Regenesis and quantum traversable wormholes

Recent gravity discussions of a traversable wormhole indicate that in holographic systems signals generated by a source could reappear long after they have dissipated, with the need of only performing some simple operations. In this paper we argue the phenomenon, to which we refer as"regenesis", is universal in general quantum chaotic many-body systems, and elucidate its underlying physics. The essential elements behind the phenomenon are: (i) scrambling which in a chaotic system makes out-of-time-ordered correlation functions (OTOCs) vanish at large times; (ii) the entanglement structure of the state of the system. The latter aspect also implies that the regenesis phenomenon requires fine tuning of the initial state. Compared to other manifestations of quantum chaos such as the initial growth of OTOCs which deals with early times, and a random matrix-type energy spectrum which reflects very large time behavior, regenesis concerns with intermediate times, of order the scrambling time of a system. We also study the phenomenon in detail in general two-dimensional conformal field theories in the large central charge limit, and highlight some interesting features including a resonant enhancement of regenesis signals near the scrambling time and their oscillations in coupling. Finally, we discuss gravity implications of the phenomenon for systems with a gravity dual, arguing that there exist regimes for which traversability of a wormhole is quantum in nature, i.e. cannot be associated with a semi-classical spacetime causal structure.

When a circuit is disconnected from its battery, the electric current flowing through it quickly stops, due to dissipation.
Using modern language, treating the circuit and its environment as a single isolated quantum system, we say the current is scrambled among other degrees of freedom of the system. Once a signal is dissipated (or scrambled), it cannot be recovered in practice, as to do that one needs to have control over the full quantum state of the system, which in practice is never possible for a system of many degrees of freedom.
In this paper we discuss a new phenomenon, based on the recent discussion of a traversable wormhole [1][2][3][4][5], where the current signal can re-appear with the need of only performing some simple operations.
Let us first describe the setup using field theory language (see also Fig. 1). Consider two identical uncoupled quantum systems, to which we will refer as L, R systems. The Hamiltonians for them are respectively H L and H R which by definition have the same set of eigenvalues {E n } with respective energy eigenstates |n L,R . Suppose L, R systems are arranged in a special entangled state such that at t = 0 it is given by a thermal field double state [6] |Ψ β = 1 Z β n e − βEn 2 |n L |n R , Z β = n e −βEn (1.1) where |n denotes time reversal of |n . |Ψ β has the property that if one operates solely in one of the systems one finds a thermal state at inverse temperature β. Consider at some time t = −t s < 0 turning on an external source ϕ R for some few-body Hermitian operator J R for a short interval. 1 In the R system there is an induced expectation value J R (t) ≡ Ψ β |J R (t)|Ψ β , but there is no response in the L system as by definition [J L , J R ] = 0. As usual J R (t) will dissipate and decay quickly to zero after ϕ R is turned off.
Now couple the two systems at t = 0, with the total Hamiltonian where g is a coupling and V is an operator involving both L and R systems, e.g. a simplest choice is for some few-body operator O(x). The surprising result from the gravity analysis of [2] is that a signal will re-appear in the L-system if t s is larger than the scrambling time t * of the L, R system. 2 Note that here O and J are generic few-body operators which do not need to have any common degrees of freedom between them.
The purpose of this paper is to argue that this phenomenon, to which we will refer below as "regenesis," is universal for generic quantum chaotic systems, to elucidate its underlying physics, to study it in detail in a class of field theories, and to discuss its gravity implications.
A general result we obtain is that in a generic chaotic system for t, t s t * : (i) J L (t) g is supported only for t ≈ t s where · · · g denotes expectation value in (1.1) with a nonzero 1 The time interval is taken to be much smaller than t s . we turn on a local coupling between L and R for a short time, which we have approximated as a delta function in time in (1.2). At t = t s , the signal reappears in the L system if t s is sufficiently large. The reappeared signal is not identical to the original signal, but related by a transformation.
g; (ii) as a function of t s , J L (t = t s , x) g has the following behavior J L (t s , x) g ≈ C(g) ϕ R (−t s , x), t s t * (1.4) where C(g) is an O(1) constant depending on g. We thus find the "input signal" ϕ R from the R system at t = −t s regroups at t = t s in the L system long after it has dissipated! 3 The result (1.4) is insensitive to the specific form of L − R interaction V . The behavior for a system with t s ∼ t * is more complicated and will be mentioned later.
The essential elements behind the regenesis behavior (1.4) are: (i) scrambling in a chaotic system makes out-of-time-ordered correlation functions (OTOCs) vanish for t t * [7,8], and (ii) the entanglement structure of (1.1) which strongly correlates an operator inserted at (−t, x) with an operator at (t, x). Compared to other manifestations of quantum chaos such 3 Note that in (1.4) signals which are input earlier in the R systems appear later in the L system, so in fact what one finds is the time reversed form of the input signal.
as the initial growth of OTOCs which deals with early times, and a random matrix-type energy spectrum which reflects very large time behavior, the regenesis phenomenon concerns with intermediate times, of order the scrambling time of a system. Instead of making the signal to appear at the same spatial location x, by considering a slight variation of (1.1) one could also make the signal from (−t, x) to regroup at (t, x + a) for some a.
One may wonder what happens if we consider the same setup in a few-body or integrable system. In general, with g = 0, some response will be generated in the L-system: interactions among each subsystem will manage to communicate the effects of ϕ R to J L . But there are two crucial differences: (1) it will not be "regenesis," as in a few-body system (or in integrable systems) there is no dissipation, so the original signal in the R-system will remain there forever; (2) the signal generated in the L-system will depend sensitively on the specifics of an individual system and the operators used. In contrast, in chaotic systems, the behavior is universal, independent of all the details.
At first sight, the regenesis phenomenon appears to be miraculous: how can a dissipated signal regroup with a very simple operation like (1.3)? If one wants to be melodramatic, we could imagine that by turning on ϕ R , we create a "cat" in the R system. The cat lives for a while, and dies. Eventually her body will be fully scrambled with the environment. Now it appears that we could bring her back to life in the L-system by simply turning on a gV ! There are two important catches here. Firstly, the success of the operation in fact requires extreme fine tuning in how we prepare the initial state at −t s when we turn on the external source ϕ R . The state should be prepared such that as the system evolves to t = 0, the system is in the thermal field double state (1.1). This is a highly nontrivial requirement as the scrambling time t * could be macroscopic for a macroscopic system, and as we will see explicitly the regenesis phenomenon is somewhat fragile: various modifications could destroy the behavior (1.4). A second catch is that as we will discuss later (in Sec. II D), at least for the regime t, t s t * , the signal (1.4) is quantum in nature, i.e. the variance is always comparable to the expectation value itself, and one cannot cut down fluctuations using macroscopic measurements.
We also study the regenesis phenomenon in two-dimensional conformal field theories (CFT) in the large central charge limit which is known to be chaotic [9]. That is, we take L and R systems to be (1+1)-dimensional and described by a CFT. We consider where c is the central charge of the system, and ∆ O , ∆ J are respectively the scaling dimensions of few-body operators O and J. In this regime we can obtain the behavior of J L (t) g in detail by applying the techniques of [10][11][12]. In addition to (1.3) we will also consider two other types of couplings (in (1.5) α denotes different operator species) which were considered in [2] in the large k limit, and For (1.5) our CFT results are fully consistent with the gravity results of [2] for a (0 + 1)dimensional holographic system.
We will refer to (1.3) as a single-channel coupling, while (1.5)-(1.6) as multiple-channel, one from multiplicity of operators, one from spatial integrations. Their local spacetime structure is chosen to maximally take advantage of the entanglement structure of (1.1).
Here is a brief summary of the main features found in explicit CFT calculations (some of these features are also present in the gravity results of [2] for (1.5)): 1. In all cases, as a function of t s , one can separate the behavior of J L (t s ) g into three different regions: (1) t s t * (sub-scrambling region), where J L (t s ) g is exponentially small and can be considered to be zero for practical purposes; given by (1.4). Here the scrambling time is given by t * = β 2π log c 6π .  3. An interesting effect for (1.5) in the transition region is that, with a choice of a sign for g, the magnitude of J L (t s ) g could be exponentially large in g, to which we refer as resonant enhancement. See Fig. 2(a) for a cartoon.
4. For (1.5)-(1.6), C(g) in (1.4) is given by is oscillatory in g. G J and G are some constants. For (1. 3) there appears no oscillation in g, and we find for large g C(g) ∝ g −1 . (1.8) Our field theory studies also have important implications for the understanding of the traversability of a wormhole on the gravity side. In [1,2], the basic picture is that the two- there are virtual particles which can propagate between them. The non-traversability can be understood as coming from perfect destructive interference between the process of a virtual particle traveling from R to L, and the mirror process of traveling from L to R. Right: turning on interaction V subtly changes the entanglement structure and gives a phase for each propagation. Now the destructive interference is no longer perfect, resulting propagations of "real" particles between the boundaries. Note that the interference is not a process between "future" and "past" as the related two boundaries are actually spacelike separated. It is possible to boost the frame such that it occurs on one spatial slice.
the traversability appears quantum in nature, i.e. cannot be associated with a semi-classical spacetime causal structure. Instead it appears to involve breaking of a delicate destructive interference, see Fig. 3 for a cartoon picture (which is appropriate for t s t * ).
The plan of the paper is as follows. In Sec. II we present a general argument for the regenesis phenomenon, explain the quantum nature of the signal, and discuss its robustness.
We also present a simple qubit model as a contrast study of this phenomenon in a fewbody system. In Sec. III we outline the main steps of the calculation of J L (t) g in a two-dimensional CFT in the large central charge limit, with details of the calculation given in Appendix C and Appendix D. Details on the CFT calculation of the robustness of the phenomenon is given in Appendix E. In Sec. IV we analyze the results obtained in Sec. III and various Appendices. In Sec. V we discuss gravity interpretation of our results, including a detailed comparison with the results of [2]. In Sec. VI we conclude with various discussions, including future directions and experimental realizations.

