JHEP04(2020)063 Springer April 9, 2020 March 2, 2020 : March 20, 2020 : : , Published Received c,d Accepted Published for SISSA by https://doi.org/10.1007/JHEP04(2020)063 and Xiao-Liang Qi 1 a,b, [email protected] , . 3 Christoph S¨underhauf 1 2002.09236 The Authors. a,b, c Quantum Dissipative Systems, Random Systems
Inspired by the Hayden-Preskill protocol for black hole evaporation, we con- , [email protected] These authors contributed equally to this work. 1 [email protected] Stanford Institute for Theoretical Physics,Stanford, Stanford California University, 94305, U.S.A. Department of Physics, StanfordStanford, University, California 94305, U.S.A. E-mail: Max-Planck-Institut f¨urQuantenoptik, Hans-Kopfermann-Str. 1, 85748 Garching, Germany Munich Center for QuantumSchellingstraße Science 4, and 80799 Technology, M¨unchen,Germany b c d a Open Access Article funded by SCOAP Keywords: ArXiv ePrint: it begins to decrease inafterwards. the middle Second, stage we of study thethe “evaporation”, the and system time decreases from needed monotonically to measurementscomputations, retrieve on we information characterize the such initially time environment injected bothconfiguration qudits. when in or the retriever not, Based has showing control on that over the explicit different initial scales numerical appear in the two cases. conserved U(1) charge and focuscally on and two numerically aspects the growth oftwo of the different the dynamics. time entanglement scales First, entropy appear: of wetime), the study while one system, analyti- the is showing other intrinsic that U(1) to depends conserved the on charge, internal we the show dynamics system-environment that (the coupling. the scrambling entanglement follows In a the Page-like behavior presence in of time: a Abstract: sider the dynamics of ament, quantum where many-body the qudit time system evolutionunitary coupled is to circuits. driven an by We external the study environ- continuous both limit of cases certain where 2-local the random unitaries are chosen with and without a Lorenzo Piroli, A random unitary circuitevaporation model for black hole JHEP04(2020)063 ] 14 – 9 ]. In this 5 – 1 25 27 23 7 11 – 1 – 16 5 24 27 29 29 ) ) ) , over all the degrees of freedom. a b c 1 22 3 ], where the authors studied how quantum information is released from a ]. In turn, this led to the development of several measures of information 6 8 , 7 scramble These considerations provided an obvious motivation for a systematic study of infor- D.2 Scenario ( D.3 Scenario ( Haar-scrambled case D.1 Scenario ( 3.1 Random Brownian3.2 circuit without conservation Random law dynamics with a conserved U(1) charge mation scrambling and the relatedsubsequent concept conjecture of by many-body Sekino quantum andin chaos, Susskind nature also that [ due black tospreading holes the are and the chaos, fastest including scramblers out-of-time-ordered correlation (OTOC) functions [ context, a particularly fruitful line ofand research Preskill was initiated [ by theblack seminal hole, work by under Hayden theTheir assumption study that suggested it that is informationthan not could the destroyed be during black released the hole inspread, evaporation lifetime, a or process. and time which related is to much the shorter time needed for localized information to 1 Introduction In the past decade,energy quantum physics, information especially ideas in connection have to become the increasingly black hole relevant information in paradox high [ C Derivation of the relevant formulasD in the Details bosonic on formalism the computation of the mutual information 5 Conclusions A Derivation of the Lindbladian forB the entropies R´enyi Derivation of the system of differential equations for the purity in the 3 The entanglement growth 4 Retrieval of quantum information Contents 1 Introduction 2 The model JHEP04(2020)063 ], ], 61 27 – 17 ], we study the ] coupled to an 6 ]. ]), and the tripar- 64 , 15 49 , 11 48 ], and their study allowed us 28 ] for an SYK model [ ], for instance in connection with aspects 47 63 – , ]. These models are generally defined in terms 35 62 – 2 – ]. 30 16 – 28 ]). However, the dynamics studied in these works is not 65 ]. As a parallel development, these ideas also had important 34 – 31 , 28 ] in the context of measurement-induced phase transitions. Indeed, in our 60 -level systems (qudits), sequentially updated by randomly chosen unitary gates ]. The U(1) charge conservation is an analog of the energy conservation. – d 52 51 , 50 In the past years, several works have appeared discussing ideas and techniques related In the rest of this paper, we focus on two aspects of the dynamics. First, we study In this work, motivated by the recent technical advances in the study of RUCs, and in- Due to the intrinsic complexity of generic many-body quantum systems, several works but there theinteractions. authors considered Next, random quantumwith global mechanical our Hamiltonians, setting evaporation were protocols with investigatedexternal in displaying no environment refs. some (see notion [ also analogy of [ Brownian, local and is analyzed by means of the Keldysh formalism. of the system. to those of thein present refs. paper. [ First,model we no projective note measurement that isis our taken, eventually and setting traced we differs consider over from instead in those an environment our studied which calculations. A similar setting was studied in ref. [ lighting the implications of conservation laws andone the is emergence intrinsic of two to different the timethe internal scales: system-environment dynamics coupling. (the scrambling Second, time), followingtime while Hayden needed the and to other Preskill retrieve depends [ informationthe on initially environment qudits, injected and in how the this system depends from on the measurements knowledge on of the initial configuration the maximally entangled state.black hole This in provides flat a space,zero for [ more which realistic the toy entropy model after the of Page evaporating timeanalytically eventually and decreases numerically to the growth of the second entropy R´enyi of the system, high- driven by the continuous limitcoupled, of but certain with 2-local only RUCs.quantitatively two These of the consist them contribution of interacting of qudits atbling the nonlocally a of environment time. and information. This internalU(1) setting dynamics charge Furthermore, allows conservation, on us we which to the evaporates consider study to scram- a a unique modified vacuum tensor state, instead network of model reaching with extensively studied in theof past entanglement few spreading years and [ thermalization in isolated systems [ spired by the Hayden-Preskill evaporation protocol,many-body we consider qudit the system dynamics of coupled a to quantum an external environment, where the time evolution is of a set of (RUCs) or time-dependentturns random out Hamiltonians that these (continuous systemsto Brownian investigate are quantitatively dynamics). typically several fast interesting scramblers featureschaotic It [ that systems are [ expected inramifications more in realistic condensed matter and many-body physics, where local RUCs have been tite mutual information defined in ref. [ on the topic relied onunitary the circuits study (RUCs), of originally a introducedand class within of continuous quantum information Brownian simplified theory dynamical dynamics [ models [ given by random (historically introduced in the context of disordered superconductors [ JHEP04(2020)063 ], 86 . We d C , chosen , interact. i,j ' j U ) j ( E . At each time and , h ) i j , corresponding to we introduce our ( S t/n and with a tunable -local circuits) were h . Finally, the most 2 k = S 5 t → ∞ ]. In particular, a Page- M , with 82 = ∆ ) ], we would like to construct – j ( E 6 1 h − 66 j -level systems (qudits), denoted t =1 of a unitary operator M j − ) j j ( S M d N t h = ) and in the presence of a U(1) conserved ⊗ and ) E i ( S 3.1 H N h , so that , placed at random positions – 3 – t and ) S ) j (the environment). The Hilbert spaces associated ( S j/n h E =1 = ( N j j t N ], and employ different techniques in our calculations. ] for an experimental implementation. Furthermore, we note 87 = ] within a quantum teleportation protocol related to the setting 84 ], based on recently-developed concepts of many-body quantum . Let us begin by considering a discrete process, and divide the S steps 83 we analyze the growth of the entanglement both in the case of , two qudits in E H 87 1 n 3 p , while we report our conclusions in section 4 and ]. However, in most of these examples the evolution is very carefully . We consider two sets of ). The retrieval of quantum information initially injected in the system (the system) and S 1 ] into 81 , S , t 3.2 are then 79 , E ], where global random unitary transformations (instead of 68 , and 85 67 We model this process byout the of action a on suitable random ensemble; S with probability Motivated by the Hayden-Preskill evaporation protocol [ The rest of this manuscript is organized as follows. In section We also mention that very recently the effects of decoherence on information scrambling It is also worth to stress that over the years many qudit models have been introduced 1. step, the system evolves according to the following rules: the physical situation where thelarger number than of in degrees the of system. freedom in the environment isa much quantum circuit which implementscoupling a between fast-scrambling dynamics for time interval [0 We start by introducingdepicted the in model figure studiedrespectively in by the restwith of this work, whichanticipate is that in pictorially our calculations we will always take the limit technical aspects of our work are consigned to a few appendices. 2 The model model, while in section Haar-scrambled local unitary evolution (section charge (section is studied in section but there the authorsto focused our work, on the aalso emergence been different of shown setup in a and ref.chaos. Page [ quantities. curve However, in in Second, a thisfrom analogously unitary work the toy we one focus model studied in on for ref. a a [ specific black microscopic hole model has which is different has been analyzed in ref. [ of this paper, see alsothat ref. the [ Hayden-Preskill protocolin with ref. a [ U(1)considered. conserved charge Finally, two has papers closely been relatedFirst, studied to a the before random present quantum article appeared circuit very model recently. for black hole evaporation was studied in ref. [ to capture aspects oflike the behavior in black time holethese for evaporation [ the process entanglement [ entropyengineered has and already allows been one observed to in only study some numerically of small system sizes. JHEP04(2020)063 . ]. S S 6 are (2.1) (2.2) 2 . The in the S ∪ E p 2 . p E display a − t and 1 → ∞ , while a swap 1 p and t p physical quan- ∆ M S − 1 a random unitary t Nλ at random positions to scale with the time = E 1 2 . We consider a system averaged p p , so that 2 t ∆ and 1 . As we will comment on again p Nt/ N M with probability and one qubit in t , S S ∆ t , 1 ), yielding a continuous dynamics for ∆ 2 Nλ λ qudits respectively. The evolution is driven by the – 4 – = = → ∞ M 1 2 . At each infinitesimal time step ∆ n p p E and , one qudit in N and 1 p S − 0 (namely 1 ≤ → 2 t of environment qudits to be p consisting of . M t are two positive real numbers. Note that while both E , they have a different dependence on t ∆ 2 2 λ λ = is applied to randomly chosen qudits in as 2 p t i,j , and . Pictorial representation of the model introduced in section U 1 λ are swapped. This models the simplest possible interaction between between a qudit in the system and one in the environment (randomly chosen) is applied with with probability In the rest of this work, we will focus on the computation of The above rule defines a quantum circuit with discrete time steps. It is convenient 2. l,m will then play thechoose role the of number acontinuous “qudit” limit. reservoir. In the discrete dynamics,tities: it at is each enough time to step this amounts to averaging over all the possible choices of pairs proportional to ∆ later, this ensures that theinteraction with internal the time environment, scales as areWith it much is the shorter assumed above that within choices, those thewell Hayden-Preskill related expectation defined protocol to limit values [ the for of ∆ Importantly, observables we computed will at be time interested in the limit of an infinitely large environment, which where is always identical in each copy. to take a continuouscomputations. limit In of order the to former, dointerval so, which ∆ we allows choose us the to probability simplify some aspects of the Note that at eachrandom time choice step of interacting the qudits system shouldsee be is later, considered not this as evolved means “fixed with once that chosen”: probability when as 1 multiple we replicas will of the system are considered, the circuit continuous limit of a randomwith quantum a circuit tunable which coupling implementsoperator a between fast-scrambling dynamics for W probability Figure 1 and an environment JHEP04(2020)063 , 1 − =0 d j (3.5) (3.2) (3.3) (3.4) (3.1) i} j {| respectively. In i
E 0 , which is naturally } )] Ψ | t ( ⊂ S S ρ . Denoting by K and K ⊗ to preserve the U(1) sectors i . )] ) t S 0 t , ( i,j ( , we will consider two distinct Ψ S | S )] U ρ } t i,j ρ . [ . Note that by construction there ) [ ( )] ]. For this reason, in the following t U S t ( K E K ( { 35 ρ 2 K ] for the case of spatially local RUCs. ) S t 2 ρ ρ , K ( )] tr t 42 ⊗ 2 K , ( ⊗ E M ρ K S ) ) log ) is proportional to the logarithm of the ⊗ 39 t t ρ tr i = 0. [ ( ( K E 0 t S 3.3 | K K K tr ρ ρ [ X [ = tr K – 5 – , we will assume that both the system and the E 2 i K E E 0 E K K h tr , log Ψ is pure, it will remain so for any realization, and its ⊗ X | S at time − { ) = tr K K − h t ( E and rewrite K ) = ) = t = tr = tr = tr S ∪ E P t ( K ( and , ) should be qualitatively the same as the averaged R´enyi-2 K t (2) S S K ( ) S\ M t , with the proper probability distribution. For a given fixed S ( (2) K = i,j 2 K S , as it was done in refs. [ U ρ ) one expects the effect of fluctuations in the disorder to be small, K j ( S E h ) is not the averaged second entropy, R´enyi as the disorder average N , which is defined as t ( ⊗ K ) K i P (2) S ( K tr h is drawn out of a Haar distribution. In the second situation, we assume a S i,j ) is the density matrix reduced to the subsystem ]. t U ( 38 K ρ , this is equivalent to averaging over all the realizations of allowed quantum circuits. t Let us now define Despite the importance of the von Neumann entanglement entropy, it is known that Finally, regarding the ensemble of two-qudit gates averaged purity However, for large so that the behaviorentropy of [ we focus on the related entropy; R´enyi-2 more precisely, we will compute We note that is taken inside the logarithm. In fact, eq. ( while we will consider different initialis product no states entanglement for between the latter is difficult to obtain in the setting of RUCs [ where a basis for theenvironment local are Hilbert initialized spaces inparticular product we states, set, denoted for finite by 3 The entanglement growth In this section we study thequantified entanglement growth by for means a subsystem of the von Neumann entanglement entropy physical situations. In the firsteach one, gate the internal dynamicslocally is conserved “maximally U(1) chaotic”, charge, namely namely wein choose the each product gate time A crucial point isparticular, that if each the individual initial realization statevon corresponds of Neumann entanglement to entropy a willover unitary remain realizations). evolution. zero for In all times (and so will its average of qudits and of gates JHEP04(2020)063 (3.9) (3.6) (3.7) (3.8) (3.12) (3.13) (3.10) (3.11) . In particular, )]. In order to t , ( 2 S ) only depends on and we may write t ρ ( , ,...,N K ⊗ K , ii 1 ) P i ] )] t , while in the last line, t ( + . ii . ( I S ) ) ) K | S t ρ k k [ ( ρ i i )) S . b a t E ⊗ ρ ( . ) S (2) S k ⊗ | ⊗ | t ⊗ ρ i ( k k ) , S + t i i , is invariant under permutation of ⊗ ( b ρ b I k ⊗ H | [ | | i S ii S ( ( ρ E + ¯ 2) H K | | ( S [ I ⊗ ⊗ ∈ 1 (0) | O K k ) ) S E ⊗ k k k ρ W =1 i N i i ⊗ H ) k O b a t − hh −L ⊗ ( I (1) S A | = S ⊗ | = ⊗ | – 6 – ρ (0) ] = K k k ¯ 2 the other two. Accordingly, the Hilbert space S i ∈ i ) (defined on the