Holographic coherent states from random tensor networks
HJE
Holographic coherent states from random tensor networks
XiaoLiang Qi 0 1 2 4
Zhao Yang 0 1 2 4
YiZhuang You 0 1 2 3
0 tum Gravity , Random Systems
1 Cambridge , MA 02138 , U.S.A
2 Stanford , CA 94305 , U.S.A
3 Department of Physics, Harvard University
4 Department of Physics, Stanford University
Random tensor networks provide useful models that incorporate various important features of holographic duality. A tensor network is usually de ned for a graph geometry speci ed by the connection of tensors. In this paper, we generalize the random tensor network approach to allow quantum superposition of di erent spatial geometries. We setup a framework in which all possible bulk spatial geometries, characterized by weighted adjacient matrices of all possible graphs, are mapped to the boundary Hilbert space and form an overcomplete basis of the boundary. We name such an overcomplete basis as holographic coherent states. A generic boundary state can be expanded in this basis, which describes the state as a superposition of di erent spatial geometries in the bulk. We discuss how to de ne distinct classical geometries and small uctuations around them. We show that small uctuations around classical geometries de ne code subspaces" which are mapped to the boundary Hilbert space isometrically with quantum error correction properties. In addition, we also show that the overlap between di erent geometries is suppressed exponentially as a function of the geometrical di erence between the two geometries. The geometrical di erence is measured in an area law fashion, which is a manifestation of the holographic nature of the states considered.
AdSCFT Correspondence; Gaugegravity correspondence; Models of Quan

xed
1 Introduction
2 General framework
3 Boundarytobulk isometry
3.1
3.2
4.1
4.2
The randomaveraged isometry condition
Fluctuations
4 Bulktoboundary isometry in code subspaces
Classical geometry and the code subspace
Local reconstruction properties
5 Overlap between di erent classical geometries
6 Conclusion and discussions
A Fluctuations and higher Renyi entropies
A.1 General results
A.2 A su cient condition for eq. (A.5)
A.3 An explicit example of states jaxyi
networks may provide a \microscopic" framework for understanding holographic duality [9,
10]. Tensor networks, or projected entangled pair states (PEPS) is an approach to construct
entangled quantum manybody states [11{16]. For a graph (see gure 1), the corresponding
PEPS is obtained by rst preparing an EPR pair for each link, and then projecting all qubits
at the same vertex to a pure state speci ed by the tensor at that vertex. This procedure
leads to a manybody state of the remaining qubits living at the end of dangling legs of
the network. The advantage of the tensor network description is that the entanglement
such states must satisfy various conditions [17, 18], such as the negative tripartite mutual
information [19]. Various tensor network models [20{23] have been proposed to incorporate
desired features of holographic duality. Among them, the random tensor networks [23]
are shown to realize many features of holographic duality naturally, including the RT
formula with quantum corrections, and the quantum error correction property of the
bulkboundary holographic mapping [24]. However, there are holographic properties that are
not reproduced by tensor networks, such as the Renyi entropy behavior [8, 25]. There
are also obviously many other open questions that have not been addressed in the tensor
network framework.
Among the open questions, an essential one is how to describe quantum superposition
of di erent geometries, as is required for a quantum gravity theory. This is also a necessary
step towards understanding Einstein equation and graviton excitations in the bulk. In this
paper we make a small progress along this direction by setting up a framework for describing
quantum superposition of tensor network states on arbitrary geometries. We generalize the
random tensor network approach in ref. [23] and de ne a linear map between geometries in
the bulk and quantum states on the boundary. A geometry is described by the adjacient
matrix axy of a weighted (unoriented) graph (with xed number of vertices and arbitrary
connectivity), which is de ned as a basis vector jfaxygi in the bulk. The linear map de ned
by random tensors then maps each such basis state to a quantum state j [faxyg]i on the
boundary, which is the holographic state that is dual to this geometry. With this linear map
{ 2 {
it is straightforward to take superpositions between di erent geometries. We prove that for
a xed size of the boundary, a large enough number of bulk vertices make such a mapping
an isometry from the boundary to the bulk. In other words, j [faxyg]i parameterized by
the weighted adjacient matrix axy form an overcomplete basis of the boundary, such that
each state in the boundary Hilbert space can be mapped to a quantum superposition of
di erent geometries. Due to the analog of boson coherent states (as will be elaborated more
in later part of this paper), we name this basis of states \holographic coherent states".
