Research Article

Spacetime from bits

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Science  09 Oct 2020:
Vol. 370, Issue 6513, pp. 198-202
DOI: 10.1126/science.aay9560

Spacetime, reconstructed

Theories of holographic duality feature a correspondence between a gravitational system and a strongly interacting conformal field theory (CFT) living on the system's boundary. Through this correspondence, the CFT encodes the geometry of spacetime in the gravitational system. Van Raamsdonk analyzed the role of entanglement in this theoretical framework. Instead of considering a single CFT, the author's starting point was a collection of CFT “bits” that are mutually entangled but do not interact with one another. The spacetime that these bits collectively encode was then shown to be arbitrarily close to the one encoded by the original CFT, suggesting that entanglement plays a crucial role in the emergence of spacetime.

Science, this issue p. 198

Abstract

In the anti–de Sitter/conformal field theory approach to quantum gravity, the spacetime geometry and gravitational physics of states in some quantum theory of gravity are encoded in the quantum states of an ordinary nongravitational system. Here, I demonstrate that this nongravitational system can be replaced with an arbitrarily large collection of noninteracting systems (“bits”) placed in a highly entangled state. This construction makes manifest the idea that spacetime geometry emerges from entanglement between the fundamental degrees of freedom of quantum gravity and that removing this entanglement is tantamount to disintegrating spacetime. This setup also reveals that the entangled states encoding spacetimes may be well represented by a certain type of tensor network in which the individual tensors are associated with states of small numbers of bits.

In the anti–de Sitter/conformal field theory (AdS-CFT) or “holographic” approach to quantum gravity (1), spacetime physics and gravitational dynamics are encoded in the states of a nongravitational quantum system, often a strongly interacting quantum field theory with invariance under scaling transformations, that is, a conformal field theory. Over the past decade, there has been increasing evidence that quantum entanglement between the degrees of freedom in this nongravitational system plays a crucial role in the emergence of spacetime [see, for example, (27), or (8) for a review]. One notable example of this phenomenon is that placing two distinct CFTs without interactions between them into a particular entangled state amounts to connecting the two encoded spacetimes via a wormhole (2). Partly on the basis of this example, it was argued that (5, 6) for any spacetime described by a CFT state, the geometry is “built up” by the entanglement present in the CFT state and that removing this entanglement destroys the encoded spacetime. However, the continuous nature of the CFT systems that encode the spacetimes makes this argument challenging. In the usual examples, part of the spacetime structure (the fixed asymptotic behavior) is directly related to the continuous geometrical space upon which the CFT is defined. Further, the local degrees of freedom of the CFT interact strongly with those around them, and completely disentangling the various parts would require an infinite amount of energy.

In this paper, I introduce a framework for holography in which the fundamental degrees of freedom are a large collection of elementary systems that do not interact with one another and have no intrinsic spatial arrangement. The individual systems are conformal field theories on spaces with boundaries; these can be thought of as “bits” of the original CFT. I show that states of a holographic CFT can be replaced by entangled states of these bits so that the system still describes a single connected spacetime that is arbitrarily close to the one encoded in the original CFT state. This construction makes it clear that it is possible to build up very generic connected spacetimes by entangling discrete noninteracting systems; because entanglement is the only thing that relates these systems, it is also manifest that the encoded spacetime completely disintegrates when the systems are disentangled.

In the bits framework, I show that there is a natural way to represent the state with arbitrary precision using a type of tensor network. This representation confirms previous work suggesting that certain tensor network states of discrete elementary subsystems (4, 9, 10) capture the qualitative features of the entanglement structure of CFT states encoding spacetimes.

Boundary conformal field theory bits

The motivation for this construction is the idea of cutting up a holographic conformal field theory into a large number of noninteracting pieces but putting the pieces into a quantum state that approximates a state of the original CFT encoding some spacetime. To make the construction precise, it is necessary to define what is meant by a CFT living on a piece of space with a boundary. Generally speaking, the boundary conditions for the fields at the edges need to be described. For any CFT, there are various choices of boundary conditions. A special subset of these conditions preserve the scaling symmetry and conformal invariance (11) and define what is known as a boundary conformal field theory, or BCFT [see, for example, (12)].

