Chaos in quantum channels

Journal of High Energy Physics, Feb 2016

We study chaos and scrambling in unitary channels by considering their entanglement properties as states. Using out-of-time-order correlation functions to diagnose chaos, we characterize the ability of a channel to process quantum information. We show that the generic decay of such correlators implies that any input subsystem must have near vanishing mutual information with almost all partitions of the output. Additionally, we propose the negativity of the tripartite information of the channel as a general diagnostic of scrambling. This measures the delocalization of information and is closely related to the decay of out-of-time-order correlators. We back up our results with numerics in two non-integrable models and analytic results in a perfect tensor network model of chaotic time evolution. These results show that the butterfly effect in quantum systems implies the information-theoretic definition of scrambling.

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Chaos in quantum channels

HJE Chaos in quantum channels Pavan Hosur 0 1 3 6 7 Xiao-Liang Qi 0 1 3 6 7 Daniel A. Roberts 0 1 3 4 7 Beni Yoshida 0 1 2 3 5 7 0 31 Caroline Street North , Waterloo, Ontario N2L 2Y5 , Canada 1 77 Massachusetts Ave , Cambridge, Massachusetts 02139 , U.S.A 2 Walter Burke Institute for Theoretical Physics, California Institute of Technology 3 476 Lomita Mall , Stanford, California 94305 , U.S.A 4 Center for Theoretical Physics and Department of Physics, Massachusetts Institute of Technology 5 Perimeter Institute for Theoretical Physics 6 Department of Physics, Stanford University 7 1200 E California Blvd , Pasadena CA 91125 , U.S.A We study chaos and scrambling in unitary channels by considering their entanglement properties as states. Using out-of-time-order correlation functions to diagnose chaos, we characterize the ability of a channel to process quantum information. We show that the generic decay of such correlators implies that any input subsystem must have near vanishing mutual information with almost all partitions of the output. Additionally, we propose the negativity of the tripartite information of the channel as a general diagnostic of scrambling. This measures the delocalization of information and is closely related to the decay of out-of-time-order correlators. We back up our results with numerics in two non-integrable models and analytic results in a perfect tensor network model of chaotic time evolution. These results show that the butter y e ect in quantum systems implies the information-theoretic de nition of scrambling. 1 Introduction 2 Unitary quantum channels 3 Butter y e ect implies scrambling Entanglement in time Scrambling Examples Chaotic channels vs. integrable channels Average over OTO correlators Early-time behavior Butter y velocity vs. entanglement velocity 4 5 6 2.1 2.2 2.3 2.4 3.1 3.2 3.3 4.1 4.2 5.1 5.2 Numerics in qubit channels Spin chains Majorana fermion fast scrambler Perfect tensor model Discussion A Haar scrambling B Entanglement propagation in CFT C Proof of eq. (3.3) D Tensor calculus Ballistic growth of operators and linear growth of I3 Recurrence time 1 Introduction Quantum information is processed in quantum circuits, or more generally, quantum channels. A useful way to characterize fault-tolerance and computational power of such channels is by whether input information remains localized or is spread over many degrees of freedom. This delocalization of quantum information by a quantum channel over the entire system is known as scrambling [1{3]. Scrambling implies that information about the input cannot be deduced by any local measurement of the output [4]. { 1 { operator W (t) = eiHtW e iHt which has an expansion as a sum of products of many local operators t 2 2! it3 3! W (t) = W + it [H; W ] For a generic H with local interactions, the kth-order nested commutator of H with W can lead to a product of as many as k local operators that acts non-trivially on a large volume of the system [5]. This implies that [W (t); V ] 6= 0 and will generically be a large operator of high weight. correlators of the form The degree of non-commutativity between W (t) and V can be measured by their group commutator: W (t) V W (t) V . In fact, the generic decay of out-of-time-order (OTO) where OA represents an operator in the input A, OD represents an operator in the output D, and SA(2C) is the second Renyi entropy of the input/output system AC. With this result, we can show that the butter y e ect implies scrambling jhOD(t) OA OD(t) OAi =0j = =) velocity" [5, 6]). This supports the idea that vB is the velocity of information in stronglychaotic systems. We will also comment on the conceptual di erences between the butter y velocity and the entanglement or tsunami velocity vE of [12{14]. Finally, we will use our enhanced understanding of the relationship between scrambling, chaos, and entanglement in time to propose a solvable model of a unitary quantum channel that exhibits scrambling. Building on the work of [12] and [15], we discuss a perfect tensor model of a chaotic Hamiltonian time evolution. This can be thought of as a toy model for an Einstein-Rosen bridge that connects the two sides of the eternal black hole in AdS. The plan of this paper is as follows. In section 2 we discuss unitary quantum channels and elaborate on the notion of entanglement in time. There, we consider the entanglement properties of such channels and introduce the tripartite information as a measure of scrambling. In section 3, we show that the decay of OTO correlation functions implies strong bounds on information-theoretic quantities, directly connecting chaos to scrambling. We provide evidence for our claims via numerical studies of qubit systems in section 4 and with a perfect tensor model of chaotic time evolution in section 5. We conclude in section 6 with a discussion of the relationship between chaos and computation. Some extended calculations, tangential discussions, and lengthy de nitions are left to the appendices. 2 Unitary quantum channels To study the scrambling properties of di erent unitary operators U by using informationtheoretic quantities, we will interpret them as states. To be concrete, let us assume that the quantum system consists of n qubits with a time independent Hamiltonian H. We will be interested in a particular unitary operator, the time evolution operator U(t) = e iHt. This will let us study a one parameter family of unitary operators indexed by t. { 3 { which we usually choose to think of in terms of a tensor with n input and n output legs, as shown in gure 1(a). However, it is also natural to map this to a 2n-qubit state by treating the input and output legs on equal footing (2.2) This is depicted graphically in gure 1(b). Clearly jU(t)i encodes all the coe cients (uij ) necessary to represent the unitary operator U(t). When t = 0, the unitary operator is simply the identity operator, and (2.1) reduces to a state consisting of n EPR pairs jIi = 1 2n=2 2n 1 X i;j=0 jiiin jiiout; where each input leg is maximally entangled with each output leg, and there is no entanglement between di erent EPR pairs. For nite t, inserting a complete set of states into (2.2) and using (2.1), we can rewrite it as 1 2n=2 2n 1 X i;j=0 jU(t)i = jiiin U(t)jiiout = I U(t)jIi; A unitary operator U(t) that acts n qubits is described by a 2n 2n dimensional matrix with I the identity operator acting on the incoming states. This o ers the following interpretation to the state jU(t)i: a maximally entangled state is created between a reference system \in" and a system of interest \out." Next, the operator U acts on \out," giving jU(t)i. This perhaps o ers a more physical interpretation of this operator-state mapping, as shown in gure 1(c). Note. In this paper we will adopt the perspective of the mapping shown in gure 1(b) and use language that refers to the entanglement properties of channels (in time) between subsystems of inputs and outputs. Additionally, we will always draw our channels from the operator perspective as in gure 1(a). It is natural to ask whether the choice of maximally entangled state jIi is arti cial. Although di erent choices of initial state jIi can be made which de ne di erent mappings from U to jU(t)i, all our discussions remain insensitive to the choice as long as jIi is a direct product of EPR pairs. The two qubits in each pair are required to be the qubits at a given real-space position in the input and output systems, which are maximally entangled with each other. This choice guarantees that all quantum entanglement between di erent real-space locations in jU(t)i are created by the unitary evolution U(t). { 4 { time (a) (b) (c) EPR pairs input U legs. (b) state interpretation jU i of the unitary operator U . By bending the input legs down, we treat input/output equally. (c) the state interpretation is equivalent to the creation of a maximally entangled state followed by acting with U on half the EPR pairs, which gives jU i. The operator-state mapping can be further generalized by considering a more generic statistical ensemble as the input state. Let fj j ig be a set of orthonormal states, and imagine that we input an initial state j j i with probability pj to a unitary quantum channel U . This means that the initial statistical ensemble is in = Pj pj j j ih j j. After time evolution, each input state evolves to j j i = U j j i, and the output statistical ensemble is given by out = Pj pj j j ih j j. The time evolution of a given input ensemble in can be mapped to the following pure state j i = X ppj j j iin j j iout = I U(t) X ppj j j iin j j iout : (2.5) j j The isomorphic state j i contains all the information required to characterize the properties of the channel. Namely, if one traces out the output system, then the reduced density matrix is the input state ( in = trout j ih j) while if one traces out the input system, then the reduced density matrix is the output state ( out = trin j ih j). The state interpretation in (2.4) corresponds to the special case of a uniform input ensemble (i.e. in = 2 nI). In general, we will simply refer to the state given in (2.5) as a unitary quantum channel. In quantum information theory, such correspondence between quantum channels and quantum states is named as the channel-state duality.1 A familiar example of a unitary channel is the thermo eld double state where Ei are eigenvalues of the Hamiltonian H and Z = tr e time t to the right output system, one obtains the following time evolved state H . Applying evolution for jTFDi = p 1 Z X e i Ei=2jii jii; 1 Z X e i jTFD(t)i = p Ei=2e iEit i j i jii: (2.6) (2.7) 1In fact, the channel-state duality in quantum information theory extends to any quantum channels with decoherence as well as those with di erent sizes of the input and output Hilbert spaces [16]. { 5 { time input A B U output C D (2.8) (2.9) (2.10) HJEP02(16)4 channels with input and output legs, when we discuss entanglement we always mean of the state jU i given by the mapping to the doubled Hilbert space as in (2.2). One can interpret this state as a quantum channel U = e iHt whose input is given by the thermal ensemble. Note this expression reduces to (2.4) for = 0. 