Furthermore, this formalism allow us to consider small uctuation around a classical
geometry, and show that such small uctuations for a \code subspace"[24] which is mapped
to the boundary isometrically. (The precise meaning of \classical geometry" and \small
uctuation" will be given later. In short, a classical geometry means all nonzero entries
of the weighted adjacient matrix axy are large, while small
uctuations correspond to
axy ! axy + axy with
axy
axy.) Such small uctuations can be considered as low
energy states of the bulk quantum
elds. The existence of bulkboundary isometry in such
subspaces guarantees that small uctuations at di erent links of the graph are independent
physical degrees of freedom. In other words, bulk locality emerges in such subspaces
even if the whole bulk theory is intrinsically nonlocal. In addition, the bulkboundary
isometry satis es the local reconstruction properties known in holographic duality. The
structure of a boundarytobulk isometry in the whole boundary Hilbert space and a
bulktoboundary isometry in code subspaces has been proposed as \bidirectional holographic
code" in ref. [22], which is schematically summarized in gure 2.
As an overcomplete basis, states j [faxyg]i for di erent geometry axy do not
correspond to orthogonal states of the boundary. However, we show that the overlap between
di erent geometries are exponentially suppressed in the large N limit. This is similar to
ordinary boson coherent states that are used in mean eld approximation of super uids and
superconductors. Di erent coherent states are not orthogonal. But because their overlap
is exponentially small, it is physically meaningful to consider them as physically di erent
states, and therefore consider the condensate wavefunction as a physical order parameter
eld. An interesting di erence of the geometrical states from ordinary coherent states
is that the overlap between two states has a \holographic" behavior. If two geometries
axy and bxy are distinct in a region R and identical outside R, we prove that the overlap
jh [faxyg]j
[fbxyg]ij is upper bounded by e cj j with
the area of a minimal surface
bounding region R. c is a constant determined by the entanglement entropy contributed
by each link crossing the boundary. The area law form of the overlap is a manifestation
of the fact that the states j [faxyg]i are consistent with the holographic principle  the
fact that the physical degrees of freedom in a region R are bounded by their area rather
than volume.
The remainder of the article is organized as follows. In section 2 we present the general
setup of our approach. In section 3 we study the condition of boundarytobulk isometry.
In section 4 we investigate the de nition of classical geometries and the code subspaces with
bulktoboundary isometry. In section 5 we study the overlap between di erent classical
geometries to show that distinct geometries are almost orthogonal. Finally, the conclusion
and further discussions are given in section 6.
{ 3 {
gure 1, one rst prepares a EPR pair of two qudits for each
link, denoted by jxyi. Then the RTN is de ned by projecting all qudits on the site x to a
random pure state jVxi. If each qudit has dimension D, and site x has k neighbors, jVxi is
a random unit vector in a Dkdimensional Hilbert space. The probability distribution of
jVxi is uniform, which means jVxi and U jVxi has the same probability for any unitary U .
Alternatively, one can de ne jVxi = U j0i with U a Haar random unitary operator and j0i
a xed reference state. For a graph G, the RTN state is expressed as
j Gi = Y
hVxj
x
Y
hxyi2G
jxyi
(2.1)
with the hxyi 2 G runs over (unoriented) edges in the graph G.
From the de nition of RTN, it is natural to see how to generalize this formalism to
include superposition of di erent geometries (graphs)The link state Qhxyi2G jxyi can be
replaced by superpositions of such states on di erent graphs. To make this wellde ned,
one needs to modify the de nition slightly to make sure the Hilbert space dimension of
each vertex is identical for di erent graphs. This can be easily achieved by de ning some
auxiliary states on links that are absent in G. For each hxyi 2= G, de ne a state jxyi0 =
jxi0 jyi0 which is a direct product state and is orthogonal to jxyi. Adding such direct
product states do not change the entanglement structure of the system. Then if we replace
Qhxyi2G jxyi by Qhxyi2G jxyi Qhxyi2=G jxyi0, the dimension of each site is DV 1 if the total
number of vertices is V . Therefore the random states jVxi can be chosen in a Hilbert
space of dimension DV 1 independent from G. Denote jPGi = Qhxyi2G jxyi Qhxyi2=G jxyi0
as the \parent state" before projection, then the superposition of two geometries G; G0
correspond to a boundary state a j Gi + b
j G0 i = Q
x hVxj [a jPGi + b jPG0 i]. In other
{ 4 {
bulk link is a threeleg tensor La , and each vertex is a random tensor. The blue links are maximally
entangled EPR pairs. The network de nes a linear map between bulk link states (red lines) and
boundary states (blue lines).