A set of “BCFT bits” associated with a CFT on some spatial geometry M is a collection of BCFTs defined on a set of “sanded” pieces M˜i of M. Here, {M˜i} are defined by cutting M into a set of simply connected pieces {Mi} and “sanding the edges,” that is, M˜i is a large subset of the interior of Mi with a smooth boundary (Fig. 1). Each BCFT is defined from the original CFT with the same choice of boundary conditions, so the BCFT-bit system is specified by the choice of {M˜i} and the choice of boundary condition.

Fig. 1 BCFT bits.

Starting with a conformal field theory on some connected space M, the associated BCFT bits M˜i are defined as a collection of conformal field theories with boundaries living on spatial geometries that approximate a set of pieces of the original space. This is illustrated for the case where M is (A) a circle or (B) a two-dimensional (2D) sphere.

The goal here is to consider some state of the CFT that corresponds to a smooth geometry and to approximate this state by a particular entangled state of the discretized BCFT-bit system. I will argue that the newly generated state still encodes a smooth connected geometry that is closely related to the original geometry.

Entangling BCFT bits via the Euclidean path integral

The Euclidean path integral gives a mechanism by which to define a quantum state of a CFT on a spatial geometry M given a spatial geometry H of one higher dimension with boundary M (e.g., Fig. 2A). The quantum state is specified by assigning a probability amplitude to each possible configuration of fields on M. This probability amplitude is given by the average of a specific quantity (the exponential of the negative of the Euclidean action) over all possible field configurations on H subject to the condition that the fields take the specified values on M. For holographic CFTs, this construction gives a natural way to define states that encode dual spacetimes with a good classical description [see, for example, (1315) and references therein]. The spacetime being described can be changed by varying the interior geometry of H and adding sources for various CFT operators on the interior of H; for example, arbitrary linearized perturbations around empty anti–de Sitter spacetime can be obtained by choosing the right geometry and sources (15).

Fig. 2 Euclidean path integrals.

The integrals defining (A) a state of a 2D CFT on a circle, (B) an entangled state of two 2D CFTs each on a spatial circle, (C) a state of a 2D BCFT on an interval, and (D) an entangled state of two 2D BCFTs.

Similarly, it is also possible to define the state of a BCFT on a spatial geometry M with boundary B given a higher-dimensional geometry H with boundary MG, where G is also bounded by B, as shown in Fig. 2C. In this case, the boundary conditions at G are taken to be those of the BCFT under consideration.

The Euclidean path integral can be used to define natural entangled states of noninteracting CFTs or BCFTs by taking the H to be a connected geometry with a disconnected boundary M (16). For example, taking H to be a finite cylinder, M will be a pair of circles, and the path integral on H defines an entangled state of a pair of CFTs encoding two spacetimes connected by a wormhole (Fig. 2B). I will use this idea to define an entangled state of BCFT bits by taking H to be a connected geometry whose boundary is the collection {M˜i}.

More specifically, I would like to define a state of the BCFT bits that is related to some state of the original CFT on M defined by the path integral over H. To do so, I define a geometry H˜ obtained from H by removing smooth “grooves” at the surface so that the part of the boundary remaining is M˜i. This is depicted in Fig. 3 for a 1+1-dimensional CFT defined on a spatial circle. The boundary of H˜ is {iM˜i}G, where G corresponds to the surface of the grooves. The state of the BCFT bits can now be defined by performing the Euclidean path integral for the BCFT over the geometry H˜, with the appropriate boundary conditions imposed at G.

Fig. 3 Entangled state of many BCFTs.

The path integral defining some state of a holographic CFT (left) is approximated by a path integral defining an entangled state of many BCFTs (right). The second path integral geometry is obtained from the first one by introducing additional small boundary components (shown in blue).