tensor product of two “replicas” ⊗ ii O ⊗ H k t ρ a ) a ) ,...,N | | ( | t S t 1 ( ( ( ¯ 1) ( S = i H ( S 1 ρ S 2 K 1 =0 + 1 =0 − ρ H ii ρ I − ¯ 1 and d X ⊗ | d X a,b K a,b ⊗ ) = = ( E t ) , and not on which sites belong to = W ( | = S t | ii S ( k : accordingly, the value of the purity e ) k H K K i ρ . In particular, we define S t i | is a product state and invariant under arbitrary permutations ii 4 ( ρ − tr ) | + i S ⊗ S I t [ I | = ( ρ S 0 | ≡ H E S ] under the continuous RUC introduced in section Ψ t ) k ⊗ ρ | d t ), and with ii d ) ( , ) , we derive in appendix t ⊗ t ( K ( 3.6 K ) S t S P ( ρ ρ | , then the initial state S → ∞ ρ S ⊗ | ). . As it will be clear from the subsequent discussion, this is also true for the ) t H M t is a swap operator exchanging the two copies of S ( ( ) is completely determined by the knowledge of e t K S H ) in eq. ( ) as a state in ( ) can be verified straightforwardly by expanding the scalar product. We note that K t ρ P | S ( [ X (2) S K E ρ S 3.11 ) = The formalism above allows us to write an equation describing the evolution of the ⊗ H t ( S k state in the limit when the initial state of qudits in qudits in evolved state the cardinality of P where Eq. ( Within this formalism one can recover the value of the purity using with corresponding to the four replicas is Finally, we also define with In the following, we label with 1 and 2 the Hilbert spaces of the two replicas associated where we introduced the maximally entangled state “tr” represents the tracethat over the entirecompute Hilbert the latter, space. it isus From convenient to this to interpret recall expression the the it operator H Choi-Jamiolkowski mapping is which clear allows where JHEP04(2020)063 , ) 1 − ) and (3.14) (3.15) (3.20) (3.16) (3.17) (3.18) (3.19) n ) is the t ( −P 3.14 n n , P P ( , n | 2 ). As we already N ) into ( j λ + j,k | + − ]. The two cases are 3.15 I 3.16 i hI h 0. Here ) to be Haar distributed j t − 42 j,k i ( , ≡ 0 ) directly. Instead, based i,j |I , +1 39 U ) 0 n t + . , ( P | 3.13 0 , which reads , ) is difficult to obtain for large +1 S − j,k 0 j,k e N ) + H U t . P − | ( 3.13 i hI 1 ⊗ 1 k − + i j,k n ∗ , with a block structure determined by ,...,N. j,k |I ) = P I t U =1 N N j ( X i,j ⊗ 2 1 = 0 ⊗ ) can be computed easily, and we have (see ⊗ | 1 i 2 U d ) + − 1 N t j λ ), which corresponds to a maximally chaotic | ) acting on ( i P j,k 2 n 3.15 d | − U = I – 7 – P ) + | i − j,k ⊗ , n S 0 j,k = 1 d Ψ ∗ j,k i | i hI grows exponentially with the system size. Luckily, in + U − U ). j,k − d j,k S E |I e (1 |I Lindbladian H (0) = 1 − qudits, while the initial conditions [corresponding to the N n 3.13 = + ≤ P | n j,k + j,k X j 5 0 t 30 25 20 15 10 n ) t ( S ( (2) for the Brownian Hamiltonian evolution of the amount of time needed before ] this was defined as the (2) n (cf. the main text), it is clear from the plot that ) S 28 ε = 1 and local dimension 25 N ] that the system ( – 8 – κN 1 ∗ ( 28 t λ . = 160 = 320 = 640 = 40 = 80 N N N N N B 8 20 ), we have that . In ref. [ 1 fixed, it was shown that 2.1 ) and its maximum possible value N t ( 6 15 (2) κN ) as a function of time, for t . Denoting by S ): entropy R´enyi-2 ( t < κ < b t N is qualitatively equivalent to a continuous Brownian Hamiltonian (2) κN 2 4 S ) represents a rare example where an explicit result for the dynamics 10 ) for large is a small positive real number, and 2 ], so that, up to prefactors, they can be identified in our model. N δ 3.19 33 ln( 2 0 5 0 5 0 . . . . . 5 0 0 1 1 2 ∼ − − − − − ). Subfigure ( κN ) κN ) t ( S (2) , where N , from which the emergence of a time scale logarithmic in κN k . entropy R´enyi dynamics for different subsystems, obtained by solving eq. ( ), we recover the same set of equations (up to prefactors) that was derived in ) ∗ ( 2 − a t ln( ( 0 ], see ref. [ ] for a Brownian Hamiltonian evolution. Thus, the internal dynamics driven by the )2 ∼ 0 δ 3.19 50 25 75 28 28 The features of the entanglement dynamics for In particular, it was shown in ref. [ It is important to comment on this result. First, we note that setting 100 150 125 ) κN ) t ( S = 0 (no coupling with the environment), (2) κN 2 ∗ ( to be also logarithmicref. [ in the system size in figure mixed state. Then, fordue 0 to the choiceso made that in eq.the ( system. Notedefined that, as following later the developments, time the needed scrambling for time OTOCs is to now decay usually to zero. However, the latter was shown evolution. This observation allowsour us model. to apply directly some of thethe results emergence of of ref. athe [ time amount scale of which time(1 is needed + logarithmic before in the purity of a subsystem of size technical, we reported it in appendix eq. ( ref. [ RUC defined in section t different times. We note that eq. ( of the entropy R´enyi can be obtained for open systems [ Figure 2 λ panel: entropy R´enyi-2 shows the difference between for increasing system sizes becomes larger than a small negative constant JHEP04(2020)063 ) s t ( = 1, 40 , the ), for (2) K 1 t < t S S λ ), from t 3.19 ( is swapped larger than (2) K 30 S j ) as a function K t ( (2) n S . As a consequence, is a qudit in ) as a function of the ): t E t j 20 ( b . When stand for “scrambling” j (2) n p . We see that for and S 4 j S\ and 10 s S\ and ) as a function of the subsystem t = 15 = 45 ( j . On the other hand, when both ), while, due to the normalization n n ) starts to increase with a constant E t (2) n N , even though the environment itself ) ( b = 10. Subfigure ( S ( S 0 2 (2) ln( K and λ S . 0 . This is due to the fact that, in each 40 30 20 10 as mainly due to the internal scrambling ∼ ). We see that there are two relevant time n ) t ( S , n S b N (2) s ): entropy R´enyi-2 s s ( t t a 3 ). – 9 – = 1 and 60 15 = 45 qudits. The parameters are chosen as t < t . ) predicts the purity of any subsystem to remain 1 3.19 − E λ between for large N 3.19 remains a pure state, since the evolution only amounts i s is not equal anymore, since the larger of the two can 50 ) for t 0 6= 0. We can immediately appreciate that the effect of | t S ( 2 K λ (2) K p ). After a time 40 t and S t S\ ( = 60. Subfigure ( i 1 for all values of ), and = 0 eq. ( K 1 | \ t 1 N ≡ 2.2 and (2) n N 30 λ S ) t K ( n , at which saturation occurs (the indices p 20 P t ) and ( ) = 2 and a , the latter becomes entanglement between ) we report the numerical values of ( a E , for different times, with d 2.1 0 25 5 0 25 5 75 ( ...... n 10 3 = 5 = 6 = 7 = 0 = 1 = 2 = 3 t t t t t t t . entropy R´enyi dynamics for different subsystems, obtained by solving eq. ( 0 5. 6= 0, the entanglement growth is non-trivial. In the following, we present our results . , for different times 0 2 n To verify this, we have computed numerically the time derivative of It is particularly interesting to follow the time evolution of a subsystem In figure Next, note that for 20 10 30 n ) t ( S (2) = 1 does not remain in a pure state, and its entanglement grows in time. We also see that , λ 1 2 can interpret the increasedynamics. of Based onchoice this in picture, eqs. we ( expect which the emergence of different regimes is manifest, cf. figure half of the system size,scales as that displayed characterize in its figure qualitativeis behavior: essentially for on short top times, of slope the entropy up R´enyi to a time and “Page” respectively: the use of these names will be justified in the next section). We with a qudit in S the entropy R´enyi of accommodate more entanglement with size the environment is to increaseconsists the of entanglement a of productinternal state. dynamics This will is generate due entanglement to between following mechanism: if constant, namely realization of the quantum circuit, to an exchange ofλ qudits based on the numerical solution of eq. ( local dimension subsystem size of time for subsystems containingλ 15 and Figure 3 JHEP04(2020)063 , ), 1 2), 8 a λ ( 4 (3.21) (3.22) (3.23) = 50 = 100 = 200 = 400 = 800 > N/ defined in N N N N N | ): indeed, 7 λ K s | . 3.19 2 λ ). ln 2 6 (with 1) 3.19 t d − ⊂ S d ( it approaches a constant ) as a function of time, for 5 t K ( (cf. the main text). In both p s 5. ) = (2) . t κN t S ( , so that only the term propor- , 0, one can make the assumption N = 1 4 directly from eq. ( S ) . Nevertheless, we can make the t P < t < t 2 2 2 ( → t λ = λ s λ d d t . 