II. A GENERAL ARGUMENT FOR THE REGENESIS PHENOMENON
In this section we present a simple argument for the regenesis behavior (1.4) for a general quantum chaotic system and discuss the robustness of the phenomenon, i.e. how it fares against imperfections of the preparation of |Ψ β . The results of this section are consistent with the gravity results of [2] for holographic systems, and will be further confirmed through explicit calculations in two-dimensional CFTs in the large central charge limit in subsequent sections.
A. More on the general setup As discussed in the Introduction we consider L and R systems in a thermal field double state (1.1). In this state, the expectation value of any set of operators which act only on one of the systems is given by the thermal average with inverse temperature β, e.g.
where on the right hand side the trace is performed in the left system and · · · β denotes thermal average at inverse temperature β. For notational simplicity we have dropped L labels. We will do this for the rest of the paper: below any quantities with no explicit labels should be understood as in the L system.
By definition any operators from L system commute with those of R system, i.e.
where x denotes spacetime coordinates, i.e. x = x µ = (t, x) and x are spatial coordinates.
Consider turning on a source ϕ R for some generic (hermitian) few-body operator J R , i.e.
perturbing the action S R of the right system by We will choose J such that its thermal expectation value is zero. Then at linear order in ϕ R we have For g = 0, there is no response in the left system due to (2.2). On general ground one expects that the thermal response function G RR for a non-conserved quantity to behave for large t − t or large | x − x | as where τ r , r are respectively relaxation time and length, both of which will be treated as microscopic, i.e. much smaller than typical scales involved in ϕ R . For a scale invariant system, they are both of O(β), see e.g. (3.1). Thus J R will quickly decay to zero in a time scale of order τ r after the source is turned off. Now with g nonzero in (1.2) we would like to see whether there is a response on the L side. We take the source ϕ R to be turned on for a short period around t = −t s < 0 such that J R (t) will have long decayed to zero before V is turned on at t = 0. From Appendix A, we find at full nonlinear level in ϕ R with |Φ defined as Expanding (2.7) to linear order in ϕ R we then find with (note we take both J and O to be Hermitian operators) The thermal field double state (1.1) has a rather specific entanglement structure between L and R systems. It can be checked that the state generated from a Hermitian operator J R acting on |Ψ β can be reproduced from the action of J L in the L system with a shift in imaginary time, i.e.
where η is the phase factor associated with time reversal on J and will be dropped subsequently as it will not play any role. Furthermore, The combination of (2.12)-(2.13) implies that where we have used L and R operators commute and (2.13) repeatedly. Note that (2.14) applies to a complex t with Im t ∈ (0, β/2). By using the above equation repeatedly we further find that where subscripts label different operators. Due to the entanglement structure of |Ψ β , we see from (2.14)-(2.15) that operators inserted at (t, x) in the R system are strongly correlated with those inserted at (−t, x) in the L system. In other words, there appears a "time reversal symmetry" between L and R systems. We will refer to such a pair of points as an entangled pair. This simple fact will play a key role in understanding the results of the paper. Moreover, the interactions (1.3) and (1.5)-(1.6) are chosen to involve couplings between operators inserted at entangled points, which as we will see makes the teleportation most efficient.
From (2.14) we have where in the second equality we have again displayed the usual behavior for a thermal twopoint function. An explicit example of (2.16) is given by (3.2) for a two-dimensional CFT, for which τ r = r = β 2π . Treating τ r , r as microscopic, we see that the two-side correlation function (2.16) is essentially nonzero only for t ≈ t and x ≈ x , and we will denote which is a constant from spacetime translational symmetries.

C. Regenesis behavior for quantum chaotic systems
Now let us examine the behavior of (2.7) (or its linear version (2.11)) for a general chaotic system. We will take x = (t, x) and x = (−t s , x ) with t s > 0. First consider g = 0. Since Thus J L (t) g (thus also G LR ) can be non-negligible only when t is of the scrambling scale t * .
Also due to the entanglement structure of (1.1), as manifested in (2.16), we expect W defined in (2.11) to be non-vanishing only when t s and t are close. So t s also has to be of order or larger than t * for G LR to be nonzero. Similarly for the full nonlinear expression (2.7).
When t, t s are of order the scrambling scale, even at linear order in ϕ R , the expression for J L g is complicated. We will study the behavior of W and G LR in detail for various choices of V in subsequent sections in two-dimensional CFTs. Here we show that when t, t s t * the behavior of full nonlinear expression (2.7) is very simple .
For clarity we will illustrate the main argument using the linear expression (2.11). Consider expanding the exponential e igV between J L and J R in power series of g, then at n-th order one gets correlation functions of the form where we have suppressed all the spatial dependence. Note that the precise form of V is not important and we have only schematically indicated that it has the form of some product of Now for a general quantum chaotic system, due to scrambling, we expect with t * the scrambling time of the system. Thus for t, t s t * , equation (2.11) should reduce to Notice that expanding the exponential e igV in (2.21) in power series of V will now give rise to only time-ordered correlation functions (TOCs). On general grounds, one expects that such TOCs to factorize at large time separations between J and O insertions, we then find 4 For ϕ R slowly varying at the scales of τ r , r , we thus see (2.9) reduces to Note that generically we expect e −igV to be complex and O(1) as already mentioned the operator V is designed to couple O L,R at entangled points.
It is interesting that the sole effect of turning on the interaction V between two subsystems is to generate a phase so that W is no longer real, resulting a nonzero G LR . Through the entanglement structure of Ψ β , information of the source ϕ R is already present in the L system, just as in the usual EPR story. Heuristically, the effect of turning on V is to turn this information into "real" physical signals.
The above discussion can be immediately generalized to the full nonlinear expression (2.7).
In fact for any few-body operator X L (t), setting all OTOCs to zero, we find in the limit where |Φ is given by (2.8) and we have introduced See Appendix B for a derivation of (2.26). For X L = J L , we then find Note that the factorization assumption can in principle be weakened or dropped. One only needs that (2.21) is complex and not small at large times.
The discussion of this subsection does not apply to t, t s ∼ t * for which we will examine in two-dimensional CFTs in Sec. IV.

