Scrambling of Quantum Information in Quantum Many-Body Systems
SScrambling of Quantum Information in Quantum Many-Body Systems
Eiki Iyoda and Takahiro Sagawa Department of Applied Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
October 23, 2017
Abstract
We systematically investigate scrambling (or delocalizing) processes of quantum information encodedin quantum many-body systems by using numerical exact diagonalization. As a measure of scrambling,we adopt the tripartite mutual information (TMI) that becomes negative when quantum information isdelocalized. We clarify that scrambling is an independent property of integrability of Hamiltonians; TMIcan be negative or positive for both integrable and non-integrable systems. This implies that scramblingis a separate concept from conventional quantum chaos characterized by non-integrability. Furthermore,we calculate TMI in the Sachdev-Ye-Kitaev (SYK) model, a fermionic toy model of quantum gravity.We find that disorder does not make scrambling slower but makes it smoother in the SYK model, incontrast to many-body localization (MBL) in spin chains.
Introduction.
Whether an isolated system thermalizes or not is a fundamental issue in statisticalmechanics, which is related to non-integrability of Hamiltonians. In classical systems, thermalization hasbeen discussed in terms of ergodicity of chaotic systems [1]. In quantum systems, a counterpart of classicalchaos is not immediately obvious, because the Schr¨odinger equation is linear. Nevertheless, it has beenestablished that there are some indicators of chaotic behaviors in quantum systems, such as the levelstatistics of Hamiltonians [2–4] and decay of the Loschmidt echo [5, 6]. More recently, the eigenstate-thermalization hypothesis (ETH) [7–11] has attracted attention as another indicator of quantum chaosin many-body systems, which states that even a single energy eigenstate is thermal. All these indicatorsof quantum chaos are directly related to integrability of Hamiltonians; non-integrable quantum systemsexhibit chaos. Such a chaotic behavior in isolated quantum systems is also a topic of active researches inreal experiments with ultracold atoms [12–14], trapped ions [15], NMR [5], and superconducting qubits [16].In order to investigate “chaotic” properties of quantum many-body systems beyond the conventionalconcept of quantum chaos, it is significant to focus on dynamics of quantum information encoded inquantum many-body systems. How does locally-encoded quantum information spread out over the entiresystem by unitary dynamics? Such delocalization of quantum information is referred to as scrambling [17–21]. Investigating scrambling is important not only for understanding relaxation dynamics of experimentalsystems at hand, but also in terms of information paradox of black holes [17], where it has been arguedthat black holes are the fastest scramblers in the universe [18]. However, the fundamental relationshipbetween scrambling and conventional quantum chaos has not been comprehensively understood.Scrambling can be quantified by the tripartite mutual information (TMI) [21, 22], which becomesnegative if quantum information is scrambled. There is also another measure of scrambling, named theout-of-time-ordered correlator (OTOC) [20, 21, 23–30]. It has been argued that the decay rate of OTOC isconnected to the Lyapunov exponent in the semiclassical limit [23]. TMI and OTOC capture essentiallythe same feature of scrambling [21], where OTOC depends on a choice of observables but TMI does not.1 a r X i v : . [ c ond - m a t . s t a t - m ec h ] O c t n the context of the holographic theory of quantum gravity, TMI is shown negative [31] if the Ryu-Takayanagi formula [32] is applied, suggesting that gravity has a scrambling property. This is consistentwith fast scrambling in the Sachdev-Ye-Kitaev (SYK) model [23, 33–40], a toy model of a quantum blackhole. Then, a natural question raised is to what extent such a property of quantum gravity is intrinsic togravity or can be valid for general quantum many-body systems.In this Letter, we perform systematic numerical calculations of real-time dynamics of TMI in quantummany-body systems under unitary dynamics, by using exact diagonalization of Hamiltonians. We considera small system (say, a qubit) and a quantum many-body system (say, a spin chain). The information ofthe small system is initially encoded in the many-body system through entanglement. The many-bodysystem then evolves unitarily, and we observe