# Holevo Information and Ensemble Theory of Gravity

Xiao-Liang Qi, Zhou Shangnan and Zhenbin Yang

###### Abstract

Holevo information is an upper bound for the accessible classical information
of an ensemble of quantum states. In this work, we use Holevo information to
investigate the ensemble theory interpretation of quantum gravity. We study
the Holevo information in random tensor network states, where the random
parameters are the random tensors at each vertex. Based on the results in
random tensor network models, we propose a conjecture on the holographic bulk
formula of the Holevo information in the gravity case. As concrete examples of
holographic systems, we compute the Holevo information in the ensemble of
thermal states and thermo-field double states in the Sachdev-Ye-Kitaev model.
The results are consistent with our conjecture.

##  1 Introduction

In recent years, various studies of low dimensional gravitational models have
suggested that a simple bulk gravitational theory may be described by an
ensemble of boundary theoriessaad2019jt ; stanford2019jt ; Saad:2019pqd ;
bousso2020gravity ; marolf2020transcending ; Stanford:2020wkf ;
Pollack:2020gfa ; Afkhami-Jeddi:2020ezh ; Maloney:2020nni ; Belin:2020hea ;
Cotler:2020ugk ; Chen:2020ojn ; Hsin:2020mfa ; Belin:2020jxr ;
Milekhin:2021lmq . More precisely, the gravitational path integral including
topology changes (Euclidean wormholes) is proposed to be the holographic dual
of an ensemble average of a family of quantum many-body systems, rather than a
single one.111Historically, the connection between wormholes and ensemble
average has been speculated by Coleman coleman1988black  and been sharpened in
AdS/CFT by Maldacena-Maoz maldacena2004wormholes .  From the quantum
information perspective, this suggests two possible different interpretations
of the bulk geometry as a quantum state. The traditional one, as in the
standard holographic dictionarymaldacena1999large , is that a given bulk
geometry and a bulk matter state is dual to a definite quantum state $\rho$ on
the boundary when a boundary Cauchy surface is given. The picture of ensemble
duality suggests another possibility: that a given bulk geometry and a
boundary Cauchy surface corresponds to an ensemble average of quantum states
$\rho(J)$, where the ensemble parameter $J$ can have a probability
distribution $p_{J}$. The parameter $J$ may be continuous or discrete. For
concreteness, we will assume it is continuous, and normalize the probability
distribution as $\int dJp_{J}=1$. These two descriptions correspond to two
different sets of rules of gravitational quantizations, and various
information paradox can arise if we are not careful enough about
distinguishing these rules. One example that illustrates these two different
descriptions is the appearance of bra-ket wormhole in a gravitational prepared
statepenington2019replica ; chen2021bra : these are global CFT states
described by summation of two Euclidean semiclassical gravitational evolutions
including the Hartle-Hawking geometry and a bra-ket wormhole geometry. On the
one hand, with the standard holographic dictionary, the boundary state is a
pure state with vanishing entropy, which can be correctly calculated by the
island rules Almheiri:2019hni . On the other hand, canonical quantization of
the bra-ket wormhole geometry predicts a mixed state of the CFT. Such a state
can be thought as an averaged description of an ensemble of different pure
states depending on some unknown parameters $J$: $\rho_{a}=\int
dJp_{J}\rho(J)$. While these two interpretations are clear in this case,
generically, the meaning of ensemble averaged states in AdS/CFT is less
explored.

Assuming that simple gravity is an ensemble theory, a natural question is the
“size" of the ensemble. There could be infinite possible values of parameter
$J$. For example in the Sachdev-Ye-Kitaev (SYK) modelsachdev1993gapless ;
kitaev2014hidden ; kitaev2015simple , the natural random parameter labeling
the states is the random coupling which is a set of independent Gaussian
random variables. (We will discuss this model in more detail later.) However,
the size of $J$ parameter space is not a physically meaningful measure of the
size of the ensemble. For example, one can consider an ensemble in which
$\rho(J)$ only depends on certain linear superposition of different $J$
parameters. An intrinsic measure of the size of the ensemble is the
information contained in the ensemble, which can be measured by the Holevo
informationholevo1973bounds . (We would like to note that Holevo information
in quantum gravity has been studied for a different kind of
ensemblebao2017distinguishability ; bao2021holevo .)

Consider an observer who carries quantum measurements on $\rho(J)$, and tries
to learn about the random parameter $J$. In general, we can consider a
projected operator valued measurement (POVM), which is defined by a set of
positive Hermitian operators $M_{a}$ satisfying $\sum_{a}M_{a}=\mathbb{I}$.
The measurement leads to an output $a$ with probability $P(a|J)={\rm
tr}\left(M_{a}\rho(J)\right)$, which corresponds to the joint probability
$P(a,J)=p(J)P(a|J)$. The mutual information $I(a:J)$ between the two random
variables $a,J$ measures how much information one can learn about $J$ from the
measurement result $a$. For example, if $J$ is a function of $a$, the mutual
information is maximal. We can define a quantum mutual information that bounds
$I(a:J)$ from above. By introducing an ancilla state $\left|J\right\rangle$
which are orthogonal for different $J$, we can define the auxiliary state

$\displaystyle\rho_{SW}=\int
dJp_{J}\rho(J)\otimes\left|J\right\rangle\left\langle J\right|$ (1)

where we denote $S$ as the system and $W$ as the ancilla. The mutual
information between $S$ and $W$ is the Holevo information, which provides an
upper bound to the classical mutual information between $J$ and the
measurement result $a$ (since measurements are quantum channels applied to
$S$, which can only reduce the mutual information). The explicit formula of
the Holevo information is

$\displaystyle H=I(S:W)=S\left(\int dJp_{J}\rho(J)\right)-\int
dJp_{J}S\left(\rho(J)\right)$ (2)

In this paper, we will study the Holevo information for the ensemble of states
in quantum gravity. In Sec. 2, we study the Holevo information for the
ensemble of random tensor network states (RTN), which are toy models of
quantum gravity. For RTN with large bond dimension, we show that the Holevo
information is determined by the difference between the generalized entropy
with trivial entanglement wedge and that with the actual entanglement wedge
(i.e. the Ryu-Takayanagi (RT) formularyu2006holographic ). In Sec. 3, we make
a conjecture on the holographic formula of Holevo information for quantum
gravity (more precisely, for asymptotically anti de-Sitter (AdS) space and AdS
coupled with non-gravitational bath). In particular, in an evaporating black
hole after Page time, the Holevo information is given by the difference
between the “Hawking entropy" computed for the original geometry without
replica wormhole, and the Page entropy which takes into account of the replica
wormhole and entanglement island penington2020entanglement ;
almheiri2019entropy ; penington2019replica ; almheiri2020replica . We also
discussed a possible generalization of the proposed Holevo information in
systems with multiple quantum extremal surfaces. In Sec. 4 we study the SYK
model to gain further understanding on the Holevo information. The random
parameters are the random coupling of the SYK model. We study two different
ensembles of states, including the thermal states and the thermofield double
(TFD) states. The Holevo information for the thermal states ensemble is given
by the difference between maximal entropy and the thermal entropy. For the TFD
states, the Holevo information is determined by the fermion equal-time
correlation between the two sides. We discuss the holographic interpretation
of these results and show that they are consistent with our conjectured bulk
formula. Finally, we provide further discussion in Sec. 5. In particular, we
emphasize that the difference between RTN and gravity theory comes from the
nonlocality of the random parameters in the latter case, and discuss how to
possibly modify the RTN model to make it closer to the gravity theory.