with a = 0; 1; 2; : : : ; DL
1 and ;
index a to indices
. In other words, states
words, now we have a linear map between di erent graphs G corresponding to di erent
EPR pair con gurations (in the same Hilbert space) to di erent boundary states.
Motivated by the discussion above, we consider a more general situation and de ne the
following tensor network. Consider a complete graph with V vertices, in which VB of them
are labeled as \boundary" vertices, and the rest of them Vb = V
VB are bulk vertices.
For each pair of vertices x; y (x 6= y), we de ne a threeleg tensor La shown in
gure 3,
= 1; 2; : : : ; D.1 This tensor de nes an isometry from
jaxyi = La j ix j iy
are orthonormal, i.e. hbxyj axyi =
ab. (Obviously this requires DL
variables axy can be considered as specifying a weighted graph. Since we want the weight
axy to label entanglement in state jaxyi, we can require the entanglement entropy between
x and y to be an increasing function of axy. For example, to be speci c we can require
D2.) The link
Sx (axy) = axy log d; with axy = 0; 1; : : : ; DL
1; dDL 1 = D
(2.3)
which means axy is the number of EPR pairs across the link, each with dimension d. The
maximal axy corresponds to a maximally entangled state.
In addition, each boundary vertex is connected with a EPR pair state jxXiB which
entangles a qudit at vertex x with one at the boundary physical site X. Then for each
con guration axy = 0; 1; 2; : : : ; DL
1, an RTN is de ned by
j [faxyg]i = Y
hVxj
x
Y
x6=y
Y
x
jaxyi
jxXiB
If we only want to incorporate superposition of RTN on di erent graphs, the simplest
choice will be DL = 2, in which case a qubit at each link determines whether the link is
1Similar link variables have been introduced in perfect tensor networks in ref. [26] for a di erent but
related purpose.
{ 5 {
(2.2)
(2.4)
connected (entangled) or not. However, it is more convenient to introduce a larger DL,
which makes it possible to de ne \small" uctuations around a classical geometry, as will
be discussed in section 4.
map by M : Hb ! HB:
dimension DVB ), which maps the basis states Q
The de nition (2.4) de nes a linear map from the bulk Hilbert space Hb spanned
by the link qubits (with dimension DLV (V 1)=2) to the boundary Hilbert space HB (with
x6=y jaxyi to j [faxyg]i. We denote this
M j bulki
hVxj j bulki
jxXiB
Y
x
Y
x
(2.5)
HJEP08(217)6
The Hermitian conjugate operator M y : HB ! Hb de nes a linear map from the boundary
Hilbert space to the bulk Hilbert space. This pair of maps M and M y can be viewed as a
holographic mapping that builds a correspondence between states (on the boundary) and
geometries (in the bulk). It is straightforward to generalize the random average technique in
ref. [23] to the current setup, which is how we will investigate properties of this holographic
mapping in the following sections.
3
Boundarytobulk isometry
In this section, we will study the holographic mapping from boundary to bulk, and show
that it is an isometry under certain conditions.
This result demonstrates that tensor
network states j [faxyg]i for all con gurations faxyg forms an overcomplete basis of the
boundary Hilbert space, so that any boundary state can be expanded in this basis.
We summarize the result rst. The isometry condition requires
B =
j [faxyg]i h [faxyg]j / I
(V
1)
X
faxyg
2 log D
log DL
Eq. (3.1) is true if the following two conditions are satis ed,
Tr
4
xy g
h 2 < Tr
4
xy g
g Tr
4
xy h
h ; 8g 6= h 2 S4
Eq. (3.2) can be trivially satis ed if V
1, while keeping log D= log DL to be O(
1
).