how the locally encoded information is scrambled over theentire many-body system. We note that temporal TMI has been investigated by using the channel-stateduality in Ref. [21], while we here calculate instantaneous TMI, with which we can study the role of initialstates.By studying quantum spin chains such as the XXX model and the transverse-field Ising (TFI) modelwith and without integrability breaking terms, we find that scrambling occurs (i.e., TMI becomes negative)for both the integrable and non-integrable systems for a majority of initial states. On the other hand, fora few initial states, scrambling does not occur (i.e., TMI becomes positive) for both the integrable andnon-integrable cases of the XXX model. These results clarify that scrambling is an independent property ofintegrability of Hamiltonians. Therefore, scrambling does not straightforwardly correspond to conventionalquantum chaos, making a sharp contrast to the level statistics and ETH. We remark that the relationshipbetween integrability and ballistic entanglement spreading has been studied [41–43], while delocalizationand entanglement spreading capture different aspects of information dynamics [44], as will be discussedlater in detail.We also consider the SYK model with four-body interaction of complex fermions, and find that disorderdoes not lead to slow dynamics but instead makes scrambling smoother than a clean case. This is contrastiveto the slow scrambling in many-body localized (MBL) phase of a spin chain [45–52]. Setup.
We consider either a spin-1 / L sites (Fig. 1). The many-body system is divided intothree subsystems B, C, D, whose sizes (the numbers of the lattice sites) are respectively given by 1, l , and L − l −
1. The lattice structure BCD is supposed to be one-dimensional for spin chains or all-connected forthe SYK model. For a single site of a spin (fermion) system, we write | (cid:105) as the spin-up (particle-occupied)state, and | (cid:105) as the spin-down (particle-empty) state. In any case, a single qubit is on a single site.We first prepare a product state 1 √ | (cid:105) A + | (cid:105) A ) ⊗ | Ξ (cid:105) BCD , (1)where | Ξ (cid:105) BCD is a product state with the state of each qubit being | (cid:105) or | (cid:105) (e.g., the N´eel state | (cid:105)| (cid:105)| (cid:105) · · · | (cid:105)| (cid:105) or the all-up state | (cid:105)| (cid:105) · · · | (cid:105) , etc). We then apply the CNOT gate on the state (1),where the control qubit is A and the target qubit is B. By this CNOT gate, information about A is locallyencoded in B through entanglement. Then, only BCD obeys a unitary time evolution with a Hamiltonian.We calculate the time dependence of TMI between A, B, C, which characterizes scrambling of the infor-mation about A that was initially encoded in B. We note that the foregoing setup is associated with athought experiment that one of qubits of an EPR pair is thrown into a black hole and then scrambled [17].We next consider quantum-information contents. Let X, Y, Z be subregions of the lattice (i.e., subsetsof the lattice sites). The bipartite mutual information (BMI) is defined as I (X : Y) := S X + S Y − S XY ,where S X := tr X [ − ˆ ρ X ln ˆ ρ X ] is the von Neumann entropy of a reduced density operator ˆ ρ X := tr X c [ ˆ ρ ] with2 c being the complemental set of X. Then, TMI is defined by [21, 22] I (X : Y : Z) := I (X : Y) + I (X : Z) − I (X : YZ) . (2)Here, TMI is negative when I (X : Y) + I (X : Z) < I (X : YZ), which implies that information about Xstored in composite YZ is larger than the sum of the amounts of information that Y and Z have individually;information about X is delocalized to Y and Z in such a case.To illustrate the meaning of TMI, let us consider three classical bits x, y, z and the following situations:(i) I ( x : y : z ) = − ln 2, if x = y ⊕ z and y, z are independent and random, where ⊕ describes thebinary sum. In this case, neither y nor z is individually correlated with x , but composite yz is maximallycorrelated with x . (ii) I ( x : y : z ) = 0, if x, y , and z are all independent and random. In this case, thereis not any correlation between x, y , and z . (iii) I ( x : y : z ) = ln 2, if x = y = z and x is random. In thiscase, the three bits form the maximum three-body correlation.While the above examples are classical, a similar argument applies to quantum situations. In fact, TMIcan be utilized to characterize nonlocal and long-ranged entanglement in topological orders [53]. Scrambling and integrability.