##  2 Tensor network models

To understand the Holevo information $H$ in quantum gravity, we first consider
the toy model of random tensor networks (RTN). An RTN is defined in the
following wayhayden2016holographic . Define two quantum systems $B$ (bulk) and
$S$ (boundary, in the case of AdS/CFT, or boundary and bath in the case of
evaporating black holedong2020effective ). Their Hilbert spaces are denoted as
$\mathbb{H}_{B}$ and $\mathbb{H}_{S}$, with the total Hilbert space
$\mathbb{H}_{BS}\equiv\mathbb{H}_{B}\otimes\mathbb{H}_{S}$. In addition,
$\mathbb{H}_{B}=\otimes_{x}\mathbb{H}_{x}$ has a tensor factorization into
different qudits, which correspond to vertices of the tensor network. An RTN
state $\rho_{s}$ in $\mathbb{H}_{S}$ is defined by a random product state
$\left|V(J)\right\rangle\equiv\otimes_{x}\left|V_{x}(J_{x})\right\rangle\in\mathbb{H}_{B}$,
and a state $\rho_{P}\in\mathbb{H}_{BS}$:

$\displaystyle\rho_{S}(J)$ $\displaystyle=p_{J}^{-1}{\rm
tr}_{B}\left(\left|V(J)\right\rangle\left\langle V(J)\right|\rho_{P}\right)$
(3) $\displaystyle\text{with~{}}p_{J}$ $\displaystyle={\rm
tr}_{BS}\left(\left|V(J)\right\rangle\left\langle V(J)\right|\rho_{P}\right)$
(4)

Here $\left|V_{x}(J_{x})\right\rangle$ is a random state in the site Hilbert
space $\mathbb{H}_{x}$. More precisely,
$\left|V_{x}(J_{x})\right\rangle=U_{x}\left|0\right\rangle$ is obtained by
applying a Haar random unitary $U_{x}$ on an arbitrary reference state
$\left|0\right\rangle$. $J_{x}$ parameterizes this symmetric manifold with a
uniform probability distribution. If we choose the normalization

$\displaystyle\int dJ\left|V(J)\right\rangle\left\langle
V(J)\right|=\mathbb{I}_{B}$ (5)

the ensemble of random states $\left|V(J)\right\rangle$ defines an ensemble of
RTN $\rho_{S}(J)$ with probability $p_{J}$. Physically, we can consider the
projection to $\left|V(J)\right\rangle$ as the consequence of a POVM, and the
state $\rho_{S}(J)$ is the state after the measurement for the qubits that are
not measuredhayden2016holographic . In this construction, $\rho_{P}$ can be a
generic state, although conventionally a tensor network state refers to the
situation when $\rho_{P}$ consists of a product of EPR pairs defined on links
of a graph.

Now we investigate the Holevo information for this ensemble, for a subsystem
$A\subset S$. According to Eq. (5), the averaged state $\overline{\rho_{A}}$
is easily determined:

$\displaystyle\overline{\rho_{A}}=\int dJp_{J}\rho_{A}(J)={\rm
tr}_{BC}\left(\rho_{P}\right)$ (6)

where $\rho_{A}(J)={\rm tr}_{C}\rho_{S}(J)$ is the density matrix of $A$ for
fixed $J$ parameter. Here $C$ is the complement of $A$ in $S$. We will
consider the situation that $\rho_{P}$ consists of EPR pairs
$\rho_{P}=\prod_{\left\langle
xy\right\rangle}\left|xy\right\rangle\left\langle xy\right|\otimes\rho_{r}$,
and the bond dimension of each EPR pair is large, with $\rho_{r}$ entropy
remains finite. In such large bond dimension limit, the Renyi entropy
$S^{(n)}(\rho(J))$ is independent of the random parameter
$J$hayden2016holographic , and is given by the RTN version of Ryu-Takayanagi
formularyu2006holographic 222More precisely, for a given small deviation
$\delta$, there exists a critical bond dimension
$D_{c}=\frac{\alpha}{\delta^{2}}e^{c_{2n}V}$ with $V$ the number of vertices
in the bulk, and $\alpha,c_{2n}$ order-one consants. For $D\gg D_{c}$, the
deviation of the Renyi entropy to the RT value is smaller than $\delta$ with a
high probability: $\left|S^{(n)}\left(\rho_{A}(J)\right)-\min_{\Sigma\subset
B}S^{(n)}_{\Sigma A}\left(\rho_{P}\right)\right|<\delta$ with probability
$P(\delta)=1-\frac{D_{c}}{D}$. For more discussions about this, see Chapter 7
of Ref. hayden2016holographic . :

$\displaystyle S^{(n)}\left(\rho_{A}(J)\right)\simeq\min_{\Sigma\subset
B}S^{(n)}_{\Sigma A}\left(\rho_{P}\right)$ (7)

If we consider the large bond dimension limit such that the equation above
holds for all integer $n$, the same formula applies to the von Neumann entropy
by analytic continuation:

$\displaystyle S\left(\rho_{A}(J)\right)\simeq\min_{\Sigma\subset B}S_{\Sigma
A}\left(\rho_{P}\right)$ (8)

The region $\Sigma$ that minimizes the righthand side is the entanglement
wedge of $A$, which we denote as $\Sigma_{A}$. Therefore we get

$\displaystyle H_{A}=S\left(\overline{\rho_{A}}\right)-\int
dJp_{J}S\left(\rho_{A}(J)\right)\simeq
S_{A}\left(\rho_{P}\right)-S_{A\Sigma_{A}}\left(\rho_{P}\right)$ (9)

If for all subregion $\Sigma$ of the bulk,
$S_{\Sigma|A}\left(\rho_{P}\right)\geq 0$, the minimization in Eq. (8) will be
given by an empty $\Sigma$, which corresponds to zero Holevo information.

We would like to note that the right-hand side of Eq. (9) is minus the
conditional entropy of $\Sigma_{A}$ and $A$:

$\displaystyle H_{A}=-S_{\Sigma_{A}|A}\left(\rho_{P}\right)$ (10)

This conditional entropy must be negative in order for an entanglement wedge
to exist. Interestingly, this is also the necessary and sufficient condition
for quantum teleportation from $A$ to $\Sigma_{A}$ bennett1993teleporting ;
witten2020mini . Our calculation shows that for random measurements, the
Holevo information of the resulting ensemble (which measures the knowledge of
the remaining quantum state about the classical measurement output) is equal
to the coherent information. In the quantum setting, coherent information is
positive if $\Sigma_{A}$ is more strongly correlated with $A$ than with the
environment, which agrees with our intuition about entanglement wedge.

Eq. (9) and (10) tells us that the information we can obtain from a subsystem
$A$ about the random parameter $J$ is equal to the entropy difference between
the contribution of trivial entanglement wedge and the actual entanglement
wedge.

Figure 1: (a) A general random tensor network, with each triangle representing
a vertex state. $\rho_{P}$ is generally a mixed state. Each line here (and
also in subfigures (b) and (c)) corresponds to a pair of indices, labeling a
complete basis in $\mathbb{H}_{x}\otimes\overline{\mathbb{H}}$. (b) and (c)
Two examples of RTN with $\rho_{P}$ factorized into bulk QFT state $\rho_{b}$
and EPR pairs (red curves). The difference between (b) and (c) is whether the
boundary is maximally entangled with its complement in $\rho_{P}$. (d) A more
specific example of RTN, where we have only drawn the EPR pairs. In this
simple case, the entropy of boundary region $A$ (purple circles on the right-
hand side) is determined by the number of links at the minimal cut
$\gamma_{A}$ (black dashed line), and the Holevo information is determined by
the difference between the area of $A$ (number of links crossing the red
dashed line) and that of $\gamma_{A}$.

To illustrate this result, we can consider two different cases, shown in Fig.
1 (b) and (c). The first example is the RTN corresponding to the AdS/CFT case,
when the boundary is connected with the bulk through maximally entangled EPR
pairs, and there is a bulk quantum field theory state
$\left|\Psi_{b}\right\rangle$. (The bulk state does not have to be pure, but
we write the formula for pure state case for simplicity.) The state
$\rho_{P}=\left|\Psi_{P}\right\rangle\left\langle\Psi_{P}\right|$ has the
structure

$\displaystyle\left|\Psi_{P}\right\rangle$
$\displaystyle=\otimes_{\left\langle xy\right\rangle\in
B}\left|xy\right\rangle\otimes\left|\Psi_{b}\right\rangle\otimes\otimes_{x\in
B,Y\in S}\left|xY\right\rangle$ (11)

Here $\left|xY\right\rangle$ are maximally entangled EPR pairs between
boundary and bulk sites. For simplicity we assume all EPR pairs have the same
dimension $D$. In this case, the averaged state $\overline{\rho_{A}}$ is
maximally mixed, and $S(\rho_{A}(J))$ is given by the RT formula with quantum
corrections, which leads to the Holevo information

$\displaystyle H_{A}$ $\displaystyle=|A|\log D-S_{\rm gen}(A)$
$\displaystyle=\left|A\right|\log D-\left(\left|\gamma_{A}\right|\log
D+S_{\Sigma_{A}}\left(\left|\Psi_{b}\right\rangle\left\langle\Psi_{b}\right|\right)\right)$
(12)

$\gamma_{A}=\partial\left(\Sigma A\right)$ is the minimal cut separating $A$
and the complement. This is illustrated in Fig. 1 (b) (for the general setup)
and (d) (for a simple example).