Eq. (3.3) is the property of the density matrix, whose details will be elaborated in the
section 3.2 and appendix A.
If we view the tensor network in gure 3 as an entangled state between boundary and
bulk link qudits, the isometry condition is equivalent to the statement that the reduced
density matrix B is maximally mixed. To study B we study its second Renyi entropy
Similar to ref. [23], we study the random average of the numerator and denominator
separately, and then study their
uctuations.
When the uctuation is small, we have
A commonly used trick in writing the Renyi entropy is to write
Tr
2
B
= Tr [XB B
B] = Tr
XB
I
B ( B
B)
region.
XB jniB
jn0iB = jn0iB
jniB.
For the state de ned in eq. (2.4), B is
with
B = j [faxyg]i h [faxyg]j the density matrix of the whole system, and XB the
swap operator acting on twocopies of the system which permutes the two copies in B
More explicitly, if we denote an orthonormal basis of B region as jniB, then
Therefore
2
2
1
DL 1
X
DL a=0
Y
x
Y
x
2
B = trb 4
jVxi hVxj
xy
jxXiB hxXjBA5
Y
x6=y
Y
x
with xy =
jaxyi haxyj
Tr
2
B
= Tr 4
XB
jVxi hVxj
= C 1
X
0
Y
x6=y
2
! 0
Y
x6=y
xy
Y
x
Here we have used the mathematical fact that the random average jVxi hVxj
/ Ix Ix+Xx,
with Xx the swap operator de ned in the same way as XB, acting on all qudits at site x.
The righthand side of eq. (3.8) is a sum over the purity of the state Q
Qx jxXiB hxXjB for di erent regions B [ R, with R running over all 2V subsets of the
V vertices. Since this state is simple, with only bipartite entanglement between di erent
sites, the purity can be explicitly computed. In the same way as in ref. [23], the sum
can be expressed as a partition function of a classical Ising model, with an Ising spin
sx =
1 de ned on each site. Each spin con guration corresponds to a region R
de ned as the spin s =
1 domain. The action of the Ising model A [fsxg] is de ned such
# which is
x6=y xy
that e A[fsxg] = Tr XB[R#
Q
x6=y xy
Qx jxXiB hxXjB
2
. Since the state on the
righthand side only contains bipartite entanglement, the Ising model action only contains
onebody and twobody terms:
with a thick red circle, and +1 elsewhere. The dashed line is the domain wall across which the spin
changes sign. The contribution to the action comes from three kinds of links, those within the spin
down region (black thick line), those between opposite spins (pink thick line) and those connecting
the spin up boundary sites to the boundary (blue thick line). These three contributions correspond
to A1;2;3 in eq. (3.10) respectively.
Here 0 < sb
log D is the second Renyi entropy of site x in the state xy, i.e. e sb =
trx(try xy)2, and the Ising coupling J is half of the second Renyi mutual information
between sites x; y in the mixed state xy. The last term in the action sums over the VB
boundary sites.
The Ising model problem is simpler than that for a generic RTN in ref. [23] because
all pairs of x; y are coupled equally. Consequently, all Vb bulk vertices are equivalent, and
all VB boundary vertices are equivalent. The action is therefore only a function of two
integers, the number of down spins in the bulk vertices nb 2 [0; Vb], and the number of
down spins in the boundary vertices nB 2 [0; VB].
A [fsxg] = A(nb; nB) = A1 + A2 + A3
A1 = log DL(nb + nB)(nb + nB
1)=2;
A2 = sb (nb + nB) (V
nb
nB) ; A3 = log D (VB
nB)
(3.10)
The three terms A1;2;3 are contributions of links within region R, links between R and its
complement, and links from R to the boundary, respectively, as is illustrated in gure 4.
With the action A(nb; nB), eq. (3.8) becomes
Tr
2
B
= C 1 X
Vb
VB
X
Vb !
nb
VB
nB
!
(3.11)
For large Vb; VB, this sum is dominated by the biggest term, which corresponds to the
minimum of S (nb; nB) = A (nb; nB)
log
log
One can show that
Vb !
nb
{ 8 {
e A(nb;nB)
VB
nB
!
.