The state |ΨH˜ and the possible dual geometry will depend on the details of H˜. However, I will now argue that if I have been sufficiently gentle with my carpentry tools—that is, if {M˜i} is close enough to {Mi}, and if H˜ is close enough to H—that the state |ΨH˜ of the BCFT bits encodes a geometry that is closely related to the original geometry described by the CFT state |ΨH.

Spacetimes dual to Euclidean path integral states

Let us now recall how to understand the spacetime geometries encoded by holographic CFT states described by a Euclidean path integral. According to the AdS/CFT correspondence, features of the encoded geometry can be deduced from the CFT by evaluating the expectation value of various local and nonlocal observables in the CFT state. For the states in this work, these expectation values are computed using a path integral over a surface H¯H that is obtained by gluing H to its mirror image H¯ along M, as shown in Fig. 4, C and I (17). The operators of interest are inserted along the junction. The AdS/CFT correspondence implies that this CFT path integral is equal to a gravitational path integral whose average over geometries is dominated by a single Euclidean geometry XH, obtained by solving the gravitational equations with boundary conditions that XH is asymptotically anti–de Sitter with boundary geometry H¯H (18).

Fig. 4 Lorentzian geometries from path-integral states.

(A) A CFT on a circle. (B) Euclidean path integral defining a holographic CFT state and (C) the path integral used to compute observables in this state. Operators can be inserted on the dashed line. (D) Euclidean gravity solution corresponding to the path integral in (C). (E) Spatial slice at the time-symmetric point serves as the initial data for the Lorentzian solution. (F) Lorentzian solution associated with our state. The interior of the causal diamond (dashed lines) is the part encoded by the CFT state at t = 0. (G to L) Equivalent constructions for the BCFT-bit states. Each BCFT bit is a boundary CFT on an interval.

The Lorentzian spacetime geometry associated with my CFT state is simply related to the Euclidean geometry XH. The geometry XH has a reflection symmetry inherited from the geometry of H¯H (Fig. 4, C and I, depicts the geometry XH sliced along the plane of symmetry). The surface left invariant under this symmetry has a geometry (XH)0. To find the spacetime associated with this state, I use the geometry (XH)0 (and the condition that time derivatives of fields vanish here) as the initial conditions for the real-time gravitational equations. The solution is a spacetime XHL corresponding to the state. More formally, XHL can be understood as an analytic continuation of XH. The CFT state at time (t) = 0 strictly encodes only the region of this spacetime that is spacelike separated from the t = 0 slice at the boundary (19), given that the spacetime outside this region can be altered by changes to the CFT Hamiltonian before or after t = 0.

The crux of my subsequent argument is that, despite the considerable differences between the original CFT and the collection of BCFT bits as physical systems, the geometry H˜ used to define the BCFT-bit state is a small perturbation to the geometry H used to define the CFT state. Thus, it might be expected that the procedure I have just outlined gives rise to a spacetime dual to the BCFT bits that is almost the same as the spacetime dual to the CFT state.

Geometry of the BCFT-bit states

I would like to understand how the Euclidean gravity solution corresponding to a BCFT on H˜¯H˜ differs from that corresponding to the CFT on H¯H (Fig. 4, C versus I). The main obstacle here is understanding how the presence of a boundary in H˜¯H˜ (geometrically described as the space G glued to a mirror image of itself along B) affects the gravity calculation. This question was considered originally in (20) and later in more detail in (21). As discussed in those papers, if the BCFT state has a geometrical dual XH˜, this dual must itself have a boundary component in addition to the asymptotically AdS boundary with boundary geometry H˜. In detailed examples using ultraviolet (UV) complete theories of gravity, this boundary component corresponds to a place where some compact internal dimensions smoothly degenerate (22, 23), similar to how a Kaluza-Klein circle shrinks to zero in a bubble-of-nothing spacetime. I expect that the specific way such a boundary is realized will depend on the boundary conditions chosen for the BCFT.