More generally, we can consider a state $\left|\Psi_{P}\right\rangle$ with $A$
not maximally entangled with the complement. For example, we can slightly
modify the state (13):

$\displaystyle\left|\Psi_{P}\right\rangle$
$\displaystyle=\otimes_{\left\langle xy\right\rangle\in
B}\left|xy\right\rangle\otimes\left|\Psi_{bS}\right\rangle\otimes\otimes_{x\in
B,Y\in S}\left|xY\right\rangle$ (13)

as is illustrated in Fig. 1 (c). In this case, $A$ contains degrees of freedom
from $\left|\Psi_{bS}\right\rangle$ in addition to EPR pairs, leading to the
entropy

$\displaystyle H_{A}=|A|\log
D+S_{A}(\left(\left|\Psi_{bS}\right\rangle\left\langle\Psi_{bS}\right|\right))-\left(\left|\gamma_{A}\right|\log
D+S_{A\Sigma_{A}}\left(\left|\Psi_{bS}\right\rangle\left\langle\Psi_{bS}\right|\right)\right)$
(14)

One example of this case is the RTN model for an evaporating black hole
coupled with a non-gravitational bathdong2020effective . When $A$ is a
subsystem of the bath with no direct connection to the gravitational part,
there is no area law term $|A|\log D$, and the only contribution to the
averaged state entropy is that of the QFT
$S_{A}\left(\left|\Psi_{bS}\right\rangle\left\langle\Psi_{bS}\right|\right)$.
More discussions about the evaporating black hole will be carried in next
section. Our discussion of the SYK model in Sec. 4 will also be related to
this case.

The RT formula (7) for RTN comes from a replica caculation of ${\rm
tr}\left(\rho_{A}^{n}\right)$ (see Appendix. A). The two terms in Holevo
information correspond to taking $n$ copies of $\rho_{P}$ and inserting cyclic
permutation operators only in $A$ (for the first term) or in $\Sigma_{A}A$.
This suggests that these two terms correspond to saddle points in
gravitational calculation where replica wormhole is absent or present,
correspondingly. We will discuss the gravity case in the next section.

##  3 Conjecture about the gravity case

Based on the tensor network results, we can make some conjecture on the Holevo
information in AdS/CFT, or AdS/CFT coupled with bath. We start from the case
of an AdS black hole coupled with a non-gravitational flat-space bath. As we
learned from recent workspenington2020entanglement ; almheiri2019entropy ;
penington2019replica ; almheiri2020replica , for a large region of the
radiation $A$, after Page time the entropy of $A$ is given by a nontrivial
quantum extremal surface which is a boundary of an Island region $I$:

$\displaystyle S_{A}=\frac{\left|\partial I\right|}{4G_{N}}+S_{\rm qft}(AI)$
(15)

If we use a replica trick and consider the calculation of Renyi entropy
$S_{A}^{(n)}$, the bulk geometry contains a replica
wormholepenington2019replica ; almheiri2020replica  that connects different
replicas in a region that becomes $I$ in the $n\rightarrow 1$ limit. In this
case, the gravitational path integral for integer $n>1$ have multiple saddle
points. Besides the leading saddle point of replica wormhole, there is always
a trivial saddle point which is $n$ disconnected copies of of the original
geometry. Considering the tensor network results, it is natural to conjecture
that after the Page time, the Holevo information is given by the difference
between the trivial and nontrivial saddle points, which leads to:

$\displaystyle H_{A}$ $\displaystyle=S_{\rm trivial}(A)-S_{\rm
actual}(A)=S_{\rm qft}(A)-S_{\rm qft}(AI)-\frac{\left|\partial
I\right|}{4G_{N}}$ (16)

Before the Page time, we get $H_{A}=0$. It should be noted that Ref.
bousso2020gravity  has already proposed that the trivial saddle point
corresponds to the entropy of the ensemble-averaged state.

Figure 2: (a) Entanglement island in an AdS black hole coupled with flat space
bath. (b) A boundary region $A$ of the AdS space, when there is no bath. (c) A
top view of the spatial slice in (b), with the regularized region
$\tilde{\Sigma}_{A}$ and its boundary $\tilde{\gamma}_{A}$ (see text).

The situation is a bit more complicated in the AdS/CFT case without a bath. If
we consider $A$ to be part of the boundary, there is no trivial saddle point
in the replica calculation of $S^{(n)}(A)$ since the boundary condition
already connects the different replicas. The Holevo information seems to be
divergent in this case. This situation is similar to the tensor network model
result in Eq. (12). Naively one expects

$\displaystyle H_{A}=S_{\rm max}(A)-S_{\rm
gen}(A)=\frac{\left|A\right|-\left|\gamma_{A}\right|}{4G_{N}}+S_{\rm
qft}\left(A\right)-S_{\rm qft}(A\Sigma_{A})$ (17)

Here $S_{\rm qft}(A)$ is the entropy of $A$ with trivial entanglement wedge.
However, since $A$ is co-dimension-$2$ in the bulk space-time, $S_{\rm
qft}(A)$ is a UV dependent quantity (just like the entropy of an infinitesimal
interval in an $1+1$-dimensional CFT), which requires regularization. To
regularize this quantity we introduce a region $\tilde{\Sigma}_{A}$ (see Fig.
2 (c)), which is a small neighborhood of $A$. For example, we can cover $A$
with balls (in the boundary space) with radius $\epsilon$, and define
$\tilde{\Sigma}_{A}$ as the union of the entanglement wedges of all the
$\epsilon$-balls. (see e.g. Ref. czech2015information  for a related
construction.) Then we can define

$\displaystyle H_{A}=S_{gen}(A\tilde{\Sigma}_{A})-S_{\rm
gen}(A\Sigma_{A})=\frac{\left|\tilde{\gamma}_{A}\right|-\left|\gamma_{A}\right|}{4G_{N}}+S_{\rm
qft}\left(A\tilde{\Sigma}_{A}\right)-S_{\rm qft}(A\Sigma_{A})$ (18)

Here $\tilde{\gamma}_{A}=\partial\left(A\tilde{\Sigma}_{A}\right)$ is a
regularized version of the area of $A$, and $S_{\rm
qft}\left(A\tilde{\Sigma}_{A}\right)$ is a regularized version of $S_{\rm
qft}(A)$.

It is interesting to discuss generalizations of our conjecture to systems
where there is a subleading saddle point corresponding to a smaller
entanglement wedge. For example, one can consider a two-sided black hole
corresponding to a thermofield double state $\left|TFD\right\rangle$, and then
apply a unitary $U_{R}$ on the right boundary system. This will preserve the
entanglement entropy between left and right, but if the unitary increases
energy density of the $R$ system, it can lead to a bigger coarse-grained
entropy, corresponding to a subleading quantum extremal surface
$\tilde{\gamma}_{R}$ that is closer to the $R$ boundary. This is illustrated
in Fig. 3(a). A simpler situation is two regions in the AdS vacuum (Fig. 3
(b)), which has a connected entanglement wedge (when the mutual information
$I(A:B)$ is nonzero to the leading order of $\frac{1}{G_{N}}$). In this case,
there is a subleading saddle point corresponding to the disconnected quantum
extremal surface $\tilde{\gamma}_{AB}=\gamma_{A}\cup\gamma_{B}$. This
corresponds to a smaller entanglement wedge
$\tilde{\Sigma}_{AB}=\Sigma_{A}\cup\Sigma_{B}\subset\Sigma_{AB}$ which is a
subset of the actual entanglement wedge.

Figure 3: Two example cases with a subleading quantum extremal surface
corresponding to a smaller entanglement wedge. (a) A two-sided black hole
where the right boundary $R$ has an coarse-grained entropy that is larger than
entanglement entropy. (b) Two regions $AB$ for which the actual entanglement
wedge is connected, and there is a subleading quantum extremal surface
$\tilde{\gamma}_{AB}=\gamma_{A}\gamma_{B}$ which is the disjoint HRT surface
of $A$ and $B$ separately.