S (nb; nB) reaches its minimum in the large Vb; VB limit at one of the corners in region
nb 2 [0; Vb]; nB 2 [0; VB]. A detailed explanation can be found in appendix A.1. The same
analysis applies to the denominator Tr [ B]2, and the only di erence is in the boundary
term A3.
and the denominator is given by nB = nb = 0, which requires
The isometry condition is satis ed if the dominant con guration for both the numerator
HJEP08(217)6
log DL
VB(VB
1)
2
V (V
1)
2
> VB log D
Condition (3.13) is simply a requirement that the bulk Hilbert space dimension DLV (V 1)=2
is larger than that of the boundary (DVB ). Condition (3.14) requires that the link state
xy is su ciently entangled. In term of coupling J = sb
requires
12 log DL, the condition (3.14)
1
Vb
J >
log D
V
1
2
log DL
Condition (3.13) and (3.14) are easy to satisfy. If we take the limit Vb; VB ! 1 with
the ratio VB=V
xed, and keep D; DL to be O(
1
), all conditions will be trivially satis ed.
The isometry condition (3.1) allows an expansion of an arbitrary boundary state j i
in this basis: j i = Pfaxyg j [faxyg]i h [faxyg]j
i = Pfaxyg
wavefunction is the analog of Wheelerde Witt wavefunction [27] of quantum gravity,
al[faxyg] j [faxyg]i. This
though here we are only taking superpositions of spatial geometries.
3.2
Fluctuations
Tr
2 2
uctuation is small. The uctuation can be studied by computing
2
B
. As has been shown in ref. [23], the random average of a quantity
2 2 , which is quartic in B, can be expressed as a partition function of a
statisB
tical model with a pseudospin gx at each site taking values in the 4element permutation
group S4. In general, any quantity in the form of Tr h
kOki, with operator Ok acting on
k copies of the system,2 is mapped to a partition function of a model with pseudospins in
kelement permutation group Sk. Similar to the Ising model analyzed above, the statistical
models for higher k is also de ned on a complete graph, which simpli es the problem. In
2For example, Tr k which determines the kth Renyi entropy can be written as Tr
A
kCAk with
CAk the cyclic permutation of the k copies of systems in A region.
(3.12)
(3.13)
(3.14)
(3.15)
appendix A we analyze these pseudospin models and obtain su cient conditions for
uctuations such as Tr
to be controlled. For bounding the uctuation of
the second Renyi entropy calculation, the su cient conditions are the following:
Tr h
k
xy g
< Tr h
k
xy g
gi Tr h
k
xy h
i
h ; 8g 6= h 2 Sk
(3.16)
(3.17)
with k = 4 in the second equation. More details of the derivation will be given in
appendix A. It is not di cult to see that conditions (3.16) and (3.17) imply the conditions we
obtain earlier in eq. (3.13) (3.15). Condition (3.16) can be easily satis ed in large volume
V . Condition (3.17) imposes addition constraints to the choice of states jaxyi and xy, but
is also not hard to satisfy, as we will discuss in more details in appendix A. We also give
an explicit example of jaxyi in appendix A.3 which satis es condition (3.17) for general k.
4
Bulktoboundary isometry in code subspaces
Since the bulk basis j [faxyg]i is generically overcomplete, the mapping from bulk to
boundary de ned by our random tensor network is not injective. However, holographic
duality requires that small uctuations around a classical geometry are independent physical
states on the boundary. For example, if we consider a dilute gas of gravitons in the bulk,
the total degree of freedom of the gas is proportional to volume. Gravitons at di erent
bulk locations should be dual to independent degrees of freedom on the boundary, since
graviton creation/annhilation operators should be mapped to independent operators on
the boundary by the dictionary of holographic duality. This requirement means that there
should be a bulktoboundary isometry in the subspace of such small
uctuations. The
bulk small uctuations are mapped to a subspace of the boundary Hilbert space, named
as the \code subspace"[21, 24]. Each geometry corresponding to a con guration a = faxyg
de nes a code subspace HC [a]. The mapping of such small uctuations to the boundary
should satisfy the following local reconstruction property: each region on the boundary
A corresponds to a minimal surface
A in the bulk that is homologous to it. The region
enclosed by A [ A is the entanglement wedge EA.3 A bulk operator acting in the subspace
of small
uctuations (the code subspace) in the bulk region EA can be reconstructed in
boundary region A. Since each bulk point can be enclosed by the entanglement wedges
of di erent boundary regions, information in the bulk can be recovered from di erent
boundary regions, making the bulkboundary map in the code subspace a quantum error
correction code [24]. The bulkboundary isometry and local reconstruction is illustrated in
gure 5.