2.1 Entanglement in time Since the state as de ned in (2.5) contains all the information concerning the inputs and dynamics of the channel, we would like to use it to establish a general measure for scrambling. We will do this by studying the entanglement properties of a unitary U via the state j i . Our setup is as follows. The input system is divided into two subsystems A and B, and the output system is divided into two subsystems C and D, as shown in gure 2. The subsystems do not necessarily have to be of the same size (i.e. it is possible that jAj 6= jBj or jAj 6= jCj), and at t = 0 the input and output partitions do not necessarily need to overlap (i.e. it could be that A \ C = ;). Additionally, despite how it is drawn, there does not need to be any spatial organization to the partitions. For example, the subsystem A could be an arbitrary subset of the input qubits. With this state interpretation of the channel (2.5), we can form a density matrix = j ih j to compute joint entropies of subsystems that include both input and output degrees of freedom. For example, the entanglement entropy SAC is given by where the notation AC means the usual partial trace Additionally, the mutual information between A and C is given by SAC = tr f AC log2 AC g; AC = trBD f g: I(A : C) = SA + SC SAC ; { 6 { and we will sometimes compute Renyi entropies Finally, let us note that, for a uniform input ensemble (or = 0), SA(NC) = 1 1 N N log tr f AC g: SA = a; SB = b; SC = c; SD = d; SAB = SCD = n (2.11) (2.12) (2.13) where a; b; c; d are the numbers of qubits on A; B; C; D respectively. These relations are true because the inputs are always maximally entangled with the outputs. Therefore, any subsystem that is only a partition of the inputs or only a partition of the outputs (including non-partitions such as AB and CD) is maximally mixed. Even if we consider the more general channel (2.5), any subsystem that does not involve both input and output systems still has an entropy that is time-independent. Therefore the scrambling e ect only appears in the entropy of regions on both sides, and (thus) the mutual information terms such as I(A : C) and I(A : D). For this reason, we will primarily be interested in the mutual information between region A and di erent partitions of the outputs. When region A is taken to be small, such as a single lattice site, the mutual information of A with part of the output system tracks how the information about local operators in A spreads under time evolution. 2.2 Scrambling Scrambling is usually considered as a property of a state. In [2], a reference state evolved with a random unitary sampled from the Haar ensemble is called \Haar-scrambled." A much weaker notion of scrambling of a state (which [2] calls \Page scrambling," or usually just \scrambling") is given by a state that has the property that any arbitrary subsystem of up to half the state's degrees of freedom are nearly maximally mixed. Said another way, a state is scrambled if information about the state cannot be learned from reasonably local measurements. Naturally (and proven in [4]), Haar scrambling implies Page scrambling. We are interested in extending the notion of scrambling to unitary quantum channels of the form (2.5). Let us try to understand the properties of scrambling channels by considering entanglement across the channel. The identity channel is just a collection of EPR pairs connecting the input to the output. An example of a channel that does not scramble is the \swap" channel, where the arrangement of the EPR pairs are simply swapped amongst the degrees of freedom in the output system. In that case, localized quantum information in the input system remains localized in the output system, though residing in a di erent particular location. Instead, for a channel to scramble it necessarily must convert the EPR pairs into a more complicated arrangement of multipartite entanglement between the input and output systems. Such local indistinguishability of output quantum states corresponding to simple orthogonal input states in quantum channels enables secure storage of quantum information: a realization of quantum error-correcting codes. Let us try to formalize this idea using our setup in gure 2. If our channel is a strong scrambler, then local disturbances to initial states cannot be detected by local { 7 { measurements on output states. This implies that measurements on a local region C cannot reveal much information on local disturbances applied to A. Therefore the mutual information I(A : C) must be small. From a similar reasoning, one also expects I(A : D) to be small when the channel is a good scrambler since D is a local region too. On the other hand, the mutual information I(A : CD) quanti es the total amount of information one can learn about A by measuring the output CD jointly. Since we are interested in the amount of information concerning A which is hidden non-locally over C and D, a natural measure of scrambling would be I(A : CD) the above quantity is equal to minus the tripartite information2 I3(A : C : D) = SA + SC + SD SAC SAD SCD + SACD I(A : C) + I(A : D) I(A : CD) : (2.15) The tripartite information I3(A : C : D) must be negative and have a large magnitude for systems that scramble. We propose that this is a simple diagnostic of scrambling for unitary channels. Scrambling in unitary channels is closely related to other notions of scrambling of states. For example, if the input to the channel is xed to be a direct product state, then tripartite scrambling implies that subsystems of the output state will be near maximally mixed. Thus, scrambling in terms of the tripartite information implies \Page scrambling" of the output state.3 In appendix A, we analyze Haar-random channels and show that Haar scrambling also implies that the tripartite information of the channel is very negative. Importantly, it should be noted that the tripartite information is a measure of fourparty entanglement, not three-party entanglement. Namely, consider a state with threeparty entanglement only, j i = j Ai j BCDi where A and BCD are not entangled. Then one always has I3(A : B : C) = 0. I3 for other choices of regions also vanish, because for pure states the tripartite information is symmetric in any partitions into four regions A; B; C; D I3(A : B : C) = I3(A : B : D) = I3(A : C : D) = I3(B : C : D): (2.16) Thus, in channels I3 is really a measure of four-party entanglement between the input system and the output system. In this paper, we will often choose to write the tripartite information as I3(A : C : D) in order to emphasize a particular decomposition. However, for unitary channels the arguments are unnecessary due to the symmetry (2.16). 2In the condensed matter community, the tripartite information is referred to as the topological entanglement entropy, which measures the total quantum dimension in a (2 + 1)-dimensional TQFT [17, 18]. 3In fact, this operator notion of scrambling is stronger than \Page scrambling" since the latter only refers to a single state. The operator scrambling implies that the information about a local operator in the input system cannot be recovered from a subsystem C of the output system (as long as it is not very close to the entire output system) even if C is bigger than half of the system. { 8 { (a) time EPR pair (b) i | i j | i swap j | i i | i i | i j | i | − i − ji |j − ii perfect tensor HJEP02(16)4 input and output qubits. (b) A swap gate and a unitary corresponding to a perfect tensor. Note the similarity to the Feynman diagrams of a 2 ! 2 scattering process for a free theory and interacting theory, respectively. Finally, note that the condition I3(A : C : D) I(A : D) is known as strong subadditivity and must always holds among entropies. On the other hand, I3(A : C : D) 0 is often referred to as monogamy of mutual information and doesn't necessarily hold for arbitrary states.4 However, for holographic systems the tripartite information must always be negative [19]. This result is usually only discussed for holographic states but it also applies to holographic channels (such the eternal black hole in AdS) [20]. It is natural to suggest that the negative I3 is related to the fact that such holographic systems are strongly-chaotic and fast-scramblers [6]. (See also [21] for a study of monogamy and other properties of entanglement in qubit systems.) 2.3 Examples scrambling. Here, we present a few examples of using the tripartite information of a channel to measure Swap channel. Let us revisit the example discussed at the beginning of this section. Consider a system of n qubits and assume that the unitary operator is the identity operator: U = I. The channel description is given by (2.3). This is a collection of EPR pairs connecting input qubits and output qubits. Since the state consists only of two-party entanglement, the tripartite information is zero. Similarly, consider a time-evolution that consists only of permutations of qubits, a \swap" channel. Namely, let us assume that jth qubit goes to aj th qubit where 1 j; aj n. Then, the isomorphic state consists of EPR pairs between jth input qubit and aj th output qubit, and the tripartite information is zero ( gure 3(a)). Permutations of qubits can be thought of as a classical scrambling. Two initial nearby classical states may become far apart after permutations of qubits, yet it is still possible to distinguish two initial states by some local measurement on the output states. 4E.g. I3(A : C : D) = 1 for the four-qubit GHZ state p12 (j0000i + j1111i). { 9 { Perfect tensor. Next, let us look at an example where the tripartite information is maximally negative. Consider a system of two qutrits (spins with three internal states j0i; j1i; j2i). We consider the following unitary evolution i j i jji ! j i j i j j i i (2.17) where addition is de ned modulo 3. It can be directly veri ed that single qutrit Pauli operators transform in the following way5 Z I ! Z Z X I ! Xy Xy I Z ! Z Zy I X ! Xy X: (2.18) HJEP02(16)4 In this unitary evolution, all local operators evolve to two-body operators. As such, information concerning local disturbances to initial states cannot be detected by any single qutrit measurements on output states. We can represent this unitary evolution as the following pure state j i = 2 1 X (jii 3 i;j=0 jji)AB ( j i j i j j ii)CD (2.19) where addition is modulo 3. It is known that this pure state is maximally entangled in any bipartition. Namely, one has SA = SB = SC = 1, SAB = SBC = SCA = 2, SABC = 1, and I3 = 2, where entropy for qutrits is measured in units of log 3. This is a so-called perfect state. In general, for any pure state ABCD one can show that I3 2 min(SA; SB; SC ; SD). Therefore, this qutrit state has minimal value of I3. Here, we note that the di erence in the depiction of this qutrit perfect tensor and the swap gate (jii jji ! jji jii) resembles the di erence in the Feynman diagrams of a 2 ! 