In such situation, the difference of generalized entropy contribution between
these two saddles do not directly corresponds to the Holevo information we
have discussed so far, since the subleading saddle point still corresponds to
nontrivial connection between different replicas. However, in random tensor
networks there is a Holevo information interpretation of this situation. In
RTN the random parameters are locally defined at each vertex. Thus we can
define a restricted ensemble by denoting $J$ as the random parameters in the
tensors in a given bulk region, rather than everywhere. For example in Fig. 3
(a), if we define $\Omega=\Sigma_{R}\backslash\tilde{\Sigma}_{R}$ to be the
region between $\gamma_{R}$ and $\tilde{\gamma}_{R}$, then averaging over the
random tensor parameters in this region, denoted by $J_{\Omega}$, corresponds
to forbidding the entanglement wedge to overlap with this region when
optimizing the location of the quantum extremal surface. Thus the optimization
will lead to the subleading saddle point $\tilde{\gamma}_{R}$:

$\displaystyle S\left(\int
dJ_{\Omega}\rho\left(J_{\Omega}\right)\right)=\frac{\left|\tilde{\gamma}_{R}\right|}{4G_{N}}+S_{\rm
qft}\left(\tilde{\Sigma}_{R}R\right)$ (19)

which implies that the restricted ensemble $\rho(J_{\Omega})$ has the Holevo
information

$\displaystyle
H_{R}=\frac{\left|\tilde{\gamma}_{R}\right|-\left|\gamma_{R}\right|}{4G_{N}}+S_{\rm
qft}\left(\tilde{\Sigma}_{R}R\right)-S_{\rm qft}\left(\Sigma_{R}R\right)$ (20)

A natural question is what the interpretation of $H_{R}$ is in quantum
gravity. The natural candidate is an ensemble of states that share the same
semi-classical geometry and are only different in the bulk region $\Omega$.
For example, we can consider different bulk matter states in $\Omega$ with the
same stress tensor. However, it is important to note that the ensemble we
discuss is much bigger than just different bulk QFT states in $\Omega$. For
example in the two-interval case in Fig. 3 (b), even if the system is in the
AdS vacuum, such that there are no other states sharing the same stress
tensor, the Holevo information is still nonzero:

$\displaystyle H_{AB}^{(\Omega)}$
$\displaystyle=\frac{\left|\gamma_{A}\right|+\left|\gamma_{B}\right|-\left|\gamma_{AB}\right|}{4G_{N}}+S_{\rm
qft}\left(\Sigma_{A}\cup\Sigma_{B}\right)-S_{\rm qft}\left(\Sigma_{AB}\right)$
$\displaystyle=I(A:B)-I_{\rm qft}\left(\Sigma_{A}:\Sigma_{B}\right)$ (21)

Our interpretation is that the random parameters are not associated with
different states in the bulk, but with different mappings from bulk to
boundary. Roughly speaking, for a given semi-classical geometry, small
fluctuations around this geometry defines a “code subspace" of states which
are mapped to the boundary with an isometryalmheiri2015bulk . The random
parameters in the tensors physically correspond to random parameters in the
bulk-to-boundary isometry. In the case of subleading saddle point, the random
tensor network model suggests that we can define a two-step isometry

$\displaystyle\Omega\longrightarrow\tilde{\gamma}_{R}\longrightarrow\text{boundary}$
(22)

and $J_{\Omega}$ are random parameters in the first isometry. Different
$J_{\Omega}$’s correspond to different mappings from the same bulk physics in
region $\Omega$ to different boundary states. One should remember that the
boundary Hamiltonian can correspondingly depend on $J_{\Omega}$, which means
the boundary dynamics can appear to be $J_{\Omega}$ independent (at least for
simple correlation functions) even if the state and the Hamiltonian depends on
$J_{\Omega}$. We will see this more explicitly in the discussion of the SYK
model in next section.

##  4 Sachdev-Ye-Kitaev model

The Majorana fermion version of SYK model describes $N$ Majorana fermions with
the $q$-body coupling

$\displaystyle
H_{J}={i^{q/2}}\sum_{j_{1}<j_{2}<...<j_{q}}J_{j_{1}j_{2}...j_{q}}\chi_{j_{1}}\chi_{j_{2}}...\chi_{j_{q}}$
(23)

where $J_{j_{1}j_{2}...j_{q}}$’s are antisymmetric in the indices, and are
independent Gaussian variables with the probability distribution

$\displaystyle p_{J}$
$\displaystyle=\Omega_{q}^{-1}\exp\left[-\frac{1}{2c_{q}\mathcal{J}^{2}}\sum_{j_{1}<j_{2}<...<j_{q}}J_{j_{1}j_{2}...j_{q}}^{2}\right]$
(24) $\displaystyle c_{q}$
$\displaystyle=\frac{2^{q-1}(q-1)!}{N^{q-1}q},~{}\Omega_{q}=\left(2\pi
c_{q}\mathcal{J}^{2}\right)^{\frac{1}{2}\small{\left(\begin{array}[]{c}N\\\
q\end{array}\right)}}$ (27)

The SYK model has approximate conformal invariant low energy
dynamicskitaev2014hidden ; kitaev2015simple ; maldacena2016remarks  and is
proposed to approximately dual to Jackiw-Teitelboim (JT)
gravityjackiw1985lower ; teitelboim1983gravitation ; Almheiri:2014cka ;
Jensen:2016pah ; maldacena2016conformal ; Engelsoy:2016xyb . The random
coupling $J_{j_{1}j_{2}...j_{q}}$ naturally defines an ensemble of states. We
will study the Holevo information in this ensemble in two different scenarios.

###  4.1 Thermal state

The first case we consider is the thermal state

$\displaystyle\rho_{\beta}(J)=Z_{J}^{-1}e^{-\beta H_{J}}$ (28)

In the large $N$ limit, the SYK partition function is self-averaged. More
precisely, for an order-$1$ integer $n$,

$\displaystyle\overline{Z_{J}^{n}}\equiv\int dJp_{J}Z_{J}^{n}$ (29)

can be written in the path integral of collective variables $G^{ab}(\tau)$ and
$\Sigma^{ab}(\tau)$, with $a,b=1,2,...,n$. The path integral is dominated by a
saddle point which is replica diagonal $G^{ab}(\tau)=G(\tau)\delta^{ab}$. The
coupling between different replicas are suppressed by a factor
$O\left(N^{2-q}\right)$ kitaev2018soft . This leads to

$\displaystyle\overline{\log
Z_{J}}=\log\overline{Z_{J}}+O\left(N^{2-q}\right)$ (30)

Thus the second term in the Holevo information $\int
dJp_{J}S\left(\rho_{\beta}(J)\right)$ self-averages, and is equal to the
thermal entropy $S_{\rm th}\left(\beta\right)$ of SYK model.

To determine the first term of the Holevo information, we need to compute

$\displaystyle\overline{\rho}=\int dJp_{J}\rho_{\beta}(J)$ (31)

$\overline{\rho}$ can be determined by symmetry. For each Majorana fermion
operator $\chi_{i}$, we have

$\displaystyle\chi_{i}\rho_{\beta}(J)\chi_{i}=Z_{J}^{-1}e^{-\beta\chi_{i}H_{J}\chi_{i}}$
(32)

$\chi_{i}$ commutes with terms in $H_{J}$ that does not contain $i$, and anti-
commutes with terms that contains an $i$. Denote
$\tilde{J}_{j_{1}j_{2}...j_{q}}$ by

$\displaystyle\tilde{J}_{j_{1}j_{2}...j_{q}}=\left\\{\begin{array}[]{cc}-J_{j_{1}j_{2}...j_{q}},&~{}i\in\left\\{j_{1},j_{2},...,j_{q}\right\\}\\\
J_{j_{1}j_{2}...j_{q}},&~{}i\notin\left\\{j_{1},j_{2},...,j_{q}\right\\}\end{array}\right.$
(35)

Apparently $Z_{\tilde{J}}=Z_{J}$ and $p_{\tilde{J}}=p_{J}$. Thus we have

$\displaystyle\chi_{i}\rho_{\beta}(J)\chi_{i}$
$\displaystyle=\rho_{\beta}(\tilde{J})$ (36)
$\displaystyle\chi_{i}\overline{\rho}\chi_{i}$ $\displaystyle=\overline{\rho}$
(37)

In other words, $\overline{\rho}$ has to commute with all $\chi_{i}$, which
means it has to be proportional to the identity. Thus
$\overline{\rho}=2^{-N/2}\mathbb{I}$ is the maximally mixed state. Therefore
we conclude that the Holevo information is given by

$\displaystyle H_{\beta}=\frac{N}{2}\log 2-S_{\rm th}\left(\beta\right)$ (38)

This formula is consistent with the random tensor network result in Eq. (12).
The first term is the maximal entropy and the second term is the actual
entropy of the state.

In the gravity dual theory, $S_{\beta}$ corresponds to the RT entropy given by
the dilaton value at the black hole horizon, plus the contribution of the bulk
fermion field (c.f. chen2019entanglement ). As is illustrated in Fig. 4 (a),
the bulk interpretation of $S\left(\overline{\rho}\right)$ is the generalized
entropy of a small interval $\tilde{\Sigma}_{R}$, given by

$\displaystyle
S_{gen}\left(\tilde{\Sigma}_{R}R\right)=\frac{\phi_{\epsilon}}{4G}+S_{\rm
qft}\left(\tilde{\Sigma}_{R}\right)$ (39)

Both terms on the right-hand side of this equation depends on the UV cutoff in
the bulk, but the sum should be independent from the choice of cutoff. $S_{\rm
qft}\left(\tilde{\Sigma}_{R}\right)$ is the entropy of a single interval for
the $N$ bulk Majorana fermions. Interestingly, if we discretize the bulk AdS2
fermion into a $1+1$-dimensional lattice fermion, and take the limit when
$\tilde{\Sigma}_{R}$ is a single lattice site, the entropy $S_{\rm
qft}\left(\tilde{\Sigma}_{R}\right)$ will reach the maximal value
$\frac{N}{2}\log 2$, which suggests that $\phi_{\epsilon}=0$ at the lattice
level. In other words, the dilaton term can be viewed as purely from
integrating out bulk UV fermion modes.

