ER = EPR postulates the remarkable equivalence between spatial wormholes in general relativity (Einstein-Rosen bridges, ER) and quantum entanglement (Einstein-Podolsky-Rosen, EPR) [1]. In essence, when two systems are quantum-mechanically entangled, a “wormhole” is conjectured to be connecting them.

The proposal is fundamentally rooted in holographic entanglement in the anti–de Sitter space-conformal field theory correspondence (AdS-CFT) [2]. There, the observation is that an honest geometric wormhole can emerge between two black holes when the microscopic quantum correlations are large enough [3]. The classic example is the thermofield double state (TFD) of two copies of the holographic CFT on R×Sd-1, |TFD⟩=1Z(β)∑ie-β2Ei|Ei*⟩L|Ei⟩R,(1)where |Ei⟩L,R are energy eigenstates of the left (L) and right (R) CFT Hamiltonians HL,R. The star in |Ei*⟩ represents the action under CRT. On the one hand, this is an entangled state between the CFTs, which canonically purifies the thermal state of a single CFT. On the other hand, the TFD is dual to a connected spatial wormhole, corresponding to the Hartle-Hawking state of an eternal black hole in AdS space [4].

The TFD example is very illuminating but, as often emphasized [1,4–7], it is rather atypical, given that the TFD is a very special entangled state. For one thing, it lies in the diagonal subspace, annihilated by HL-HR. Additionally, its wave function has large correlations between the two systems, captured geometrically by a short wormhole with a bifurcate Killing horizon. Such fine-tuned correlations are dynamically destroyed by small perturbations injected at a scrambling time in the past [5].

If ER = EPR is to represent a general principle, one might ask: is there a wormhole for a typical entangled state of two black holes?

To motivate a definition of “typical EPR” that makes sense for CFTs in thermal equilibrium, we first consider a toy example consisting of two collections of N qubits, prepared in the maximally entangled state |I⟩=1D∑i=1D|i*⟩L|i⟩R,(2)where the |i⟩≡|i1⟩⊗…⊗|iN⟩ form an orthonormal basis and the dimension of each Hilbert space is D=2N. The state (2) is the infinite-temperature TFD, in which the systems share N Bell pairs.

We will define the ensemble of typical entangled states, ℰEPR, between these systems as the collection of states obtained by applying a random single-sided unitary to the TFD, |ΨEPR⟩=UR|I⟩, or, more explicitly, |ΨEPR⟩=1D∑i,j=1DUij|i*⟩L|j⟩R,(3)where U is a Haar random unitary in U(D). The state (3) is a random purification of the maximally mixed state ρL=ρR=(1/D)1.

Now, to define the typical entangled states of two black holes, we will proceed by analogy and generalize the notion above to finite temperature. In AdS-CFT, an equilibrium black hole has an associated equilibrium density matrix ρ of the CFT. For a neutral black hole, the state ρ is ensemble equivalent to the canonical Gibbs state ρβ=e-βH/Z(β) in the large-N limit of the CFT.

We define the ensemble of typical equilibrium entangled states, ℰEPRρ, as the collection of states of the form (ρ⊗ρ)|ΨEPR⟩, or explicitly, |ΨEPRρ⟩∝∑i,jρiρjXij|Ei*⟩L|Ej⟩R.(4)Here, ρi is the eigenvalue of ρ in the state |Ei⟩, and we have introduced X=DU for the Haar random U∈U(D). With this normalization, we can safely take the infinite Hilbert space dimension limit D→∞. In this limit, the coefficients Xij converge to i.i.d. Gaussian random variables with zero mean and unit variance. In Appendix A, we outline some of the properties of the ensemble ℰEPRρ. For example, on average, the reduced state is the equilibrium density matrix, E[ρL,R]=ρ.

Our question becomes, do such typical equilibrium entangled states have wormholes associated with them? Our approach in this Letter is to explicitly construct ensembles of states with wormholes that better and better approximate such typical EPRs.

Random states and unitaries are ubiquitous in quantum physics, but implementing them in practice is computationally hard. In the present context, the objective is to, even in principle, construct a typical-looking entangled state of two black holes starting from the TFD.

A natural possibility is to consider the time evolution of the TFD state |TFDt⟩=e-itHR|TFD⟩, or explicitly, |TFDt⟩=1Z(β)∑ie-(β2+it)Ei|Ei*⟩L|Ei⟩R.(5)As we elaborate in Appendix B, |TFDt⟩ will never become a random EPR, even if the Hamiltonian is completely structureless. The main reason is the conservation of energy along the trajectory, which prevents the state vector from exploring most of Hilbert space.

