Building up spacetime with quantum entanglement II: It from BC-bit

Abstract

In this note, we describe how collections of arbitrary numbers of "BC-bits," distinct non-interacting quantum systems each consisting of a holographic boundary conformal field theory (BCFT), can be placed in multipartite entangled states in order to encode single connected bulk spacetimes that approximate geometries dual to holographic CFT states. The BC-bit version of a holographic CFT state corresponds to a geometry that can be made arbitrarily similar to the associated CFT-state geometry within a "causal diamond" region defined by points that are spacelike separated from the boundary time slice at which the state is defined. These holographic multi BC-bit states can be well-represented by tensor networks in which the individual tensors are associated with states of small numbers of BC-bits.

Figures (6)

• Figure 1: Euclidean path 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 d) an entangled state of two 2D BCFTs e) a state of a 3D CFT on a sphere f) a state of a 3D BCFT on a disk.
• Figure 2: a)-c) BC-bits defined from a 2+1 dimensional CFT on a spatial sphere. d) Geometry $\tilde{H}$ used to define a state of the BC-bits. e) Cross section of $\tilde{H}$.
• Figure 3: Lorentzian geometries from path-integral states: a) A CFT on a circle. b) Euclidean path integral defining a holographic CFT state. c) Path integral used to compute observables in this state. Operators can be inserted on dashed lines. d) Euclidean gravity solution corresponding to this path integral. e) Spatial slice at time-symmetric point serves as initial data for 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 $t=0$. $\tilde{a}$) - $\tilde{f}$) Equivalent construction for the BC-bit states. Each BC-bit is a boundary CFT on an interval.
• Figure 4: Euclidean geometries associated with path integrals for states of two BC-bits, using the end-of-the-world brane model for holographic BCFTs. Each shows half of the Euclidean geometry, up to the slice that becomes the initial data for the Lorentzian geometry. a) When $\tilde{H}$ is sufficiently close to $H$, the ETW branes are disconnected and localized near the corresponding boundary components. The Lorentzian initial data slice is connected and similar to that for the single CFT path integral state defined by $H$. b) As the modifications defining $\tilde{H}$ become too severe, we can have a transition in which the ETW brane topology changes. Here, the Lorentzian initial data slice becomes disconnected, though there will still be entanglement between the matter in the two components of spacetime. c) A path integral defining a pure state of the same two BC-bits as in b). This corresponds to two disconnected spacetimes with no entanglement between the matter in the different components.
• Figure 5: a) An extra circular boundary component is added to the interior of $\tilde{H}$ in the path integral for a six BC-bit state. b) The path integral can now be decomposed into a product of path integrals defining three BC-bit states. The field configurations for the BC-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.