###  4.2 thermofield double state

Now we can consider a different scenario where for each $J$ we are given the
thermofield double state rather than the thermal state. The thermofield double
state is a purification of the thermal state, defined as follows. First we
choose a maximally entangled state $\left|I\right\rangle$ between two copies
of the SYK model, and then define

$\displaystyle\left|TFD_{\beta}(J)\right\rangle=2^{N/4}\rho_{\beta
L}^{1/2}\left|I\right\rangle$ (40)

Here $\rho_{\beta L}^{1/2}$ is the square root of the thermal density operator
acting on the left system. In the limit $\beta\rightarrow 0$, we get
$\left|TFD_{0}(J)\right\rangle=\left|I\right\rangle$. It is convenient to
choose $\left|I\right\rangle$ to be the state satisfying

$\displaystyle\left(\chi_{iL}+i\chi_{iR}\right)\left|I\right\rangle=0$ (41)

For this choice, the two-point functions satisfy

$\displaystyle\left\langle
TFD_{\beta}(J)\right|i\chi_{iL}\chi_{iR}\left|TFD_{\beta}(J)\right\rangle={\rm
tr}\left[\chi_{i}\left(\frac{\beta}{2}\right)\chi_{i}(0)\rho_{\beta}\right]\equiv
G\left(\frac{\beta}{2}\right)$ (42)

The average state is defined as

$\displaystyle\overline{\rho}=\int
dJp_{J}\left|TFD_{\beta}(J)\right\rangle\left\langle TFD_{\beta}(J)\right|$
(43)

The Holevo information is equal to the entropy of $\overline{\rho}$, since
$\left|TFD_{\beta}(J)\right\rangle$ is a pure state and the second term
vanishes.

Figure 4: The holographic dual interpretation of the Holevo information, for
two different ensembles of states. (a) For the ensemble of SYK thermal states,
the Holevo information is the generalized entropy of a small interval
$\tilde{\Sigma}_{R}$, minus the thermal entropy. (b) For the ensemble of
thermofield double states, the Holevo information is given by the generalized
entropy of two small intervals $\Sigma_{L}\Sigma_{R}$. (c) An illustration of
$\Sigma_{L},\Sigma_{R}$ in higher dimensions.

To determine $\overline{\rho}$ we notice that in the large-$N$ limit,
correlation functions of $\overline{\rho}$ satisfies the Wick theorem (which
is the consequence of the saddle point approximation of SYK model). Therefore
$\overline{\rho}$ is approximately a Gaussian state, which is thus determined
by the two-point function

$\displaystyle{\rm
tr}\left(\overline{\rho}i\chi_{iL}\chi_{iR}\right)=G\left(\frac{\beta}{2}\right)$
(44)

This condition determines $\overline{\rho}$ to be

$\displaystyle\overline{\rho}$
$\displaystyle=Z_{a}^{-1}e^{-\lambda\sum_{i}i\chi_{iL}\chi_{iR}},~{}\text{with~{}}\tanh\lambda=G\left(\frac{\beta}{2}\right)$
(45)

Thus the Holevo information is

$\displaystyle
H=S\left(\overline{\rho}\right)=-N\left(\frac{1+G\left(\beta/2\right)}{2}\log\frac{1+G\left(\beta/2\right)}{2}+\frac{1-G\left(\beta/2\right)}{2}\log\frac{1-G\left(\beta/2\right)}{2}\right)$
(46)

There is also a more precise way to obtain this result, which we discuss in
Appendix. B.

Now we discuss more about the behavior of Holevo information (46) and its
gravity interpretation. In the limit $\beta\rightarrow 0$, $H$ vanishes
because $\left|TFD_{0}(J)\right\rangle=\left|I\right\rangle$ is independent
from $J$. In the low temperature limit $\beta\rightarrow\infty$, $H$
approaches the maximal value $N\log 2$:

$\displaystyle H\simeq N\log 2-NG\left(\frac{\beta}{2}\right)^{2}$ (47)

The correction term is proportional to
$\frac{N}{\left(\beta\mathcal{J}\right)^{4/q}}$maldacena2016remarks .

In the holographic dual interpretation, naively one would thought the
expectation is $S\left(\overline{\rho}\right)=N\log 2$ reaching the maximum,
just like in the thermal state case. However, one should remember that in SYK
model the bulk theory contains $N$ Majorana fermions (which is required to
reproduce the $N$ boundary Majorana fermions). Therefore the correction to
area law (i.e. the dilaton value) is order $N$. We can view the entropy
$S(\overline{\rho})$ as the generalized entropy of the two small intervals
$\Sigma_{L},\Sigma_{R}$ in Fig. 4 (b). (For clarity, a higher dimensional
picture of $\tilde{\Sigma}_{L},\tilde{\Sigma}_{R}$ is illustrated in Fig. 4
(c).) The generalized entropy is given by

$\displaystyle
S_{gen}\left(\tilde{\Sigma}_{L}\tilde{\Sigma}_{R}\right)=S_{bulk}\left(\tilde{\Sigma}_{L}\tilde{\Sigma}_{R}\right)+2\phi_{\epsilon}$
(48)

Here $\phi_{\epsilon}$ is the dilaton value at the boundary of
$\tilde{\Sigma}_{L}$. We can write
$S_{gen}\left(\tilde{\Sigma}_{L}\tilde{\Sigma}_{R}\right)$ in term of mutual
information:

$\displaystyle
S_{gen}\left(\tilde{\Sigma}_{L}\tilde{\Sigma}_{R}\right)=S_{gen}\left(\tilde{\Sigma}_{L}\right)+S_{gen}\left(\tilde{\Sigma}_{R}\right)-I_{bulk}\left(\tilde{\Sigma}_{L}:\tilde{\Sigma}_{R}\right)$
(49)

with
$S_{gen}\left(\tilde{\Sigma}_{L}\right)=\phi_{c}+S_{bulk}\left(\tilde{\Sigma}_{L}\right)$
and similar for $S_{gen}\left(\tilde{\Sigma}_{R}\right)$. The mutual
information term is only contributed by the bulk fermion. In the limit when
the size of $\tilde{\Sigma}_{L}$ approaches zero,
$S_{gen}(\tilde{\Sigma}_{L})$ is the entropy of the averaged thermal state
(Fig. 4 (a)), which should reproduce the maximal entropy $\frac{N}{2}\log 2$.
The mutual information
$I_{bulk}\left(\tilde{\Sigma}_{L}:\tilde{\Sigma}_{R}\right)$ given by bulk
fermion is expected to approach $N\log 2-H$.

It is interesting to note that at large $\beta$, the TFD state is also the
ground state of the global AdS2 system, which is dual to two SYK models
coupled by a bilinear termmaldacena2018eternal . Therefore the Holevo
information of the ensemble of TFD states can be viewed as an example of the
fact that even the AdS vacuum corresponds to a nontrivial ensemble, as we
discussed in Sec. 3.

###  4.3 $G-\Sigma$ action approach to the thermal state

In this subsection, we will revisit the SYK thermal state discussion in Sec.
(4.1) with a different approach. We consider the large-$N$ effective action,
with collective fields $G,\Sigma$. This approach will provide us further
understanding on the gravitational saddle point that contributes to the Renyi
entropy calculation of the averaged state.

