We will confirm this intuition in section 5.1.1.

Entanglement velocity from two-sided perturbations
In this section we compute the two-sided mutual information both in the unperturbed geometry and in the presence of a shock wave. As explained in the introduction, the disruption of the mutual information in the second case characterizes the butterfly effect in holographic theories.
In order to compute the two-sided mutual information we consider a strip-like region A on the left boundary of the geometry and an identical region B on the right boundary. The mutual information is defined as where S A is the entanglement entropy of region A, and so on. The above entanglement entropies can be computed holographically using the HRRT prescription (2.10). The first two terms, S A and S B , are given by the area of the U-shaped extremal surfaces γ A,B whose boundary coincide with the boundary of A and B, respectively. These surfaces lie outside the event horizon, in the left and right regions, respectively. The last term, S A∪B , is given by the area of the extremal surface whose boundary coincides with the boundary of A ∪ B. There are two candidates for this extremal surface. The first one is the surface γ A ∪ γ B , while the second one is a surface γ wormhole that stretches through the wormhole JHEP11(2018)072  Figure 2. Schematic representation of the t = 0 slice of (a) the unperturbed two-sided black brane geometry and (b) the two-sided black brane geometry in the presence of a shock wave. We assume that the shock wave is sent at some time t 0 < 0, therefore, it effectively increases the size of the wormhole at t = 0. In both cases the blue curves represent the U-shaped extremal surfaces γ A (in the left side of the geometry) and γ B (in the right side of the geometry). The red curves represent extremal surfaces γ 1 and γ 2 connecting the two sides of the geometry. The extremal surface γ wormhole defined in the text is given by the union of these two surfaces, γ wormhole = γ 1 ∪ γ 2 .
connecting the two boundaries of the geometry. See figure 2 for a schematic illustration. If the surface γ A ∪ γ B has less area than the surface γ wormhole , then we have I(A, B) = 0, because Area(γ A ∪ γ B ) = Area(γ A ) + Area(γ B ). On the other hand, if γ wormhole has less area than γ A ∪ γ B , then we have that Area(γ A ∪ γ B ) < Area(γ A ) + Area(γ B ), which implies a positive mutual information I(A, B) > 0.
Before proceeding further, let us explain the general expectations. In the unperturbed geometry, the mutual information must be zero if the regions A and B are small enough, and become positive for large regions. The presence of the shock wave should decrease the amount of mutual information at a given time slice t > t 0 . Eventually, the mutual information must drop to zero as we move the shock wave farther into the past t 0 → −∞. As explained in the introduction, the positive mutual information characterizes the special left-right pattern of entanglement of the TFD state, and the fact that I(A, B) decreases (and eventually vanishes) in a shock wave geometry shows that this pattern of entanglement is sensitive to arbitrarily small perturbations sent in the asymptotic past.

Mutual information in the TFD state
We will discuss two cases, the "commutative strip" and the "non-commutative strip", as defined in reference [88].
Commutative strip: the region dubbed as the "commutative strip" are the set of points with 0 ≤ x 1 ≤ and −L/2 ≤ x 2,3 ≤ L/2, with L → ∞. In this case the appropriate embedding is X m = (0, x(r), x 2 , x 3 , r, θ i ), where θ i are the angles of the five-sphere. The