D. Quantum nature of the regenesis signal
In this section we show that the regenesis signal (2.25) is quantum in nature. 5 We do this by comparing (2.28) and the corresponding variance with those in a standard response setup.
For this purpose, let us first recall the standard response story, 6 where U L is the unitary operation for turning on some source ϕ L in the L-system.J is the corresponding signal and δJ is the variance. Note that since both U L and J L belong to the L-system, (2.29) just reduce to thermal averages. In this context we will thus suppressed index L. We also denote the variance and fourth moment of J in the thermal state (recall J β = 0) as To make the discussion explicit, let us imagine a lattice system of interacting spins.
Suppose J is given by some operator at a single site, say σ z , then clearly bothJ and δJ are order O(1), and one needs multiple measurements to detect the effects of U . One can make life easier by measuring the average polarization, say, The content of this section is developed from discussions with Juan Maldacena, Douglas Stanford and Zhenbin Yang. The main conclusions have also been anticipated by them. 6 In a field theory we assume J is suitably smeared such that both J and J n are bounded operators with a finite norm and considering a source which acts on all spins U = P e i j ds ϕ L (s)Z j (s) (2.32) where Z i is σ z at i-th site and N is the total number of sites. Putting (2.31)-(2.32) into (2.29), and assuming there is no long-range spin correlation, one then finds, in the thermodynamic limit N → ∞, the following scalings The signalJ is then much larger than fluctuations δJ, and thus it is enough to make one single measurement. In other words, the signal is macroscopic or "classical." Also note pure thermal fluctuations have the scaling Similar scaling behavior can also be obtained in a large N matrix-type theory (including two-dimensional CFTs in the large central charge limit). In this case take J to be a singletrace operator and U ∼ e iN Jϕ . We then find scalings and thus the signalJ is again "classical." Now coming back to (2.28), using the Cauchy-Schwarz inequality we find that In other words, up to an O(1) constant J L (t) g is bounded from above by the variance of J in the thermal ensemble. Now consider the variance of J L with g = 0. Using (2.26) with various choices of X(t), we find where we have again used Cauchy-Schwarz in the last step. Given thatJ g ∼ J 2 , and in general J 4 ∼ J 2 , all terms in (2.37) are of order J 2 2 . We thus find that modulo miraculous cancellations δJ g ∼J g , i.e. the variance is always of the same of order as the signal regardless of the choice of J and U R .
More explicitly, let us consider the three situations mentioned above for the standard response story. For J to be a spin operator at a single site, again all quantities are of order O(1). For J L of the form (2.31) with U R of the form (2.32), then from (2.36) and (2.34) we find thatJ In fact in this case considering the average polarization not only does not help to reduce the fluctuations, but also reduces the signal itself. One might as well just measure a single spin. Note that if one considers linear order in ϕ R as in (2.10)-(2.11) one may conclude that J g ∼ O(1) instead of (2.39). This suggests that the linear response analysis forJ g could be potentially misleading in this regime.
Finally for J given by a single-trace operator in a large N matrix-type theory, the counterparts of (2.35) areJ with again reduced signal.
To summarize, the regenesis signalJ g in the L-system due to U R and coupling V is intrinsically quantum in all situations!

E. Robustness of the regenesis phenomenon
Let us now consider the robustness of the regenesis phenomenon, i.e. how it fares against imperfections in the preparation of the initial state at the time −t s when we turn on the external field ϕ R . For simplicity, we will restrict to our discussion to linear order in ϕ R , i.e. (2.10)-(2.11). The arguments presented below generalize straightforwardly to full nonlinear level.
Here we consider two types of "small" perturbations. One type is that at t = 0 instead of |Ψ β we get where is a small parameter. Physically this means that in preparation of the system at t = −t s to aim for |Ψ β at t = 0, the aiming is not perfect, but misses a bit. Such perturbations may result if at −t s in addition to ϕ R there are some other "small" sources present (whose strengths are characterized by ). With (2.41) in (2.11) instead of |Ψ β , the corrections are clearly controlled by , and thus the qualitative conclusion above should not be modified.
Another possibility is that at t = 0 instead of |Ψ β we have with some t 0 for some few-body operator γ(x) (suitably smeared so that (2.42) is normalizable). Heuristically, this describes a state obtained adding a "γ L -particle" to the thermal field double at time t 0 . One could consider similar states obtained by acting with some operator in the R-system, but from (2.14) that is equivalent to an operation in the L system with inverted time, so (2.42) covers all cases. Strictly speaking, (2.42) is not a small perturbation of |Ψ β as it is orthogonal to |Ψ β since a generic few-body operator γ will have negligible expectation value in |Ψ β . There is, however, a physical sense that such perturbations are "small": at t = t 0 , it is hard to tell the difference between (2.42) and |Ψ β by making measurements using generic few-body operators, such as J, as they generically commute with γ.
With (2.42), we should replace (2.11) by where the spatial coordinates are suppressed. Consider first g = 0, then which is an OTOC. Equation (2.44) is small whenever in which case insertion of γ L will destroy the correlation between points (t, x) and (−t s , x) in |Ψ β , and destroy regenesis even without worrying about possible effects of γ on the interaction between two subsystems. For example, for t ∼ t s ∼ t * , equation (2.45) means any t 0 0 or t 0 2t * . The latter can be more intuitively understood as insertion of γ R of order t * before we send the signal at time −t * . Now look at (2.43) with g = 0. Notice that the ordering between γ L (t 0 ) and any V insertions are also out-of-time-ordered. From the same argument we then expect the effects of V will be destroyed when |t 0 | t * (destroy the coupling between two systems) . (2.46) Hence we expect regenesis is no longer present in (2.42) for t 0 t * and t 0 0. See Fig. 4 for regions of insertion of γ which will destroy the regenesis phenomenon.
We will confirm the above qualitative expectations in Sec. IV E by explicit calculations in two-dimensional CFTs. Our conclusion is also qualitatively consistent with gravity expectations discussed in [2], which we will elaborate more in Sec. V.

F. A contrast study: "regenesis" in a qubit model
Here we study the regenesis phenomenon a simple qubit model to help sharpen some essence aspects of the phenomenon in a quantum chaotic many-body system.
Consider a system consists of four qubits: L 1,2 and R 1,2 , with the Hilbert space H = We will write 2 by 2 identity matrix and Pauli matrices as σ µ = {I, X, Y, Z}. We take the Hamiltonian to be like an Ising model For simplicity we will consider the thermal field double (1.1) with β = 0, which is then giving by the following state of 2 EPR pairs as one can readily check that each component |i A general hermitian J L operator on L 1 site is a µ σ µ , where a µ is a real vector. The corresponding operator J R acting on R 1 is then (note β = 0) We choose V to act on site L 2 and R 2 , i.e. it commutes with J L and J R as they act on different sites, so as to model the situation described in the many-body context that V and J are generic few-body operators whose degrees of freedom do not overlap. We will take O = X, and therefore which as expected is a function of only t − t s . Since the above expression is real we have both G RR = G LR = 0. That even G RR = 0 is an artifact of that we are considering a β = 0 state. G RR is nonzero for other values of β. Now turn on V at t = 0, we the have G RR (t, −t s ) is given by the same expression as above with t − t s replaced by t + t s .
This simple example provides an interesting contrast which highlights some key elements of the regenesis phenomenon for a chaotic many-body system: (1) for a few-body system, there is no dissipation, and thus G RR does not dissipate, i.e. even with ϕ R turned off, the signal will remain in the R-system forever (turning on g only modifies the signal somewhat).
(2) With g = 0, the signal also appears in the L-system. The effect of V is not regenesis, more like "double genesis." The reason is of course trivial: interactions among degrees of freedom within each subsystem will manage to communicate ϕ R to the L-system through V .
(3) The response in the L-system depends sensitively on time, choice of the specific operator J, the Hamiltonian H L,R of the subsystems, and choice of interaction V .
In other words, in a few-body system or an integrable many-body systems, some kind of signal in the L-system will be generated by turning on V . But it is not "regenesis," and the signal will depend on all the specifics of an individual system and the operators used. In contrast, in chaotic systems, the behavior is universal, independent of all the details.

G. A generalization: regenesis between spatially separated points
With the understanding of the entanglement structure, the set can be trivially generalized to be between any spatial points. Instead of Ψ β we could use a one-side spatially shifted thermal field double state e i P · a |Ψ β where P is the spatial translation operator in either left or right system. Two choices are equivalent as P L + P R is a symmetry of |Ψ β , and for definiteness we take P = P R . The entangled pair of points for the shifted state are (t, x) and , and the regenesis is now between them. We will also modify the interaction V between L and R accordingly, e.g. replacing (1.3) by The story is then exactly same as before, with equation (2.11) becoming where we have used e −i P R · a V a e i P R · a = V .

III. EXPLICIT COMPUTATIONS IN LARGE c CFTS
To calculate (2.7) explicitly for a general quantum many-body system is a difficult task.
In [2] it was calculated at leading order in ϕ R (i.e. (2.10)) for a (0+1)-dimensional holographic system by summing over scattering diagrams on gravity side. In this paper we will compute it in (1 + 1)-dimensional CFTs in the large central charge limit, again restricting to (2.10).
This will enable us to obtain the behavior for J L for t, t s ∼ t * which one could not access using general arguments of Sec. II.
We will take O to be a scalar primary operator with conformal dimension ∆ O = 2h O , and J to be a scalar operator with dimension ∆ J = 2h J . Furthermore, for convenience of calculation we will consider the regime This regime is natural physically. We do not want the coupling V to change the UV behavior of the system, i.e.
would like to take it to be a relevant operator, and thus ∆ O ∼ O(1). h J should be much smaller than c as c is a measure of total number of degrees of freedom of a CFT. In our calculation we will neglect terms suppressed by 1 c and h O c while keeping all dependence on h J . 7 Here we will outline the main steps and results, leaving technical details to Appendix C.
Readers who are only interested in the final expressions can skip this section.
A remark on notation: below all x's refer to spatial coordinate in (1 + 1)-dimension, although earlier we have used it as a shorthand for spacetime coordinates.