The $n$-th Renyi entropy of the nonaveraged state $\rho_{\beta}(J)$ is given
by the replica calculation333Here we do not include $Z_{J}$ in the denominator
since at large N it factorizes out.:

$\int dJp_{J}\text{Tr}e^{-n\beta H_{J}}=\int D\Sigma
DG~{}\text{Pf}^{N}(\partial_{t}-\Sigma)e^{-{N\over
2}\int_{0}^{n\beta}dt_{1}\int_{0}^{n\beta}dt_{2}\Sigma(t_{1},t_{2})G(t_{1},t_{2})-{J^{2}\over
q}G^{q}(t_{1},t_{2})}.$ (50)

Notice that since $t_{1,2}$ runs from $0$ to $n\beta$, $\Sigma(t_{1},t_{2})$
and $G(t_{1},t_{2})$ represent correlations between all the $n$ copies of the
replicas. The saddle point of this path integral is just a Euclidean disk with
period $n\beta$ which has nonvanishing correlation between different replica
copies. Now let’s look consider the case of the averaged state. We have:

$\begin{split}&\int dJ_{1}...dJ_{n}p_{J_{1}}...p_{J_{n}}\text{Tr}e^{-\beta
H(J_{1})}...e^{-\beta H(J_{n})}\\\
=&\int\prod_{i=1}^{n}D\Sigma_{i}DG_{i}~{}\text{Pf}^{N}(\partial_{t}-\Sigma)e^{-{N\over
2}\sum_{i}\int_{0}^{\beta}dt_{1}\int_{0}^{\beta}dt_{2}\Sigma_{i}(t_{1},t_{2})G_{i}(t_{1},t_{2})-{J^{2}\over
q}G_{i}^{q}(t_{1},t_{2})}.\end{split}$ (51)

where $\Sigma=\text{diag}(\Sigma_{1},\Sigma_{2},...,\Sigma_{n})$ is a block
diagonal matrix of all the $\Sigma_{i}$s. However, $\partial_{t}$ term is not
diagonal between replicas.:

$\text{Pf}(\partial_{t}-\Sigma)=\int D\psi e^{-{1\over
2}\int_{0}^{n\beta}dt\psi(t)\partial_{t}\psi(t)+{1\over
2}\sum_{i}\int_{0}^{\beta}dt^{i}_{1}\int_{0}^{\beta}dt^{i}_{2}\Sigma_{i}(t_{1},t_{2})\psi(t^{i}_{1})\psi(t^{i}_{2})};~{}~{}~{}t^{i}=t+(i-1)\beta$
(52)

The action (51) can be viewed as a modification of the non-averaged one (50),
with $J^{2}$ in $J^{2}G^{q}(t_{1},t_{2})$ replaced by
$J^{2}\theta\left(t_{1},t_{2}\right)$. Here $\theta(t_{1},t_{2})=1$ or $0$ if
$t_{1},t_{2}$ belongs to the same or different replica, respectively. As a
consequence, at the saddle point $\Sigma$ is always replica diagonal even if
$G$ is not diagonal. The Schwinger-Dyson equation is:

$\begin{split}G(t_{1},t_{2})&=\left(\partial_{t}-\Sigma\right)^{-1};\\\
\Sigma_{i}(t_{1},t_{2})&=J^{2}G(t_{1}^{i},t_{2}^{i})^{q-1}.\end{split}$ (53)

Notice that $\left(\partial_{t}-\Sigma\right)^{-1}$ is a matrix with time
range in $(0,n\beta)\times(0,n\beta)$ and the second equation only uses the
diagonal blocks of $G$ with $t_{1}^{i},t_{2}^{i}$ belonging to the same
replica. It is easy to check that the solution of the SD equation is given by:

$G(t^{i}_{1},t^{i}_{2})=G_{\beta}(t_{1}^{i},t_{2}^{i}),~{}~{}~{}G(t^{i}_{1},t^{j}_{2})=-2G_{\beta}(t^{i}_{1},0)G_{\beta}(t^{j}_{2},0),~{}~{}~{}i<j.$
(54)

where $G_{\beta}(t_{1},t_{2})$ is the thermal correlator. One way to
understand this result is that after average of $J$ the operator $\int
dJp(J)e^{-(\beta-t)H(J)}\chi_{i}e^{-tH}$ needs to be proportional to
$\chi_{i}$ due to O(N) symmetry. The proportional constant is uniquely fixed
by the thermal two point function:

$\int dJp(J){1\over
Z_{J}}e^{-(\beta-t)H(J)}\chi_{i}e^{-tH(J)}=2G_{\beta}(t,0)\chi_{i}.$ (55)

Combing this with the fact that $\int dJp(J){1\over Z_{J}}e^{-\beta
H(J)}=2^{-N/2}\mathbb{I}$, one directly gets:

$\langle\chi(t_{1}^{i})\chi(t_{2}^{i})\rangle=G_{\beta}(t_{1}^{i},t_{2}^{i});~{}~{}~{}\langle\chi(t_{1}^{i})\chi(t_{2}^{j})\rangle=-2G_{\beta}(t^{i}_{1},0)G_{\beta}(t^{j}_{2},0),~{}~{}~{}i<j.$
(56)

Taking the strong coupling limit, the above solution suggests a replica
geometry in JT gravity shown in figure 5.

Figure 5: Illustration of the Euclidean bulk geometry for $n$ replica of the
averaged thermal state. The red circles represent the boundary of each replica
with perimeter $\beta$. $n$ replica of hyperbolic disks (with $n=3$ in the
figure) are glued at a branching point (red cross) at the boundary. The points
connected by the red dashed lines are identified. The green and blue curves
are geodesics that contribute to the same-replica two-point function
$G\left(t_{1}^{i},t_{2}^{i}\right)$ and the across-replica two-point function
$G\left(t_{3}^{i},t_{4}^{j}\right)$, respectively.

On such a geometry, correlators with the same replica index (the green
geodesic) is the same as the thermal correlator. Meanwhile, correlators
between different replicas (the blue geodesic) will necessary factorize into
product of two two-point functions since the bulk geodesic has to pass through
the origin to enter a different replica system.444In pure JT such correlators
will be exponential small due to the infinite distance of the origin. In SYK
there is a natural IR cutoff of the boundary, which leads to a finite
correlations between the off diagonal replicas.  Clearly, this is like
freezing the disk geometry in JT gravity and only allowing the region near the
boundary to glue together, which is consistent with the tensor network
picture. In the $n\rightarrow 1$ limit, the quantum extremal surface is close
to the boundary which leads to the maximum entropy in the system (see
Fig.4(a)).

To evaluate the Renyi entropy directly from $G-\Sigma$ action, we can use the
following trick:

$\partial_{J^{2}}\text{Tr}\overline{\rho}^{n}=\partial_{J^{2}}({1\over
Z^{n}}\int dJ_{1}...dJ_{n}p_{J_{1}}...p_{J_{n}}\text{Tr}e^{-\beta
H(J_{1})}...e^{-\beta H(J_{n})})=0$ (57)

The variation of $\int dJ_{1}...dJ_{n}p_{J_{1}}...p_{J_{n}}\text{Tr}e^{-\beta
H(J_{1})}...e^{-\beta H(J_{n})})$ with respect to $J$ is equal to ${N\over
2q}\sum_{i}\int\int G_{i}^{q}(t_{1},t_{2})$ by equation of motion. Using the
on-shell solution that $G_{i}(t_{1},t_{2})=G_{\beta}(t_{1},t_{2})$, this is
equal to ${nN\over 2q}\int\int G_{\beta}^{q}(t_{1},t_{2})$. This cancels
exactly the variation of $Z^{n}$ with respect of $J$ which is also equal to
${nN\over 2q}\int\int G_{\beta}^{q}(t_{1},t_{2})$. Therefore the Renyi entropy
is independent of $J$ and we can simply evaluate the Renyi entropy in the
limit of $J=0$, which gives

$\text{Tr}\overline{\rho}^{n}=2^{-(n-1)N/2}.$ (58)

This confirms our earlier conclusion that $\overline{\rho}$ is a maximally
mixed state.

The $G-\Sigma$ action is a direct analog of the Einstein Hilbert action in
gravity path integral. In the language of Euclidean gravitational path
integral, allowing nontrivial correlation between different replica copies
corresponds to allowing short Euclidean paths between replica systems. We see
that in the Renyi entropy calculation of the nonaveraged state, correlations
between different replica systems are allowed and gives a nontrivial action.
In the Renyi entropy calculation of the ensemble averaged state, correlations
between different replica systems do not contribute to the action.

The $G-\Sigma$ action approach to the averaged TFD state will be left for
future work.

##  5 Discussion

In this work, we explore the meaning of ensemble average in holographic system
from a quantum information perspective. Our basic assumption is that a simple
bulk geometry can be represented as an ensemble of microstates $\rho(J)$
parameterized by ensemble parameter $J$. This assumption is true both in the
random tensor network model, where the parameter $J$ is related by the local
random projections in the bulk, and in the SYK model and JT gravity where $J$
is a random coupling in the Hamiltonian. Based on random tensor network model,
we conjectured that the Holevo information of a boundary region is given by
the difference between the generalized entropy of bulk regions with and
without entanglement wedge (equation 17):

$\displaystyle
H_{A}=S\left(\overline{\rho_{A}}\right)-\overline{S\left(\rho_{A}\right)}=\frac{\left|A\right|-\left|\gamma_{A}\right|}{4G_{N}}+S_{\rm
qft}\left(A\right)-S_{\rm qft}(\Sigma_{A}A).$ (59)

Here $A$ is the boundary region, and $\Sigma_{A}$ is the usual entanglement
wedge with quantum extremal sruface $\gamma_{A}$. From the perspective of
entanglement wedge recontruction, this formula suggests that all the operators
in the entanglement wedge are ensemble dependent boundary operators. Notice
that even for the operators in the causal wedge, the HKLL reconstruction
depends on the ensemble parameter $J$ through the dependence of the boundary
Hamiltonian. We give an explicit check of our conjecture with two SYK examples
including the thermal state and the thermofield double state.