2 scattering process between a free theory and an interacting theory (see gure 3(b)(c)). We will comment on this in much greater detail in the context of conformal eld theory in appendix B. Black hole evaporation. Another interesting example is the thought experiment by Hayden and Preskill [1] as shown in gure 4. They considered the following scenario. Alice throws her secret (A), given in the form of some quantum state of a = jAj qubits, into a black hole (B) with the hope that the black hole will scramble her secret so that no one can retrieve it without collecting all the Hawking radiations and decoding them. Bob tries to reconstruct a quantum state of Alice by collecting some portion of Hawking radiation (D) after a scrambling unitary evolution U applied to the black hole, consisting both of Alice's secret A and the original content of the black hole B. The remaining portion of the black hole after the Hawking radiation is denoted by C. So, as usual, this channel is split into four segments A; B; C; D as shown in gure 4. First, assume that Bob only knows the dynamics of the black hole (i.e. the operator U ). In order for Bob to successfully reconstruct Alice's system A, the mutual information between Alice's secret and the Hawking radiation must be I(A : D) 2a. (Recall that 5Pauli operators for p-dimensional qudits are de ned by Xjji = jj + 1i and Zjji = !jjji with ! = ei 2p for all j where addition is modulo p. time Alice’s secret Black hole A C Black hole Hawking radiation I(A : D)=2 is roughly the number of EPR pairs shared by A and D.) However this is possible only when c 0 since the channel is maximally entangled along any bipartition due to the assumption of U being a scrambling unitary. Namely, I(A : D) 0 as long as D is smaller than B. Next, let us assume that Bob not only knows the dynamics U , but also knows the initial state of the black hole B. This is possible in principle if Bob has been observing the black hole since its formation. In this case, Bob has an access to both B and D. Then for d > a, the mutual information between A and BD becomes nearly maximal; I(A : BD) 2a because BD contains more than half of the entire qubits in channel ABCD. In this case, the tripartite information is given by I3(A : B : D) = I(A : D) + I(A : B) I(A : BD): (2.20) Since I(A : D); I(A : B) 0, we nd I3 indeed learn about Alice's secret.6 I(A : BD) 2a which implies that Bob can Holographic channels. In the nal example, we will consider thermal systems with Einstein gravity bulk duals. Under the holographic duality, the unitary quantum channel representing the CFT time evolution operator is geometrized as the black hole interior or Einstein-Rosen bridge [5, 12]. Such holographic states are already known to be fast scramblers [6], so here we will simply con rm that holographic channels scramble in the sense of I3. It's a trivial extension of the ideas in [12] and [6] to calculate the tripartite information across the eternal AdS black hole (the holographic state dual to thermo eld double state [25] of two entangled CFTs) so we will be brief. For simplicity, we will take A to be aligned 6The rewall paradox of [22{24] is related to the fact the scrambled channel with near maximally negative I3 cannot allow bipartite entanglement I(C : D) between the evaporating black hole C and the recently evaporated Hawking radiation D. This is a consequence of the monogamy of entanglement, which is captured by the negativity of the tripartite information. channel given by the time evolution operator of the dual CFT. Left: Penrose diagram for the eternal AdS black hole geometry with a spacelike slice (blue) anchored on the left boundary at the middle of the diagram anchored at time t on the right boundary. Right: geometric depiction of the spacelike slice through the Einstein-Rosen bride (ERB). The spatial coordinates on the boundary CFT are represented by '. The renormalized length of the ERB is proportional to t. For small t, the RT surface used to compute the entanglement entropy SAC goes across the ERB (red). After a time proportional to the size of A or C, the disconnected RT surface (blue) is preferred and the entanglement entropy is a sum of disjoint contributions (SAC = SA + SC ). with C across the Einstein-Rosen bridge, and A; B; C; D to have the same size, as shown in gure 5. For simplicity, we will consider time evolution only on the right boundary U(t) = e iHt. For any nite time, the mutual information I(A : D) is always zero for any nite regions A; D. The Ryu-Takayanagi (RT) surfaces used to compute the entanglement entropy are always disconnected. The only interesting behavior is from I(A : C), which was computed in this setup in [12]. The initial RT surface extends across the Einstein-Rosen bridge, and I(A : C) begins equal to the nite part of SA + SC , since the nite part of SAC is vanishing. Under time evolution, the nite part of SAC will increase linearly in time with a characteristic entanglement velocity vE [12], and I(A : C) will decrease linearly to zero. Since I(A : CD) = SBH the Bekenstein-Hawking entropy of the black hole, after a time of at most O(SBH=2), we nd I3(A : C : D) = SBH; (2.21) which is its minimal possible value. Haar-random channels. In appendix A, we analyze Haar-random unitaries. Using those results, we can bound the tripartite information in a Haar random channel I3; Haar 2 min(SA; SB; SC ; SD) + 1 + : : : : (2.22) The tripartite information of a random channel is near maximally negative plus one \residual" bit of information (independent of the overall system size). As we mentioned before, Haar scrambling implies tripartite scrambling. Chaotic channels vs. integrable channels In this section, we will focus on the aspects of unitary quantum channels built from timeevolution operators that can be used to di erentiate chaotic systems from integrable systems. A system, de ned by a time independent Hamiltonian H, can be chaotic or integrable. A unitary operator U(t) = e iHt can scramble. In section 3, we will learn that channels U(t) that scramble for most values of t must be built from chaotic systems. To be concrete, we will take integrable systems to be those that have a quantum recurrence time that is polynomial in their number of degrees of freedom n. On the other hand, chaotic systems will generally have recurrence times that are doubly exponential in n, O(een ). The important point about scrambling in channels is that all bipartite mutual informations between the input subsystems and output subsystems become small. As an extreme example of an integrable system, let's return to the swap channel ( gure 3(b)). As a reminder, for all t this channel has I(A : C) + I(A : D) = I(A : CD) or I3(A : C : D) = 0. Swap gates preserve bipartite entanglement and cannot create multipartite entangled output states. As such, an input localized at one lattice site can only be moved around to n possible locations. The recurrence time must scale like O(n). In order for the tripartite information of the channel to vanish for all times, the decrease of I(A : C) must be exactly compensated by the increase of I(A : D). If we take C to contain A at t = 0, the initial values of mutual information are I(A : C) = 2a and I(A : D) = 0. If I3 vanishes for t > 0, I(A : C) must decrease in order for I(A : D) > 0. If I(A : C) returns to the initial value 2a at a later time (i.e. there is a recurrence), in the meantime we must have I(A : D) = 0. Therefore the signature of an integrable system is a sharp peak in I(A : D) = 0 (or equivalently, a dip in I(A : C)). The sharpness of the peak is determined by the relative sizes of the systems A and C. In chaotic systems for times shorter than the Poincare recurrence time O(een ), I(A : D) and I(A : C) will asymptote to the channel's Haar-random value (see appendix A). As we will see numerically in section 4, for unequal divisions of the input (a b) and output (c d) the signature of an integrable system is a spike in I(A : D) which occurs at the time signals from A arrives at D. In such systems the tripartite information might become negative, but it will never become close to the Haar-scrambled value, and it will quickly return near zero. However, for equal divisions of input and output, the tripartite information of an integrable channel will tend to a constant much greater (i.e. less negative) than the Haar-scrambled value. This equal-sized subsystem con guration appears to be the most robust measure of scrambling. This discussion of the spike in I(A : D) highlights an important point: in integrable systems, such as the transverse Ising model, it is still possible to have early time exponential decay and ballistic growth of operators [5] (see the top-middle panel of gure 7), and in integrable CFTs some (but not all!) OTO correlators can decay at late times [8]. However, for these systems the growth of operators (or the decay of I(A : C)) must always be followed by a later decrease in size (or a recorrelation in the OTO correlator) as the system exhibits a recurrence.7 We will see this behavior explicitly in our numerics in section 4. 3 Butter y e ect implies scrambling Now, we will show that the generic decay of OTO correlators of the form hW (t) V W (t) V i implies that the mutual information between any small subsystem in the inputs and any partition of the output should be small. We will provide an exact formula relating the operator average of OTO correlators in di erent size subsystems to the second Renyi entropy for a subsystem consisting of both inputs and outputs. Using tripartite information as our diagnostic of scrambling in a unitary channel, we will show the butter y e ect implies For simplicity of discussion, we consider a system consisting of qubits. We consider a complete basis of Hermitian operators Di in subsystem D, which satis es the orthonormal tr fDiDj g = 2d ij : Similarly, we de ne an orthonormal basis Ai in subsystem A. As a reminder, our state lives in a 2n-dimensional Hilbert space, and AB and CD are two di erent decompositions of that Hilbert space. Hilbert space A is 2a-dimensional, and the operators Ai act on A. A similar statement holds for D. Both D and A are setup as in gure 2, and SD = d, SA = a. If a = 1, then one possible basis choice for Ai the three Pauli operators X; Y; Z; and the identity I. In general, there are 4a independent operators in A. We can think of this as choosing one of the four operators I; X; Y; Z at each site. If the Hilber space decompositions A and D do not overlap, then [Ai; Dj ] = 0 for all i; j. However, As a measure of OTO correlation functions for generic operator choices, we consider the following quantity (3.1) (3.2) jhOD(t) OA OD(t) OAi j : = hDi(t) Aj Di(t) Aj i = 1 1 4a+d 4a+d X ij Z 1 X tr fe ij H Di(t) Aj Di(t) Aj g; where the sums i; j run from 1 to 4d; 4a, respectively. Here, h i represents a thermal average, and j j represents an operator average over the complete bases Di; Aj . Additionally, we will take the in nite temperature limit = 0 so we don't have to worry about the Euclidean evolution. For t = 0, every correlator is unity hDi(0) Aj Di(0) Aj i =0 = 1 due to the orthonormal condition and the fact that Aj and Di(0) commute. 