We now discuss scrambling in the XXX model in one dimension with andwithout an integrability breaking term. The Hamiltonian is given byˆ H spin := (cid:88) (cid:104) i,j (cid:105) J σ i · σ j + (cid:88) (cid:104)(cid:104) i,j (cid:105)(cid:105) J (cid:48) σ i · σ j , (3)where i and j are indices of sites, and (cid:104) i, j (cid:105) and (cid:104)(cid:104) i, j (cid:105)(cid:105) mean that i and j run within nearest neighbor(n.n.) and next nearest neighbor (n.n.n.), respectively. The Pauli matrices for a spin are written as σ αi ( α = x, y, z ), and we define σ i := ( σ xi , σ yi , σ zi ). Let J >
0. This model is integrable if J (cid:48) = 0, while isnon-integrable if J (cid:48) > J (cid:48) = 0 . J in Eq. (3), where the parameters are taken sothat the level statistics is the Wigner-Dyson distribution [4], implying that the system is fully chaotic inthe sense of conventional quantum chaos. Figure 2 shows the time dependence of TMI with the initialstate being (a) N´eel or (b) all-up, along with BMI I (A : BC) (inset). At initial time t = 0, BMI is givenby 2 ln 2 because of the entanglement between A and B. As time increases, BMI decays for all the cases.In (a), the decay is much smoother, where BMI saturates at zero for l = 1 and at ln 2 for l = L/ − l = 1, TMI decreases from zero, goes through a minima, and gradually returns to zero. This means thatinformation is scrambled inside ABC in a short time regime, and then completely disappears from BC in alonger time regime. For l = L/ −
1, TMI monotonically decreases and saturates at a negative value. Thismeans that information is scrambled but is not totally lost from BC even in a long time regime. Theseresults are consistent with the behaviors of BMI.On the other hand, as shown in Fig. 2 (b), TMI is positive when the initial state is all-up. In this case,information is not scrambled, but a three-body correlation forms among A, B, and C. On the other hand,as shown in Supplemental Material, entanglement spreads ballistically even in this case. This clarifiesthat what TMI characterizes is delocalization of quantum information, rather than ballistic spreading ofentanglement.We next discuss the integrable case with J (cid:48) = 0 in Eq. (3). Figure 3 shows the time dependence ofTMI for the initial state being (a) N´eel or (b) all-up. The qualitative behavior of TMI is similar to thenon-integrable case; scrambling occurs in (a) but does not in (b). We do not observe recurrence inducedby integrability, because our system size is sufficiently large.3e therefore conclude that scrambling occurs independently of integrability. We note that the timerange of our numerical simulation is sufficiently long to see the role of non-integrability. In fact, thelevel spacing at the peak of the Wigner-Dyson distribution corresponds to J t (cid:39) in our non-integrablemodel [54]. We also note that our numerical simulation is not restricted to the low-energy states whichcan be effectively described by the integrable field theory [55].To study the initial-state dependence of scrambling more systematically, we calculated the XXX modelwith all possible product states | Ξ (cid:105) BCD . We label 2 L product states by bit sequences from | · · · (cid:105) to | · · · (cid:105) . Figure 4 shows the initial-state dependence of the maximum and minimum values of TMI in0 ≤ J t < , written as I max3 and I min3 respectively, for (a) non-integrable and (b) integrable cases. Thehorizontal axis shows the labels of | Ξ (cid:105) BCD in decimal. We see that scrambling occurs ( I min3 <
0) for mostof the initial states.On the other hand, there are only four initial states with which scrambling does not occur ( I min3 = 0) forboth of (a) and (b). These four states are | (cid:105)| (cid:105)| (cid:105) · · · | (cid:105) , | (cid:105)| (cid:105)| (cid:105) · · · | (cid:105) , | (cid:105)| (cid:105)| (cid:105) · · · | (cid:105) , and | (cid:105)| (cid:105)| (cid:105) · · · | (cid:105) .The reason why these four states are exceptional is that the Hamiltonian (3) conserves the total magneti-zation in the z direction. This confines the dynamics into a much smaller subspace of the Hilbert space,which leads to the absence of scrambling.We have also calculated TMI for the TFI model with and without an integrability breaking term. Thenumerical results are shown in Supplemental Material, where scrambling occurs for both the integrableand non-integrable cases and for all of the initial states. The reason why scrambling occurs for the initialall-up state is that the total magnetization in the z direction is no longer conserved in the TFI model, andtherefore quantum information is mixed up in a huge subspace. Sachdev-Ye-Kitaev model.