A. Some useful expressions
Here we first mention some standard results on two-point functions in the state Ψ β for a two-dimensional CFT, which we will use later. The Wightman function for two J's in the same subsystem is given by where 12 < 0 assigns the ordering of two J R operators and avoids singularity. The response function (2.4) is obtained from the imaginary part of (3.1). The two-point function of J's from different subsystems is given by where C J is a constant and x 12 = x 1 − x 2 . Note that in (3.1)-(3.2), the correlators decay exponentially for (t 2 , x 2 ) lying outside the region (t 1 ± β 2π , x 1 ± β 2π ), as indicated earlier in (2.6) and (2.16). The form of (3.2) is a manifestation of the entanglement structure of (1.1) discussed in Sec. II B: the two systems are entangled in such a way that an operator inserted at point (−t, x) in R system is highly correlated with the same operator inserted in a region of size β 2π around (t, x) in L system. Similarly we have In our discussion below we will also use the following notations B. More elaborations on W Equation (2.11) is the central object that we would like to calculate and analyze. Here we elaborate a bit further on its structure. We can expand it in an infinite series (for definiteness using (1.6) as an example) where we have suppressed superscripts L for operators in L system. More explicitly, where we have used (2.15) repeatedly and introduced short-hand notations Note that allÕ's commute with one another.
C. Evaluating W : part I We will proceed by first evaluating (3.11) and then performing the sum (3.6). The thermal correlation functions (3.11) are in turn obtained by analytic continuation from those in the Euclidean signature. Let us first describe how to compute a multiple-point function of the in the Euclidean signature, i.e. with all the t = −iτ understood as being pure imaginary.
Following the standard procedure, we first perform a conformal transformation to map the cylinder (τ, x) (τ is periodic in β) to the full complex z plane. Note that for pure imaginary t, z,z are complex conjugates of each other, but are independent variables for general complex t. The calculation of (3.13) on the z-plane is still nontrivial. Fortunately, in the regime h O h J c, one could do it by applying techniques developed recently in [11].
For example, at the level of 4-point function we find that where for notational simplicity we have used the subscripts to denote the positions of operators, and (ū is defined as u with z's replaced byz's) More generally, for (3.13) we have Appendix C for details.
We now analytically continue the above expressions to Lorentzian signature to obtain (3.11). Correlation function of Lorentzian operators with a specific ordering can be obtained from continuation of the corresponding Euclidean correlation function by assigning appropriate i 's [13]. For example, where the left hand side denotes Lorentzian correlation function of a specified order, while the right hand side denotes Euclidean correlation function with the time argument t = −iτ i for each operator replaced by t = t i + i i , and i ordered as indicated. This i -prescription instructs how one continues through possible branch cuts encountered when analytically continuing from imaginary to real times.
Therefore for each term in (3.9)-(3.11) we just need to continue (3.18) by assigning different orderings of i 's. For example, from (3.15), we find that (recall (3.12)) where we have used (3.5) and For simplicity we will take t s = t and x s = x, which as discussed earlier is the most relevant case. By tracking the motions of u 1 ,ū 1 as one varies t, we can write A(u 1 ,ū 1 ) more explicitly as where V 1 (u) and V 2 (u) denote respectively the values of (3.16) along the negative real axis on its first and second sheet (V(u) has a branch cut along (1, +∞)) and for convenience we have slightly redefined u 1 ,ū 1 as The explicit evaluation of (3.23) is given in Appendix C 3.

D. Evaluating W : part II
For general W n , let us first look at the case of V given by (1.5), for which where subscripts denote different types of operators all inserted at t, x = 0. Applying (3.18) to a term obtained by expanding commutators in (3.27), we see that there are two types of contractions among O's: two-sided contractions between a O α i and aÕ α j which are given by O α iÕ α j β = Gδ α i α j (recall (3.5)), and same-sided contractions between O's (or betweenÕ's) which are in fact divergent. We will assume that O andÕ are smeared such that same-sided contractions are finite. The two-sided contractions can be further separated into contractions among operators in the same sums or different sums. Note there is an enhancement factor k if in a sum each O α i is contracted with the correspondingÕ α i from the same sum [2]. Thus in the large k limit, this type of contractions will dominate over all others, including same-sided contractions. Also note that for various terms obtained by expanding commutators of (3.27) only orderings between O and J matter (all the O andÕ commute with one another). We then conclude that to leading order in large k At finite k, which includes (1.3) as a special case with k = 1, one has to keep track of all other contractions, which is very complicated. The detailed derivations are given in Appendix D. The final result can be written in a form where V 1 , V 2 were given before in (3.24)-(3.25), V −1 is the corresponding value on −1 sheet, given by (3.31) In (3.30), the arguments of V-functions are defined as where 0 < < β is a regulator which makes same-sided contractions finite. A couple of further comments on (3.29). In the limit η → ∞, W becomes real and thus G LR is zero in that limit. In large k limit, B 0 terms cancel out in the exponential and recovers (3.28).
Now finally consider (1.6), which we will take L to be much larger than β. 9 The discussion here is similar to the large k story described above, with the sums over indices α replaced by integrations over x. The counterpart of k is L β . In the large L β limit we will need to include contractions between O i andÕ i which belong to the same integral. Parallel discussion as (3.28) then leads to leading order in β

A. General remarks
We first note that in all cases (i.e. (3.28), (3.29) and (3.33)) W is proportional to . As commented earlier that the form of (3.2) is in turn determined by the entanglement structure of thermal field double state |Ψ β . Thus possible spacetime points (t, x) to which one could send signal from (−t s , x s ) is determined by the entanglement structure with β 2π characterizing the size of the window for possible nonzero signal. Now consider (2.9) which we copy here for convenience, Since G LR (t, x; t s , x s ) ∝ ImW falls off rapidly outside the window |t − t s | < ∼ β 2π and |x − x s | < ∼ β 2π , for sources ϕ R (t, x) which are slowly varying in spacetime at the scale of β 2π we can approximate (4.2) as . Below without loss of qualitative features, instead of considering the averagedḠ LR (t, x; −t, x), we will simply examine the behavior of G LR (t, x; −t s , x s ) for t s = t and x s = x. In this case we then have which may be interpreted as the effective coupling between two systems.
For notational simplicity we will write G LR (t, x; −t, x) as G LR (t, x), and then from (2.10) we find for (3.28) and for (3.33) The expression of G LR for (1.3) can be straightforward obtained by taking the imaginary part of (3.29) with k = 1, but the formal expression is not very illuminating, so we will not write it explicitly.
Now recall that A is the normalized four-point function with A 0 are given by setting x 1 to zero. The commutator upstairs is the difference between which can also be checked explicitly from (3.23). 10 We then find that G LR goes a constant, e.g. for (4.6)-(4.7) Similarly for (3.29) we find as t → ∞, B 0 → √ 2, and thus (4.12) Note that while (4.11) is oscillatory in g eff with period 2π, for k = 1 it is not, and goes to zero as g −1 eff for large g eff . From (4.3) we then find that in all cases with C(g) a constant, which is consistent with (2.23)-(2.24) deduced on general ground.
We will now proceed to understand the behavior of G LR at general times in more detail.

B. A scaling limit
Based on general discussion of Sec. II we expect the nontrivial behavior of G LR to arise when t becomes of order of the scrambling time t * which for a CFT at large c is proportional to log c [9]. Since in general we do not expect x − x 1 to scale with c, thus for values of t of interest we should have t |x − x 1 | for which 1 . (4.14) Denoting V 1,2 (u) = U 1,2 (u) 2h O and expanding in small u we find from (3.24) We are interested in h J c, i.e. α ≈ 1. It can be readily checked all the coefficients of higher powers of u in U 1 going to zero as α → 1 (U 1 = 1 for α = 1). Expanding for small (4.16) Now we notice that as α → 1, each coefficient becomes singular. There is a scaling regime in which one can resum the whole series (4.16) Q can be written more explicitly as where we have introduced We then find Similarly, in the scaling limit, from (3.30) we find B 0 can be written as Note that in contrast to A(u 1 ,ū 1 ) and A 0 , B 0 is always real.
In the large c limit, t * t J , |x − x 1 |, the scaling limit helps us to focus on the time scale t ∼ t * during which Q is O(1) (in terms of large c scaling) and the commutator (4.9) between generic few-body operators O and J becomes sizable. This defines t * as the scrambling time.
Note that it is curious that at leading order in (4.21)-(4.22) the only dependence on h J is through a time shift t J .