7Another relevant di erence between chaotic and integrable systems is in terms of the expansion (1.1) for the time evolution of a simple operator. Due to ergodicity, such an expansion will have a number of terms exponential in the size of the system for chaotic dynamics. Integrable systems are not ergodic, so the expansion will only have a linear number of terms. Under chaotic time evolution, Di(t) = eiHtDie iHt will evolve into a high weight operator and cease to commute with Aj . (To see this, consider the BCH expansion for Di(t) as in (1.1). As t increases, the later terms with high weight will become important. These terms will no longer be con ned to subspace D. Under chaotic evolution the Di(t) will grow to reach A.) This will lead to the decay of the OTO correlation functions hDi(t) Aj Di(t) Aj i =0 for generic i; j (as long as the Di or the Aj are not the identity in which case hDi(t) Aj Di(t) Aj i =0 = 1 for all t) and all such partitions A; D. At early times, the average will be very close to unity. With chaotic time evolution, the butter y e ect will cause most of the correlation functions in the average to decay exponentially. Using standard techniques, one can relate (3.2) to the second Renyi entropies of the time evolution operator considered as a state jhOD(t) OA OD(t) OAi =0j = 2 n a d SA(2C) ; (3.3) where SA(2C) is the second Renyi entropy of AC de ned in (2.11), 2n is the dimension of the input or output Hilbert space, and the subsystems A and D have dimension 2a and 2d, respectively. A proof of this result and its generalization to nite temperature is given in appendix C. At rst glance, it is a little surprising that a Renyi entropy appears here: the entropy determines mutual information, which bound two-point functions, and chaos is distinctly measured by OTO four-point functions. However, the key point is that when A and D are small (so that they only contain approximately local operators), B and C contain highly nonlocal operators covering almost the entire input and output systems, respectively. As a result, SA(2C) is sensitive to correlations between the few operators in A and the complete (and nonlocal and high weight) set of operators in C. The OTO average (3.3) is an operator-independent information-theoretic quantity that is constrained by chaotic time evolution. To understand its behavior, let's consider its maximum and minimum values. The average will be the largest at t = 0, when all the correlators are unity. On the other hand, the average will be minimized when the Renyi a is maximal: max SA(2C) = min(a + c; d + b). Let us assume for the rest of this section that d, so max SA(2C) = a + c. This means the OTO average is bounded from below by 4 a. (It's worth mentioning that the Haar-scrambled value of the average is generally larger than this lower bound.)8 Therefore, we see the OTO average is bounded by 4 a jhOD(t) OA OD(t) OAi =0j 1: (3.4) Now, we will recast this result in terms of mutual information in order to make a connection to our scrambling diagnostic. Let's assume that after a long time of chaotic time evolution the OTO average asymptote to a small positive constant . This means 8Since (3.2) includes 4d + 4a 1 terms where Aj = I or Di = I (one for each term where Aj = I or Di = I, and minus one to prevent overcounting when they both are), if all the non-identity correlation functions decay the OTO average will be 4 a + 4 d 4 a d > 4 a . Using the results from appendix A, we can show that this is larger value is exactly the Haar-scrambled value of the OTO average. To get lower value, some of the correlation functions need to cross zero so that the operators are negatively correlated. that the entropy SAC is bounded: SAC SA(2C) = n a R d log2 ; R where in the rst part we used the fact that S(i) > S(i+1) for Renyi entropies, and in the second part we used (3.3). In terms of mutual information, we have where here we have used the fact that SA and SC are always maximally mixed. Eq. (3.6) is one of our main results. At t = 0, I(A : C) = 2a. The information about the input to the channel in A is 2a bits and that information is entirely contained in the output subsystem C. Since there are 4a linearly independent basis operators in A's Hilbert space, we can interpret these 2a bits as the information about which of the 4a operators was input into the channel. (For instance, if a = 1, it takes two bits to index the operators I; X; Y; Z.) Under chaotic time evolution, 1, and the mutual information between A and C becomes small. The smallest possible value for is 2 2a, which occurs when I(A : C) = 0. In practice, there is always residual information between AC. Using the results from appendix A, we see that for Haar scrambling the mutual information can be bounded as I(A : C)Haar 1 + log(1 which corresponds to all the non-identity terms in the OTO average decaying to zero. If the information-theoretic quantities constructed from the second Renyi approach their Haar-scrambled value, then all the nontrivial OTO correlators hDi(t) Aj Di(t) Aj i =0 must approach zero. Next, we note that (3.6) implies I(A : C ) 2a + log2 ; SAC + SAC SAC ; for any partitioning of C = C [ C . This can be seen from subadditivity and the de nition of mutual information. (This is also intuitive: any information contained about region A in region C must necessarily be more than or equal the information in a partition of C.) Therefore, we learn that in chaotic channels, local information in the input must get delocalized in the output (i.e. cannot be recovered in an output subsystem smaller in size than the total system n). Since the partitions A and C were completely arbitrary, we conclude that the decay of OTO correlators implies that all bipartite mutual informations are small. If the OTO correlator average is given by after a long time, then we have (3.5) (3.6) (3.7) (3.8) (3.9) approaches the minimum value min = 2 2a, I3 approaches the most negative value I3;min = 3.2 In this section, we will attempt to connect the universal early-time behavior of OTO correlators in strongly chaotic systems with the information-theoretic quantities we use to diagnose scrambling. In strongly chaotic systems, all OTO correlation functions of operators with nontrivial time evolution will decay to zero. However, the behavior of the OTO correlation function hW (t) V W (t) V i as it asymptotes to zero is not universal. The approach will depend on the speci c choices of operators W; V . For instance, in two-dimensional CFTs with large central charge and a sparse low-lying spectrum at late times the OTO correlator decays as where hv is the conformal weight of the V operator, and it is assumed 1 hv On the other hand, at early times the behavior of hW (t) V W (t) V i usually takes a certain form. The initial decay is known to t the form where in analogy to classically chaotic systems L has the interpretation of a Lyapunov exponent [26].9 In [10], it was shown that quantum mechanics puts a bound on L L 2 ; with saturation for strongly-interacting conformal eld theories that have holographic descriptions in terms of Einstein gravity. This Lyapunov exponent is expected to be universal | independent of the choice of operators W; V |and as such any model that saturates the bound (3.13) is expected to be a toy model of holography [11, 26]. (In section 4, we will use numerics to explore the Majorana fermion model proposed in ref. [11] that in a certain limit is expected to have this property.) Since the partitioning of the inputs into A; B and outputs into C; D was entirely arbitrary, let's rst consider channels that operate on 0-dimensional systems, e.g. fast scramblers in the sense of [2], such as the Majorana fermion model of Kitaev [11] (a simpli cation of the Sachdev-Ye model of N SU(M ) spins [27], see also [28]) or a large N strongly interacting CFT holographically dual to Einstein gravity near its Hawking-Page point.10 These systems still have low-weight k-local Hamiltonians wth k N , but each degree of freedom interacts with every other degree of freedom. For these systems, the OTO correlation functions decay as hW (t) V W (t) V i = f0 N 2 f1 e Lt + O(N 4); (3.14) 9However, the analogy is imprecise. In weakly coupled systems, L has a semiclassical analog that does not map onto the classical Lyapunov exponent. Despite this, we will follow convention and refer to L as a Lyapunov exponent. We are grateful to Douglas Stanford for emphasizing this point. 10We require this limit so we can think of the black hole as unit sized and not yet worry about the operator growth in spatial directions. (3.11) (3.12) (3.13) where N 2 = n is the total number degrees of freedom, and the constants f0 and f1 depend on properties of the W; V operators (e.g. their CFT scaling dimensions) [10]. The decay of the correlator is delayed by the large number of degrees of freedom at time t = L1 log N 2. This is usually referred to as the scrambling time [2]. Plugging (3.14) into (3.6), we nd that at early times the mutual information between A and C is bounded as 2a #e L(t t ) + : : : : (3.15) information constructed from the second Renyi SA(2C) . Thus, the information between A and C must begin to decay by the scrambling time t = L1 log N 2. This inequality would be an equality if we instead considered the mutual Now, let's consider systems arranged on a spatial lattice but do not have a large number of degrees of freedom per site, e.g. spin chains. For these systems, the butter y e ect implies ballistic growth of operators in spatial directions [5]. For local operators W and V separated by large distance jxj OTO correlation functions decay as and numerical investigations of one-dimensional spin chains) strong chaos implies that , in many known examples (such as holography hW (t)V W (t)V i = f10 f20 e L(t jxj=vB) + O(e 2jxj L=vB ); (3.16) with additional constants f10 ; f20 that depend on the details of the operators.11 In this case, the early-time decay of the correlator is suppressed by the large spatial separation between the operators. Under chaotic time evolution, the operator W (t) will grow ballistically with some characteristic \butter y" velocity we denote vB.12 Thus, W (t) and V will commute until a time t > jxj=vB when V enters the \butter y" light cone of W . Let's focus on a lattice of spins in d-spatial dimensions. We will pick our subsystem A to be a ball of a sites surrounding the origin with a radius ra. We will pick C to also be a ball surrounding the origin with a radius rc such that rc ra = jxj. Then, after a scrambling time of t = vBt, the mutual information between A and C must begin to decay I(A : C) 2a #e L(t jxj=vB) + : : : : Since we will study this quantity in section 4, we note that we can directly equate (rather than bound) the behavior of the second Renyi entropy to the Lyapunov behavior of the OTO correlators. Let's restrict to a one-dimensional spin chain, and take A to be the rst spin of the input and D to be the last spin of the output. In that case, if we assume a form of the correlator (3.16), plug into (3.2) and compute the average, then we nd SA(2C) (t) = SA(2C) (0) + #e Lt + : : : ; 11As emphasized in [10], the butter y e ect is relevant for systems with a large hierarchy of scales. For the 0-dimensional systems we just considered, the hierarchy is provided by the parametric di erence between the thermal time and the fast scrambling time log N 2. In the present case, the role of the large scale is instead being played by the large spatial separation jxj between the operators. 