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components of the induced metric are the following: The area functional to be extremized is given by = Ω 5 L 2 R 8 drr 3 1 = Ω 5 L 2 R 8 dr L(x, x ; r) , (4.8) where L 2 = dx 2 dx 3 , and Ω 5 is the volume of a unit S 5 . The above functional does not depend on x, and so there is a conserved quantity associated to translations in x where in the last equality we computed p at the point r = r m at which x → ∞. By solving the equation (4.9) for x we get (4.10) Using equation (4.10) we can write the on-shell area of the surface as so, the entanglement entropies of the subregions A and B are (4.12) The entanglement entropy S A∪B is computed from the area of the extremal surface γ wormhole connecting the two sides of the geometry where the factor of 4 comes from the fact that we have two sides in the geometry and two disconnected surfaces, at x 1 = 0 and x 1 = , respectively. The entanglement entropy of A ∪ B is then given by (4.14) From the above expressions we can compute the mutual information, which is a function of the turning point r m . We can plot the mutual information as a function of the strip's width by writing the later quantity as a function of r m , and then making a parametric plot of I(A, B) versus . Note, however, that both quantities I(A, B) and are independent of the non-commutative parameter "a". This means that the results for the commutative strip are the same as for a strip in a 5-dimensional AdS-Schwarzschild geometry. The plot of I(A, B) for the commutative strip is shown in figure 3, and corresponds to the curve labeled by a = 0 (black curve).
Non-commutative strip: the "non-commutative strip" is given by the set of points with 0 ≤ x 2 ≤ and −L/2 ≤ x 1,3 ≤ L/2, with L → ∞. In this case the appropriate embedding is X m = (0, x 1 , x(r), x 3 , r, θ i ) and the components of the induced metric are g 11 = G 11 , (4.17) The area functional to be extremized is given by = Ω 5 L 2 R 8 dr L(x, x ; r) , (4.24) where L 2 = dx 1 dx 3 , and Ω 5 is the volume of a unit S 5 . The above functional does not depend on x, and so there is a conserved quantity associated to translations in x where in the last equality we computed p at the point r = r m at which x → ∞. By solving the equation (4.25) for x we get (4.26)

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Using equation (4.26) we can write the on-shell area of the surface as so, the entanglement entropies of the subregions A and B are (4.28) The entanglement entropy S A∪B is computed from the area of the extremal surface γ wormhole connecting the two sides of the geometry where the factor of 4 comes from the fact that we have two sides in the geometry and two disconnected surfaces, at x 2 = 0 and x 2 = , respectively. The entanglement entropy of A ∪ B is then given by From the above expressions we can compute the mutual information, which is a function of the turning point r m . We can plot the mutual information as a function of the strip's width by writing the later quantity as a function of r m , and then making a parametric plot of I(A, B) versus . Both quantities I(A, B) and depend on the non-commutative parameter "a", because they have factors of h(r) = (1 + a 4 r 4 ) −1 . Also, note that we can recover the expressions for the commutative strip by setting a = 0 (or equivalently h = 1). In figure 3 we plot I(A, B) as a function of the strip's width for several values of the non-commutative parameter at a fixed temperature. As expected from the results of mutual information for the one-sided black brane [88], increasing in the non-commutative parameter a reduces the critical length = crit and hence lowers the threshold for the phase transition of mutual information. This implies that non-commutativity introduces more correlations between two sub-systems as compared to the commutative case.

Disruption of mutual information by shock waves
Let us now study how the two-sided mutual information changes in the presence of a shock wave geometry. In the following, we will specialize to the case of a homogeneous shock wave, in which the shock wave parameter has the form α = constant × e 2πt 0 /β . The mutual information in a shock wave geometry will be denoted as where we have indicated that S A∪B generically depends on the shock wave parameter α, while S A and S B do not. This can be easily understood, since the corresponding extremal surfaces γ A and γ B remain outside the horizon, while the shock wave only affects quantities that probe the black hole interior. As expected on general grounds, the entanglement entropy S A∪B (α) has various αindependent divergences. In practice we find it convenient to define a regularized entanglement entropy S reg  In the following, we will consider cases where > crit so that the mutual information is positive in the unperturbed geometry, i.e., I(A, B; α = 0) > 0. That means that the extremal surface stretching between the two sides of the geometry γ wormhole has smaller JHEP11(2018)072 area than the two extremal surfaces lying outside the black brane γ A and γ B . When α > 0, the wormhole becomes longer and the area of the extremal surface probing the interior also increases, resulting in a decrease of the mutual information. As α increases the mutual information eventually drops to zero, signaling the total disruption of twosided correlations.
Commutative strip: the appropriate embedding in this case is X m = (t, 0, x 2 , x 3 , r(t), θ i ). The components of the induced metric are 38) and the functional to be extremized is Since the above functional is invariant under t-translations, there is an associated conserved quantity, where in the last equality we computed E at the point r 0 at whichṙ = 0. By solving (4.42) forṙ we obtainṙ 2 = r 2 f 2 1 + E −2 f r 6 . (4.43) Using the above result we can write the on-shell area as (4.44) and the time coordinate t along the extremal surface as Since these expressions do not depend on the non-comutative parameter a, we can expect the disruption of mutual information to be the same as for the commutative SYM theory. However, we will proceed with the analysis for illustrative purposes. The entanglement entropy S A∪B is given by It is convenient to divide the region of integration of the above integral into three regions, I, II and III, as shown in figure 4. Since the regions II and III have the same area, we can write I∪II∪III = ∞ r H +2 r H r 0 . The entanglement entropy S A∪B can then be written more explicitly as where the extra factor of 2 accounts for the two sides of the geometry. The effect of the shock wave on S A∪B is controlled by the turning point r 0 ≤ r H . The shock wave is absent when r 0 = r H (or, equivalently, E = 0), and its effects become stronger as one decreases r 0 . In terms of this parameter, the regularized entanglement entropy S reg A∪B can be written as Finally, the shock wave parameter α can also be written as a function of the turning point r 0 (see appendix B for details). The final result reads where (4.52)