C. Three regimes of G LR
Let us now look at the behavior of G LR more closely, using (4.6) for (1.5) as the main example, and will comment on the differences at finite k. Equation (4.7) for (1.6) will be discussed in next subsection.
At leading order in 1/c, A has simple dependence on Q which in turn is given by a simple exponential. So the behavior of (4.6) is straightforward to obtain. One immediate thing to notice is that G LR is a function of t − |x| only. From (4.3), points with the same t − |x| then get multiplied by the same factor in going from source to signal. The behavior of A and G LR can be separated into three distinct regimes: 1. Sub-scrambling regime: for t − |x| t * − t J (i.e. t − |x| large and negative) we have Q 1 and 11 β (t−|x) + · · · (4.23) 11 Note that the first term is not pure imaginary when including h J /c corrections. and Note that (4.23) is exponentially increasing with time with Lyapunov exponent 2π β and butterfly velocity v B = 1 [9], but the leading behavior is pure imaginary and does not contribute to G LR . In this regime, the signal one sent in at time −t just started getting scrambled before we set up the communication channel V at time 0. The signal is very weak in this regime and can be considered as being approximately zero for practical purpose. G LR for a finite k has very similar behavior.
2. Transition regime: for a narrow window of size β 2π around t − |x| = t * − t J , both real and imaginary parts of A 0 are O(1), and G LR also becomes O(1). In this window, the exponential factor e g eff ImA 0 in (4.6) can enhance the magnitude of G LR significantly when g eff has the right sign and not too small. Note as can be explicitly checked from the expression of A 0 (see Fig. 5(a) for an example), for ∆ O ∼ O(1), ImA 0 is always positive and smaller than 1. 12 Thus enhancement requires g eff to be positive. See In contrast, at a finite k including the case for (1.3), from the imaginary part of (3.29) we find there is no enhancement in the transition region for generic values of regulation parameter ∼ O(β). See Fig. 5(c) for some examples.
One may understand the exponential enhancement in the large k limit as coming from constructive interference of different channels.
3. Stable regime: as we further increase t beyond the transition regime, i.e. for t − |x| t * −t J (t−|x| large and positive), Q quickly grows to be Q 1, for which A 0 approaches to 1 exponentially (quasi-normal behavior) and for G LR we have To conclude this subsection we should emphasize that the regimes described above are not evolutions; they correspond to different types of behavior when we vary the time separation between the time of turning on the source and the time we turn on interaction V between L and R systems.

D. Multiple channel from integration
Let us now examine the behavior of (4.7)-(4.8). We will consider t * L β as for L < ∼ β the story is essentially the same as that of single-channel. There are some new elements in (4.7) compared with (4.6). Firstly due to the integration over x 1 , G LR is no longer a function of t − |x| only. Secondly, as we will see the transition regime can be significantly lengthened.
For illustration let us consider x = 0 for which we have to leading order in 1/c curves are for t s = t * + 3, t * + 8, t * + 13, t * + 18 respectively.
The sub-scrambling regime is for t 0 such that Q(x 1 ) is exponentially small for the whole integration range. The stable regime is for t L 2 , for which Q(x 1 ) is exponentially large for the whole integration range. The behavior of A for these regimes is completely parallel to the corresponding regimes of A 0 discussed in last subsection (with only differences in some constant prefactors), and thus the behavior of G LR is also parallel to those of (4.6). Things are more interesting for t in the window t ∈ (0, L 2 ) (i.e. t ∈ (t * − t J , t * − t J + L 2 )) for which as x 1 changes from 0 to L, Q(x 1 ) varies from exponentially large to exponentially small. 13 To find A for such values of t we note that (4.27) can in fact be exactly integrated, yielding where B(x, a, b) is the incomplete beta function. Using that Q(0) is exponentially large and Q( L 2 ) exponentially small we find that where c 0 is a numerical constant. This behavior is extremely simple with linear dependence on t and a constant imaginary part, leading to Thus we see that for (4.7), the size of transition regime is extended to a region of size L 2 (in contrast to β 2π for (4.6)), but the imaginary part of A is down by an order β L compared with A 0 . Thus while the resonant enhancement is extended to a much larger range of time period, the enhancement effect is more moderated. See Fig. 6 for various numerical plots of A and the corresponding G LR .
The behavior for general x is qualitatively similar. The only difference is that the transition regime is now from t * + θ(|x| − L/2)(|x| − L/2) to t * + L/2 + |x| , which has maximal length of L when |x| ≥ L/2. Here θ(x) is step function. See Fig. 7. Finally let us note that when L → ∞, the behavior (4.29)-(4.30) will last forever and one never reaches the stable regime.

E. Robustness of regenesis from CFT calculations
We now turn to the explicit calculation of (2.43) for CFTs in the large c limit. For simplicity we will take t s = t and consider the regime c ∆ γ ∆ J ∆ O ∼ O(1), for 13 We will be concerned about t's in the middle of the window (0, L 2 ), i.e. not close to either edges. which we will be able to confirm explicitly the conclusion of Sec. II E. We will present only the results here leaving details to Appendix E.
For simplicity we will consider only (1.5) in the large k limit and (1.6) when L β. For these cases we find where for notational simplicity we have suppressed x, x 0 in the arguments of various functions. FunctionÃ 0 (t, t 0 ) is obtained fromÃ(t, t 0 ; x 1 ) by setting x 1 = 0 and G 0 is obtained from G in the same way.
By comparing (4.31) with (3.28) and (3.33), we see that the following three differences between the corresponding expressions for (2.42) and Ψ β which reflect three distinct aspects how an insertion of γ in Ψ β affects the regenesis phenomenon : 1. The prefact factor JJ is multiplied by another function J , which modifies correlation between J L (t, x) and J R (−t, x).
2. the effective coupling g eff is multiplied by a function G, which modifies correlation between O L (0) and O R (0) and thus the effective coupling between L and R systems.

the function A is replaced by another functionÃ which reflects how "interactions"
between O and J operators are modified due to presence of γ.
Note that item (1) and (2) can be interpreted as coming from modification of the entanglement structure of Ψ β . Now let us look at the explicit expressions of J , G andÃ. Note that if the spatial location x 0 is sufficiently far away, e.g. if |x 0 − x| t − t 0 , clearly from causality γ cannot have any effect on J. Similar statement applies to O. Now from Appendix E we find that (assuming x − x 0 is much smaller than t * ) where γ is a UV regulator need to make (2.42) normalizable. Note that J ≤ 1, and J → 0 when |t − t 0 | t * as anticipated in (2.45). From Appendix E, G has the form Again G ≤ 1 and we see that G → 0, and thus the effective coupling is destroyed, when |t 0 | t * , which confirms the expectation (2.46).
The behavior ofÃ is much more difficult to work out explicitly. This is also easy to understand physically: to see how the presence of γ modifies "interactions" between O and J is warranted to be complicated in a strongly interacting system. Fortunately the conclusion does not depend on the detailed formÃ. We expectÃ ≈ A if |t 0 | t * and |t − t 0 | t * , i.e.
the effect of γ on J −O correlation functions will be small if γ excitation does not have enough time to grow. Outside this region, the form ofÃ is expected to be complicated, but we do not really care as from (2.45)-(2.46) and (4.32)-(4.33), outside this region, the correlations between two systems already become too weak to have the regenesis phenomenon.

V. GRAVITY INTERPRETATION
In this section we compare with the gravity discussion of [1,2] and discuss implications of our results for wormhole physics in the context of holography. Other recent papers on traversable wormholes from gravity perspective include [14][15][16][17][18]. Note that our twodimensional CFT calculation in the large c limit may be considered as describing a BTZ black hole.
In the gravity description, thermal field double (1.1) is described by an eternal black hole which has two asymptotic boundaries connected by a non-traversable wormhole (see Fig. 8(a)). The regenesis phenomenon corresponds to the statement that with a coupling like (1.3), one could send signals between two boundaries, i.e. the wormhole becomes traversable.
Turning on the source ϕ R for a short time on the right boundary generates bulk excitations dual to J. These excitations fall toward and are absorbed by the black hole membrane, which corresponds to the dissipation of J R . The process happens very fast, with time of order O(β), as also seen in our earlier CFT calculation. When we briefly couple the two boundaries at t = 0, the physical picture of [1,2] is that, the interaction V deforms the bulk geometry, especially the causal structure, making the wormhole traversable, as indicated in Fig. 8(b).
It is important in the discussion of [1] that only one sign of the coupling g, i.e. g > 0 which generates negative bulk energy, allows for the traversability.
Below we will first compare our CFT results with that of [2] obtained from gravity scat- we enlarge ∆ J as ∆ J = 3. Blue, yellow, green and red curves are for t s = t * + 0.5, t * + 3.5, t * + 6.5, t * + 9.5.
terings. We then discuss implications of the regenesis phenomenon on gravity side. In particular, we will argue that there are other scenarios for wormhole traversability in addition to that suggested by Fig. 8(b). For example, the regime for t s t * should correspond to a "quantum traversable wormhole."