In the following we will explain how our formalism of uctuating geometry allows the
de nition of small uctuations and code subspaces. In section 4.1, we obtain the condition
of code subspaces in which global reconstruction (i. e. bulktoboundary isometry) can be
3More precisely, EA here is the intersection of the entanglement wedge and the spatial slice. Since we
will always be dealing with a spatial slice, we neglect this di erence and call EA the entanglement wedge.
from the whole bulk subspace to the boundary is an isometry. Furthermore, the local reconstruction
property requires that degrees of freedom in EA which is the entanglement wedge of A can be
reconstructed in A, which means an isometry is de ned from EA to A for arbitrary states in EA
and A. (b) A small region in the bulk (orange disk) can be reconstructed in di erent boundary
regions such as A; B.
de ned. A su cient condition is given in eq. (4.7) and (4.8). In section 4.2, we discuss the
condition of local reconstruction on a subregion of the boundary.
4.1
Classical geometry and the code subspace
Each con guration faxyg corresponds to a \geometry" (i.e. a weighted graph), but if axy
takes arbitrary values, one cannot de ne what uctuations are considered \small". With a
large link variable dimension DL, one can de ne a classical geometry as one with all
nontrivial links (axy 6= 0) contributing a large entropy / DL, and then de ne small uctuations
as uctuations of axy that are small compared to DL.
a0
For concreteness, we pick a value of link variable a0 with 0 < DL 1 < 1, and take the
a0
limit DL; D ! 1 with DL 1
xed. We de ne a classical geometry by a state j [faxyg]i
with all axy equal to either a0 or 0.4 For such states, we can de ne an adjacient matrix K
with Kxy = 0; 1, such that axy = Kxya0.
Now de ne a range of small uctuation
a0. In the limit DL !
1,
is kept
nite. Then we de ne small uctuations around the classical geometry Kxya0 as all states
j [faxyg]i satisfying
(axy 2 [a0
axy 2 [0; 2 ];
; a0 + ]; if Kxy = 1
4It is straightforward to generalize the following discussion to states with di erent a0 on di erent links
as long as all of them are taken to in nity with the ratio a0=(DL
1) xed.
the black thick lines and grey thin lines are connected links with axy = a0 and disconnected links
with axy = 0, respectively. The uctuations are encoded by uctuation of link quantum number a
around the classical value in a small range.
This range of axy de nes a subspace of the bulk, which is mapped to the boundary by
the random tensor network. The de nition of the classical geometry and small uctuation
subspace is illustrated in gure 6.
To study whether the bulkboundary map is an isometry, we carry the same
calculation as in section 3 to evaluate the second Renyi entanglement entropy between bulk and
boundary. An isometry is de ned if the bulk subspace is maximally entangled with the
boundary. The calculation is exactly parallel to that in section 3, except that the bulk link
state xy in eq. (3.6) is replaced by
ja0 + axyi ha0 + axyj ; if Kxy = 1
j axyi h axyj ;
The Ising action is changed correspondingly to
(4.2)
(4.3)
xy = <
8
>
>
>>>> 1 =
>
>>>> 2 =
>
:
2
2
1
1
X
2
X
+ 1 axy=
+ 1 axy=0
A [fsxg] = A0 [fsxg] +
A [fsxg]
A0 [fsxg] =
Here J1;2 are half the Renyi mutual information of the states 1;2 on the connected and
the disconnected links respectively.
In the limit of J1
J2, log D ! 1 with hC and
nite, A0 is the leading term in
the action, and
A is a subleading correction.
The analysis of this action is essentially the same as the original RTN case in ref. [23].
The boundary term prefers sx =
1, while the bulk pinning eld hC prefers sx = +1.