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As expected, the shock wave parameter α = α(r 0 ) increases monotonically as one decreases r 0 (with α(r H ) = 0) and it diverges at some critical radius r c = r H 3 1/4 (see appendix C for details). Indeed, both K 3 and t(r 0 ) ∝ t 0 diverge at r 0 = r c . This means that the region r < r c and, in particular, the singularity cannot be probed by I(A, B; α), even in the limit t 0 → ∞. The results for the commutative strip can be found by setting a = 0 in the results for the non-commutative strip (which will be presented below). The plots for α(r 0 ), as well as for S reg A∪B (α) and I(A, B, α) for the commutative strip are shown in figures 5 and 6, respectively, and correspond to the curves labeled by a = 0 (black curves).
Non-commutative strip: the appropriate embedding in this case is X m = (t, x 1 , 0, x 3 , r(t), θ i ). The components of the induced metric are (4.53) 56) and the functional to be extremized is Since the above functional is invariant under t-translations, there is an associated conserved quantity, where in the last equality we computed E at the point r 0 at whichṙ = 0. By solving (4.60) forṙ we obtainṙ 2 = r 2 f 2 1 + E −2 r 6 f h . (4.61) Using the above result we can write the on-shell area as and the time coordinate t along the extremal surface as The entanglement entropy S A∪B is then given by Again, we divide the region of integration of the above integral into three regions, I, II and III, as shown in figure 4. Since the regions II and III have the same area, we can write I∪II∪III = ∞ r H +2 r H r 0 . The entanglement entropy S A∪B can then be written more explicitly as (4.65) where the extra factor of 2 accounts for the two sides of the geometry. The effect of the shock wave on S A∪B is controlled by the turning point r 0 ≤ r H . The shock wave is absent when r 0 = r H (or, equivalently, E = 0), and its effects become stronger as one decreases r 0 . In terms of this parameter, the regularized entanglement entropy S reg A∪B can be written as  in a shock wave geometry and how this results in the disruption of the two-sided mutual information I(A, B; α). As we can see from these plots, the disruption of the mutual information occurs faster as we increase the non-commutative parameter a. In the next section we will quantify this statement more clearly by the calculation of the so-called entanglement velocity.