12The velocity vB can depend details of the theory that do not a ect L. For instance, it is modied in Gauss-Bonnet gravity [5] and for certain Einstein gravity theories can even acquire a temperature dependence [29]. (3.17) (3.18) showing that at early times the Renyi entropy can grow exponentially with characteristic Lyapunov exponent L.13 We will roughly see this behavior in gure 7. Butter y velocity vs. entanglement velocity There are two nontrivial velocities relevant to the growth of information-theoretic quantities in unitary quantum channels arranged on a lattice. The butter y velocity vB [5, 6] is the speed at which the butter y e ect propagates. It is the speed at which operators grow under chaotic-dynamics. Such behavior is reminiscent of the Lieb-Robinson bound on the commutator of local operators separated in time for systems with local interactions [30{32] and suggests identifying vB with the Lieb-Robinson velocity. The butter y velocity is often di cult to compute directly, but in holographic theories with Einstein gravity duals it is known to be [6] vB = s 2(d d 1) ; (Einstein gravity); (3.19) where d is the spacetime dimension of the boundary CFT. This value is modi ed in GaussBonnet gravity [5] and for certain theories can even acquire a temperature dependence [29]. The entanglement velocity (sometimes called the tsunami velocity) studied in [12{ 14, 33, 34] is often described as the rate at which entanglement spreads. It is rate of growth of entanglement entropy after a quench, and in holographic systems dual to Einstein gravity it can be computed directly [12{14] vE = pd(d [2(d 1 1 2) 2 d 1)]1 d1 ; (Einstein gravity); (3.20) a di erent nontrivial function of the CFT spacetime dimension d. In these theories, vE vB, and vB = vE = 1 for d = 1 + 1. In the context of unitary quantum channels, these velocities have a very speci c interpretation in terms of di erent mutual informations, see gure 6. Consider a lattice of n degrees of freedom and divide the input up such that a = b = n=2, the output such that c = d = n=2, and such that all subsystems are contiguous. If A and C are aligned such that at t = 0, I(A : C) = n, then under time evolution (for chaotic and integrable systems!) it is expected that for a long stretch of time that the mutual information will decrease linearly as I(A : C) = n vEst; (3.21) until near when it saturates at I(A : C) = 0. This is often referred to as a quench, which we discuss in depth in appendix B in the context of CFT. Here, s is the \entropy density," which converts vE from a spatial velocity (with units lattice-sites/time) to an 13This assumes that L is independent of the choice of operators W and V and that the ansatz (3.16) is the correct form for the initial decay of the correlator. Both of these assumptions are not necessarily true for some spin systems. Additionally, if the constant in front of the exponential is not small (for example, in holographic systems), then the expansion will not be valid and one cannot see the exponential growth behavior. We thank Tarun Grover and Douglas Stanford for emphasizing these points. channel (blue) and Haar-channel (black) with n = 7 spins; input subsystems of a = 1, b = 6 spins; output subsystems of c = 6, d = 1 spins. Top left: the average of OTO correlators decays immediately, showing the butter y e ect. Top right: SAC (solid) and SA(2C) (dotted) shows roughly the same behavior as the OTO average. Bottom left: I(A : C), a trivial function of SAC , show that for an initial 2 bits of information between the subsystems in the fermion channel gets delocalized so that at late times only a small amount (0:59 bits) remains. Bottom right: the negative of the tripartite information normalized by its maximum value (2 bits) is a simple diagnostic of scrambling. 5 Perfect tensor model Now that we understand the relationship between strong chaos and the scrambling behavior of quantum channels, we will present a tensor network model of a scrambling channel with ballistic operator growth.16 This model serves two purposes. First, it is useful as a tractable model of ballistic scrambling. The network implements the expected entanglement structure of chaotic time evolution with a (discretized) time independent Hamiltonian. It also serves as a concrete toy model to study the growth of computational complexity in scrambling quantum channels consisting of local quantum circuits.17 established [37]. 16See also [34] for a similar recent tensor network model of ballistic entanglement propagation. 17In fact, there is a well-de ned notion of the complexity of randomness, called unitary t-designs, and lower bounds on the complexity growth under random quantum circuits in this sense have been rigorously Left CFT AdS e m i t ERB AdS each node. We will consider a network of perfect tensors. Second, it provides a model for the interior of the eternal AdS black hole [25]. In [12], it was proposed that the interior connecting the two asymptotic regions can be represented by a at tensor network whose length is proportional to the total time evolution on the boundary. In [38] and [5], this proposal was explored in a larger variety of black hole states that were perturbed by shock waves. Here, we provide a concrete model of such a network (i.e. we specify the tensors). This is in the spirit of previous work on the AdS ground state: in [39] it was suggested that the ground state of AdS can be represented by a hyperbolic tensor network (such as MERA [40]), and then an explicit tensor network model was proposed in [15] (see also [41]).18 Before we begin, let us review the proposal of [12]. The tensor network representation of the thermo eld double state is shown in gure 10. At the left and right ends, we have a hyperbolic network, representing the two asymptotically AdS boundaries. This network extends in nitely from the UV into the IR thermal scale at the black hole horizon. Then, the middle is at representing the black hole interior. The entire network grows as t grows by adding more layers in the middle at region. We would like to further elaborate on this proposal of tensor network representation of the black hole interior. We will study networks of perfect tensors and demonstrate chaotic dynamics by nding ballistic growth of local unitary operators and the linear growth of the tripartite information until the scrambling time. For the rest of discussion, we take the in nite temperature = 0 limit so we can ignore the hyperbolic part and focus in on the planar tiling of tensor networks representing the interior. 18This model has the additional nice property of implementing the holographic quantum error correction proposal of [42]. 5.1 Let us review the de nition of perfect tensors. Consider a tensor T with 2n legs and bond dimension v. A tensor can be represented as a pure state j i = X i1;:::;i2n Ti1;:::;i2n ji1; : : : ; i2ni; SA = n; 8A s.t jAj = n; with a proper normalization. We call a tensor T perfect if it is associated with a pure state j i, called a perfect state, which is maximally entangled along any bipartition. Namely, where for tensors of bond dimension v we measure entropy in units of log v. The qutrit tensor eq. (2.19) is an example of a perfect tensor. There are known methods for constructing perfect tensors via the framework of quantum coding theory. Also, a Haar random tensor becomes a perfect tensor at the limit of v ! 1. Growth of local operators. Imagine a at planar tiling of 4-leg perfect tensors as shown in gure 10 which may be thought of as a discretized time-evolution by a stronglyinteracting Hamiltonian. We can examine time evolution of a local unitary operator V and observe linear growth of spatial pro les of operators V (t) by using a basic property of perfect tensors. Let j i be a 4-spin perfect state and denote 4 legs by a; b; c; d. Consider a single-body unitary operator Ua 6= I acting exclusively on a. Since ab and cd are maximally entangled, there always exists a corresponding operator V Ucd 6= I acting exclusively on cd such that Uaj i = Ucdj i ; or in the tensor representation, we have t) Ua = Ucd ; (5.1) (5.2) (5.3) (5.4) where a gray square represents a four-leg perfect tensor. One can prove that Ucd must act non-trivially both on c and d. Namely, if Ucd were a single-body operator acting only on c (i.e. Ucd = Uc), then one would have UaUcyj i = j i. However this contradicts with the fact that ac and bd are maximally entangled. To see the contradiction, one can simply use UaUcyj ih j = j ih j and take a partial trace over b; d on both sides of the equation. If a; c is maximally entangled with b; d we obtain UaUcy = I is the identity operator, which is not possible. The conclusion is that, due to the perfectness of the tensors, each twoqudit unitary associated with perfect tensors always expands a single-body operator to a two-body operator. This observation is consistent with linear ballistic propagation of entanglement for single connected regions predicted for chaotic systems [43]. The implication of this ballistic expansion of unitary operators under perfect tensors is quite interesting. The size of the region of nontrivial support for V (t) increases linearly time (0) V (t) HJEP02(16)4 radius of the operator V grows with the butter y velocity as vBt, and the vertical depth of the circuit grows as vEt. as shown in gure 11. At t = L=2, for a lattice of linear size L, a local operator will evolve into a global operator supported over the entire lattice. The growth of OTO correlation functions originates from this linear growth of spatial pro les of local operators. Namely, for a local operator W which is separated in space from V (t = 0), the commutator [V (t); W ] becomes non-negligible after t = L=2 indicative of the butter y e ect.19 Growth of tripartite information in time. Let us then compute the tripartite in formation for a network of perfect tensors. The entire system is split into four regions A; B; C; D of equal size as in gure 12(a). The growth of entanglement entropy can be exactly calculated by using a method developed in [15]. Recall that, for a perfect state j i with four spins, there always exist a two-qubit unitary operator Dab such that In other words, Dab disentangles a perfect state into two decoupled EPR pairs as graphically shown below Dabj i = jEPRiac jEPRibd: Dab| = a c b d : (5.5) (5.6) A key observation is that the process of nding a minimal surface by local updates can be viewed as entanglement distillation by applications of disentanglers. This led to the proof of the Ryu-Takayanagi formula for single intervals in networks of perfect tensors [15]. In general, calculation of entanglement entropies for disjoint regions is challenging even for networks of perfect tensors. Indeed, the veri cation of the Ryu-Takayanagi formula is 19A qualitatively similar behavior occurs when Haar-random unitary operators are used instead of perfect tensors, which we checked numerically. For an analytical discussion a random tensor network in the context of a holographic state rather than a channel, see [44]. (See also [45].) given only for single intervals for a network of perfect tensors in [15]. Here we assume that the planar tensor network is translationally invariant in both time and spatial directions. To be speci c, we also assume that the network consists of the qutrit perfect tensor introduced in (2.19). For such a perfect tensor tiling, an analytical calculation of the tripartite information is possible for time t shorter than the scrambling time t = L=2. Namely, one can prove the following: I3(A : B : C) = 2t; 0 L=2; (5.7) where L is the linear length of the system and as a reminder for qutrits we measure entropy in units of log 3. Below, we sketch the derivation. For a network of perfect tensors, entanglement properties can be studied by applying local disentanglers which correspond to distillations of EPR pairs. The disentanglers map each region unitarily to the minimal surface bounding it, as is shown in gure 12. For time t L=2, one observes that minimal surfaces for A; C collide with each other, and similarly for B; D. Let us distill entanglement as shown in gure 12(b) by applying some appropriate local unitary transformations on each region and remove decoupled spins. Regions AC and BD possess EPR-like entanglement along the collided surface of geodesic lines. In the middle of the network, we nd square regions which are responsible for fourparty entanglement among A; B; C; D. Such regions, which are not included inside causal wedges of boundary regions, are referred to as residual regions [15]. These become essential in understanding entanglement properties behind the horizons of the multi-boundary black holes considered in [46]. At the time step t, there will be a pair of square residual regions with linear length t=2 as shown in gure 12(b). In appendix D, we study multipartite entanglement for rectangular residual regions. Namely, we show that each residual region contributes to the tripartite information by t. We thus obtain eq. (5.7). 