We next consider the SYK model [23, 33–40] with complex fermions:ˆ H SYK := 1(2 L ) / (cid:88) i,j,k,l J ij ; kl c † i c † j c k c l , (4)where c † i ( c i ) is the creation (annihilation) operator of a fermion at site i . The coupling in the SYK modelis all-to-all and four-body (4-local), and random: J ij ; kl is sampled from the complex Gaussian distributionwith variance J , satisfying J ij ; kl = − J ji ; kl = − J ij ; lk = J ∗ lk ; ji . We also consider a clean SYK model withoutdisorder (i.e., J ij ; kl ≡ J for i > j , k > l ) in order to clarify the role of disorder.Figure 5 (a) shows the time dependence of TMI for the SYK model with the random coupling, wherethe initial state is N´eel (i.e., fermions are half-filled), the ensemble average is taken over 16 samples, andthe error bars represent the standard deviations over the samples. In particular, TMI for l = L/ − | (cid:105) · · · | (cid:105) ), where TMI is positive andscrambling does not occur for both the disordered and clean cases (see Supplemental Material). This isa consequence of the conservation of the fermion number, as is the case for the XXX model with theconservation of the initial magnetization. Concluding remarks.
We have systematically investigated scrambling dynamics of quantum informa-tion in isolated quantum many-body systems, where we have adopted TMI as a measure of scrambling.We summarize the foregoing numerical results in Table I. We have observed that scrambling occurs inde-pendently of integrability, where an overwhelming majority of initial states exhibit scrambling for both the4 crambled ( I <
0) Not scrambled ( I > J (cid:48) (N´eel) XXX+ J (cid:48) (all-up)TFI+ h z (N´eel, all-up)Integrable XXX (N´eel) XXX (all-up)TFI (N´eel, all-up)Clean SYK (N´eel) Clean SYK (all-up)Disordered MBL (N´eel) MBL (all-up)Disordered SYK (N´eel) Disordered SYK (all-up) Table 1: Summary of our numerical results.integrable and non-integrable cases. Although the connection between TMI and scrambling has alreadybeen established in previous works [21, 22], our work has newly revealed that scrambling is a separateconcept from conventional quantum chaos.We have also investigated the SYK model. We have found that disorder makes scrambling smootherin the SYK model, which is contrastive to the case of the MBL spin chain. We postpone more detailedanalysis of the origin of this feature of the SYK model [56]. Here we only note that the clean SYK modelis integrable as shown in Table I [56].We remark that experimental realizations of the SYK model have been theoretically proposed withultracold atoms [38] and a solid state device [57]. Furthermore, OTOC has experimentally been measuredwith trapped ions [58]. By using such state-of-the-art quantum technologies, scrambling dynamics ofquantum many-body systems can be investigated, and our results can be experimentally tested, which isa future issue.T. S. is grateful to M. Rigol for a valuable discussion. E.I. and T.S. are supported by JSPS KAKENHIGrant Number JP16H02211. E.I. is also supported by JSPS KAKENHI Grant Number 15K20944. T.S. isalso supported by JSPS KAKENHI Grant Number JP25103003.