This example nevertheless provides a more general instance of ER = EPR. From the spacetime perspective, the wormhole grows in time [1,8]. The geometric volume of the wormhole is conjectured to encode a suitable notion of “quantum complexity” of the time evolution [9].

The way to continuously reach random-looking entangled states is to instead consider time-dependent Hamiltonians driving the evolution. For the sake of clarity, we shall first describe this in the qubit system. We will consider a general time-dependent Hamiltonian of the form H(t)=∑α=1Kgα(t)Oα,(6)where the Oα form a collection of K Hermitian operators, normalized so that Tr(Oα2)=D, with associated time-dependent couplings gα(t). The time-evolution operator now becomes U(t)=T{exp(-i∫0tdsH(s))}.(7)Applying it to the TFD generates states |Ψt⟩=U(t)R|I⟩, or, explicitly, |Ψt⟩=1D∑i,j=1DU(t)ij|i*⟩L|j⟩R.(8)In this Letter, we will select the couplings from a collection of white-noise correlated Gaussian random couplings, E[gα(t)]=0,E[gα(t)gα′(t′)]=Jδαα′δ(t-t′).(9)Such couplings define an independent random Hamiltonian at each time step. Upon exponentiation, U(t) follows a multiplicative Brownian motion in U(D), which corresponds to the continuous-time analog of a random quantum circuit. In particular, at each time step, the time evolution implements an infinitesimal “random gate” exp[-iδtH(t)]. At the level of the states (8), this Markov process defines an ensemble of states ℰt.

Under general assumptions for the drive operators, and in the absence of symmetries, it is possible to show that the measure of the ensemble of states ℰt converges weakly to the uniform measure at infinite times [10,11], limt→∞ℰt=ℰEPR,(10)or, equivalently, that U(∞) is Haar random.

For black holes, we will consider a holographic random quantum circuit selecting a collection of perturbations Oα of the CFT associated with low-energy matter fields in the bulk. An obvious problem that arises here is that applying U(t) to a finite-temperature TFD will eventually heat up the corresponding black hole, and this process will not generate a typical equilibrium EPR of the form (5). To stabilize the energy of the black hole, we will instead gradually cool down the random circuit applying the operator [12], U(Γt)=e-δβH0U(n,δt)e-δβH0⋯e-δβH0U(1,δt),(11)where n=t/δt. Formally, this defines an operator U(Γt)=P{exp(-i∫ΓtdsHc(s))}(12)by path ordering on a contour Γt, shown in Fig. 1. We follow an infinitesimal cooling-and-evolving process, with δt=δβ→0, where the relative magnitude between δt and δβ has been absorbed into the definition of J in (9).

We will utilize the gradually cooled random circuit to define the ensemble of states ℰΓt at fixed time, |ΨΓt⟩∝∑i,je-β4(Ei+Ej)U(Γt)ij|Ei*⟩L|Ej⟩R.(13)At t=0, the state (13) corresponds to the TFD (1). As we will later explain, when t→∞, U(Γ∞) converges weakly to a random matrix ρ0Xρ0 for Gaussian random X and a density matrix ρ0. The state ρ0 is determined by the steady state of the random quantum circuit [12]. Accordingly, the ensemble ℰΓt converges to the ensemble of typical entangled states of the two black holes, limt→∞ℰΓt=ℰEPRρ,(14)for the equilibrium state ρ=e-β4H0ρ0e-β4H0.(15)Thus, the random circuit allows us to continuously connect the TFD and a typical EPR of two black holes.

We will now elucidate the geometric properties of the semiclassical duals to |ΨΓt⟩, using the gravitational path integral to prepare them. While the precise geometry of individual states will be difficult to characterize, we will argue that they include two black holes connected by an Einstein-Rosen (ER) caterpillar [13]: a spatial wormhole with a large number of matter inhomogeneities and geometric features qualitatively similar to those illustrated in Fig. 2.

Unlike the multiple-shock spacetimes of [6], the caterpillars are not spherically symmetric and are constructed by introducing matter from the Euclidean section directly into the black hole interior, thereby preserving equilibrium and allowing the wormhole to become much longer.

In order to construct them, we start from the norm of the right-hand side of (13), which is proportional to the correlation function ZΨ(t)=Z(β)-1Tr[e-(β/2)H0U(Γt)e-(β/2)H0U(Γt)†]. This quantity can be alternatively expressed as the survival amplitude [12], ZΨ(t)=⟨TFD|U(Γt)1⊗U(Γt)1¯*|TFD⟩.(16)Here, we have introduced the TFD state between forward (1) and backward (1¯) contours at inverse temperature β. In this form, ZΨ(t) is computed by the CFT path integral along a complex time contour Ct, shown in Fig. 3. The time orientation of the backward contour is reversed, as U(Γt)1¯* is applied on this contour. The un-normalized state (13) is prepared by the lower half of this contour.