Entanglement velocity
Before saturation, the entanglement entropy S A∪B (α) grows linearly with log α, and this implies that it grows linearly with the time t 0 at which the system was perturbed (α = JHEP11(2018)072 const×e 2π β t 0 ). From this linear behaviour we can define the so-called entanglement velocity, which is a quantity that characterizes the spread of entanglement in chaotic system. In the following, we will specialize to the case of the non-commutative strip. The results for the commutative strip can be obtained simply by setting a = 0 in all the formulas below.
As shown in the previous section, the function α(r 0 ) increases monotonically as we decrease r 0 ≤ r H and diverges at a critical radius r 0 = r c given by (4.71). In the vicinity of r c , one can show that Since the shift α grows exponentially with time, α = constant × e 2πt 0 /β , the above result implies that S A∪B grows linearly with t 0 . The rate of change of S reg A∪B with the shock wave time is dS reg where G  of light already at some value of ar H of order ar H ∼ O(1) and grows without bound in the limit of strong non-commutativity. This behavior is in qualitative agreement with the results obtained for the entanglement entropy for a free scalar field on the fuzzy sphere following a quantum quench [96]. Finally, we note that v E,x i ≤ v B,x i generically, for any value of the non-commutative parameter. This implies that the conjecture made in [14] holds for our non-commutative setup, and suggests that it might indeed be true for any (possibly non-local) quantum system.

Butterfly velocity from one-sided perturbations
In this section we present an alternative derivation of v B that does not rely on the shock wave results. This alternative way of computing v B is based on entanglement wedge subregion duality [99], and it was first proposed in [14]. Here we extend their results for the kind of anisotropic metrics that we consider in this paper.

Infalling particle and entanglement wedge
The derivation goes as follows. Consider the application of a localized bulk operator V in a black brane geometry. This operator creates a one particle state in the bulk theory which eventually falls into the black hole and thermalizes. As the particle falls into the black hole,

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V gets scrambled with an increasing number of degrees of freedom and, as a result, the operator effectively grows in space. This is consistent with the standard intuition from the holographic UV/IR connection [100][101][102], which implies that the information of the particle gets delocalized over a larger region as it falls deeper into the bulk. The proposal of [14] is that, at late times, the rate of growth of this region is controlled by the butterfly velocity.
In this context, the butterfly velocity can be calculated using the entanglement wedge subregion duality. According to this duality a certain subregion A of the boundary theory can be completely described by a subregion in the bulk geometry, which is called the entanglement wedge of A. In the following, we compute the butterfly velocity by requiring that the entanglement wedge of a certain region A contains the particle created by V .
Let us assume a generic black hole metric of the form where (t, x i ) are the boundary theory coordinates, r is the holographic radial coordinate and i, j = 1, . . . , d − 1. We assume the boundary is located at r = ∞ and the horizon at r = r H . We now consider a fixed time slice of the geometry, at a long time after the application of V , such that this operator is delocalized in a very large region A. This limit simplifies the analysis for two reasons. First, the equations of motion defining the entanglement wedge linearise, because the corresponding RT surface lies very close to the black hole horizon. Second, the particle created by V also lies very close to the horizon, having a simple description in terms of Rindler coordinates. We assume the following near-horizon expressions In terms of c 0 and c 1 , the inverse Hawking temperature reads It is convenient to go to Rindler coordinates, ρ 2 = (r − r H ) 2π β 2 1 G tt (r H ) , in which terms the above metric becomes The infalling particle gets blue shifted as it falls into the black hole and, at late times, it approaches the horizon exponentially Now we proceed to calculate the position of the RT surface defining the entanglement wedge of A. Considering the embedding X m = (0, x i , ρ(x i )), the area functional can be written as

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where we have assumed G ij to be diagonal. The equations of motion that follows from the above functional are where In order to solve this equation we define the new coordinates σ i = x i / G ii (r H ), in which terms the equation of motion becomes The solution of the above equation is [14] ρ(σ i ) = ρ min Γ(a + 1) In this formula, ρ min is interpreted as the radius of closest approach to the horizon and I a is a modified Bessel function of the second kind. As explained in [14], when ρ exceeds β, the surface exits the near horizon region and reaches the boundary very quickly. It is then possible to determine the size of the region A in terms of ρ min by solving the equation where R σ is the size of the region A in the σ-coordinates. The approximate solution for this equation at large R σ is ρ min ≈ e −M Rσ . (5.12)

Butterfly velocity
For an anisotropic system, v B is different along the different directions and so is the size of the region A. Let us say that R i is the size of the region A along the x i direction. This is related to its size in σ-coordinates by the equation R σi = G ii (r H )R i . Requiring the infalling particle created by V to be contained in the entanglement wedge implies 13) or, equivalently where the butterfly velocity v B,x i along the x i -direction is calculated as .