References [1] G. Gallavotti,
Statistical Mechanics: A Short Treatise (Springer, 1999).[2] H-J. St¨ockmann,
Quantum Chaos — an introduction (Cambridge University Press, 1999).[3] Ph. Jacquod and D. L. Shepelyansky,
Phys. Rev. Lett. , 1837 (1997).[4] L. F. Santos and M. Rigol, Phys. Rev. E Phys. Rep. , 33 (2006).[6] A. Goussev, R. A. Jalabert, H. M. Pastawski, and D. Wisniacki, arXiv:1206.6348 (2012).[7] M. Srednicki,
Phys. Rev. E , 888 (1994).[8] M. Rigol, V. Dunjko, and M. Olshanii, Nature , 854 (2008).[9] G. Biroli, C. Kollath, and A. M. L¨auchli,
Phys. Rev. Lett. , 250401 (2010).[10] L. D’Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol,
Adv. in Phys. , , 239 (2016).[11] E. Iyoda, K. Kaneko, and T. Sagawa, Phys. Rev. Lett. , 100601 (2017).512] T. Kinoshita, T. Wenger, and D. S. Weiss,
Nature
Science
Nature Physics , 325 (2012).[15] G. Clos, D. Porras, U. Warring, and T. Schaetz, Phys. Rev. Lett. , 170401 (2016).[16] C. Neill, P. Roushan, M. Fang, Y. Chen, M. Kolodrubetz, Z. Chen, A. Megrant, R. Barends, B.Campbell, B. Chiaro, A. Dunsworth, E. Jeffrey, J. Kelly, J. Mutus, P. J. J. OMalley, C. Quintana, D.Sank, A. Vainsencher, J. Wenner, T. C. White, A. Polkovnikov, and J. M. Martinis,
Nature Physics , 1037 (2016).[17] P. Hayden and J. Preskill, JHEP
120 (2007).[18] Y. Sekino and L. Susskind,
JHEP
065 (2008).[19] S. H. Shenker, D. Stanford,
JHEP , 067 (2014).[20] J. Maldacena, S. H. Shenker, and D. Stanford, JHEP , 106 (2016).[21] P. Hosur, X-L. Qi, D. A. Roberts, and B. Yoshida, JHEP , 004 (2016).[22] N. J. Cerf and C. Adami, Physica D , 62 (1998).[23] A. Kitaev, Talks at KITP, April 7, 2015 and May 27, 2015 (2015).[24] I. L. Aleiner, L. Faoro, L. B. Ioffe,
Ann. of Phys. , 378 (2016).[25] F. M. Haehl, R. Loganayagam, P. Narayan, and M. Rangamani, arXiv:1701.02820 (2017).[26] D. A. Roberts and B. Yoshida,
J. High Energ. Phys. (2017) 2017: 121.[27] I. Kukuljan, S. Grozdanov, and T. Prosen, arXiv:1701.09147 (2017).[28] E. B. Rozenbaum, S. Ganeshan, and V. Galitski,
Phys. Rev. Lett.