Isomtion is sx =
etry condition is satis ed in the limit log D ! 1; J1 ! 1 if the lowest action con
gura1 everywhere, which corresponds to an entropy hC V = V (V2 1) log (2
+ 1) =
log (dimHC ). In order for this con guration to have the lowest action, one requires that
creating any spin up domain R costs a positive action. Denoting the action of a spin
conguration with sx = +1 in R and sx =
1 elsewhere as AR, the isometry requirement is
AR
A; = (J1
+ log D jR \ Bj
hC jRj > 0; 8R
bulk
(4.4)
where the rst two terms are action cost from the twobody interaction terms, the third
term is the action cost from boundary pinning
elds, while the last term is the action
saved by the external eld term hC . jRj is the number of vertices in R and j@Rj is the
number of links connecting R and its complement in graph K (excluding the boundary
links). j@R \ Bj is the number of links connecting R with boundary, i.e. the number of
boundary sites in R.
For su ciently large J1; log D and
nite J2; hC , condition (4.4) is satis ed. To
obtain a more explicit understanding on the requirements, in the following we derive a
su cient condition which guarantees that the isometry condition (4.4) is satis ed for all
classical geometries.
Denote N and M as the number of interior sites and boundary
sites in R, respectively, such that jRj = N + M and jR \ Bj = M . The action cost
A
AR
A; is a function of N; M and the graph dependent parameter j@Rj. If we are
considering a particular given graph, j@Rj is not independent from N and M . However,
simpli cation occurs when we require condition (4.4) to hold for all R and for all classical
2
h
0; (N+M)(V N M) i). Therefore we can view the action cost
geometries. By varying the graph, one can always vary j@Rj of a given region R in the range
A as a function of three
independent variables N; M; j@Rj. This simpli ed the problem of minimizing
A, because
the function in eq. (4.4) does not have local minimum in term of N; M and j@Rj. Thus
the minimum can only occur at corners of the parameter space. Given that J1
A = J2 (N + M ) (V
N
J2 > 0,
M ) +
A at the four corners N = 0 or Vb and M = 0 or VB
M log D
hC (N + M ). Evaluating
leads to two nontrivial conditions:
HJEP08(217)6
A(Vb; VB) > 0 ) VB log D >
A(Vb; 0) > 0 ) J2VbVB >
V (V
Vb(V
2
2
1)
1)
log(2
+ 1)
In summary the two su cient conditions are
log D >
J2 >
log(2
+ 1)
log(2
+ 1)
1)
V (V
(V
2VB
1)
2VB
(4.5)
(4.6)
(4.7)
(4.8)
Physically, the rst condition (4.7) is simply the requirement that the bulk code subspace
has smaller dimension than the boundary. The second condition requires that even weak
links with coupling J2 provide strong enough entanglement to propagate information from
bulk to boundary isometrically. It should be noted that condition (4.7) requires D to grow
exponentially with volume V (if we x the ratio VB=V ). This is necessary since the bulk
code subspace dimension grows with (2
+ 1)V (V 1)=2. Besides, eq. (4.8) only requires J2
to be a O(
1
) number in this limit. If we consider a limit V ! 1 with large but nite D, it
will be impossible to faithfully represent all link variable uctuations axy to the boundary.
However, it is probably still possible to de ne a code subspace with lower bound dimension,
which contains bulk excitations with a low enough density. (An example of such kind of
code subspace was discussed in ref. [22].) Such a code subspace which is not a direct
product of Hilbert spaces of each link is probably closer to the code subspace in AdS/CFT,
consisting low energy bulk quantum eld theory excitations.
Local reconstruction properties
Now we further investigate the local reconstruction properties of the bulkboundary
isometry. The local reconstruction requirement can be phrased in an entanglement entropy
calculation. In the old setup of tensor networks with
xed geometry, shown in
gure 5,
one can view the bulkboundary map as a quantum state that contains four partitions
A; A; EA; EA. The requirement that A contains all information about EA is equivalent to
the statement I(EA : A) = S (EA) + S(A)
S(EAA) = 0. In the following we will evaluate
the second Renyi entropy version of the mutual information. In the large D limit when the
uctuation of Renyi entropies are small, we expect the von Neumann entropy to be equal
to the Renyi entropy. Before proceeding, we would like to note that in the current setup
the bulk degrees of freedom are de ned on links, so that the bulk Hilbert space do not
factorize into di erent regions. For a boundary region A, one can de ne an entanglement
wedge EA, such that all links with both ends contained in EA can be reconstructed from
A. In the following we will demonstrate that under certain conditions, the region enclosed
by a boundary region A and the minimal surface A homologous to A is the entanglement
wedge EA. This is illustrated in gure 7.