As a simple model, the suggestion in (20, 21) can be used to introduce an explicit end-of-the-world (ETW) brane with constant tension and Neumann boundary conditions. In that case, as long as a given boundary region of H˜ is small compared with other geometrical features of H˜ (such as the distance to other boundary components) the ETW brane ending on that boundary component stays localized to the vicinity of that boundary component, as depicted in Fig. 4, J and K (blue surfaces). The ETW brane may source bulk fields and affect the rest of the geometry, but for a fixed location away from the ETW brane, these effects will become negligible in the limit where the boundary component is taken to be small.

For 1+1-dimensional CFTs, a more direct argument can be given that does not rely on a particular holographic model for BCFTs. In this case, H˜¯H˜ is a two-dimensional Euclidean geometry, as in the example of Fig. 4I. Here, each component of G can be taken to be a small circle, as shown in Fig. 4I. If the circle is sufficiently small relative to the distance to other boundary components and operator insertions, it is expected that (as in the operator product expansion) its insertion is equivalent to the insertion of a sum of local CFT operators. In the limit where the circle becomes small, this operator sum approaches the identity operator (24), so the insertion of the boundary G has a very small effect.

According to these arguments, the Euclidean geometry associated with H˜¯H˜ should be almost the same as that associated with H¯H. But this is not quite true for the corresponding Lorentzian geometries. The reason is that no matter how small the boundary components of H˜¯H˜ are, they still change the asymptotic geometry of the slice that serves as the initial data for the Lorentzian evolution (Fig. 4J). The result is that in the Lorentzian picture, the initial data slice is almost the same as that for the original CFT state, but because of the differences in asymptotics at the boundary, there may be some type of shockwave evolving forward and backward from each introduced boundary component. In the ETW brane picture, this shockwave can be understood as a Lorentzian ETW brane whose worldvolume is part of a hyperboloid (the analytic continuation of the hemispherical ETW branes in the Euclidean picture of Fig. 4J). The Lorentzian spacetime is depicted in Fig. 4L. In a limit with very many BCFT bits and very many small boundary components, this shockwave or ETW brane will propagate outward from the full asymptotic boundary of the initial data slice. Thus, the BCFT-bit version of a holographic state faithfully reproduces the region of the original spacetime whose points are spacelike separated from the t = 0 slice of the boundary.

A necessary consequence of these shockwaves is that it is no longer possible to move causally between asymptotic regions associated to different BCFT bits; such causal paths would imply interactions between the bits that are not present in this study’s setup. In this sense, the modified spacetime has causal properties similar to a two-sided black hole, where it is not possible to move between the two asymptotic regions. As with a black hole, the spacetime considered here may also contain a future singularity, which might be avoided by reintroducing some interactions between neighboring BCFT bits.

Emergence of the radial direction and quantum error correction

It is of interest what part of the dual spacetime can be reconstructed having access only to density matrices for subsystems of a number (n) of adjacent BCFT bits. In general, the region encoded by a subsystem has been shown (2528) to be the region bounded by the minimal-area surface enclosing a portion of the geometry that includes the subsystem A at the boundary; the areas of such surfaces give the entropies of the corresponding subsystems (3). As shown in Fig. 5, the region for a subsystem of n adjacent bits extends farther into the bulk as n is increased; standard calculations (3) show that the proper distance into the bulk increases as the logarithm of n (29). To learn about regions deeper within the geometry, knowledge of entanglement structure and correlations at longer scales is needed. This relation between the geometrical depth and the scale of entanglement aligns with the well-known connection between the radial direction in the geometry and renormalization group flow in the field theory: physics deeper within the bulk geometry corresponds to physics further into the infrared in the dual quantum theory (1, 30). The setup in this study also manifests the recently explained feature (31) that the encoding of geometry into holographic systems is qualitatively similar to a quantum error-correcting code: Local physics near the point P in Fig. 5 can be recovered with knowledge of the state of bits 1 to 6 or bits 4 to 9, but it cannot be recovered with knowledge of only bits 4 to 6, that is, the intersection of these systems. Thus, the information about physics near point P is encoded nonlocally and redundantly in the BCFT bits in the way that a logical qubit is encoded nonlocally in the physical qubits of a quantum error-correcting code.

