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JHEP05(2019)052

Published for SISSA by Springer

Received: February 9, 2019

Accepted: May 2, 2019

Published: May 9, 2019

Holographic Renyi entropy from quantum error

correction

Chris Akers and Pratik Rath

Center for Theoretical Physics and Department of Physics, University of California,

Berkeley, CA 94720, U.S.A.

Lawrence Berkeley National Laboratory,

Berkeley, CA 94720, U.S.A.

E-mail: cakers@berkeley.edu, pratik rath@berkeley.edu

Abstract: We study Renyi entropies S

in quantum error correcting codes and compare

the answer to the cosmic brane prescription for computing

≡ n

∂



n−1



. We ﬁnd

that general operator algebra codes have a similar, more general prescription. Notably, for

the AdS/CFT code to match the speciﬁc cosmic brane prescription, the code must have

maximal entanglement within eigenspaces of the area operator. This gives us an improved

deﬁnition of the area operator, and establishes a stronger connection between the Ryu-

Takayanagi area term and the edge modes in lattice gauge theory. We also propose a new

interpretation of existing holographic tensor networks as area eigenstates instead of smooth

geometries. This interpretation would explain why tensor networks have historically had

trouble modeling the Renyi entropy spectrum of holographic CFTs, and it suggests a

method to construct holographic networks with the correct spectrum.

Keywords: AdS-CFT Correspondence, Gauge-gravity correspondence, Models of Quan-

tum Gravity

ArXiv ePrint: 1811.05171

Open Access,

 The Authors.

Article funded by SCOAP

https://doi.org/10.1007/JHEP05(2019)052

JHEP05(2019)052

The RT formula naturally appears when one computes the von Neumann entropy of an

encoded state in a quantum error-correcting code. We demonstrate this now in the context

of the type of code we will be using throughout: an operator-algebra quantum error-

correcting (OQEC) code with complementary recovery. These appear to be the best codes

to model AdS/CFT [9], and we will explain their details in section 2. For now, it suﬃces

to say that we consider a ﬁnite-dimensional Hilbert space H = H

⊗ H

, and a Hilbert

space H

code

⊆ H. Furthermore, these Hilbert spaces have the following decompositions

= ⊕

⊗ H

) ⊕ H

, (1.2)

code

= ⊕

⊗ H

¯a

) , (1.3)

with dim H

= dim H

. In these codes, a “logical” density matrix ρ acting on H

code

encoded in a “physical” density matrix acting on H. Moreover, we say that A encodes a if

there exists a unitary operation U

on H

such that

eρ

= U

(⊕

⊗ χ

)) U

†

, (1.4)

and the density matrices ρ

act on H

and correspond to the state ρ

that we wished

to encode in eρ

. I.e. ρ

acts on H

in the same way that ρ

does on H

. The density

matrices χ

act on H

and correspond to the extra degrees of freedom that help encode.

The von Neumann entropy is

S(eρ

) = tr (eρ

) −

log p

S(ρ

) (1.5)

where the “area operator” is deﬁned as

≡ ⊕

S(χ

¯a

. (1.6)

Compare this to eq. (1.1). The ﬁrst term on the r.h.s. of both are the expectation value

of linear operators evaluated in the state of interest. The other terms are naturally the

algebraic von Neumann entropy of the logical state [9]. This is the basic connection between

RT and quantum error-correcting codes.

In AdS/CFT, there is a natural generalization of the RT formula that we now describe.

The von Neumann entropy S(ρ) has a well-known generalization called the Renyi entropies,

(ρ) ≡

1−n

log tr ρ

. While the Renyi entropy equals the von Neumann entropy for n = 1,

it is widely believed that the Renyi entropy does not satisfy anything qualitatively similar

to the RT formula for n 6= 1 [11, 12]. However, a related quantity called the reﬁned Renyi

entropy

(ρ) ≡ n

∂



n − 1

(ρ)



(1.7)

also reduces to the von Neumann entropy for n = 1 but in fact does satisfy a generalized

version of the RT formula [11]. One computes

(ρ

) holographically via the so-called

“cosmic brane prescription,” which works roughly as follows. Into the state ρ

, insert an

extremal codimension-2 cosmic brane in AdS homologous to the boundary region A, and

– 2 –

JHEP05(2019)052

let this brane have tension T

n−1

4nG

. The reﬁned Renyi entropy is related to the area of

this brane:

(ρ

) =

brane

n,bulk

, (1.8)

where

n,bulk

is the reﬁned Renyi entropy of bulk matter ﬁelds in a particular state ρ

which

is deﬁned naturally by an n-sheeted path integral on the boundary (for details see [13]).

We describe this prescription in detail in section 4. As one might expect, the cosmic brane

prescription limits to the RT formula as n → 1, because in that limit the tension vanishes

and the cosmic brane reduces to an extremal surface.

Because the cosmic brane prescription generalizes RT, it behooves us to investigate

whether the connection between RT and quantum error-correcting codes can be generalized

to the cosmic brane prescription. Let us formulate this question more precisely. Does the

reﬁned Renyi entropy of eρ

from eq. (1.4) satisfy some formula like

(eρ

)

= tr(eρ

brane,A

) +

n,logical

, (1.9)

for some state eρ

brane,A

, where

n,logical

represents the reﬁned Renyi entropy of the logical

state? If eρ

indeed satisﬁes such a formula, what determines the state eρ

brane,A

, and why

does this state manifest in AdS/CFT as inserting a brane with a particular tension into eρ

Our main result is to answer these questions. In short, yes: one can derive the cosmic

brane prescription within the formalism of OQEC, and we do this in section 4. Notably,

this derivation requires that the AdS/CFT code has certain special properties, which we

now explain by discussing reﬁned Renyi entropy in general OQEC codes.

For a general OQEC code, the reﬁned Renyi entropy of eρ

from eq. (1.4) is

(eρ

) =

(n)

(χ

) +

n,logical

, (1.10)

where

(n)

= tr



eρ

tr(eρ

)



(1.11)

and P

|α, ii hα, i| is a projector onto a particular value of α. While this equation

bears some resemblance to eq. (1.9), they do not match in general.

Indeed, there are two aspects of the code that need to be true for eq. (1.9) to hold.

First, it must be the case that

(n)

(χ

) = tr



eρ

(n)



(1.12)

It is important to note that ρ

is diﬀerent in general than the original bulk state dual to ρ

. It is the

state of the matter ﬁelds prepared by a path integral on a bulk geometry with an n-sheeted boundary and

that satisﬁes the gravitational equations of motion everywhere. This matches our results below, in which

n,bulk

comes predominantly from the state of the bulk ﬁelds on the saddle geometry dual to the n-sheeted

boundary state.

– 3 –

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