5.2 Recurrence time We have shown that the network of perfect tensors, as shown in gure 10 and gure 11, serves as a toy model of scrambling dynamics. A naturally arising question is whether such a system stays scrambled after the scrambling time t = L=2. In this section, we study the recurrence time of the planar network of perfect tensors. For concreteness, we will restrict our considerations to those with qutrit perfect tensors. We assume periodic boundary conditions in the spatial directions of the network. Imagine that we inject some Pauli operators from the top of the tensor network and obtain output Pauli operators on the bottom. We are interested in the minimal time step necessary for a network to output the initial Pauli operators again. To nd the recurrence time, we inject two-body Pauli Z operators from the top left corner of the tensor network and compute the output Pauli operator on the bottom. We de ne the recurrence time trec to be the minimal time step trec necessary for the network to output the initial two-body Pauli Z operators. Recall that the tensor network based on stabilizer tensors maps Pauli operators to Pauli operators. Since Pauli operators can be treated as classical variables, one can e ciently nd the recurrence time via numerical methods. A C D t t/2 t/2 HJEP02(16)4 residual region tensor networks. Square-like tensor networks are responsible for the tripartite information. The recurrence time crucially depends on the system size L as shown in the plot in gure 13. Note that the plot uses a logarithmic scale. When the system size is L = 3m, the recurrence time grows only linearly: trec = 4L. This expression can be analytically obtained. The linear growth is due to the fact that the qutrit tensor can be viewed as a linear cellular automaton over F3 which has scale invariance under dilations by factor of 3. For such special system sizes, the trajectories of time-evolution of Pauli operators form short periodic cycles. This is similar to the classical billiard problem where trajectories of a billiard ball are not ergodic for ne-tuned system sizes and ne-tuned angles. Yet, the billiard problem is ergodic for generic system sizes. Likewise, the perfect tensor network has longer recurrence time for generic values of system sizes. When L is a prime number, the growth is rather fast, and seems exponential as shown in gure 13. (We do not have an analytic proof of this statement.) Assuming the exponential growth of the recurrence time for prime L (trec ekL), let us nd out the growth for typical values of L. For typical values of L, we expect that trec grows faster than any polynomial functions. This is because given a positive integer n, the probability for its largest prime factor to be larger than, say pn, is nite.20 Assuming that L is not a prime number, let us decompose it as L = L1L2. Then, due to the translation invariance, one can show that trec(L) trec(L1); trec(L2). As such, the recurrence time trec(L) will be lower bounded by trec(p) where p is the largest prime factor of L. This argument implies an exponential growth of the recurrence time for typical values of L. 20In general, the probability for the largest prime factor to be larger than 1=nu is given by the Dickman function [47]. (blue) and non-prime size (black). The recurrence time of the perfect tensor network is much longer than that of integrable systems, but is much shorter than that of chaotic systems. By construction, perfect state network based on the stabilizer formalism can have at most exponential recurrence time. This is essentially because unitary circuits implemented by stabilizer-type tensors belong to the so-called Cli ord group which is a subgroup of unitary transformations that map Pauli operators into Pauli operators.21 Quantum circuits solely consisting of Clifford operators are classically simulable since transformations of Pauli operators can be e ciently characterized by pairs of classical bits. In this sense, the stabilizer perfect tensor network exhibits marginally chaotic behaviors. The classical simulability enables us to study chaos and scrambling behaviors in quantum channels at relatively early times in a computationally tractable manner. This highlights an important point about perfect tensor networks as models of holography. In many cases they can exhibit key features expected of holographic systems (such as the error correcting and bulk reconstruction properties of the model presented in [15]). However, since the recurrence time of the perfect tensor network is exponential in the system size and not doubly exponential, it's clear that it fails to capture a very important feature: the possibility of exponential computational complexity. In particular, the (comparatively) quick recurrence means that the longest minimal perfect tensor networks are far less complex than the degree of complexity that generic holographic states are expected to reach. by (1.1)). 21This means that under time evolution a simple Pauli operator X, Y , or Z can only grow into a product of Pauli operators (rather than a sum of products of Pauli operators as would be generically expected One possible resolution is to modify the stabilizer perfect tensor by applying some single qudit non-Cli ord rotation, such as a rotation around the Z axis by some angle . An inclusion of a single non-Cli ord operator to the full Cli ord group enables us to e ciently approximate an arbitrary unitary operator, an important result known as the Solovay-Kitaev theorem [16]. As such, we speculate that non-Cli ord modi cation of perfect tensors would create a tensor network with doubly exponential recurrence time. This resolution is along the spirit of the billiard problem since the Cli ord transformations are ne-tuned operations. Another possible resolution of this is that to reach the more complicated states (which are not at all understood holographically, see e.g. [7, 22, 24, 48]), one needs to consider superpositions of such tensor networks which do not have a geometric description and thus would not be expected to have a semiclassical bulk interpretation. Regardless, a network of perfect tensors is very capable of scrambling. This observation leads us to envision that a certain measure of complexity can be attached to each tensor in the network, in particular, I3 up to proper normalization. This would represent the complexity of forming the four-leg perfect tensor from a product state. It would be interesting to see if some kind of upper bound on the gate complexity can be imposed by considering an integral of I3 over all the tensors in the network. 6 Discussion In this paper, we have shown that the butter y e ect | as expressed by the decay of out-of-time-order (OTO) correlation functions | implies the information-theoretic notion of scrambling. The butter y e ect is manifested by the growth of simple operators under time evolution to complicated operators of high weight. These time-evolved operators will then have large commutators will all other operators in the system. If we think of the initial simple operator as an input to a unitary quantum time-evolution channel, then the output will be an operator spread over the entire system. All information associated with the input will be delocalized; the output system is scrambled. The method of characterizing scrambling/chaos via the framework of quantum channels may also nd interesting applications in studying thermalization in many-body quantum systems. We have already demonstrated the usefulness of our approach by studying the tripartite information in several di erent examples: numerical results in integrable/nonintegrable spin chains and the nonlocal interacting Majorana fermion model of Kitaev, and both analytical and numerical results in a perfect tensor network model of discretized time evolution. It would be interesting to study many-body/single-body localization and delocalization transitions in the setup of quantum channels. A closely related question may concern the information-theoretic formulation of the Eigenstate Thermalization Hypothesis (ETH). The state interpretation of the channel is able to consider a set of initial states as well as to probe o -diagonal elements in the Hamiltonian. In order for quantum information to really be processed, it has to interact with the other information distributed across the system. Said another way, to process information the channel has to be capable of scrambling. This suggests that there is a strong connection between quantum chaos and computation. As a surrogate for a de nition of computation, let's consider the computational complexity of the quantum circuit or channel. For tensor network models, this is simply the number of tensors in the minimal tensor network. As a simple example, let's consider the quantum channel that only contains swap gates. The channel doesn't scramble, and information can only be moved around. As discussed multiple times, the swap channel has a quick recurrence and can never get very complex. The only output states accessible are those related to permutations of the input, all the multipartite states cannot be accessed. For a system of n qubits, the complexity can only ever reach O(n) (the complexity of swapping localized information from one end of the system to the other using local swap operations). The maximal complexity of a state of n qubits is O(2n); thus, for the simple swap channel most of the possible output states are entirely inaccessible. It is essentially only capable of classical computation. Quantum computation requires interaction, and strong chaos is a signature of a strongly interacting system. Thus, in some sense, we speculate chaos must be the capacity for a system to do computation. This suggests that strongly chaotic-systems must be fast computers. In fact, in [49, 50] it was recently hypothesized that black holes are the fastest computers in nature. Given that black holes are already known to be nature's densest hard drives [51, 52] and most chaotic systems [10], it seems reasonable to suspect that a system's computational power must be limited by its degree of chaos. It would be interesting to try and make this dependence on chaos for computation more direct. Acknowledgments This work began at KITP, and the authors would like to acknowledge the KITP programs \Entanglement in Strongly-Correlated Quantum Matter" and \Quantum Gravity Foundations: UV to IR." We would also like to thank Tarun Grover, Aram Harrow, Patrick Hayden, Matt Headrick, Isaac Kim, John Preskill, Steve Shenker and Douglas Stanford for discussions, and Douglas Stanford for comments on the draft. PH and XLQ are supported by the David and Lucile Packard foundation. XLQ is also partially supported by the Templeton foundation. DR is supported by the Fannie and John Hertz Foundation and is also very thankful for the hospitality of the Stanford Institute for Theoretical Physics during a stage or two of this work. DR also acknowledges the U.S. Department of Energy under cooperative research agreement Contract Number DE-SC0012567. This paper was brought to you by the butter y e ect. BY is supported by the David and Ellen Lee Postdoctoral fellowship and the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation. In this paper, we've generally