(5.15)
This formula is in complete agreement with the ones obtained from shock wave calculations (3.28)-(3.29).

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Before closing this section we would like to offer some intuition about the boundary picture of the bulk operator V and the corresponding one particle state, in our noncommutative setup. Without loss of generality, we can imagine inserting V directly at the boundary, and follow the evolution of the created particle as it falls into the black brane. In ordinary AdS/CFT this would mean that we are turning on an operator localized in space, which can in turn be interpreted as a local quench, see e.g. [103]. However, in noncommutative gauge theories there are no local operators in position space, so it is necessary to explain how the above prescription works in the present case.
As explained in section 2.2, there is a natural set of gauge invariant operators that can be defined in non-commutative gauge theories, which can be obtained by smearing the ordinary gauge covariant operators O(x) over a Wilson line W , according to (2.6). The size of this Wilson line W scales with the momentum k, roughly as W θk. Let us now imagine having a very large Wilson line along one of the directions, say the x 1 -direction. This can be achieved by taking either θ or k 1 to be very large, such that W → ∞. In this approximation the information about the perturbation is initially localized along the x 1 -axis. As the system evolves in time, the information will get delocalized in a cylindrical region around this axis. In the bulk, the information about this perturbation will be contained inside the entanglement wedge, which will also display a cylindrical symmetry. The derivation of v B goes as before, except that now the entanglement surface will not depend on x 1 . In particular, the butterfly velocity will be the same as before and will describe how fast the information about the smeared operator gets delocalized inside a 'cylinder' whose radius along the x-direction is v B (φ) t, where φ is the angle between x and the x 1 -direction. In more general cases, when the Wilson line is not very large, we expect v B (φ) to describe the expansion of the operator in a region around the Wilson line that defines it.

Conclusions and outlook
In this paper we have studied shock waves in the gravity dual to N = 4 non-commutative SYM theory. From the shock wave profiles, we extracted several chaos-related properties of this system, namely, the butterfly velocity, the scrambling time, and the Lyapunov exponent. As expected on general grounds, we find that the Lyapunov exponent saturates the chaos bound, λ L = 2π/β, while the scrambling time scales logarithmically with the entropy of the system, t * = β 2π log S. Since neither the temperature nor the entropy are affected by the non-commutativity, both λ L and t * are exactly the same as the corresponding values in ordinary SYM theory.
In contrast, the butterfly velocity is largely affected by the non-commutative parameter θ, specially in the UV. The results for v B as a function of a r H = πλ 1/4 √ θT are shown in figure 1. Since the non-commutativity is introduced along the x 2 − x 3 plane, i.e. [x 2 , x 3 ] ∼ iθ, the gravity dual is hence anisotropic, G 11 = G 22 = G 33 . This causes v B to depend on the direction of the perturbation. For simplicity, we only computed the components v B,x 1 and v B,x 2 , where the first one is the butterfly velocity along the x 1 -direction, and the latter one corresponds to the butterfly velocity along the x 2 -and x 3 -directions. We observe that,