Using the AdS-CFT dictionary, ZΨ(t) can be evaluated semiclassically by the gravitational path integral with sources for the bulk fields gα(t) given by the time-dependent couplings of U(Γt). For single instances of the circuit, evaluating the gravitational path integral in a saddle point approximation is highly nontrivial. A trick that we will use is to average over the time-dependent couplings and evaluate the average norm, Z(t)≡E[ZΨ(t)].(17)Given that the perturbations that we have chosen are semiclassical for single realizations of the couplings, Z(t) will give us information about the average geometry over the ensemble of states. This is justified because the norm of the states—and thus, the semiclassical geometry that computes it—is self-averaging over ℰΓt [12]. At late times, Std[ZΨ(t)]/Z(t)∼e-S2(ρ) where S2(ρ)=-logTr(ρ2) is the second Rényi entropy of the equilibrium state (15).

The average over the Brownian couplings can be performed exactly, using the identity E[U(Γt)1⊗U(Γt)1¯*]=exp(-tHeff,1)(18)for the effective time-independent Hamiltonian Heff,1=H01+H01¯+J2∑α=1K(Oα1-Oα1¯*)2.(19)That is, the disorder average over the Brownian couplings effectively produces local-in-time interactions between both contours in the form of a time-independent Hamiltonian Heff,1. The quadratic interaction in (19) arises from the Gaussianity of the random couplings.

Inserting the identity (18) into (17), we find that the average norm corresponds to the matrix element Z(t)=⟨TFD|exp(-tHeff,1)|TFD⟩.(20)This is considerably simpler because (20) is a purely Euclidean object and so is the gravitational saddle point manifold Mt that evaluates it semiclassically, Z(t)∼Zgrav[Mt]as  N→∞,(21)where Zgrav[Mt]=e-Igrav[Mt]Z1-loop, with Igrav[Mt] the Euclidean gravitational action and Z1-loop the one-loop partition function for the bulk fields on Mt.

The relevant properties of Mt follow from an assumption for the effective Hamiltonian of the random circuit:

At large N, the effective Hamiltonian Heff,1 (19) is gapped, and it contains a unique ground state |GS⟩ with a semiclassical description as a connected spatial wormhole.

We will later motivate specific choices of the drive operators Oα, which lead to an effective Hamiltonian satisfying this assumption, but we keep the argument general here. Under the assumption above, the operator exp(-sHeff,1) is approximately proportional to the projector |GS⟩⟨GS| for s≥t⋆. What this means for Z(t) is that there is a Euclidean time-translation symmetry of the CFT path integral Z(t)≈⟨TFD|exp(-sHeff,1)|GS⟩⟨GS|exp[-(t-s)Heff,1]|TFD⟩, and the bulk saddle point Mt must inherit this isometry. The onset time of this symmetry is t⋆∼Egap-1logN*, where Egap is the gap and N* is the number of first excited states of Heff,1. As illustrated in Fig. 4, this symmetry deforms the geometry of the disk and makes it very long. Cutting the path integral open at a constant Euclidean time prepares the un-normalized ground state, represented in the bulk as a connected spatial wormhole (blue slice). For this reason, the metric on Mt is that of a Euclidean eternal traversable wormhole [12]. The relevant slice that determines the average geometry of the ensemble of ER caterpillars is the red reflection-symmetric slice on Mt. Because of the isometry, this slice contains an approximately cylindrical spatial wormhole of length scaling linearly with t-t⋆.

Since the effective Hamiltonian is gapped, the lightest particle excitations on this wormhole will correspond to the first excited states, with energy Egap. These excitations serve as a bulk “clock” to measure the wormhole length ℓ(t) and relate it to the asymptotic circuit time. The precise relation is [12] ℓ(t)ℓΔ=Egap(t-t⋆),(22)where ℓΔ is the correlation scale of the lightest excitations on the wormhole. The quantities Egap and ℓΔ are determined by quantizing the bulk fields on Mt with suitable boundary conditions and reading their energy and the decay of the bulk two-point function.

For explicit constructions of the ER caterpillars, we could specify the drive operators Oα to be local single-trace relevant conformal primaries, in which case α labels spatial points. In this case, the interaction (19) corresponds to an eternal version of the Gao-Jafferis-Wall double-trace perturbation [16]. Coincidentally, the effective Hamiltonian (19) was analyzed in detail in the Sachdev-Ye-Kitaev (SYK) model by Maldacena and Qi, with an interaction term formed by a large number K∼N of relevant Majorana bilinears [17]. In that case, the lesson is that the low-temperature phase of the coupled systems is gapped, with a unique ground state that corresponds to a connected spatial wormhole, satisfying our assumptions. The ground state is similar to the TFD at some coupling-dependent temperature β(J) (in units of the SYK couplings). Moreover, it has O(1) overlap with it for large-q SYK and for 1/N≪J≪1 in the regular q=4 SYK (in units where the SYK couplings J=1). In [12], we provided a totally explicit construction of the ER caterpillars of the SYK model based on this construction. A similar picture is expected to hold in higher dimensions, provided that the Hamiltonian H0 is sufficiently chaotic [18].