considered scrambling by a one-parameter family of unitary operators U(t) = e iHt and found that for chaotic systems, increasing time t leads to more e cient scrambling. Instead, we will now take U to be a Haar random unitary operator. This is useful as a baseline for scrambling. We expect that the late-time values of entropies and informations computed from scrambling operators U(t) will asymptote to Haar random values. Additionally, we will see that Haar-random values of quantities such as I(A : C) and I3(A : C : D) are not necessarily maximal, but exhibit \residual" information regardless of system size. At its fastest, Page scrambling has a complexity of n log n gates, while Haar scrambling is nearly maximally complex requiring en gates. In [4], it was proven that Haar scrambling implies Page scrambling. As we will now show, Haar scrambling also implies the tripartite scrambling. However, the implication does not work in the other direction: since the late-time values of I(A : C), I(A : D), and I3(A : C : D) approach Haar-scrambled values, entanglement is not enough [53] to diagnose typicality (in the Haar-random sense). This possibly has a strong bearing on the paradoxes of [22, 24] as discussed in [48]. To proceed with this analysis, we will consider an expectation over density matrices constructed from Haar-random states. These tools were used by Page to analyze the entropy of subsystems for random states [4], and our approach will be similar to [1] and appendix A of [6]. In fact, our calculation is very similar to that in [6]. However, beware that the results do not simply carry over; since we are pairing together input and output subsystems of possibly di erent size (in our notation, the fact that a 6= c), we will nd a very di erent result. Our setup will be the usual division into subsystems ABCD, with the state given by (2.4), and U a random 2n 2n unitary matrix taken from the Haar ensemble. The Haar average lets us consider expectations over a number of unitary matrices and is non-zero only when the number of U s equals the number of U ys. For instance, with two U s and two U ys, the formula for the average is Z dU Ui1j1 Ui2j2 Ui01j10 Ui02j20 = 1 22n 1 1 2n(22n 1) i1i01 i2i02 j1j10 j2j20 + i1i02 i2i01 j1j20 j2j10 (A.1) i1i01 i2i02 j1j20 j2j10 + i1i02 i2i01 j1j10 j2j20 : This formula will let us compute the average over the trace of the square of the density matrix AC HJEP02(16)4 Z Z 2 dU tr f AC g = 22n dU Uk`mo Uk0`m0o Uk0`0m0o0 Uk`0mo0 ; (A.2) where as in appendix C, k = 1 : : : 2a are A indices, ` = 1 : : : 2b are B indices, m = 1 : : : 2c are C indices, and o = 1 : : : 2d are D indices. In applying the average (A.1) to (A.2), note that both k and ` in (A.2) are \i"-type indices in (A.1), and similarly m and o are \j"-type indices. After a quick game of delta functions, we nd dU tr f AC g = 2 a c + 2 b d 2 2 a d n 2 b c n : (A.3) As of now, we have been completely general about the size of the subsystems. However, in this paper we've primarily considered the case where a + c = n and b + d = n. Without loss of generality, let's also take a b and d subleading pieces, we get c.22 Simplifying and throwing away exponentially Z 2 dU tr f AC g ' 2 1 n 1 2 a d 1 : Using this with an appropriate caveat,23 we can compute the Haar average of the Renyi entropy SA(2C) Haar = n 1 This is rather interesting: the maximal value for SA(2C) is n. Therefore, the Haar-scrambled state never reaches this maximal value. On the other hand, this is not unexpected. The corrections to Page's entropy of a subsystem formula for the divisions we are considering (A.4) (A.5) this using the fact that SAC > SA(2C) , and we nd Next, let us use (A.5) to put a bound on the mutual information I(A : C). We can do I(A : C) 1 + log2(1 22Ref. [6] neglects the bottom line of (A.1) as subleading. For the subsystems we consider, the rst term on the second line of (A.1) is actually the same order (in n) as the terms in the rst line and cannot be neglected. 23To use this result to compute the Haar average of the Renyi entropy Z dU SA(2C) = Z dU log2 tr f 2ACg; we need to commute the Haar average with the log. This can be checked numerically and holds as long as n is su ciently large (which in this case \large" means about n = 4). For large n, the system self-averages so that any single sample is extremely likely to be at the mean value. This lets the average commute with the log. 24One might have thought that it would be possible to make a better bound with the Fannes-Audenaert inequality [54, 55] and using the 2-norm to bound the 1-normal as in [1] and [6]. However, such an approach actually leads to a bound that's actually much looser than the simple one given by (A.6). indicating the possibility of residual information between A and C that is independent of n, even in the Haar-scrambled limit.24 By considering more equal partitions of the input a b), there will be more residual information between A and C, though the fraction of residual information I(A : C)=2a decreases. Let's complete our discussion by trying to bound I(A : D). Following the approach outlined above, we nd Z dU tr f ADg = 2 a d + 2 b c 2 2 a c n 2 b d n which is the same as (A.3) with c , d. Taking as before a + c = n and b + d = n, with a b and d c, we see that the three latter terms are exponentially smaller in n than the rst term. We nd and we can bound the mutual information as SA(2D) Haar = a + d + O(2 2n+a+d); I(A : D) 0 + O(2 2n+a+d): (A.6) (A.7) (A.8) (A.9) Entanglement propagation in CFT B gure 3(b), we pointed out a strong resemblance to Feynman diagrams of a 2 ! 2 scattering process. The swap gate resembles a diagram that contribute to a noninteracting theory: the only allowed operation is that the particles can swap locations between the inputs and the outputs (or they can do nothing). On the other hand, the perfect tensor resembles a Feynman diagram that contributes to a scattering process in an interacting theory. This is not a coincidence; the strength of chaos should be related to the strength of the coupling, see e.g. [10]. With this point of view, let us consider entanglement propagation in CFT. The general setup considered in CFT is a global quench; the system is preprepared in a groundstate of a Hamiltonian H0 and then the Hamiltonian is suddenly changed to a di erent Hamiltonian H such that the system is now in a nite energy con guration. The system is then evolved with the new Hamiltonian and certain entanglement entropies saturate at their thermal values. This often referred to as thermalization. For two-dimensional CFT, the entanglement entropy of a single connected region after a global quench was shown to grow linearly in time, saturating at its thermal value at a time of order half the length of the region [56{58]. To explain this, [56] proposed that entanglement is carried by pairs of entangled noninteracting quasi-particles that travel ballistically in opposite directions.25 The quasi-particles would travel at the entanglement velocity vE , which is equal to unity in two-dimensional CFT. Entanglement entropy increases as the entangle pairs are split between the region and its complement. This model of entanglement propagation is su cient to explain the pattern of entanglement growth after a quench of a single interval (in fact, the result is universal [56{58]), but gives puzzling results for the entanglement entropy of two separated disjoint intervals in interacting (e.g. holographic) systems [33, 59{62].26 Let us label the two intervals as F and G, both of size L, and the rest of the system as H. We will take F and G to be separated by a distance D, and all the scales are taken to be much greater than the thermal correlation length L; D; the left-hand side of gure 14. . Additionally, we require D > L. This setup is depicted in In the quasi-particle model, after a quench SF G will grow linearly for a time D and then saturate at its thermal value. However, at time of order (D + L)=2 it will exhibit a dip. In the quasi-particle picture, entangled pairs created in the region between F and G are beginning to enter F and G, respectively, causing the entanglement entropy of F G with the rest of the system H to dip. For holographic systems, this is known not to happen: after SF G saturates, it remains saturated [33, 59{62]. This puzzle was explored in depth in the context of two-dimensional CFT in [62]. There, it is shown that the quasi-particle picture cannot be universal and must depend on the spectrum. Indeed, [62] concludes that for interacting CFTs \entanglement scrambles" | meaning there's no memory e ect or dip in SF G. Here, we would like to put these results in the context of unitary quantum channels. We will show that \entanglement scrambling" 25See also [34] for a generalization of the non-interacting quasi-particle model to interacting systems and a discussion of the entanglement velocity. 26See also [12{14] for holographic investigations of entanglement growth after a global quench. L separated by distance D > L. Left: for two intervals in the noninteracting quasi-particle model SF G has a dip at time t (D + L)=2. Right: a simpli ed description in terms of two entangled CFT involves one partner of the EPR pair traveling in the left CFT and one partner traveling in the right CFT. This can be reinterpreted as a quantum channel. To make contact with the notation in the paper, we relabel as F 0 ! A, G0 ! D, and H0 ! BC. is precisely scrambling as diagnosed by the tripartite information. Furthermore, we will argue that the cause of such entanglement scrambling is chaotic dynamics. Strong chaos implies a picture of strongly interacting quasi-particles. To make the connection, one simply has to realize that a global quench can be simply understood as the time evolution of the thermo eld double state [12]. That is, a global quench is the channel jTFD(t)i given by (2.7). This was also pointed out in [62], where the thermo eld double state was used to simplify the setup of the two-interval calculation while retaining the basic puzzle.27 In this new setup, the puzzle is a memory e ect between an interval on the left CFT F 0 and interval on the right CFT G0, where the spatial separation D and interval sizes L are large. After a time of order (D + L)=2, the quasi-particle model predicts a dip in the entanglement between F 0G0 and the rest of the system H0. This new setup is shown in the right-hand side of gure 14. Now, let us relabel the subsystems: F 0 ! A, G0 ! D, and H0 ! BC. With the perspective of the unitary channel setup ( gure 2), the memory e ect is simply a question of whether I(A : D) has a spike. Integrable systems will have a spike and can be modeled by entanglement carrying quasi-particles (the swap gate in gure 3(b) provides an explicit cartoon of such noninteracting quasi-particles). Chaotic systems are strongly interacting, and the quasi-particle picture breaks down (the perfect tensor in gure 3(b) provides the cartoon for the interacting system). This memory e ect was shown explicitly in the bottommiddle panel of gure 7 for the integrable spin chains we studied in section section 4. This connection to the work of [62] allows us to probe scrambling and chaos in particular CFTs. For instance, the results of section 4:4 in [62] suggest that the D1-D5 CFT at the \orbifold point" does not scramble (in the sense of tripartite information) as expected for a free theory. It would be interesting to make additional connections between scrambling/chaos and CFT results. 