In the RTN model, the random parameters are local, associated with each bulk
vertex. However, in SYK and JT gravity the $J$ parameter cannot be defined
locally in the bulk. Here we would like to discuss a bit the meaning of
locality of the $J$ parameters in the bulk. In a random tensor network, the
random projections $|V(J)\rangle$, after ensemble average, introduces
permutations among the different bulk legs which share the same $J$ parameter.
Therefore if the $J$ parameters are local, namely they are independently
random at different bulk locations, then the averaging only introduces
permutations among different replica systems, without introducing any
nontrivial connections within a single copy. Such permutations among replica
systems can be thought of as a direct analog of replica wormholes in quantum
gravity. The replica wormholes do not change simple correlation functions in a
single copy of the system, but do have nontrivial effects on the bulk Hilbert
space by decreasing the number of independent states.

If we modify the RTN ansatz by taking the same random state
$\left|V(J)\right\rangle$ on different vertices, then after ensemble average
there will be nontrivial permutations within a single copy of the system. Such
permutations change the geometry of the averaged state and correspond to more
general types of spacetime wormholes. One type of such wormhole is the one
that brings a particle inside the horizon to outside at late timeSaad:2019pqd
. Such wormholes are direct consequences of the random matrix behavior of the
Hamiltonian, and they describe the system’s memory of the initial condition.
Even after ensemble averaging, such wormholes have observable consequences in
a single copy system, such as the late time behavior of the two point
functions.

A natural question is whether the nonlocality of $J$ parameters lead to
nontrivial modification of our conjecture. In the RTN model, we expect that
the additional permutations due to the nonlocality of $J$ parameters are non-
perturbatively small corrections to the averaged density matrix because of the
large bond dimension of the local EPR pairs in the state $|\Psi_{p}\rangle$.
Therefore we can safely ignore their contributions in the calculation of the
Holevo information, and the conjectured formula does not change. This is
consistent with the bulk picture that even in the presence of non-local $J$
parameters, there is an approximate notion of locality due to the non-
perturbative nature of the wormholes. What we learn from the tensor network
story is that the suppression of such non-perturbative contributions stems
from the entanglement structure of the parent state $\rho_{P}$.

We would like to conclude the discussion with a few words about the situation
when such non-local effects become significant. This occurs when the wormholes
proliferate. Consider a late time thermofield double state, whose interior can
be approximated by a very long cylinder. When the length of the cylinder is
longer than exponential of the transverse area, the wormhole will have large
fluctuations by splitting and joining different locations of the cylinder.
This completely breaks the notion of locality in the interior. To say it
differently, the interior geometry at such late time is no longer well defined
and one cannot trust the classical geometric description. As a toy model,
consider in the RTN model a geometry with $L$ identical projections
$|V(J)\rangle$ with bond dimension $D$. This can be understood as a cylinder
with length $L$ and transverse area $\log D$. The average of tensor product
$\left|V(J)\right\rangle\left\langle V(J)\right|^{\otimes L}$ results in a
projection to the permutation-symmetric subspace of the $L$ bulk vertices.
When each site has Hilbert space dimension $d$, the permutation symmetric
subspace has the dimension

$D_{L}=\binom{L+d-1}{L}\approx\begin{cases}d^{L};~{}~{}~{}L\ll d;\\\
L^{d};~{}~{}~{}d\ll L.\end{cases}$ (60)

In the simple model without bulk field, the vertex dimension is $d=D^{2}$. In
the limit of $L\gg D^{2}$, the permutation symmetric subspace dimension
$D_{L}$ is polynomial in $L$, which is much smaller than the dimension before
projection. This implies that the ensemble averaging has a strong effect to
single copy physical properties, although the argument is not rigorous. Notice
that this is a truncation of the naive bulk Hilbert space in the averaged
state, which is different from the truncation of the black hole Hilbert space
from replica wormhole. For instance, the effects discussed here will lead to
an order-one change of simple correlation functions and the experience of an
infalling observer. It will be interesting to study if there exists a new
geometric description of the averaged state at late time, which is related to
the typical state firewall paradoxMarolf:2013dba . The breakdown of the
interior geometry at an exponentially late time is also expected as a
consequence of the complexity-equals-to-volume conjectureSusskind:2015toa ;
Brown:2015bva ; Brown:2015lvg . For a recent discussion about the behavior of
the volume in JT gravity, see Yang:2018gdb ; Iliesiu:2021ari .

## Acknowledgements

We thank Ahmed Almheiri and Henry Lin for informing us their parallel
workalmheiri2021 . We also noticed another recent work on a related
topicrenner2021black . This work has been presented by one of us on a recent
workshopqi2021geoflow . We thank Arvin Shahbazi Moghaddam for useful
discussions. This paper is supported by the National Science Foundation under
grant No. 2111998 (XLQ and ZS), and the Simons Fundation (XLQ and ZY). This
work is also supported in part by the DOE Office of Science, Office of High
Energy Physics, the grant de-sc0019380.

##  Appendix A Some more details about random tensor networks

In this appendix, we will discuss some more details about the replica
calculation of the averaged entropy $\int dJp_{J}S\left(\rho_{A}(J)\right)$
for the ensemble of RTN given in Eq. (3) and (4). For convenience we write
these equations again here in a slightly different way:

$\displaystyle\pi_{S}(J)$ $\displaystyle={\rm
tr}_{B}\left(\left|V(J)\right\rangle\left\langle V(J)\right|\rho_{P}\right)$
(61) $\displaystyle p_{J}$ $\displaystyle={\rm tr}\left(\pi_{S}(J)\right)$
(62) $\displaystyle\rho_{S}(J)$ $\displaystyle=p_{J}^{-1}\pi_{S}(J)$ (63)

Here we have defined the unnormalized state $\pi_{S}(J)$.

Using the standard replica trick

$\displaystyle S(\rho)=-\left.\frac{\partial}{\partial n}\log{\rm
tr}\left(\rho^{n}\right)\right|_{n\rightarrow 1}$ (64)

we have

$\displaystyle\int dJp_{J}S\left(\rho_{A}(J)\right)$
$\displaystyle=-\left.\int dJp_{J}\frac{\partial}{\partial n}\log{\rm
tr}\left(\rho_{A}(J)^{n}\right)\right|_{n\rightarrow 1}$
$\displaystyle=-\left.\int dJp_{J}\frac{\partial}{\partial n}\log{\rm
tr}\left(p_{J}^{-1}\pi_{A}(J)^{n}\right)\right|_{n\rightarrow 1}$
$\displaystyle=-\left.\int dJp_{J}\frac{\partial}{\partial n}\left[\log{\rm
tr}\left(\pi_{A}(J)^{n}\right)-n\log p_{J}\right]\right|_{n\rightarrow 1}$
$\displaystyle=-\left.\int dJp_{J}\left[\frac{1}{{\rm
tr}\left(\pi_{A}(J)^{n}\right)}\frac{\partial{\rm
tr}\left(\pi_{A}(J)^{n}\right)}{\partial n}-\frac{1}{p_{J}^{n}}\frac{\partial
p_{J}^{n}}{\partial n}\right]\right|_{n\rightarrow 1}$ (65)

Taking $n\rightarrow 1$ the denominator in both terms of the last line is
$p_{J}$, so that

$\displaystyle\int dJp_{J}S\left(\rho_{A}(J)\right)$
$\displaystyle=-\left.\frac{\partial}{\partial n}\left[\int dJ{\rm
tr}\left(\pi_{A}(J)^{n}\right)-\int dJ\left({\rm
tr}\left(\pi_{A}(J)\right)\right)^{n}\right]\right|_{n\rightarrow 1}$ (66)

Define

$\displaystyle Z_{A}^{(n)}$ $\displaystyle=\int dJ{\rm
tr}\left(\pi_{A}(J)^{n}\right)$ (67) $\displaystyle Z_{\emptyset}^{(n)}$
$\displaystyle=\int dJ\left({\rm tr}\left(\pi_{A}(J)\right)\right)^{n}=\int
dJp_{J}^{n}$ (68)

we notice that $Z_{A}^{(1)}=Z_{\emptyset}^{(1)}=1$. Thus we can further
transform Eq. (66) into