We will now derive a quantitative relation between the average length of the wormhole ℓ(t) and a microscopic feature of the ensemble of states ℰΓt. Let us explain this notion in the qubit system first. The idea is that, at finite but large circuit time t, the states (8) are not fully typical EPRs because ℰt and ℰEPR are statistically distinguishable. The approach to typical is quantified by the notion of a quantum state k design: an ensemble of states that reproduces the first k moments of ℰEPR. More precisely, the kth moment superstate of a general ensemble of states ℰΨ is defined as ρk(ℰΨ)=EℰΨ[(|Ψ⟩⟨Ψ|)⊗k].(23)The state ρk(ℰΨ) is defined on 2k replicas of the original Hilbert space. It is generally a mixed state, given the correlations between replicas that arise from the average over ℰΨ. Given this, a quantum state k design is defined by the condition ρk(ℰΨ)=ρk(ℰEPR). For ℰΨ=ℰt, this is equivalent to the condition that the random circuit U(t) forms a unitary k design: an ensemble of unitaries that reproduces the first k moments of the Haar ensemble.

Note that these notions are physically very strong, since a k design is indistinguishable from ℰEPR by any statistical k-copy measure. In practice, only approximate notions of k designs make sense at finite time. For these, one typically bounds some quantum distance between moment superstates.

We define a (ρ equilibrium) quantum state k design by the condition ρk(ℰΨ)=ρk(ℰEPRρ). For ℰΨ=ℰΓt, we will compute the distance to k design Δkρ(t)≡∥ρk(ℰΓt)-ρk(ℰEPRρ)∥2∥ρ1(ℰEPRρ)∥2k,(24)and the corresponding time to ktk=min{t:Δkρ(t)≤ϵ}. We are now ready to state the main result of this Letter.

The ensembles of caterpillars constructed in this Letter provide a window into the generic structure of the black hole Hilbert space in any theory of gravity with low-energy matter. The construction and main result of this Letter support a vastly more general form of ER = EPR and seem to be in some tension with arguments against semiclassicality of typical interiors [7,19–22]. We believe that resolving this tension could be relevant for the typical-state firewall paradox. We think caterpillars can be useful for this purpose because, from the randomness perspective, we see no reason why they should have their properties dramatically altered before exponential times, even though we cannot show this with semiclassical methods. In future work, this extrapolation of the semiclassical analysis could be addressed in explicit UV complete models at finite N using numerical or even experimental methods. Another open question is whether caterpillars offer any insight into weaker notions of firewalls in typical states, such as highly boosted matter shocks near the horizon [6,23–27]. Naively, since the matter in the caterpillar begins at rest, the states do not seem to exhibit this type of firewall. Past-evolved caterpillars, evolved over a few scrambling times, do, however, because the matter becomes highly boosted in the frame of the infaller. At the level of ensembles of microscopic quantum states, both the caterpillars and the past-evolved caterpillars form approximate quantum state k designs under the 2-norm definition used in this Letter. While, at the technical level, this points to the need for a stricter 1-norm definition of an approximate k design, it conceptually suggests that distinguishing states with firewalls from those without is extremely difficult. In fact, if we could extrapolate our result to exponentially large values of k, the indistinguishability would be so strong that the very question of whether a state has a firewall might become meaningless. It would be interesting to know whether this limitation is what the overlap analysis of [24–26] is indicating. It would also be worth exploring whether the state-dependent interior reconstruction of [28] plays any role here, given that our conclusion is largely state independent.

Finally, in the spirit of the “quantum gravity in the lab” program [29], it would be interesting to construct and study these states in the lab as a way of directly probing the black hole interior. Physically instantiating these states is not trivial since they involve nonunitary elements due to the gradual cooling, but such evolutions can be implemented inefficiently using postselection or a variety of other methods, e.g., the recently discussed double-bracket flow approach [30]. Alternatively, by constructing an isometry that embeds the eS low-energy states of the black hole into the microscopic Hilbert space (e.g., as was effectively done in [31] for a noninteracting fermion system), one might be able to combine this isometry with a conventional random circuit in the low-energy Hilbert space to produce random low-energy states embedded in the microscopic Hilbert space.