A. Explicit comparison with gravity results
Here we compare our explicit expression for W calculated from two-dimensional CFTs with that of [2] calculated for a (0 + 1)-dimensional boundary theory from gravity. By using graviton scattering amplitudes between JJ and OÕ near horizon, for V given by (1.5) they derived the following expression for W (below β has been set to 2π, and we have rescaled p and used notations introduced in Sec. IV A) where p is a bulk momentum of the J-quantum, and To compare (5.1) with our results, it is convenient to deform the integral contour to be along the imaginary p axis from 0 to i∞ (note the integral along the arc from +i∞ to +∞ vanishes). We then find (after a scaling p → ip) for t s = t which has exactly the same form as (3.28) with A 0 given by (4.21) (with x = x 1 = 0). 14 To make sure our Gaussian approximation is valid, we need the exponential term involving g eff in (5.1) to be slowly varying within the variance of the Gaussian distribution, i.e.
The above equation is satisfied for all values of Q if in the limit t, t s → ∞, it reduces to (2.22). Another important aspect of our discussion is the reversed time ordering between the input signal ϕ R and output signal J L as indicated for example in (4.3). It can be checked that (5.1) also has this property, as can be seen explicitly from the plots of the resulting G LR in Fig. 9.

B. A semi-classical regime
We now elaborate on a "semi-classical" regime of (5.1) which was identified in [2]. 15 Consider smearing J-operator so that its high energy component is suppressed. One can represent such a smearing by inserting a Gaussian factor e − p 2 2σ 2 in the momentum integral of (5.1). Then one finds that, in the large g eff limit (with σ and ∆ J , exists a regime corresponding to the picture of Fig. 8(b).
More explicitly, consider expanding inQ in the exponent of (5.1) and keeping only the linear term inQ. Equation (5.1) can then be approximated by Note that the prefactor e −ig eff has now been canceled. Without the Gaussian factor in (5.10), the integral in (5.10) would yield which has a "light-cone" singularity at Thus we can view (5.10) as the propagator of a smeared field in a spacetime where points on two boundaries satisfying (5.12) are connected by light rays, as indicated in Fig 8 (b).
Let us make some further remarks: 1. For the approximation in (5.10) to be valid, we need for the range of p allowed by the Gaussian factor, the terms of O(Q 2 ) and higher in (5.9) are suppressed, i.e.
Equation (5.12) (which corresponds to g eff ∆ O ∆ JQ = 1) lies within the range (5.13) provided that (recall we have β = 2π which sets the unit) √ g eff σ 1 . (5.14) 2. Note that equation (5.12) has a solution only for g eff > 0 and and for t s satisfying (5.15), a corresponding t l (t s ) always exists. Note dt l d(−ts) > 0 when (5.15) is satisfied, which means the light-cone structure is such that the earlier a signal (smaller −t s ) is sent, the earlier (smaller t l ) the arrival of the signal at the other boundary. In particular, as −t s → −∞, t l (t s ) → t c which is the earliest possible arrival time, and as −t s → −t c , t l (t s ) → +∞. This is consistent with the heuristic picture of Fig. 8(b), but is sharply different from the behavior exhibited in (5.6) or for general ∆ O , ∆ J , g eff , for which the signal sent from −t s arrives at t ≈ t s . We contrast these two different types of behavior in Fig. 10. 3. For σ ∼ √ g eff or larger, terms with quadratic and higher powers inQ in the expansion of (1 + pQ 2∆ J ) −2∆ O become important. Note sinceQ ∝ G N , these terms may be understood as due to backreaction of J-quanta on the geometry. Without the momentum suppression factor e − p 2 2σ 2 (i.e. σ → ∞), such backreactions are always important.

C. Old cats never die
Connecting the boundary and bulk pictures for the regenesis phenomenon in the semiclassical regime discussed in the last subsection raises leads to a rather amusing scenario.
To make the contrast of the two pictures a bit sharper, let us imagine by turning on ϕ R , we create a "cat," which contains only "low energy" constituents compared with the coupling g eff which we will eventually turn on (i.e. with their bulk momenta satisfying (5.14)). From the boundary picture, the cat lives for a while, but eventually her body gets more and more scrambled with the environment. We wait until her body is fully scrambled, turn on the interaction gV . From normal standards, it should be safe to say by this time the cat has long died (in other words, her body should have long been decomposed). From the genesis phenomenon, after another scrambling time, the cat is reborn in the other universe. As emphasized in the Introduction this process requires extreme fine tuning of the initial state at the time we created the cat. Now let us refer to the bulk dual of the cat as the bulk cat, which we suppose is also a living object. Then in the bulk picture, the bulk cat travels in the deformed geometry created by the interaction gV . She never "dies" 16 , just sailing through the bulk geometry.
The journey should be smooth as no regions of large curvature will be encountered. In this picture, the reborn cat simply corresponds to the arrival of the bulk cat through a wormhole.
In fact if she travels close to the light cone, the proper time that the cat experiences might not be too long.
From the holographic duality, these two pictures must be equivalent. In particular, since all boundary events should have a bulk version, the bulk cat should be able to "see" her own funeral on the boundary. This is the bulk way to say that the regenesis cat in fact contains the "memory" of her previous life.

D. Quantum traversable wormholes
In Sec. V B we discussed in detail that the semi-classical scenario of Fig 8(b) corresponds to the parameter region of (5.14) and k → ∞, as well as the ranges of t, t s satisfying (5.13).
Outside these parameter regions, how do we interpret the traversability of a wormhole?
Here we offer some qualitative discussions by combining the general discussion of Sec. II, the explicit CFT results of Sec. IV, as well as discussion of (5.1). One can identify the following different physical scenarios/regimes: 1. A most straightforward scenario is that the picture of Fig. 8(b) still qualitatively applies, but with the deformed geometry interpreted as including both effects of gV and the backreaction of J itself. Since the deformed geometry now depends on the quanta propagating through it, strictly speaking, one can no longer talk about a background causal structure as in the case of a linear wave moving in a fixed geometry. Nevertheless, the essential physics picture remains qualitatively similar. So we will still refer to this scenario as the semi-classical scenario. For example, consider (5.1) (which is k → ∞ 16 By dying here we refer to something a bit more general, i.e. the body remains as a whole. limit) with a large g eff and a momentum suppression factor e − p 2 2σ 2 , and slowly increase σ beyond the range of (5.14). As we increase σ, the backreactions of J become more and more important. One should be able to include them perturbatively, say first including theQ 2 term in (5.9) but neglecting higher order terms, and then theQ 3 term, and so on. 17 2. When t ∼ t s t * , the traversability should arise from a distinct physical picture from that of Fig. 8(b). In this regime, independent of the details of any theory (and for general g, k and V ), we discussed in Sec. II that the sole effect of gV is to generate a complex factor in W proportional to e −igV with no dependence on the quantum numbers of J at all. There is no scattering between J and O quanta. For a CFT, the conclusion also does not depend on the large c limit (as far as the theory is chaotic).
Let us also highlight some features which are sharply different from those of the semiclassical regime discussed above: (a) Even in the k → ∞ limit, the regenesis phenomenon, and thus traversability of Thus the traversability for this regime does not appear to be associated with any spacetime causal structure at all, and is in a sense driven by entanglement. In other words, one has a "quantum traversable wormhole." 17 In the limit σ → ∞ (or σ g eff ), one finds that for certain range of t around t * the integral is dominated by the contribution of a (real) saddle-point with p saddle ∝ g eff (see Appendix B of [2]). Such a saddle point may have an interpretation in terms of bulk Einstein equations. 18 Again we are treating relaxation time scales as microscopic.
We do not yet have a precise way to characterize the "quantum geometry" associate with such a traversable wormhole. Here we offer some preliminary thoughts. Heuristically, the transmission of a signal from the R to L boundary feels like a tunneling process across the horizon mediated by gV interaction. Directly translating the discussion of Sec. II C to the bulk gives the following picture. Consider first g = 0, for which the wormhole in non-traversable. Nevertheless, the bulk Wightman and Feynman functions between L and R for the bulk field dual to J is nonzero. We may interpret the vanishing of [J L , J R ] = J L J R − J R J L as perfect destructive interference between the process of a virtual particle traveling from R to L, and the mirror process of traveling from L to R, as indicated in the left plot of Fig. 3. Turning on a nonzero g gives a phase shift to each propagator, and in general the destructive inference is no longer perfect, resulting propagation of real particles. See right plot of Fig. 3. In (4.11) we saw that G LR is periodic in g eff , thus as one dials the value of g, perfect destructive interference can be again reached at various special values.
3. Now for general k, g and σ, and t, t s ∼ t * , the picture is no longer so sharp. There is a continuous spectrum in going from the semi-classical regime of item 1 to the quantum regime of item 2.
For example, for large, but finite k, while the bulk stress tensor induced from turning on V will have a finite spread, for g eff , σ satisfying (5.14), we expect the physics should still be close to the semi-classical picture.
On the other hand, for (3.28) and (4.6) (or its (0 + 1)-dimensional counterpart (5.6)), both feature (a) and (b) listed in item 2 also apply. So it appears reasonable to expect the traversability is governed by the mechanism of item 2 except that in general there are also scatterings between J and O involved. Note that from (5.1), equation (5.6) is obtained from a pure imaginary "saddle" p saddle ≈ i2∆ J , which also suggests that the underlying physics cannot be understood straightforwardly in terms of classical scatterings of O and J quanta. 19 In this paper we presented a general argument for the regenesis phenomenon in a manybody chaotic system and studied it in detail in two-dimensional CFTs in the large central charge limit. We also discussed the implications of these field theory results for wormhole physics.
Here we end with some further discussion, including future directions:

Teleportation?
As discussed in [1,2] (see also [5]), the coupling V between L and R system is reminiscent of the operations in a teleportation process. During the time evolution of the system, the effect of having the gV δ(t = 0) term in the Hamiltonian can be considered as being equivalent to the process of performing some measurements in the R system, communicating the results to the L system, and then performing operations on the L system. But we would like to stress that the regenesis phenomenon is in fact very different from quantum teleportation in the usual sense. In teleportation one would like to send an unknown state to another party. Here while the signal from the R system re-appears in the L system, in general there is no state teleportation. It can be readily checked in the qubit model of Sec. II F that general H L,R and coupling V , do not implement teleportation of a state. Thus the regenesis phenomenon can at most be considered as a "signal teleportation". This conclusion is also supported by the discussion of [5] that the operation for a state teleportation involve a much larger complexity than that of the regenesis setup. As emphasized in [2] the regenesis setup also shares some similarities with that of Hayden-Preskill [19], but as we emphasized in the Introduction regenesis requires extra fine tuning in the preparation of the initial state.
Thus for Hayden-Preskill, the decoding requires much higher level of complexity [20].
It is an interesting question for further study, say if one wants to send some known signals (one knows the input signal ϕ R one is applying) from R to L, whether the the contour to imaginary p, which also indirectly suggests that (5.10) and (5.6) are controlled by very different physics. current protocol is an efficient one (see [21] for a discussion of treating the wormhole setup as a quantum channel).

Nature of quantum traversable wormholes
It is important to have a more precise bulk picture for "quantum traversable wormholes" which we argued in Sec. V D. In particular, it would be ideal to have a real-time evolution picture for it. Equation (2.28) can be reproduced from (2.7) by replacing Φ of (2.8) by 20 |Φ ≈ a * |Ψ β + U R |Ψ β . Note that this cannot be a true identity as the right hand side does not have the right normalization (and it does not reproduce (2.26)). Nevertheless, this expression is suggestive as it indicates that Φ is a superposition of two macroscopic states, one from acting U R on TFD, while the other corresponding to multiplying a TFD by a complex number. Equation (2.28), and thus traversability, arises from interference between them. How should we think the bulk geometry corresponding to Φ? Is there a firewall at the horizon?

Other systems
It is clearly of interest to study this phenomenon in other systems like spin chains or using random unitary circuits (see e.g. [22]) which have generated lots of insights into chaotic systems. Note that since regenesis concerns with time scales of order the scrambling time, thus it should be insensitive to the early time behavior such as whether the system has a nonzero Lyapunov exponent.

Using effective field theories (EFTs)
The computation of the behavior G LR in the transition regime (i.e. for t s ∼ t * ) in twodimensional CFTs is rather complicated and technical even in the large c limit. While there are reasons to believe that the qualitative behavior we obtained should apply to generic chaotic systems, it would be good to understand it in a system-independent 20 We thank J. Maldacena, D. Stanford and Z. Yang for this observation.
way. Recently a class of EFTs which aims to capture scrambling of general operators in chaotic systems (at least for those close to being maximally chaotic) has been proposed in [23]. In particular, the EFT for two-dimensional CFT in the large c limit has been obtained in [24,25]. The EFT approach could provide a simpler and systemindependent way to study many aspects of the regenesis phenomenon and wormhole physics. We will leave this for future investigation.

Experimental realizations
It would be interesting to observe the regenesis phenomenon experimentally. For example, one could imagine setting up the protocol in bilayer graphene or quantum hall systems. Experimentally realizing a thermal field double state in a many-body system appears difficult. 21 If one could realize (1.1), to fine tune the state at t = −t s is similar to realizing an OTOC, as one needs to run the system "backward" in time, turn on the source, and then move forward in time. Recently there has been significant progress in realizing OTOCs in the lab (see e.g. [27]), so perhaps such tuning is not that far-fetched. In the spirit of ER = EPR [28], realizing regenesis experimentally may be interpreted as creating a quantum traversable wormhole in the lab! where We consider to linear order in H 2 while to full nonlinear order in H 1 , i.e.
where we have used that the support of ϕ R (x) is much earlier than that of f (x). Now for simplicity we will take we will the have We thus find that

Appendix B: An identity
For an operator X in the L-system, consider and the source ϕ R (t) is supported near t = −t s , while V is supported at t = 0. In the limit t, t s t * , with OTOCs set to zero, X g can be greatly simplified. More explicitly, we have (suppressing Ψ β ) Note that X commutes with U R . Now as in (2.22) we factorize parts of a correlator which are widely separated in time (with e −igV X(t)e igV = X ), which then gives which maps z a → w a = ∞, z b → w b = 1 and 0 → 0. This map has a branch cut in z plane from z b to z a . The Jacobian is The relation between 4-pt function in z and w planes is where the subscript denotes the coordinate. Note that J a and J b are vanishing, so the above formula should be regarded as taken in a proper limit since the LHS is also vanishing as w a → ∞.
The advantage of this conformal transformation is that in w plane, 4-pt function J(w a )J(w b )T (w) does not depend on h J explicitly. In large c limit, this amounts to the leading order approximation in 1/c expansion. To be more precise, by Ward identity, the 4-pt function in z plane is Since stress tensor is not primary field, it has an extra Schwarzian term under conformal transformations. Given the transformation (C1), the stress tensor transformations as Hence we find that in w plane where the Schwarzian cancels the term that is proportional to h J . In w plane, the whole Virasoro block is summing over all Virasoro descendents just like z plane. We can Taylor expand T (w) around w = 0: One can show that the commutation between L n and general primary operator X(w) obeys the same rule as in z plane: and Virasoro algebra still holds for all L n : Indeed, above relations always hold when the Jacobian J(z) is nonsingular around w.
The relation between L n , the Virasoro mode of T (z), and L n can be solved explicitly by which implies that all L n are linear combinations of L m with m ≥ n. This immediately gives an important result L n |h = 0, n ≥ 0 (C11) for any primary |h . However, since the expansion (C7) is not convergent around infinity due to the existence of branch cut from 1 to ∞, we should not expect L † n = L −n . In other words, the radius of convergence of series (C7) is bounded by the location of branch cut.
Formally we can define a "w-primary state" h w | as with h w | L −n = 0 for n ≥ 0 and normalization h w | h = 1. The whole Virasoro block in w plane can be calculated as insertion of projection P T k between J a J b and O 1 O 2 , where P T k ≡ L −n 1 · · · L −n k |h h w | L n k · · · L n 1 h w |L n k · · · L n 1 L −n 1 · · · L −n k |h (C13) Note that these P T k 's are not orthogonal, and in general we need to take all overlaps between different projectors into account. Let us take h = 0 for identity Virasoro block for now.
There are a few features of this construction. First, 0 w |L n 1 · · · L n k |0 w = 0|L n 1 · · · L n k |0 z because L n and L n obey the same algebra. Therefore, we can simply estimate the denominator of (C13) in large c limit. For n i ≥ 2, Second, 0 w | L n k · · · L n 1 O(w 1 ) · · · O(w n )|0 is the same as that in z plane because of the same algebra (C8) and (C12). In particular, the two point function is Note that this is different from conformal transformed version of 0|O(z 1 )O(z 2 )|0 z to w plane. Physically, this means that ignoring the neighborhood of branch cut, we regard all other regions in w plane the same as ordinary CFT on complex plane. Third, 0| J a J b L −n 1 · · · L −n k |0 is not the same as z-plane CFT due to the branch cut, but restricted to the conformal transformation rules from z-plane to w-plane. In particular, the two point function obeys The advantage of this special conformal transformation is that in w plane, for all n ≥ 0 due to (C6). This method can be generalized to multiple T insertion in (C6) and one can show that the leading order of h J vanishes in w-plane. Indeed, notice that T is different from primary field only by a central term. Therefore, for multiple T insertion in a correlation function of primaries X , there is a induction relation: where hat means omiting T (z i ). From above formula, the k insertion should have the following expansion where z n denote coordinates collectively. In c h J limit, the expansion has the orders from high to low as Note that if we are considering L −n rather than L −n , the powers of c does not contribute to 0| J a J b L −n 1 · · · L −n k |0 . But transforming to w plane, these powers are leading contributions to 0| J a J b L −n 1 · · · L −n k |0 . On the other hand, terms involving O 1 and O 2 after inserting Therefore, the highest order terms in first a few orders of P T k insertion are where above explicit terms are from k = 0 to k = 5.
We are interested in cases with even number O insertion. In leading order, this can be calculated as where {(s 2i , s 2i+1 )} is the collection of contractions between s 2i -th and s 2i+1 -th operators.
In the second line we ignored higher orders of 1/c, and in third line we used large N (large c) ansatz to factorize all O's in two point functions. Associated with antiholomorphic part, (C28) becomes (3.18).