2 N s is a positive small number. We see s ) λ b P K ( ε 2 = 1, 3 1) λ 6 4 2 0 0 8 1 ) ln 2 ...... ln(2) 1 1 1 1 1 2 1 t λ − κN /dt ) t ( dS ( d (2) d 1) N ( 2 d P − , where , while for to be the amount of time needed in order for λ ε , and also for general 40 ) is dominated by the term proportional to d − – 10 – p 2 N ( , so that t + λ = 2 and ) n a = 40 = 60 = 80 . = 2 ' − ( + d 3.19 ), as we also verified with a quantitative fit. On the ) = s λ t ) the system is almost maximally scrambled. Then, N N N 2 t N ) s ( N t λ and ) is in perfect agreement with the numerical solution to − N ( s N t d 1 N ln( 30 P d . The constant dotted line corresponds the value λ p ln( ' P 2 3.21 t N ∼ d ) ∼ t s ), we consider the case log ( t s n t d d P 3.21 t 20 precisely, by defining it as the amount of time needed in order for − t > t ) is non-vanishing. In the limit s ) that ): time derivative of the entropy R´enyi-2 ): same data shown in the region close to t 4, local dimension b b a / ( ) = t ( 3.19 4 = 3 (2) N , so that indeed κ 10 . It is not straightforward to compute S t N d d to become smaller than in eq. ( to be smaller than a small positive constant, and as it is clear from figure one would get ∼ the absolute value of the latter becomes comparable to the term proportional 2 → ∞ p n . Subfigure ( s λ ) for arbitrary values of t t ). Subfigure ( /dt /dt ) ) N 0 and both contribute in a non-negligible way to t t ∼ ( ( 2 1 0 6 5 4 3 3.19 We can estimate 3.21 2 κN as /dt ) t ( dS t λ (2) (2) (2) K K 2 λ dS clearly from figure other hand, one candS define analogously one has Remarkably, we found that eq. ( eq. ( suggesting that it should be possible to derive it rigorously from eq. ( for large and so In order to motivate eq.tional ( to that that after a time s while at short timesfor the r.h.s. of eq.to ( conjecture different values of the systemeq. size ( figures we chose the derivative is large and increases with Figure 4 JHEP04(2020)063 ) 3.24 (3.24) (3.25) (3.26) ]). 67 . ] for the case N 42 , depends on the , ∼ p t p 39 t 2 Haar-random unitary × we have ]: namely it initially grows but , 2 51 is initialized in the product state E =2 , Q ), while the other, j U . q N ! =1 ln( Q X ∈S∪E ∼ 0 0 0 1 U j s t = – 11 – =0 = Q This difference between a black hole and random j U q S∪E ] we consider gates of the form 1 1 and the second block is a 2 Q , with s × 42 t = , ], this was introduced by hand, moving qubits from black hole to 86 39 i,j U have some special structure, as done in refs. [ i,j U , and for the RUC constructed in section E = 2. Then, following [ d grows always monotonically with time, even if . On the other hand, in a unitary black hole evaporation process, one expects that We implement a dynamics with a U(1) conserved charge by imposing that the two- It is difficult to introduce energy conservation in tensor network models, but it is In summary, the above analysis shows that in the presence of both internal dynamics M In some toy models such as ref. [ ⊂ S ⊗ 1 i 0 on each site. the bath at each time step. defines a dynamics conserving the charge where the charge operator is where the first and last blocksmatrix. are 1 Since the interaction with the environment is driven by swap gates, eq. ( qudit unitary gates of spatially local circuits.namely For the rest of this section, we will focus on the case of qubits, possible to introduce abath U(1) is infinitely charge large conservation,black and which hole initialized charge plays will in gradually a decrease the similarzero-charge and zero approach state role. zero (i.e. is in lowest) the unique, When charge finallimit the state. “vacuum” the (for black state, As a hole long the as very entropy the different will approach eventually that vanish achieves in similar the phenomena, long-time see ref. [ tensor networks originates fromlong-time the limit, absence the of blackradiation, energy hole while conservation returns the in to random the aentanglement tensor vacuum latter. entropy state network between model In since the system approaches its the and a energy the random leaves bath. with state the with large In the previous subsectionK we have seen| that the second entropy R´enyi the for entanglement follows a a subsystem “Page-like”starts behavior in to time decrease [ inthe the black middle hole evaporates stage completely. of the evaporation, and eventually vanishes when and system-environment interaction, two distinct timewith scales emerge: the one internal can scrambling be time interaction associated with 3.2 Random dynamics with a conserved U(1) charge JHEP04(2020)063 . ] ], so ii 39 34 6 (0) C (3.33) (3.30) (3.31) (3.32) (3.27) (3.28) (3.29) S ρ = 1 and ' . 2 1 ⊗ d eff j − h π = (0) i } 0 S D d ρ . | | live in a single local s D , where , , {z n − i QQ k eff N k , while D h Q Q } | = i ⊗ · · · ⊗ | = i hI 1 δ δ , D + + 0 | s QQ β β { and . , , δ δ |I j 1 i i i i ⊗ · · · ⊗ Q Q Q C 1 | = = d k γ γ eff 1 , 0110 0011 1111 ⊗ · · · ⊗ | i Q + + h j i α α } | − B s Q δ δ | 0 ) introduces additional computational = and of the initial state k k I , i ≡ | s i i i ≡ | i ≡ | QQ i L . 1 eff S B D | 6 A hI 3.24 ED | | e | 0 H k C n {z , γδδγ γγδδ Q i | | j ⊗ j j s 1 ) has to be replaced by QQ s Q | i i 6 I i ⊗ · · · ⊗ | – 12 – , , , I , C 0 i i i i | | 3.16 k 0 differential equations for the purity, and a different | 1 π Q ααββ αββα can be written in terms of the following 6 states [ d N | | 1 − 1100 1001 0000 j N 1 S i ) efficiently. Luckily, one can exploit an observation of Q ∈ k 2 Q t X ( π d d ], eq. ( Q X X j αβγδ αβγδ ! k (2) i ≡ | i ≡ | i ≡ | K Q 39 D Q Q 0 = = S A X C n | 6= |I | | ! j X E E C Q ) makes the computations considerably more involved. In par- k k = i n X s ! Q Q D j j = B X + s − + Q Q 3.27 n , n ! I I = C A 1 n ! , n j,k 1 B U n ! are states living in the tensor-product of two local sites of the four replica , n 0 A i n ! k , n Q N is a matrix acting on the space j 1 eff S Q √ ]. Namely, the states e i,j , n H |I 0 U = 39 effectively takes place in a Hilbert space We start by introducing the following class of permutationally invariant states on the The above consideration becomes particularly powerful when combined with the un- The form of eq. ( Averaging over unitary gates of the form ( n | = 2. j,k 1 space derlying permutational symmetry of the operator Indeed, this allows usand to obtain exploit a an logic efficientexact which scheme fashion. is to similar compute to the the evolution one of developed in the ref. system [ in a numerically Once again, we stressspace that of the the states fourU replicas. This meansthat that the evolution dictated by the averaged gates ref. [ where, for the case ofd qubits, the Greek indices take value in ticular, one can notstrategy derive is a set needed of to obtain Here space. In terms of single-site states, they can be written as difficulties with respect to theing case of the Haar-distributed results operators. derived in In particular, ref. by [ exploit- JHEP04(2020)063 , ), perm ⊂ S 3.18 (3.34) (3.35) (3.36) D K , ) reaches i t ( Ω | , we see from (2) κN D p t n S , the initial state † D the entropy R´enyi a L . In practice ), and then comput- 6 N C ]. After n at which 3.34 = 80. This was done by 51 is a vacuum state. One p † C t a N i , Ω ) | B 4 ) admit a simple expression in . n N j ( ) for different subsystems i t † B 3.12 1 ( O a | are invariant under arbitrary permu- + (2) K , while = 80. In rest of this section, we report L A 5 S 2+1 n the Page-time [ j,k N N N δ N/ p O ) displays another qualitative difference. t † A t ), is a fixed point for the internal dynamics. = ( = j a = in the vector basis ( j ) is only due to system-environment coupling (2) i K t 1 i 3.18 L 0 ( S n | † k – 13 – 2 + 5 5 (2) K , a † 1 defined in eq. ( =1 j S j a N/ O . We call N solving the system of differential equations encoded a considered in the previous subsection. Nevertheless, 2 ii . h ) is a non-monotonic function. For a given subsystem = t 0 κ ii K ( n N i ) = t S 0 W N/λ ( (2) K | ) form a basis of the Hilbert space in which the dynamics † 0 Ψ S S a | 1], we have verified that the time ρ perm , ! 