Phys. Rev. D , 046003 (2013).[32] S. Ryu and T. Takayanagi, Phys. Rev. Lett. , 181602 (2006).[33] S. Sachdev, and J. Ye, Phys. Rev. Lett. , 3339 (1993).[34] S. Sachdev, Phys. Rev. X , 041025 (2015).[35] B. Michel, J. Polchinski, V. Rosenhaus, and S. J. Suh, arXiv:1602.06422 (2016).[36] J. Maldacena and D. Stanford, Phys. Rev. D , 106002 (2016).[37] W. Fu and S. Sachdev, Phys. Rev. B , 035135 (2016).638] I. Danshita, M. Hanada, and M. Tezuka, arXiv:1606.02454 (2016).[39] J. S. Cotler et al. , arXiv:1611.04650 (2016).[40] S-K. Jian and H. Yao, arXiv:1703.02051 (2017).[41] P. Calabrese and J. Cardy, J. Stat. Mech. , P04010 (2005).[42] A. M. L¨auchli and C. Kollath, J. Stat. Mech. , P05018 (2008).[43] H. Kim and D. A. Huse,
Phys. Rev. Lett. , 127205 (2013).[44] A. Bohrdt, C. B. Mendl, M. Endres, and M. Knap, arXiv:1612.02434 (2016).[45] A. Pal and D. A. Huse,
Phys. Rev. B , 174411 (2010).[46] J. H. Bardarson, F. Pollmann, and J. E. Moore, Phys. Rev. Lett , 017202 (2012).[47] M. Serbyn, Z. Papi´c, and D. A. Abanin,
Phys. Rev. Lett. , 127201 (2013).[48] D. J. Luitz, N. Laflorencie, and F. Alet,
Phys. Rev. B , 081103 (2015).[49] Y. Huang, Y-L. Zhang, and X. Chen, arXiv:1608.01091 (2016).[50] Y. Chen, arXiv:1608.02765 (2016).[51] R. Fan, P. Zhang, H. Shen, and H. Zhai, arXiv:1608.01914 (2016).[52] B. Swingle and D. Chowdhury, arXiv:1608.03280 (2016).[53] A. Kitaev and J. Preskill, Phys. Rev. Lett. , 110404 (2006).[54] M. L. Mehta, Random Matrices (Academic Press, 2004).[55] L. Samaj, and Z. Bajnok,
Introduction to the Statistical Physics of Integrable Many-body Systems (Cambridge University Press, 2013).[56] E. Iyoda, H. Katsura, and T. Sagawa, in preparation.[57] D. I. Pikulin and M. Franz,
Phys. Rev. X , 031006 (2017).[58] M. G¨arttner, J. G. Bohnet, A. Safavi-Naini, M. L. Wall, J. J. Bollinger, and A. M. Rey, Nature Physics , 781 (2017). 7 Entangled C DB
Figure 1: (color online). Schematics of our setup. Initially, qubit A is maximally entangled with qubitB, while C and D are not correlated with A, B. Then BCD evolves unitarily with a Hamiltonian that iseither integrable or non-integrable, and either clean or disordered. We calculate real-time dynamics of TMIbetween A, B, C. -1-0.5 010 -2 -1 -2 -1 -2 -1 (a)(b) -2 -1 Figure 2: (color online). Time dependence of TMI for the non-integrable XXX model with parameters L = 14 , J (cid:48) = 0 . J , and l = 1 or L/ −
1. The initial state is (a) N´eel or (b) all-up. (Inset) Time dependenceof BMI with the same parameters. 8 -2 -1 -2 -1 (a)(b) Figure 3: (color online). Time dependence of TMI for the integrable XXX model with parameters L = 14 , J (cid:48) = 0, and l = 1 or L/ −
1. The initial state is (a) N´eel or (b) all-up. -1-0.5 0 0.5 0 1024 2048 3072 4096-1-0.5 0 0.5 0 1024 2048 3072 4096 (a)(b)
Initial state
Figure 4: (color online). Initial-state dependence of the maximum (purple) and the minimum (green)values of TMI for the XXX model with parameters L = 12, l = L/ −
1. (a) Non-integrable case ( J (cid:48) = 0 . J ),(b) integrable case ( J (cid:48) = 0). 9 -2 -1 -1-0.5 010 -2 -1 (a)(b) disordered SYKclean SYK Figure 5: (color online). Time dependence of TMI for the SYK model with parameters L = 14, and l = 1or L/ −
1. The initial state is the N´eel state. (a) Disordered SYK model. (b) Clean SYK model andtypical samples of the disordered SYK model. 10 upplemental Material: Scrambling of Quantum Information in Quantum Many-Body Systems
Eiki Iyoda , Takahiro Sagawa [1] Department of Applied Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656,JapanIn this Supplemental Material, we show supplemental numerical results on the bipartite mutual infor-mation (BMI) and the tripartite mutual information (TMI). Some of the following results are mentionedin the main text. Figure S1 shows the time dependence of BMI for the integrable XXX model. -2 -1 -2 -1 (a) (b) Figure S1: Time dependence of BMI for the integrable XXX model with parameters L = 14 and J (cid:48) = 0.The initial state is (a) N´eel or (b) all-up.In addition, we show numerical results on ballistic entanglement spreading for the integrable and non-integrable XXX models. In particular, we consider the initial all-up state, where scrambling does not occur.In this case, we can numerically access a much larger size ( L = 128), because the total magnetization isconserved in the XXX model. Figures S2 (a) and (b) show the time dependence of BMI for several l .Figure S2 (c) shows the decay time τ , at which I (A : BC) becomes a ln 2 (0 < a <
2) for the first time.The constant a can be arbitrarily chosen and we here set a = 1 .
