Kotaro Tamaoka,

^{3}

We argue that the entanglement of purification for 2D holographic conformal field theories (CFT) can be obtained from conformal blocks with internal twist operators. First, we explain our formula from the viewpoint of a tensor network model of holography. Then, we apply it to bipartite mixed states dual to a subregion of AdS

In the anti-de Sitter/conformal field theory (AdS/CFT) duality [

One such measure is the entanglement of purification (EoP) [

In this paper, we propose a formula of EoP for 2D holographic CFT [

More information about the blocks

In particular, our computation in

This section describes a heuristic justification for Eq. (

The holographic code model is a toy model of the AdS/CFT duality constructed by the tensor network. It captures the important relations between the bulk geometry and entanglement structures of the boundary theory; e.g., the Ryu–Takayanagi formula and he quantum error-correction feature of the boundary dual of bulk local operators in low-energy states (Hamilton–Kabat–Lifschytz–Lowe (HKLL) bulk reconstruction [

In the holographic code model, the duality map from bulk states to boundary states is an isometry map constructed by putting the so-called perfect tensors on the uniformly tiled 2D hyperbolic space. This isometry map is known as a “holographic code”. The features of a perfect tensor ensure that this code is just a quantum error-correcting code that embeds the bulk Hilbert space into a boundary one. The non-uniqueness of the reconstruction of the bulk local operators in the boundary theory can also be understood as the well known property of the quantum error-correcting code against erasure errors. Therefore, we think of this model as a toy model of the low-energy sector of the AdS/CFT with classical geometries. Then, we assume that the bulk state is the vacuum state and the quantum bulk degrees of freedom (d.o.f.) of the tensor correspond to the d.o.f. of the quantum fields in the semiclassical theory. Although the Hamiltonian is not specified in this model, we consider the bulk vacuum state as a product state so that the Ryu–Takayanagi formula holds without quantum corrections^{1}

First, we discuss the EoP in this model. Let us consider a bipartite mixed state

(a) Part of the boundary within the tensor network (TN) of

(i) Choose the original boundary pure state

(ii) Act the Hermitian conjugate of an isometry matrix ^{2}

(iii) After procedure (ii) ends, divide the boundary subregion

(iv) Iterate procedure (iii) with a different division of

It is very important that each step of the iterative procedure (ii) exactly corresponds to the steps of the “greedy algorithm” defined in Ref. [

The boundary space along

As a result of the previous subsection, the optimally purified state ^{3}

In this section, we argue that the right-hand side of Eq. (

Firstly, we extract the right-hand side of Eq. (

Hereafter we will assume that the scaling dimensions of the ^{4}

We have a boundary cylinder at ^{5}

Let us take the semiclassical limit, i.e., the large-

Eventually, we obtain

It is worth stressing that we have extracted the entanglement wedge cross section within the CFT framework.

Next, we derive the entanglement wedge cross section of a BTZ black hole [

It is well known that the above metric can be obtained from the global AdS

Let us first consider the case when the entanglement wedge does not cover the black hole (see the left-hand side of

Left: an entanglement wedge (shaded region) for a BTZ black hole. In this situation, the wedge does not cover the black hole horizon (black point in the center). Here we consider a time slice

In what follows, we will be more precise about our setup. We are now considering the semiclassical heavy–light Virasoro CBs (heavy–light CBs, for brevity) associated with the following six-point correlator:

Since we are considering the time slice of the boundary of Eq. (^{6}

This is a generic argument for heavy–light CBs, but let us focus on the OPE channel in which the

There is another phase of the entanglement wedge that covers the horizon. In this case, a rigorous GWD expression of the six-point CB has not yet been produced. On the other hand, it is known that the heavy–light CBs can be obtained from a different method, called the world-line approach [

For sufficiently large subsystems, an entanglement wedge covers the black hole horizon. In this case, the entanglement wedge cross section becomes the blue solid lines

Here the ^{7}

This situation is just what we want to consider. In the end, we get

The right-hand side of Eq. (

Before closing this section, we briefly mention the case of the two-sided eternal black hole [

In this paper, we have proposed a formula (

From the argument in

We have focused on 2D CFT and its bulk dual. One may be curious about its extension to higher dimensions. Since the twist operators in higher dimensions become non-local, generalization of our argument is not so straightforward. At least the

Since our insight and optimization were based on the holographic code model, we are still assuming some holography. In particular, we do not say that our argument proves

It is also promising to see the connection of EoP to the kinematic space [

We are grateful to Tokiro Numasawa, Tadashi Takayanagi, and Satoshi Yamaguchi for valuable comments and discussions. K.T. would like to thank the organizers of the workshop “Holography, Quantum Entanglement and Higher Spin Gravity II”, where part of this work was presented.

Open Access funding: SCOAP

In this appendix, we note some explicit forms of the quantities displayed in

Here

The minimum length between two geodesics

Here

^{1} In the holographic code model, the HKLL-like property holds independently of the bulk state. On the other hand, if the bulk state is entangled, the Ryu–Takayanagi formula receives quantum corrections, which is the EE of the bulk state.

^{2} To be precise, more effort is needed to justify the second equality in Eq. (

^{3} Strictly speaking, in the holographic code model,

^{4} In unitary CFT, the twist operators have the lowest scaling dimension in the sectors with twist number

^{5} Note that we are not discussing conformal partial waves but rather conformal blocks.

^{6} We can relate

^{7} Our convention of the Virasoro conformal blocks is different from Ref. [