3.34 ⊗ , defined in eq. ( [0 D = 0). For this reason, we consider in figure D = 0 and ) S t n ∈ 2 i ( ! i λ S κ † k C D ρ . We immediately see that the qualitative behavior is different from n | ! , a , n 5 † j , to which we refer the interested reader. B ) decreases and approaches zero exponentially fast, with an exponent C and a t n h ! ( C , which would be unfeasible for 1 , n and the vector A L B (2) κN κN n -operators. Since we won’t make use of them in the following, we report ! ii S a 1 , n ] = = k n ). Note that in this way we did not need to diagonalize exactly the matrix A (0) | ! 0 S , a , n K ρ j n | 1 that 3.13 a √ ⊗ , n ] ), for the initial state 5 t 0 Besides its non-monotonic behavior, We first consider the case where the system is initialized in the product state ( Since both the initial state and the Lindbladian = ( 34 n (0) | , with S (2) K ρ (there is no evolution if S that does not appear to depend on Indeed, the initial state of Hence, at shortscrambling, times, since one the initial can growth not of distinguish clearly the contribution of the internal as reported in figure the Haar-scrambled case, since K its maximum grows linearly with figure ing the evolved state in eq. ( associated with the numerical results obtained by following the above procedure. and study the time evolution of the entropy R´enyi and thus grows onlyis polynomially still (rather very than large exponentially)we for with were the able values to of implementing perform the numerically matrix exact corresponding calculations to up to tation of qubits, the statestakes ( place. Crucially, the corresponding dimension is where [ of the advantages of| the bosonic representationterms is of that the thethem operator in appendix as [ Importantly, we can rewrite these states by introducing a set of bosonic creation operators JHEP04(2020)063 , ): i 0 the | and 20 200 200 S 3.36 200 = 45 = 15 = 50 = 60 = 70 N e (cf. the H n n N N N s t 15 150 t 150 15 = 45 qubits. 100 t 10 ). The evolution − N 3.36 = 60 and subsystems 5 t t 0 100 N 100 4 2 0 2 4 = 2. The plots correspond − − n ) t ( S ln . The evolution parameters 2 0 (2) λ 0 5 0 . . . N 5 2 0 κN /dt ) t ( dS 50 (2) 50 = 1 and ) 1 b ) ( b λ ( 0 ): time derivative of the entropy R´enyi-2 0 b 0 6 4 2 0 30 20 10 κN n ) t ( S /dt ) t ( dS (2) ) for different values of the system size (2) t ( ) as a function of time for t (2) κN – 14 – 120 5 5 ( . . 200 S (2) n = 0 = 0 = 1 = 1 ): ) S κ κ κ κ a a 10 and increasing values of ( . = 15 = 45 / 100 = 60 and subsystems containing 15 and n = 2. Subfigure ( n n = 30, = 60, = 30, = 60, 2 N = 7 150 λ N N N N κ 80 ) as a function of time for ) for t t ( ( . It is straightforward to see that in the four replica space (2) t n (2) n i t 100 60 = 1 and S S ). Subfigures ( 1 = 2. The inset shows the same plot in the time region close to (obtained after averaging over the initial configurations) is indeed | 1 ): entropy R´enyi 2 ): 15 = 45 qubits. The plot corresponds to a random evolution in the presence a λ λ b ii 3.18 − 40 (0) N 50 S ρ ) for the same values of t ⊗ = 1 and ( 20 1 ) (2) n λ (0) . Subfigure ( a . entropy R´enyi ( S 0 S 1. Subfigure ( , ρ | 0 5 ) as a function of time, for 0 . 5 0 t 20 10 30 κN n 15 10 30 25 20 S ) t ( ( ) t ( S (2) (2) = 0 (2) κN namely, we average over alland the product rest states are where setstate to half of the qubitspermutationally are invariant, and initialized we to can employ the approach explained above. main text). Note that this state isa not protocol invariant under where not permutation only of wean qubits. sample average over Accordingly, different we over realizations consider all of the the RUC, initial but we product also states take obtained by permuting the qubits in ( containing 15 and of a global U(1)parameters conserved are charge, set while to S the system is initializedare set as to in eq. ( Figure 6 to a randominitialized evolution as in in the eq.κ presence ( of aInset: global ln U(1) conserved charge, while the system is Figure 5 JHEP04(2020)063 N = 2, ln 0 2 -axis). (3.37) (3.38) . 0 ∼ λ x s t ). Subfigure 2 . 0 . If the system 3.37 = 1 and S ). Although the 1 t ( λ 4 . 0 ). (2) t κN /N ) ( t S ( S S Q 6 Q . 0 ) defined in eq. ( t . 8 ( . 0 S j Q ) q b ( . coupled linear differential equations, ) (note the inverted scale on the 0 ∈S t 2 . j X ) ( 1 t S ( )] 0 t 20 10 30 Q S N N +1) ) t ( S ( (2) ) as a function of S Q t N ρ ( [ ) a separation of time scales for the initial (2) E N a effects, we can see the same qualitative behavior – 15 – S ) = ( t S ( N 200 6 BH ). The evolution parameters are set to ) = tr D ) depends on the initial state chosen for t t ( ( 3.18 S 150 Q BH D ), then the effective Hilbert space dimension will first increase, is the Page time. ) can be carried out using the very same techniques outlined above t t 100 N ( = 60. 3.18 . Of course, the evolved system state will have nonzero projection ) the time derivative of the entropy R´enyi-2 ) as a function of the charge S t b ): time evolution of the system charge ( ∼ ( N Q , where we also show a N Q for the Haar-scrambled dynamics. In particular, after a time p (2) 6 N t 7 4 S 50 ) a ( ). In order to make this more transparent, and following the previous section, , where p . Subfigure ( 0 3.36 2 0 8 6 4 0 ...... Q S 0 0 0 0 0 1 /N ) t ( We note that the dimension of the Hilbert space associated with a given integer value Finally, in order to push further the analogy between our model and a unitary black As expected, we see from figure of the charge is < t < t ): entropy R´enyi s b “black hole” Hilbert space dimension associated with the averaged charge as We see that the behavior of is initialized as in eq. ( the details of the computationdisplayed here, in and figure only report our final numerical results.Q These are onto different sectors of the charge at a given time. Nevertheless, we can define an effective The computation of for the second entropy. R´enyi In fact,only we involve two note replicas, that instead the ofcan calculations four. be are In obtained simpler, particular, as since it the solution turns they which to out can a that be system the of average easily charge ( treated numerically. Since no additional complication arises, we omit t hole evaporation process, it is interesting to study the time evolution of the system charge state ( we report in figure results are now plagued bydisplayed larger in finite- figure the derivatives appear to approach a plateau, and it remains approximately constant for The plots correspond to athe random evolution system in is the presence initializedand of as a the in global system U(1) eq. size conserved ( charge, is while Figure 7 ( JHEP04(2020)063 = 0, (4.1) t by ac- C ) we used ], and allows 4.1 6 is initialized in a as the black hole, E ): in this case, Bob S a ( provides a microscopic 8 2 , ) for the initial state of the ) is captured quantitatively by into the system at time t ( 3.2 t , which is initialized in a given A S . Note that in eq. ( C∪E S . As usual, due to computational S 8 − 2. On the other hand, if the system is ) t and evolved by the RUC introduced in ( E N/ E S . Depending on the initial configuration of is initialized, instead, in a random product S ) is close to its maximal value, then Bob can E t ) + ( t ] – 16 – ( E , C C [ S , ) a ( I ) = t ( ] E ) plays the role of the “global” vacuum state. However, it , C [ , ) reaches the value ) 3.2 ), the effective Hilbert space decreases monotonically, as one t a ( ( I S (Charlie), holds a reference qudit which is maximally entangled 3.36 Q C is quantified by the mutual information = 0. The capability of recovering information on the injected qudit t E ], and then proceed to present our results. 6 . In particular, if E ], we then imagine that Alice injects a qudit 6 ], so that one can ask what is the minimum amount of time that is needed ) to distinguish the two settings in figure 2 a . Finally, we imagine that Bob wants to recover information on the injected qudit consists of all its exterior degrees of freedom (hence including Hawking radiation). 