Fig. 5 Reconstructing spacetime from adjacent bits.

The density matrices for bits 1 to 6, 4 to 9, and 4 to 6 encode the geometry of regions A, B, and C, respectively, bounded by the minimal area extremal surfaces enclosing these bits. Physics at P can be recovered from bits 1 to 6 or bits 4 to 9 but not from bits 4 to 6.

Although this section has focused on the encoding of the spatial geometry, I emphasize that the construction described in the previous section leads to a full spacetime dual to the BCFT-bit state. However, it is an interesting open question how the emergent bulk time and, more generally, the local bulk physics at various spacetime points can be understood from the BCFT perspective.

Tensor networks for holographic BCFT-bit states

The BCFT-bit construction of holographic states bears a close resemblance to tensor network toy models of holography (4, 9, 10) in that there are explicit multipart entangled states of a discrete set of noninteracting constituents. I will now demonstrate that, as in the toy models, the states considered herein may be represented arbitrarily well by a type of tensor network, where the tensors correspond to states of small numbers of BCFT bits. The construction involves additional small changes to H˜.

I have argued previously that the introduction of a small boundary component to H˜ has a vanishingly small effect on the dual Euclidean geometry in the limit that the size goes to zero, except very close to the insertion locus. Thus, if additional boundary components are introduced away from M, these should have a negligible effect on the BCFT-bit state as the newly introduced boundary components are taken to be infinitesimal.

By introducing these extra boundaries (for example, by drilling narrow holes through H˜), it is possible to represent the newly fabricated state as a tensor network, by decomposing the path integral into chunks (Fig. 6). Here, each individual tensor corresponds to the state of a small number of BCFT bits, which are the original BCFT bits for the external legs or newly constructed internal BCFT bits. The internal edges of the tensor network correspond in path integral language to identifying the field configurations on the two BCFT bits joined by the edge and integrating over these. In quantum language, this corresponds to projecting the state of the pair of BCFT bits onto a maximally entangled state. The full set of these projections joins up the tensor network, defining what is known as a projected entangled pair state [see, for example, (32)].

Fig. 6 Tensor network representation.

(A) An extra circular boundary component is added to the interior of H˜ in the path integral for a six-BCFT-bit state. (B) The path integral can now be decomposed into a product of path integrals defining three BCFT-bit states. The field configurations for the BCFT bits connected by dashed lines are equated and integrated over the path integral equivalent of the pair projection that connects a tensor network. (C) The tensor network representation of the state.

There is a great deal of freedom in how to build up a tensor network representation: The placement of the boundary components can be chosen, as can the surfaces along which to break up H˜ to give the internal BCFT bits. So, as expected, it is possible to have many tensor networks that represent the same state. The networks here apparently have a closer connection to the geometry of the path integral defining the state rather than the geometry of the space being encoded; such a connection was emphasized recently in (33) [see also (34, 35)]. However, other recent work (36, 37) suggests that by a particular optimization of the path-integral geometry over the geometries related by conformal invariance, the path-integral geometry actually becomes the geometry of the bulk spatial slice. Similarly, it may be that some optimization of the geometrical slicing produces a tensor network similar to those (9, 10) where the network geometry has a close connection to the bulk spatial geometry.

The limit of many BCFT bits

For simplicity, the various examples shown in the figures involve only a small number of BCFT bits. In this case, the individual BCFT bits might carry information about a substantial portion of the dual geometry. However, the faithful reproduction of the original spacetime applies equally well when the number of BCFT bits is taken to be very large, so long as the modifications leading from H to H˜ are kept small (e.g., each M˜i is still a large subset of Mi). In the limit where the BCFT bits are all small compared with any scale associated with M or the original CFT state, I expect that the individual bits carry almost no information about the geometry being represented by the collection of BCFT bits, apart from the asymptotic behavior of the bulk fields at a single boundary location corresponding to the bit. Thus, the spacetime geometry is almost entirely encoded in the entanglement structure of a multipart BCFT-bit system. In this system, it is manifestly true that disentangling the bits causes the corresponding spacetime to disintegrate, as suggested for continuous CFTs (6). I emphasize that in the BCFT-bit system, the individual bits have no intrinsic location relative to one another, so it is not only the radial direction of the spacetime that “emerges.”