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for a r H ∼ 1, both v B,x 1 and v B,x 2 become larger than the speed of light. For large values of the non-commutative parameter, v B,x 1 saturates a constant value, while v B,x 2 increases indefinitely.
The fact that the butterfly velocity exceeds the speed of light in the regime of strong non-locality is not surprising. Indeed, Lorentz invariance is explicitly broken by the noncommutative parameter θ so the standard notions of causality do not apply. Nevertheless, this result is remarkable in the context of quantum information theory, since it represents a novel violation of the known bounds on the rate of transfer of information. We comment, though, that in this limit the information is highly delocalized due to the UV/IR mixing, 9 so the implementation of a local protocol to retrieve the information might require an exponentially longer time than the commutative case. As a result, an increase on v B due to the non-commutativity is necessarily compensated by an increase in the "computational cost" or a decrease on the amount of "useful information" at fixed time. It would be interesting to understand this phenomenon better.
Finally, we also computed the entanglement velocity v E by studying the disruption of the two-sided mutual information in the presence of homogeneous shock waves. In figure 6 we show the results for S A∪B and I(A, B) for various values of the non-commutative parameter. In general, we find that the mutual information is reduced in the presence of the shock wave, and eventually vanishes as one let t 0 → ∞. Right before the transition, the entanglement entropy of the two sub-systems S A∪B grows linearly, with a slope given by v E . In figure 7 we show the behavior of the entanglement velocity as a function of the non-commutative parameter. We considered two geometries. For a "commutative strip" (strip with finite width along the x 1 -direction), the results are the same as for an AdS black brane, while for a "non-commutative" strip (strip with finite width along the x 2or x 3 -direction) the entanglement velocity increases with the non-commutative parameter. Eventually, v E,x 2 exceeds the speed of light in the limit of strong non-locality. This behavior is in qualitative agreement with the results obtained for the entanglement entropy for a free scalar field on the fuzzy sphere following a quantum quench [96].
We also confirmed the expectation based on the conjecture proposed in [14], namely that v E,x i ≤ v B,x i in general quantum systems. Indeed, we find that this is valid in our setup for any value of the non-commutative parameter suggesting that the conjecture might indeed be true for any (possibly non-local) quantum system. It would be interesting to test further this conjecture in other non-local theories, for example, in the gravity dual of the dipole deformation of N = 4 super Yang Mills [45] or in the gravity dual of the so-called little string theory [46]. It will also interesting to compute v B directly in a noncommutative field theory (using perturbation theory) and compare with the strong coupling results obtained in this paper.

A Momentum space correlator and the butterfly velocity
In this appendix we study shock wave geometries and present a detailed derivation of the formula (3.27). We start from the solution of the shock wave profileα at finite momentum, α(t, k) = e 2π(t−t * )/β G ii k 2 i + M 2 . (A.1) The above function has a pole at G ii k 2 i + M 2 = 0. By writing the momentum components in spherical coordinates the position of the pole can be specified as k = iM . Interestingly, at the pole, the modulus of k gives us the ratio of the Lyapunov exponent and the butterfly velocity k 2 = k 2 1 + k 2 2 + k 2 3 = −λ 2 L µ 2 G 11 cos 2 φ + G 22 sin 2 φ sin 2 φ 2 + G 33 sin 2 φ cos 2 φ 2 , where λ L = 2π/β is the Lyapunov exponent and v B (φ, φ 2 ) = 1 µ G 11 cos 2 φ + G 22 sin 2 φ sin 2 φ 2 + G 33 sin 2 φ cos 2 φ 2 , (A.6) is the butterfly velocity along an arbitrary direction. Note that, in the most general case, G 11 = G 22 = G 33 , the butterfly velocity is completely anisotropic and depends on the two spherical angles φ and φ 2 . If we assume isotropy in the x 2 − x 3 plane, i.e. G 33 = G 22 , the above formula simplifies to v B (φ) = 1 µ G 11 cos 2 φ + G 22 sin 2 φ , (A.7) which leads to (3.27). Note that v B still depends on φ. This is a consequence of the anisotropy in the x 1 -direction, i.e. G 11 = G 22 . The fact that the pole of C(t, k) gives the Lyapunov exponent and the butterfly velocity is implicit in other holographic calculations (see for instance the appendix C of [85]). In our setup, we can confirm that the quantity appearing in the pole ofα(t, k) can indeed be identified with the butterfly velocity. We do so by looking at the limit k → 0 (or | x| > > The coordinate V 2 can be written as The shift α can then be computed by considering the variation in the V -coordinate along the segment III Finally, after some simplifications we find that the parameter α can be expressed as α(r 0 ) = 2 e K 1 (r 0 )+K 2 (r 0 )+K 3 (r 0 ) , (B.9) where the K i 's are given by equations (4.68)-(4.70).