2 E First, let us consider the setting pictorially depicted in figure We recall that the information stored in a black hole is emitted in the form of Hawking Before leaving this section, we comment on the choice ( which tells us howcessing much those information in can befaithfully extracted recover from the the informationan initially reference injected index qudit into ( has no control over theproduct initial state configuration at of time theby system measurements on with the former. Thesection system is in contactby with only performing measurementsthe outside system, of the ability tothe faithfully mutual do information so between after different a sets given of time qudits, as we now explain. before such information canblack be hole. recollected In fromwhile order measurements to performed make outside contactFollowing of ref. with [ the our model,and we that interpret a third party model for the information-retrieval protocolus studied to by Hayden investigate and quantitatively Preskill severalthe [ aspects setting of of the ref. latter. [ We start by briefly reviewing radiation [ Page time. 4 Retrieval of quantum information In this section we finally discuss how the RUC introduced in section is natural to wonderstate. what In would this happen case, if interval, a so non-vanishing one charge would would expect besimilar that pumped as the into in the qualitative the behavior evolution systemnumerical without of at calculations the each the entropy U(1) time that R´enyi would conserved be this charge.random is We product have indeed verified state by the we explicit case. observe no In monotonic particular, decrease if of the entropy R´enyi after the initialized in the statewould ( expect in a more realistic unitary black holeenvironment. evaporation As process. we haveevaporation mentioned, process, this is where motivated ( by an analogy with the black hole and then decrease after JHEP04(2020)063 (4.3) (4.4) (4.5) (4.2) (in the is initially A 3.2 (representing E A ), but rather its . In the first case is in a maximally 4 4.1 S , where the dynamics . S ) e , . H t ), for both RUCs without ( ) ) , RUC t t ) B ( ( t 4.3 ( (2) (2) S S S S (2) C∪E∪ (2) C∪E ) initially holds a copy of the black ). S S − − B ) and ( ) ) − 3.3 − t t ) ( ( ) t 4.2 t ( ( B (2) (2) S∪C S∪C B (2) ), we imagine instead that the black hole E S S (2) E∪ b S ( S 8 ) + ) + = 0. The system is in contact with is injected, the black hole t t ( ) ) holds a reference system, namely an ancilla which ) + ( ( t ) + t C t ( – 17 – A ( (2) (2) )], the retriever ( C C b (2) C S S (2) C S S (the “black hole”) in a product state. A qudit , is defined in eq. ( S ) = ) = ) = t t K ) = t ( ( ] ] t ( ] ( A E B ] , E , C B [ C , E∪ [ , ) , (for the maximally chaotic RUC) and in section ) E∪ C a is equal to that of its complementary one (with respect to the , [ (2) ( a , (2) ( C ) I ). Accordingly, analogously to the previous case, in the following [ b I , t (2) ( K ) 3.1 ( b I ] (2) ( , and his capability to recover the initially injected information is )]. In this case, Bob can also access these qudits, together with those I B b E RUC ( E∪ , 8 C [ , ) b ( is initialized in a maximally entangled state with a set of ancillary qudits. I ), for a given system S t )], we initialize the system ( a in figure . Pictorial representation of the two settings considered in section (2) . In the second setting [subfigure ( K ( ) B S 2 It turns out that the formalism introduced in the previous section is adequate to In the second setting, displayed in figure Each individual entropy in thethe r.h.s. of approach the in above equations section presence can of be a computed by conserved exploiting U(1)onto the charge). computation In of particular, the in time each evolution in case a we four can replica map space the problem and with a conservedentropy of U(1) a charge. subsystem whole To space), see and this, rewrite we can exploit the fact that the R´enyi compute numerically the mutual information in eqs. ( entangled state with the previously[region emitted radiation system, which isin under Bob’s the control environment quantified by we will compute its counterpart R´enyi-2 where formed long ago,since. and Accordingly, that by Bob the time has the been qudit collecting its emitted Hawking radiation ever hole, namely limitations, in the following version,R´enyi-2 namely we will not compute the quantity in eq. ( Figure 8 [subfigure ( injected into the black hole,is and maximally a entangled third with party the ( formerthe at exterior time of the blacksection hole, including Hawking radiation) and evolved by the RUC introduced in JHEP04(2020)063 . ) t ( ] 12 ) dis- t → ∞ E∪B , ( = 10 = 20 = 40 = 80 = 160 ] C , and in [ , B N N N N N N ) . For both 10 b (2) D ( E∪ , N I , which have C | [ , ) ). Since these K b 8 (2) ( )] for the setting I W 4.5 hh 4.2 t 6 ) for increasing values ) and t t ( ] ( ) and ( ] . We can interpret this as B E (without conserved charges), 4 , C 4.4 N E∪ 2 [ , , ) C [ a , (2) ( ) b I 2 ): mutual information (2) ( b I )]. In particular, from the plot we ) ) [defined in eq. ( b b ( ), and increasing values of t ( b ( 0 ] ( 8 E ) and , 8 0 5 0 5 0 t C . . . . . E∪ C [ ( 2 1 1 0 0 , ] [ ) ( B , , b ] ) t ( I ) E ) correspond to the two different settings (2) a , (2) ( b C . Subfigure ( I [ , ) N a ). The only difference with respect to the steps (2) ( – 18 – I . Interestingly, we see that after a time scale of the = 20 = 40 = 60 = 80 , the mutual information reaches a small non-zero 3.14 ) and ( N ) that the information retrieval is much faster in the N N N N = 2. a N b ( d , where we report data for the maximally chaotic RUC , namely the scrambling time. 60 ln 9 9 N ∼ . We see that the functions s t 10 , the value of the mutual information after the scrambling time 40 t 8 ): mutual information = 2 and chose a 2 ), and increasing values of λ a ( )] for the setting displayed in figure 20 8 = 1, ) of figure 4.3 1 a λ ) a . Subfigure ( ( 0 0 5 0 5 0 . . . . . E C . In both cases, the mutual information has a monotonic behavior, although with 0 1 1 0 2 ] [ ) ( , , a Conversely, we see from figure We have repeated the same calculations for a RUC with a conserved U(1) charge, and We begin by discussing figure ) t ( I (2) N be explored by the system, hence generally increasing the knowledge on its state. reported our results in figure play the same qualitative features. Itto is subfigure interesting to ( note that, inis the setting larger corresponding than thatThis in is intuitive: the maximally the presence chaotic of case, conservation although laws still constrains vanishing the for Hilbert space that can case Bob holds aclearly copy see of that the the blacklogarithmic mutual with hole information the [cf. reaches system figure its size maximum value after a time which is order of the scramblingvalue, time which, however, is seen tofollows: decrease after with the the system scramblinginitially size time, injected Bob information, is and ablesize needs to in to only order wait reconstruct to for retrieve a a all small time of amount proportional it. of to the the black hole (no conserved charge).discussed above, Subfigures and ( display respectively of qualitative differences. In theclearly proportional first to case, the it system size reaches its maximum value in a time which is is driven by the Lindbladianpresented operator in ( the previous sectionto is in be the modified initial state forcalculations and each purity do vector individual not term presentthe in additional rest difficulties, the of we r.h.s. this report section of we them eqs. present in ( our appendix final results. displayed in figure [defined in eq. ( plots, the evolution is driven by thewhere maximally we chaotic RUC set of section Figure 9 JHEP04(2020)063 , ) t A ( ] E 10 (4.6) , C [ = 10 = 20 = 40 = 80 , ) N N N N a (2) ( I 8 ), a 6 corresponds to N t ). b 4 = 2). , and a fixed system size d n ). t 2 ( ] ) has a non-monotonic behavior, . E , j C i ) [ , b = 2 (and ) at late times. The plot shows the 1 ( ) 3.38 t | increases: in subfigure ( a 2 2. As we have already pointed out, in ( (2) 0 ( ] λ I 1 E n +1 , − 0 5 0 5 0 N/ n C . . . . . E∪ C [ ), namely over all the product states with 2 1 1 0 0 , O ] [ ) ( B , , b N = (where the last qubit ) t ( I ) j = 1, (2) a i (2) ( j 4.6 1 1 i I | λ 0 | in the product state – 19 – 60 n =1 ) to S j O = 10 = 30 = 50 = 70 n , for different values of has been proven for a particular regularized version N N N N = − 50 i 11 1 . The evolution is now driven by the RUC with a