A universal entanglement–gravity duality?

Because the entanglement structure is playing such a key role here, and because the individual bits carry almost no information about the interior of the spacetime, it is of interest what properties of the BCFT bit are really required here. If the BCFT bits are replaced with some other type of bit but the entanglement structure is kept the same, does this still encode the same spacetime and gravitational physics? This might seem too optimistic; the specific theory of gravity encoded in a CFT state has a certain number of dimensions and a certain set of fields in addition to the metric, and these are related to the dimensionality and the operator content of the CFT. If an attempt is made to replace BCFT bits from one CFT with BCFT bits from another CFT, perhaps of a different dimensionality, it may seem impossible that the same gravitational physics is being described, given that the new CFT may correspond to a completely different theory of gravity.

But there is a way around this obstacle: The optimistic scenario could work if there is fundamentally only one theory of quantum gravity. This is in line with expectations from string theory (38), where the study of string dualities suggests that various UV complete gravitational theories in various dimensions can be understood as descending through compactification and dualities from 11-dimensional M-theory. Thus, even if the gravitational physics in one region of spacetime has the fields and interactions associated with a particular low-energy theory of gravity, the fields and interactions in another region of spacetime could correspond to a different low-energy theory if there is some transition region in between where the properties of the compact extra dimensions change.

In the example where BCFT bits from one CFT are replaced with BCFT bits associated with another CFT, the asymptotically AdS regions very close to the BCFT bits (near the diamonds in Fig. 4L) will clearly have different physics. But moving inward in the radial directions, there may be a transition (e.g., with some compact dimensions changing shape or topology) such that inside a radial position associated with a boundary scale containing a large number of BCFT bits, the physics is that of the spacetime described by the original CFT. If the BCFT bits are replaced with more general quantum systems (e.g., collections of qubits or macaroni), it may be that the asymptotic region no longer has a geometrical description, but the same interior region emerges.

The idea that it is possible to obtain a precise description of quantum gravitational physics starting from sufficiently many copies of an arbitrary quantum system is intriguing but certainly does not follow from any of the arguments in this paper. Nevertheless, it is fascinating that the possible unity of gravitational theories as suggested by string theory leaves open the possibility of such an exact and universal entanglement–gravity duality.

References and Notes

  1. For example, the vacuum state of the CFT on a half space with these boundary conditions at the edge preserves an SO(d-1,2) of the SO(d,2) conformal symmetry.
  2. Any complex sources on H should be conjugated in H¯, but I will restrict to the case of real sources; the resulting geometry will then have a time-reversal symmetry.
  3. The asymptotic behavior of other fields in the geometry is fixed by the sources for the corresponding operator in the path-integral action.
  4. This is the domain of dependence of the region (XH)0.
  5. To understand this limit, consider inserting the circular boundary in a disk path integral. The resulting geometry is conformally equivalent to a finite cylinder; in the limit where the circle becomes small, this cylinder becomes infinitely long, so the path integral gives the vacuum state. In this limit, the inserted boundary is equivalent to inserting the identity operator.
  6. An interesting point is that the areas and entropies are finite here, because they end on ETW branes rather than at the asymptotic boundary of the spacetime.
Acknowledgments: Funding: This work was supported in part by a Simons Investigator award, by the Simons Foundation grant “It from Qubit: Simons Collaboration on Quantum Fields, Gravity and Information” and by the Natural Sciences and Engineering Research Council of Canada. Competing interests: The author has no competing interests.
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