9. Figure S2 (c) clearly shows the lineardependence of τ on l , which implies that entanglement spreads ballistically. Figure S3 shows BMI versustime and l , from which we again see the ballistic entanglement spreading. The Hamiltonian of the transverse field Ising (TFI) model is given byˆ H TFI := (cid:88) (cid:104) i,j (cid:105) J σ zi σ zj + (cid:88) i h x σ xi + (cid:88) i h z σ zi . (S1)1 (a) (b) (c) nonintegrableintegrable Figure S2: Time dependence of BMI for (a) non-integrable and (b) integrable XXX models. The systemsize is L = 128, and the initial state is all-up. (c) Decay time τ versus subsystem size l .Figure S3: BMI versus time t and subsystem size l for (a) non-integrable and (b) integrable XXX models.The system size is L = 128 and the initial state is all-up.The transverse and longitudinal magnetic fields are h x = J and h z = 0 for an integrable case, and are h x = 2 . J and h z = 1 . J for a non-integrable case.Figures S4 and S5 show the time dependence of BMI and TMI for the TFI model, respectively. FigureS5 shows that scrambling occurs independently of integrability or the initial state.Figure S6 shows that the initial-state dependence of I max3 and I min3 , where the initial states are repre-sented by decimal from | · · · (cid:105) to | · · · (cid:105) , as in Fig. 4 of the main text. Scrambling occurs for both(a) non-integrable and (b) integrable cases for all of the initial states without exception. The Hamiltonian of disordered spin chains is given byˆ H := (cid:88) (cid:104) i,j (cid:105) J σ i · σ j + (cid:88) i h i σ zi , (S2)where h i is a random magnetic field and is generated uniformly from [ − h, h ] ( h ≥ -2 -1 -2 -1 -2 -1 -2 -1 (a)(b) (c)(d) Figure S4: Time dependence of BMI for the TFI model with L = 14. The integrability and the initialstate are (a) non-integrable/N´eel, (b) non-integrable/all-up, (c) integrable/N´eel, and (d) integrable/all-up.Figure S8 shows the time dependence of BMI and TMI for the disordered XXX model with the initialall-up state. As shown in Figs. S8 (c) and (d), scrambling does not occur both in the ergodic and MBLphases. Figure S9 shows the time dependence of BMI for the disordered SYK model with the initial N´eel state.Figure S10 shows the time dependence of BMI and TMI for the disordered SYK model with the initialall-up state. As shown in Fig. S10 (b), scrambling does not occur as TMI is positive.3 -2 -1 -1-0.5 010 -2 -1 -1-0.5 010 -2 -1 -1-0.5 010 -2 -1 (a)(b) (c)(d) Figure S5: Time dependence of TMI for TFI with L = 14. The integrability and the initial state are (a)non-integrable/N´eel, (b) non-integrable/all-up, (c) integrable/N´eel, and (d) integrable/all-up. -1.5-1-0.5 0 0.5 0 1024 2048 3072 4096 -1.5-1-0.5 0 0.5 0 1024 2048 3072 4096 Initial stateInitial state (a) (b)
Figure S6: Initial-state dependence of the maximum (purple) and the minimum (green) values of TMIfor the TFI model with parameters L = 12 and l = L/ −