Explicit expression of A
Here we give an explicit derivation of (3.23) from (3.21) for t s = t and x s = x. The discussion of this subsection has some parallel to that of [9] for OTOCs in a large c CFT.
From (3.16), V(u) is real for u < 1 and has a branch cut along u ∈ (1, ∞). With where we have introduced FIG. 11. Plots of u 1 (plot (a)) andū 1 (plot (b)) as a function of t. Blue and yellow are for V + and V − respectively. One can see that u 1 stays on first sheet for V + (u 1 ), but moves to second sheet for V − (u 1 );ū 1 stays on first sheet for both V ± (ū 1 ).
Recall that V + is defined with ordering 1 < J <˜ J <˜ 1 , while V − with ordering J < 1 < J <˜ 1 . Let us look at the behavior of A(u 1 ,ū 1 ) as we increase t from 0 while keeping x − x 1 fixed (assuming x − x 1 is not exactly zero). For sufficiently small t, regardless of the sign of x − x 1 , (0, x 1 ) and (t, x) are spacelike separated, with u 1 ,ū 1 < 0. In this case 's do not matter, and thus V + = V − , leading to A = 0. This of course can be deduced from (3.20) without doing any calculations as the commutator of spacelike separated operators must vanish.
x − x 1 < 0, for which u 1 starts being negative at t = 0 and is again negative for large t, but in between u 1 undergoes nontrivial motions in the complex plane as the lightcone y + = 0 is crossed. For V (+) , one finds that u 1 remains on the first sheet throughout the process, while for V (−) , u 1 crosses the branch cut from upper half u-plane and moves to the second sheet. In contrast,ū 1 always remains real and negative. See Fig. 11. We thus find that for sufficiently large t, where V 1 (u) (V 2 (u)) denotes the value along the negative real axis on the first (second) sheet, Similarly for x s − x 1 > 0 we find that, u 1 always remains real, negative, butū 1 moves nontrivially in the complex plane as the light cone is crossed. Again one finds that V (+) remains on the first sheet, while V (−) moves to the second sheet from above. We thus find One consistent check is that (C31) and (C34) agree when x = x 1 . We can write (C31) and (C34) in a unified way as with now u 1 andū 1 defined as Appendix D: Full k-dependence in multiple operator species In this appendix, we will use the following notation: where we suppressed all spacetime coordinates. In order to calculate W in the case of multiple operator species, we need to note that there are three types of contractions, V 2 (−µ)V 1 (−μ) and V 1 (−µ)V 1 (−μ) respectively (see Fig. 12). ForH type contraction, there is only one ordering: ab21 , which is V 1 (µ)V 1 (μ). Here the subscript i of V i means the value on i-th sheet given by (3.24), (3.25) and In all above statements, the cross ratios are u = u 1 in (C36) with x 1 = 0 and where is the difference of -prescription in the time ordering of O 1 and O 2 and it now plays a role as UV regulator 22 . 22 In order to evaluate the Lorentzian correlation function in (3.18) one has to assign i for each O i before continuation. Since all the O's commute, the relative values of their 's do not matter. Therefore we can assign ordering for each pairing such that for each Virasoro block is the same. Physically it is natural to take of order O(β), and in the main body of this paper, we will choose = β/2 for definiteness.
For any given choice of contractions, the number of H contraction must be the same as H.
To be more precise, with q contractions of H, then number ofH is q and the number of G is n − 2q. In such a contraction, the scaling of (3.18) is H qH q G n−2q . If we track all commutators in W n , we will find that the contribution from this contraction is where C q is a constant. A consistent check is that in (D5) there are in total 4 q · 2 n−2q = 2 n terms, which is the same number of terms in the expansion of commutators (3.11 Hence, evaluation of W n boils down to the problem of calculating C q for various contractions.
In the following, since H =H by definition, we will not distinguish them and only use H.
To count C q is somewhat tricky. Due to the factor 1/k n in W n , any O α contracting with its dualÕ α contributes with α δ αα /k = 1, but contracting with any other operators contributes with only 1/k. Keeping this in mind, we will use the following procedure: 4. In step 1, completing a chain becomes S n−1 . In step 2, i) becomes S n−2 , ii) becomes T n−3 , and iii) becomes U n−3 . In step 3, i) becomes S n−2 , ii) becomes T n−3 , and iii) becomes U n−3 . For any S # cases, do step 1, for any T # cases, do step 2, and for any U # cases, do step 3. This process ends when we finish all contractions.
Assuming c h γ h J , we could first do a conformal transformation with respect to γ L by introducing a branch cut from z A to z B with a Jacobian Similar to (C28), (E4) becomes For the expectation value in v plane 0 v |J a J b O 1 · · · O 2n v , since 0 v | behaves exactly as ordinary vacuum state, we can apply the same technique to simply it. Previous result (C28) applies with a simple coordinate replacement z → v. After some manipulation, it turns out that w γ,n ≈ J a J b z U η (u J ) where the new variable v = v(z) is given by (E5). The antiholomorphic part is parallel with above discussion.
It is clear that from (E3), the relative ordering among γ, J andJ is always fixed as γ A Jγ BJ , and that among γ, O andÕ is also fixed as γ A O i γ BÕi . The commutators in (E3) will only be affected by the relative ordering among O, J andJ, and this leads (E3) to where A α is the difference of two orderings OJJÕ and JOJÕ It follows that for large k and large L cases, we have where subscript 0 means taking x 1 = 0. In these formula, V η can be absorbed into the definition of coefficient g (at least in large k case), and the prefactor U η controls the overall traversability. In the main body of this paper, we use a simpler notation J (t, t 0 ) ≡ U η (u J )U η (ū J ) and G(t, t 0 ; x 1 ) ≡ V η (u 1 )V η (ū 1 ).
To be more precise, let us study the value of these conformal blocks in -prescription. For Similar for u 1 andū 1 by setting x = x s = x 1 and t = t s = 0 with corresponding . In general, we need to take all 's to zero at the end of calculation, which seemingly leads to trivial u J andū J . However, we could smear it for some small range in time for γ L that equivalently sets AB small but finite. For t s − t 0 − |x s − x 0 |, t − t 0 − |x − x 0 | β, it is clear from (E14) and (E15) that u J → −i(e − 2π β (x+t) + e − 2π β (xs+ts) )e This shows that for finite AB , u J andū J are suppressed by |t 0 − t s | and |t 0 − t| exponentially.
Set t = t s and x = x s , and check the contour of u J andū J . For simplicity we will take x 0 − x > 0 from now on. We find that u J is on −1-th sheet when t − t 0 > x 0 − x, and on first sheet when t − t 0 < x 0 − x. On the other hand,ū J is on second sheet when t 0 − t > x 0 − x, and on first sheet when t 0 − t < x 0 − x. The value of U η on either sheet is given by Taking large c but cu J , cū J fixed limit and using (E16) and (E17), we see that