27In fact, in [62] the memory e ect was diagnosed by considering properties of the second Renyi entropy for the intervals in question. is probably most easily understood diagrammatically as shown in gure 15. For simplicity of discussion, we assume a system consisting of qubits while our discussion straightforwardly generalizes to a system consisting of qudits by considering generalized Pauli operators. Thus, this proof applies to lattice systems with a nite-dimensional Hilbert space at To proceed with the proof, we need to make use of an operator identity. Consider a partition of a system AB with Aj a complete basis of operators in A. Then, for any operator O on the entire system AB, we have X Aj O Aj = jAj IA trA fOg; j where the sum i runs over the entire basis, and jAj is the size of the Hilbert space of A. The set of all qubit Pauli operators supported in A forms a complete basis of orthonormal operators, and (C.2) can be easily veri ed by decomposing O into that basis. A diagrammatic depiction of this identity is shown in gure 15(a). We would like to use this to evaluate the averaged correlator 2 2a 2d n X tr fDi(t)Aj Di(t)Aj g = 2 2a 2d n X tr fU yDiU Aj U yDiU Aj g: (C.3) ij ij Proof of eq. (3.3) The proof of the relation jhOD(t) OA OD(t) OAi =0j = 2n a d SA(2C) ; (C.1) (C.2) (C.4) (C.5) Here the prefactor 2 2a 2d is the inverse of number of operators in A and D, and 2 n is the normalization factor such that the quantity equals to 1 if all operators are identity operators. Let's apply (C.2) to DiU Aj U yDi to do the sum over i. This gives 2 2a d n X tr fU Aj U y trD fU Aj U yg IDg; j where note that we have made use of the cyclicity of the trace. At this point, it's useful to adopt indices. We will use k = 1 : : : 2a for A indices, ` = 1 : : : 2b for B indices, m = 1 : : : 2c for C indices, and o = 1 : : : 2d for D indices. This lets us rewrite (C.4) as j 2 2a d n X Uk1`mo(Aj )k1k10 Uk10`m0oUk2`0m0o0 (Aj )k2k20 Uk20`0mo0 ; 28In a continuum limit, we would need some notion of the operator identity (C.2), which is the completeness condition for a basis of operators. Naively, due to the in nite Hilbert space dimension, (3.3) is trivially true; the Renyi entropy is UV-divergent and the correlation function average is vanishing due to normalizing by the total number of operators. However, the connection between our results and entanglement propagation in CFT (see appendix B) suggests that perhaps a recasting of the relation (3.3) in terms of mutual information might lead to a sensible continuum limit. suppressed. (a) The operator identity eq. (C.2). (b) AC for the unitary channel. (c) Calculation of the average. In the nal panel, the lines at the top and the bottom are appropriately connected. to give where repeated indices imply summation. Now, we apply (C.2) again, speci cally to (Aj )k1k10 Uk10`m0oUk2`0m0o0 (Aj )k2k20 . This sets k10 = k2 and k20 = k1 (and multiplies by 2a) Now, we remember how to express the density matrix of our channel (see gure 15(b)) in gure 15(c). De ne Applying this to (C.6) and then using the de nition of the second Renyi entropy (2.11) gives our desired result (3.3). This whole proof, up to factors of normalization, is shown Finite temperature. It is easy to generalize this formula for nite temperature > 0. 2 a d nUk1`moUk2`m0oUk2`0m0o0 Uk1`0mo0 : = 2 n Uk`mo Uk0`0m0o0 : Z( ) := tr(e H ); j ( )i := jTFD( ; t)i: (C.6) (C.7) (C.8) To get an expression in terms of an entropy, we need to distribute the operators around the thermal circle jhOD(t) OA OD(t) OAi j Z( ) 1jtr fOD(t i =4) OA OD(t i =4) OAgj: Here, rather taking a thermal expectation value we are evolving the operators in D in Euclidean time (and then renormalizing by Z( )). The trace of these Euclidean-evolved correlators is expected to be related to the thermal expectation of the original OTO correlators as long as the temperature is high enough. Following our proof gure 15 but with the time argument for the unitary operators as U(t i =4), we nd Z( =2)2 Z( ) 2 a d SA(2C) ( =2); (C.10) where SA(2C) ( =2) is evaluated for the state j ( =2)i de ned in (C.8). Higher order OTO correlators. Finally, we will brie y comment on another possible generalization. The OTO correlation functions we studied here are observables for the chaotic dynamics of a thermal system perturbed by a single operator. In [7], chaos is studied in holographic thermal systems that are perturbed by multiple operators. For two perturbations, the relevant observable is a six-point OTO correlation function of the form hW (t1) V (t2) Q V (t2) W (t1) Qi ; (C.11) where W; V; Q are all simple Hermitian operators. (For simplicity, we will consider the case where = 0.) This observable is related to the e ect of simple perturbations W; V made at times t1; t2 on measurements of Q at t = 0. This correlation can be simpli ed by summing over a basis of operators in three regions associated with the W; V; Q as we did for four-point functions earlier in this appendix. However, it's easy to see that one cannot get something as simple as a Renyi entropy: since there's two explicit times t1; t2, we can form density matrices (t1), (t2), and channel of time evolution by U(t) = e iHt. The averaged six-point function will be related t2), where (t) = jU(t)ihU(t)j is the density matrix associated with the quantum to contraction of these density matrices with a complicated permutation. This may be considered as a more generic entanglement property of the system, which is beyond Renyi entropies. In the case of nite temperature, for the four-point functions we were able to evolve the operators in Euclidean time in order to symmetrize the time arguments of the unitary operators, as is discussed around (C.9). However, cannot do that for the six-point functions since we need all three operators to be separated from each other in Lorentzian time. D Tensor calculus In this appendix, we provide more details about the perfect tensor calculation sketched in gure 12. We have seen that the two rectangular residual regions, which are not contained in any of causal wedges, are responsible for multipartite entanglement arising in a network Q S (D.1) (D.2) (D.3) of perfect tensors. To calculate I3, we typically need to consider rectangular residual regions. In this appendix, we study multipartite entanglement in a rectangular network of perfect tensors as shown in gure 16 where tensor legs are split into four subsets P; Q; R; S. We assume that P; R contain r legs and Q; S contain t legs. In [15], it was shown that, for any planar network of perfect tensors with non-positive curvature, the Ryu-Takayanagi formula for single intervals holds exactly. Keeping this in mind, let us summarize properties of entanglement in an arbitrary rectangular tiling of perfect tensors: SP = r; SQ = t; SR = r; SS = t; SP Q = r + t; SQR = r + t; SRS = r + t; SP S = r + t; where as a reminder, for tensors of bond dimension v, we measure their entropy in units of log v. Thus, the tripartite information I3 is given by Note that the above statement holds for any perfect tensors and is not restricted to qutrit perfect tensors. But the value of SP R is non-universal for networks of perfect tensors since P R consists of two spatially disjoint intervals. Below, we will prove that I3 = SP R: I3 = 2 min(r; t); for a network of four-leg qutrit perfect tensors. The qutrit tensor network discussed in section 5 can be described by the stabilizer formalism [16], and analytical calculations of entanglement entropies are possible. Let us recall a useful formula for entropy calculations for stabilizer states. Consider an n-qutrit pure state j i speci ed by a set of n independent Pauli stabilizer generators gj such that gj j i = j i for j = 1; : : : ; n with [gi; gj ] = 0. The stabilizer group S stabilizers S (stab) = hfgj g8j i. Therefore we have gj i = j i for all g 2 S (stab) consists of all (stab). We are interested in entanglement entropy of j i for some subset A of qutrits. A useful formula SA = jAj log3 jSA (stab) ; j Z Z Z Z where jAj is the number of qutrits in A and SA (stab) is the restriction of S onto A (i.e. a group of stabilizer operators which are supported exclusively on A). Note, log3 jSA be understood as the number of independent stabilizers supported on A. The stabilizer generators for the qutrit tensor are given by Z X Z X Z X Zy Xy I I Z; X; where Xy = X2 and Zy = Z2. Graphically, stabilizer generators are given by Z X Z Z I; I; † X X † X X Z Z † I Z Z† Z †ZX X † IX X† † X ;X † I † X X † I where Zjji = !j jji and Xjji = jj + 1i with ! = e i23 . Observe that stabilizer generators commute with each other. Also observe that there is no two-body stabilizer generator. This implies, from eq. (D.4), that entanglement entropies for any subsets of two qutrits are two, and thus this stabilizer state is a four-leg perfect state. We need to nd the number of stabilizer generators which can be exclusively supported on P R. Let us rst consider a contraction of two perfect tensors (i.e. r = 2 and t = 1). There are stabilizer generators supported only on upper and lower tensor legs Z Z Z Z X X† † X X (D.4) (stab)j can (D.5) (D.8) (D.9) (D.10) Z Z† Z Z † X X † X X So, SP R = jP j + jRj 2 = 2. Next, let us consider the case where t = 1 and r > 2. Since X-type and Z-type stabilizers are separable, one can treat them separately. We want to nd all the stabilizer operators that are supported on P R. Here we consider an input Pauli operator X(f ) supported on P where f is a degree r 1 polynomial over F3. That is, for a polynomial we de ne the Pauli-X operator as f = c0 + c1x + c2x2 + : : : + cr 1xr 1; cj 2 F3; X(f ) = Xc0 Xc1 : : : Xcr 1 ; where Xj acts on the j-th leg on P for j = 0; : : : ; r polynomial f0 over F3, one can write f0 as follows 1. Given an arbitrary degree r † I † I † I † I : (D.7) f0 = (2 + x)g0 + h0; where g0 is some degree r 2 polynomial while h0 is some degree 0 polynomial (in other words, a constant). Note that X(2 + x) is the Pauli X operator on P in (D.7), whose \output" on R is given by X(1 + x). Let us then look for a stabilizer operator whose action on the upper leg is given by X(f0). When t = 1, the output Pauli operator is supported exclusively on R if and only if h0 = 0. Namely, the output operator can be written as X(f1) f1 = (1 + x)g0: Similar analysis holds for Z-type stabilizers. Therefore, there are in total 2r 2 independent stabilizer generators supported on P R. Thus, SP R = r + r HJEP02(16)4 consider the cases where t > 1. For this purpose, we think of decomposing fj recursively fj = (2 + x)gj + hj ; fj+1 = (1 + x)gj : The output has supports exclusively on R if and only if hj = 0 for j = 0; : : : ; t implies that there are in total 2(r t) stabilizer generators supported on AC for t there is no such stabilizer generator for t > r. Thus, one has I3 = 2 min(r; t). In fact, the aforementioned result applies to a larger class of perfect tensors. Notice that stabilizer generators of the qutrit tensor can be written as tensor products of Pauli Z or X operators only. Such a stabilizer state is often referred to as a CSS (CalderbankShor-Steane) state, and a number of interesting quantum error-correcting codes belongs to this class. Let us assume that four-leg perfect tensors are based on CSS stabilizer states. Then, one is able to prove that I3 is always given by I3 = 2 min(r; t) as long as the bond dimension v is a prime number. 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Pavan Hosur, Xiao-Liang Qi, Daniel A. Roberts. Chaos in quantum channels, Journal of High Energy Physics, 2016, 4, DOI: 10.1007/JHEP02(2016)004