conserved 0 S πT ), we actually consider averages over all the initial states − 9 Ψ ) we report data for decreasing values of the initial charge | ). This allows us to exploit the permutational symmetry in b , where we set N 40 C 3.36 3.2 and ( t 30 ) and ( i a 0 | 20 ) we report instead the logarithm of the difference between the maxi- c ]. The analog of temperature dependence in our model is the charge 13 increases, while the opposite happens in subfigure ( , respectively larger or smaller than 10 n n ) − a . Same plots as in figure ( ), but we initialize the system 1 0 a , and an upper bound of 2 ( − T 0 5 0 5 0 8 . . . . . E C N 0 1 1 0 2 ] [ ) ( , , a In subfigure ( We report our results in figure In energy conserved systems, the Lyapunov exponent measured by OTOC growth ) t ( I (2) = 60. In subfigures ( = qubits initialized to realistic unitary evaporation protocol. Thistwo plots is display reflected a in different the qualitativedecreases fact behavior as as that, at short times, the mum value 2 of the mutual information and N Q the former case the effectivewhereas Hilbert space in dimension the ( latter case it is monotonically decreasing, as one would expect in a more As we have clarified after eq.obtained ( by permuting different qubits inn eq. ( and is entangled to the ancilla the four-replica space, and proceednumerically following exact the data very for same the steps mutual outlined information above to obtain figure of the OTOC [ dependence of the information retrieval0 time. or 1, We expect the Hilbert thatin space when energy size the is conserved charge smaller, is system.for leading closer to to states For a this with similar purpose, effect different as we charge. reducing study temperature To the this mutual end, information growth we first consider the protocol depicted in U(1) charge defined in section generically depends on temperature.ature Usually a slower scrambling occurs at lower temper- Figure 10 JHEP04(2020)063 . E 2 t 60 = 1, (4.7) . As =59 =49 =39 =29 =19 . The Q Q Q Q Q 1 8 ) shows λ c 50 12 )], so that ) saturates a 2 ( t , where we set ( t ] 40 10 E , 3.2 C , [ , ). We report our ) ) 2 c ( t 30 I 2. Subfigure ( + D.3 1 = 39, while smaller than the Page ) 1 c ( t ) for increasing values of 1 N/ − ( t 2 ) for different values of the 20 t 0 2 4 , so a retriever accessing 25 [cf. figure E C − ( − − ] [ ) ( , , a N ] )] t ( I ln[2 E 4.6 (2) (2) . This has a non-vanishing C∪E E ∼ , S = C B ) at late times. For all plots, the [ p )] for the setting displayed in fig- , t ). In this case, the initial state ) Q − and c b Q (2) ( 4.2 ( 50 ) I 2 S and t 8 . This is depicted in figure S + S 40 1 . Thus we study the mutual information t ( E is dictated by the same RUC for a time : this allows us, once again, to rely on the is approximately a maximally mixed state 30 t (2) E A S S 20 E ∪S ). In this case, we see that = 2). – 20 – ) + a d 2 ( =29 =19 =9 ) [defined in eq. ( t Q Q Q t 10 ( ) b ] , which we now discuss. ( 13 + E , 1 C t 0 [ 14 , 0 5 0 5 0 ( . . . . . E C ) 2 1 1 0 0 ] [ ) ( , , a a I ) t ( = 40 and initial charge (2) ( (2) (2) C I S and ) correspond to the initial state ( N b , respectively larger and smaller than n 13 50 ) = − 2 t 1 ( ] . After the Page time, 40 ) and ( E , E ) displays the mutual information − a = 60 (while we chose C a [ ) here is used to distinguish this protocol from those in figure , ( N ) c N for the same state (and different values of 30 c t (2) ( and i 13 I = ) t has more control over the configuration of the “black hole” when the extra S ( 20 ] Q . After that, the dynamics of 1 E t , C , we introduce a new qubit which is maximally entangled with an ancillary one, C =59 =49 =39 =29 [ . Mutual information = 2 and 1 , ). For this reason, we consider a different setting, which maintains some of its Q Q Q Q = 10 ) t ) a 2 a a t (2) ( ( ( for the data reported in figure λ , the RUC increases the entanglement between ). Subfigures ( I = p a p 8 t 0 t ( − 0 5 0 5 0 First, figure Next, we consider the protocol reported in figure . . . . . E C 0 1 1 0 2 ] [ ) ( , , a 2 ) t ( I = 2. For this choice of the parameters, the Page time is 8 = 1, < t (2) h . The plot corresponds to 2 1 1 1 t faster as time increases,time which is whatat we time expect. Indeed,qubit for is injected. permutational symmetry in thedeveloped four-replica so space, and far exploit tonumerical the efficiently results exact in simulate same figures the techniques dynamics (cf. appendix t λ where the index ( usual, we average over the choice of the qudit state between in a certain chargetime sector (with chargedenoted decreasing by in time),We as are interested we in discussed the retrieval earlier. of this qubit At in is given byprojection a maximally over all entangledfigure the state charge between sectors,features, so but we allows can usidea not to is tune vary to the arbitrarily initialize initial its the charge system charge, of in as a for product state, and let the RUC generate an entangled λ emergence of an exponentialinitial decay, charge which takes longer starts to first evaporate). for smaller initial charge (a larger Figure 11 ure initial charge ln evolution is driven by the RUC with a conserved U(1) charge defined in section JHEP04(2020)063 : in 60 = 39 = 39 = 39 = 39 = 39 = 39 and fixed ,Q ,Q ,Q ,Q ,Q ,Q decreases → ∞ = 39 and 0 0 0 0 0 0 ...... 50 p 1 E t Q , where we set = 30 = 35 = 40 = 45 = 50 = 25 > t 1 1 1 1 1 1 t t t t t t 1 40 and t . We initialize the 3.2 is evolved with the . For all plots, the 4 S p A t 2 ) for t . For this reason, we 30 2 t N ( ] E ∪ S \ E , randomly chosen, which is , 20 C A [ , is evolved with the same RUC, )] for the setting displayed in ) c (2) ( , leading to an extremely slow 4.7 I S 10 A A E ∪ S ) b ) is the effective dimension defined in ( 0 p . The system t C ( 0 5 0 5 0 . . . . . E 2 1 1 C 0 0 ] [ ) ( , , c 2 ) t ( I BH will be extremely close to the vacuum, and RUC (2) increases the mutual information saturates RUC D 1 1 t . After that, t 1 t = 2). – 21 – ) [defined in eq. ( = 60 2 d t t ) and ( = 39 = 39 ] = 39 = 39 p E t , ,Q ,Q ( C 0 0 ,Q ,Q . . [ 50 0 0 , . . ) c BH (2) ) the mutual information ( = 0 = 5 = 10 = 15 b I at time 1 1 1 1 D ( t t t t ) show the mutual information for initial charge 40 b S 13 = ln 2 will be maximally entangled within a given charge sector. Thus, S t 30 ). ) and ( E 2 . However, since the charge is conserved, the portion of the Hilbert a t s ( t , respectively smaller and larger than the Page time = 40 (while we chose ] 1 E 20 t , and N C [ , where , . The plot shows that as ) S S c (2) ( , Q I p 10 t log . . In this respect, it is particularly simple to understand the limit 2 . Mutual information ∼ ) . Pictorial representation of the third settings considered in section = 2 and t p ∝ a (the “black hole”) in a product state, except for a qubit ( 2 1 ). Unfortunately, we can not reach large enough system sizes to test this statement . Subfigures ( 0 s t λ t S > t 0 5 0 5 0 12 . . . . . E C 1 0 1 1 0 2 3.38 Next, we report in figure At ] [ ) ( , , c 2 ) t ( I t = 1, (2) 1 more slowly, which isfor due to the factthis case that the the configuration entanglementthere of between will be essentiallysaturation no of scrambling of information in space that can beexpect explored during theeq. dynamics ( is smaller thatquantitatively. 2 initial charge the retriever should bethe able scrambling to time faithfully recollect information on the injected qubit after Figure 13 figure increasing values of evolution is driven byλ the RUC with a conserved U(1) charge defined in section for a time Figure 12 system maximally entangled with an ancillaryRUC one, with denoted a by U(1) conserved charge for a time JHEP04(2020)063 40 , for Q ). Here = 39 = 29 = 19 = 9 1 t ( ,Q ,Q ,Q ,Q 0 0 0 0 . . . . 2. Subfigure . (2) N ) is defined as , where we set 30 1 S = 25 = 25 = 25 = 25 t ( = 0 1 1 1 1 t t t t s 3.2 δ t . For all plots, the 1 t ) is a monotonically 39 and a given time 1 2 t t 20 ( , namely s ). Here 1 t 1 t t ( Q < (2) N and fixed S Q 10 at which the mutual information will be small, leaving the retriever 2 t . In the plot, we chose E δ ) b ( − 0 0 0 5 0 5 . . . . . and E C 0 2 1 1 0 ] [ ) ( , , c 2 ) t ( I when the extra qubit is injected. (2) S ) as a function of S 1 t = 2). – 22 – , which is what we expect: if the initial charge ) for different values of the initial charge ( , as we already discussed above. 