$\displaystyle\int dJp_{J}S\left(\rho_{A}(J)\right)$
$\displaystyle=-\left.\frac{\partial}{\partial n}\left[\log Z_{A}^{(n)}-\log
Z_{\emptyset}^{(n)}\right]\right|_{n\rightarrow 1}$
$\displaystyle=-\left.\frac{\partial}{\partial
n}\log\frac{Z_{A}^{(n)}}{Z_{\emptyset}^{(n)}}\right|_{n\rightarrow
1}=-\left.\frac{\partial}{\partial
n}\frac{Z_{A}^{(n)}}{Z_{\emptyset}^{(n)}}\right|_{n\rightarrow 1}$ (69)

We prefer the last expression because $Z_{A}^{(n)}$ and $Z_{\emptyset}^{(n)}$
have the interpretation of a classical statistical model partition function,
and

$\displaystyle-\log\frac{Z_{A}^{(n)}}{Z_{\emptyset}^{(n)}}$ (70)

is the free energy cost of the spin model caused by changing the boundary
condition in $A$ region. The spin model comes from the fact that $\pi_{S}(J)$
is linear in $\left|V(J)\right\rangle\left\langle V(J)\right|$, and

$\displaystyle\int dJ\left|V(J)\right\rangle\left\langle V(J)\right|^{\otimes
n}=c_{n}^{-1}\sum_{g\in S^{n}}X_{g}$ (71)

with $g$ an element of the permutation group $S^{n}$, and $X_{g}$ is the
permutation $g$ acting naturally by permuting different replica. $c_{n}$ is a
normalization constant. The explicit expression of $Z_{A}^{(n)}$ and
$Z_{\emptyset}^{(n)}$ ishayden2016holographic

$\displaystyle Z_{A}^{(n)}$
$\displaystyle=c_{n}^{-V}\sum_{\left\\{g_{x}\right\\},~{}g_{x}\in S^{n}}{\rm
tr}\left(\rho_{AB}^{\otimes n}\otimes_{x\in B}X_{g_{x}x}\otimes X_{An}\right)$
(72) $\displaystyle Z_{\emptyset}^{(n)}$
$\displaystyle=c_{n}^{-V}\sum_{\left\\{g_{x}\right\\},~{}g_{x}\in S^{n}}{\rm
tr}\left(\rho_{AB}^{\otimes n}\otimes_{x\in B}X_{g_{x}x}\right)$ (73)

It is interesting to note that the derivation so far has not used any specific
property of $\rho_{P}$. The averaged entropy $\int dJp_{J}S(\rho_{A}(J))$ is
directly related to the analytic continuation of the free energy of the spin
model. On comparison, the quantity discussed in Ref. hayden2016holographic  is
the average $\int dJS^{(n)}\left(\rho_{A}(J)\right)$ which only corresponds to
$-\frac{1}{n-1}\log\frac{Z_{A}^{(n)}}{Z_{\emptyset}^{(n)}}$ in the large bond
dimension limit when the numerator and denominator are self-averaging. In
other words, going beyond the self-averaging limit, the averaged entropy $\int
dJp_{J}S(\rho_{A}(J))$ provides a physical interpretation of the free energy
cost $-\log\frac{Z_{A}^{(n)}}{Z_{\emptyset}^{(n)}}$.

##  Appendix B Averaged state of the thermofield double state of SYK model

In this appendix, we carry a more detailed discussion on the averaged state of
the SYK model TFD state. It can be seen that $\overline{\rho}$ is invariant
under $SO(N)$ rotation that rotate $\chi_{i}$ as a vector $\chi_{i}\rightarrow
O_{ij}\chi_{j}$, since such rotation is equivalent to $SO(N)$ transformation
of the coefficients

$\displaystyle J_{i_{1}i_{2}...i_{n}}\rightarrow
J_{j_{1}j_{2}...j_{n}}O_{j_{1}i_{1}}O_{j_{2}i_{2}}...O_{j_{n}i_{n}}$ (74)

which preserves the probability $p_{J}$. The only $SO(N)$ scalar operators are

$\displaystyle\hat{L}=\sum_{i}\frac{i\chi_{iL}\chi_{iR}+1}{2},~{}F_{L}=i^{N/2}\chi_{1L}\chi_{2L}...\chi_{NL},~{}F_{R}=i^{N/2}\chi_{1R}\chi_{2R}...\chi_{NR}$
(75)

In addition, Eq. (36) translates to a property of the TFD state:

$\displaystyle\chi_{iL}\chi_{iR}\left|TFD(J)\right\rangle\left\langle
TFD(J)\right|\chi_{iR}\chi_{iL}$
$\displaystyle=\left|TFD(\tilde{J})\right\rangle\left\langle
TFD(\tilde{J})\right|$ (76)
$\displaystyle\Rightarrow\left[i\chi_{iL}\chi_{iR},\overline{\rho}\right]$
$\displaystyle=0$ (77)

This symmetry condition further excludes $F_{L}$ and $F_{R}$, so that
$\overline{\rho}$ is a function of $\hat{L}$. $\overline{\rho}$ can be written
in the eigenstate subspaces of $\hat{L}$ as

$\displaystyle\overline{\rho}$
$\displaystyle=\sum_{n=0}^{N}p_{n}\frac{\hat{\Pi}_{n}}{D_{n}}$ (78)

with $\hat{\Pi}_{n}$ the projector to the subspace of $\hat{L}=n$, with the
dimension $D_{n}=\left(\begin{array}[]{c}N\\\ n\end{array}\right)$. Thus the
entropy is given by

$\displaystyle S\left(\overline{\rho}\right)=\sum_{n=0}^{N}\left(-p_{n}\log
p_{n}+p_{n}\log D_{n}\right)$ (79)

$p_{n}$ can be computed using a generating function:

$\displaystyle F(\theta)$ $\displaystyle\equiv\sum_{n}p_{n}e^{in\theta}={\rm
tr}\left(\overline{\rho}\hat{\Pi}_{n}\right)e^{in\theta}={\rm
tr}\left(\overline{\rho}e^{i\hat{L}\theta}\right)$ (80)

If we analytically continuate $\theta$ to $i\mu$, $F(i\mu)$ can be computed as
a partition function of the coupled SYK model:

$\displaystyle F(i\mu)=\left\langle
TFD(J)\right|e^{-\mu\hat{L}}\left|TFD(J)\right\rangle=\frac{\mathcal{Z}_{\mu}}{\mathcal{Z}_{0}}$
(81)

which is self-averaging. $F(i\mu)$ can be computed explicitly in the large-$q$
limit of SYK modelqi2019quantum 555See Eq. (5.1) of Ref. qi2019quantum .
Without going into detail of $F(i\mu)$, we know that in the large-$N$ limit it
has the form

$\displaystyle F(i\mu)=e^{-N\mathcal{A}(\mu)}$ (82)

with $\mathcal{A}(\mu)$ a finite function of $\mu$. Thus according to Eq. (80)
we have

$\displaystyle
p_{n}=\int_{0}^{2\pi}\frac{d\theta}{2\pi}F(\theta)e^{-in\theta}=\int_{0}^{2\pi}\frac{d\theta}{2\pi}e^{-N\mathcal{A}(-i\theta)-in\theta}$
(83)

In the large-$N$ limit, we can expand $\mathcal{A}(\mu)$ into second order of
$\mu$:

$\displaystyle\mathcal{A}(\mu)$ $\displaystyle\simeq
p\mu-\frac{1}{2}\alpha\mu^{2}$ (84)

with

$\displaystyle p$
$\displaystyle=\frac{\left\langle\hat{L}\right\rangle}{N}=\frac{1}{2}\left(G\left(\frac{\beta}{2}\right)+1\right)$
(85) $\displaystyle\alpha$
$\displaystyle=\frac{1}{N}\left(\left\langle\hat{L}^{2}\right\rangle-\left\langle\hat{L}\right\rangle^{2}\right)$
(86)

This leads to

$\displaystyle p_{n}\simeq
C\exp\left[-\frac{N}{2\alpha}\left(p-\frac{n}{N}\right)^{2}\right]$ (87)

In the large-$N$ limit, $p_{n}$ is strongly peaked, which has the entropy (79)

$\displaystyle S\left(\overline{\rho}\right)$ $\displaystyle\simeq\log
D_{pN}+O(1)\simeq N\left[-p\log p-(1-p)\log(1-p)\right]+O(1)$ (88)

This agrees with the Gaussian state result (46).

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