2 s d p t Q t ( 5 ] ) . a E > t ( , 22 C [ 1 , t ) is some small positive number. This is reported in fig- c 0 (2) 2. From the plot it is clear that ( . ) for various initial charges . δ 2 I 20 ) reaches the value 2 t 2 ( ] t ) for ( = 0 E 1 ] , 5 t . E C δ , ( [ on the system entropy at R´enyi-2 time ) , 17 1 ) is defined as the value of C ) [ t 1 c , 1 ( (2) (2) N ( ) t t , where ) shows c (2) N I ( (2) S ( ): scrambling time δ 0 b S s . I a ( t ) we can also extract the dependence of the scrambling time for infor- = 40 (while we chose 15 − b ) is decreasing with ( 14 2 N t = 39) (the Page time depends on the initial charge). In this case, we 5 . ( 13 ] 12 E Q , at which ( 2 C . [ p , 2 t 0 ) t . = 39), the entanglement between c = 0 = 40 (2) = 2 and ( . Subfigure ( 10 δ I N ∼ 2 ), where we chose Q λ ( a 1 ( p t 14 22 20 18 16 Finally, figure From figure 1 s t ( t ) 14 = 1, > t ): mutual information 1 1 b distributed or with aentanglement conserved displays U(1) a charge. Page-likemiddle In behavior stage the in latter of time, case,microscopic the we realization where “evaporation”. of have it shown the Finally, begins Hayden-Preskillquantitatively that the protocol we the to time for have evolution decrease information shown of retrieval, in the that studying mutual the our information between model different provides subsystems. a In this work, wecoupled to have an considered external the environment,limit where dynamics of the of certain time a 2-local evolutionsecond is quantum random entropy R´enyi driven many-body unitary displays by circuits. two qudit theinformation continuous different system scrambling We time and have scales the showncharacterized interaction that that the with are the the qualitative related growth environment. differences of to that Furthermore, the the we emerge internal have choosing the unitaries to be Haar- t with little control over the configuration of 5 Conclusions decreasing function of fixed see that is small, then the corresponding Page time is short. So, for mation injected at time the scrambling time reaches the value 2 ure the value of ( evolution is driven byλ the RUC with a conserved U(1) charge defined in section Figure 14 JHEP04(2020)063 ] ) ); t j ( 14 t – ( U 9 (A.1) U j,k . U ) = t . Choosing . ) = 2 ,...,N t E 1 . To this end, +∆ i j ii + t +∆ ) ( I t j ⊗ | ( t U ( S o o ) ) ρ U t t ⊗ ) t + ∆ + ∆ ( j j , so that S , we have three possibilities: t t j ρ ( ( t | 2 † † and one qudit in ). Then, we have j ρU ρU S t ) ) ( t t U + ∆ + ∆ is applied, so that j j t t k ( ( ) by applying a suitable unitary operator. j U U t , and the implications of conservation laws ( n n and N U E E j – 23 – tr tr ⊗ ⊗ S S H H 1 1 which we denote by no unitary is applied at time = 2 p ii ) − t S ∪ E 1 a swap exchanges one qudit in p a unitary between ) is obtained from 2 − + ∆ 1 t p p j t ( + ∆ S ρ j ], our model provides an ideal playground where numerical and analytic t ( ⊗ 83 U ) t fixed, we focus on an individual realization of the circuit. This defines a global + ∆ j j t t ( with probability 1 with probability with probability S Finally, when compared to holographic duality, our model gives us a toy model for the Next, it would be extremely interesting to consider the growth of local operators [ The RUC considered in this work can be enriched in a number of ways. For instance, we • • • ρ | The operator In particular, according to the evolution described in section We wish to writewe down start an with evolution the equation discretea version for of time the the state quantum circuitunitary introduced transformation in section on National Science Foundation underpart grant by the No. Department of 1720504, Energy the under grant Simons No. Foundation, DE-SC0019380. andA in Derivation of the Lindbladian for the entropies R´enyi Acknowledgments We are very grateful tooration Ignacio on Cirac, this Vedika project Khemani,Miyaji, and Norbert Zhenbin valuable Schuch, Yang, comments for Shunyu on early Yao,edges this collab- Shangnan support manuscript. Zhou from We for thank the helpful Masamichi Alexander discussions. von LP Humboldt acknowl- foundation. XLQ is supported by the boundary dynamics. It wouldbulk be degrees interesting of to freedom, use and a study tensor the network entanglement approach wedge to structure. describe number of qudits, possibly with a non-trivial internal dynamics. in our setting. While thethe effect literature of [ decoherence onresults the latter can has be been alreadyexplored derived considered in for in detail. large We plan values to of go back to these questions in future investigations. study the charge dependence of scrambling dynamics. have always considered the limitthat where are the environment non-interacting hasfeatures with an described infinite one in number another. of this qudits, work One are modified could by wonder considering whether an the environment with qualitative a finite The conserved U(1) charge provides a tunable effective Hilbert space size, and allow us to JHEP04(2020)063 . , 1 t E p ∆ and / (B.1) (A.5) (A.2) (A.3) (A.4) j j . n Nt ) to obtain . ] (A.6) . 3 ) defining the M ] ii C 3.13 ) ii 2 j , commutes with ) t p 3.11 j ( t S + ( , ∈ S ρ S 2 ] ρ , and hence having no C ⊗ and the environment ii + 1 coupled differential k 1 ) ) i j, k ⊗ p S j j 0 ) . Assuming t t N | . j ( j ( + t t 0 we recover the differential S S ii ( 1 ) ρ ρ , for S | t C . Indeed, let us denote by [ ) of ρ ( ) → . | ⊗ j,k )) for subsystems of size 2 [ S E ] ) t ) p U ρ j E ii t L 3.19 ) − ( t ⊗ → ∞ 3.11 j S 1 ) t j,k ∆ t ρ ( p | ( M ) results in ) into the equation ( U [ S − S − ρ E ρ ⊗ (see ( 2.2 | . Under this assumption (which becomes j ) ⊗ | 3.14 E ∗ ii j,k ) + in the environment has interacted before with j S I U t ] = (1 −L ρ h k ] = (1 ( ( – 24 – | ii j ⊗ S ii ) i ⊗ ) and ( n ρ ) t 0 | . In the limit ∆ t S [ j,k , W L 2.1 ρ 0 U E | hh , n + ∆ 0 ). Putting all together and scaling the probabilities + ∆ ⊗ = j = , W j t 0 1 ) as t ( ) | ∗ 3.8 j,k ( hh t C S ( S U ρ to be in its initial configuration =1 N n = ρ dt j 3.14 X P ⊗ E k n ⊗ d ), it is straightforward to compute 1 ) P N ) t t j hh hh h hh 2 n 1 ) = 1 − ( d N 1 1 − ( − λ − n λ N 2 2 −L 2 2 ( 2 ) can be computed. Using the identities N N 1 ( N ( λ λ N d | n N n − − − − − − − 3.16 ’th site of W k = hh as can be explicitlythis, checked by is comparing follows the action of the two sides on any state. From In this section we discuss ina more set detail of the formulas formalism thatto introduced are write in needed section operators for in numerical terms implementations. of We the start bosonic by showing how by considering separately theand three sets ofrespectively. terms The in differential equation the ( sum C Derivation of the relevant formulas in the bosonic formalism from ( and keeping in mindthis that results for in Next, the action of JHEP04(2020)063 ∈ i , z (C.5) (C.6) (C.3) (C.4) } ,... A . I -operators. i i # k a k i ) , ) ,...,N | k | 1 1 1 ) z z I | − z i h # i h π , x ∈ { x i h | 0 | ( , x ( I i ··· | i ) ( ⊗ # A , ∗ ⊗ Ω I | j ⊗ j i,z ) ⊗ ) | A j i c | t · · · | I ) t =: ( | A # t | i h ,A ) i h y k A y † A i h | ,A is antisymmetric under simul- | I | ( t a y c A | ( I 1 , z,t c − + ( ,A 1 ⊗ h 1 1 Λ I k + ( k A I j − ) c I ) | | k | ⊗ 0 c π t x,y , z ) t i , 1 | ) h 0 I t 1 I i h is instead symmetric, the sum is zero. | k i h c # y ) z,t y i h 1 | ) | ··· | , t ( y † 1 ( Λ i 1 z,t ··· | I a + ! ( i h Ω Λ ⊗ c ( ⊗ | i A y 1 j i 0 | , ⊗ ) j I I k ) x,y ( . ) 1 | A i j | I ⊗ | – 26 – ) z y !# i c z i | ··· ⊗ t 0 1 0 z i h I j a | I i h i,A + ) ⊗ | z x c # | x 0 i h | , | a ) j z † A ( !# x 0 ( i † y † 0 + | h I 0 a h a ( x a i h i c I | are symmetric matrices. We can rewrite ( † x h i . Since Γ x } z,t ), in terms of bosonic operators, together with the states i,A 0 z,t | a t 1 , !# c ( | h Λ z,t x,y 0 Λ t,z I ...N 1 N Γ Λ + c 1 z,t ↔ i, z,t 3.27 x,y . One can now prove a general formula, which can be directly x,y { } c † 1 Λ p Γ Λ x,y Γ = Λ = z a } + Γ 1 , ...N x,y x,y i i, ∪··· z,t 1 y X i Γ X c Γ { ...N x,y,z,t A ). X X 0 1 I x,y,z,t = x,y,z,t | { . + X ∪ ↔ x,y,z,t 0 1 = † 0 C.2 j