Since we need to compute Renyi entropy of regions including both bulk and boundary,
we should not trace over bulk link variables to obtain a reduced density matrix xy. Instead
we treat the whole RTN with bulk and boundary indices as a state, and map the second
Renyi entropy calculation to an Ising model partition function. All dangling ends of the
tensor network in bulk and boundary correspond to xed external spins that couple to the
dynamical Ising spins de ned on bulk vertices. We denote the dynamical Ising spins as sx,
and the external spins as mX on the boundary and mxy on bulk links. sx; mX ; mxy all take
values of
1. When we compute the Renyi entropy of a bulk region EA and a boundary
region A, the external spins are de ned as
mX =
(
1; X 2 A
+1; X 2= A
(
1; xy 2 EA
+1; xy 2= EA
;
mxy =
(4.9)
red circles are the entanglement wedge EA, enclosed by A and the minimal surface A. The code
subspace that can be locally reconstructed in region A are labeled by links with both ends in EA,
marked by red bulk lines. (For clarity we have only drawn a few of the unconnected (grey) links.)
For small uctuations around a graph K considered here, we have
Tr
2
A[b
= const:
J1 and J2 are the same as in eq. (4.3). The earlier calculation of the entropy of entire
boundary in eq. (4.3) corresponds to the special case mxy = +1; 8x; y and mX =
1; 8X.
The constant prefactor is not important as it is the same for all con gurations, and does not
a ect normalized quantities such as Tr
can be computed by the same action with di erent boundary conditions.
2A[b =Tr [ A[b]2. Similarly, Tr [ A]2 and Tr [ b]
2
The mutual information is determined by the correlation between external spins
mediated by the dynamical spins. We denote the e ective action Ae
the e ective action with boundary condition (4.9), with
labeling the sign of external
spin in A and EA respectively. Similarly Ae
Ae
++ = Tr [ ]2 is the normalization constant. Then
+ =
log Tr h 2 i; Ae
A
+
=
=
log Tr h 2
i
as
A[EA
log Tr h 2 i, and
EA
I(2)(EA : A) = S(2) + SE(2A)
A
SA(2E) A ' Ae
+
+ Ae
+
Ae
++
Ae
(4.11)
is determined by the \energy cost" of the external spins in A and EA being antiparallel.
The requirement of zero mutual information is equivalent to the requirement that the two
external spins are completely uncorrelated. It is easy to see that this is true in the limit
we consider, with J1; log D ! 1 and J2;
nite. In this limit, the spin con guration
sx is completely determined by boundary external spins mX , and thus I(2)(EA : A) = 0.
For nite J1; log D, the local reconstruction condition depends on more detailed properties
of the classical geometry. Although it is possible to write down some su cient condition
by taking J1 and log D to be very large, we feel these conditions are not so useful to
include here.
5
Overlap between di erent classical geometries
In the discussion above we have shown that each classical geometry labeled by a graph
K is accompanied with a code subspace that satis es bulkboundary isometry and local
reconstruction properties. The next question is whether the code subspaces for di erent
classical geometries are truely independent subspaces of the boundary Hilbert space. Since
the basis j [faxyg]i is overcomplete, di erent geometries are generically not orthogonal,
but in the following we will show that states in the code subspace of di erent classical
geometries have exponentially small overlap.
For this purpose we study the overlap Cab = h [faxyg]j [fbxyg]i between two generic
geometries axy and bxy. Using the de nition (2.4) we have
2
x
Cab = D VB Tr 4
Y jVxi hVxj
Y jbxyi haxyj5
x6=y
Carrying the random average one obtains
Cab = D (V 1)Vb V VB ab
It is essential to go to the second order and study the uctuation around the average value,
so that we evaluate jCabj :
2
jCabj2 = D 2VB Tr 4
Y jVxi hVxj
Y jbxyi haxyj jaxyi hbxyjA5
= D 2VB
Y jbxyi haxyj jaxyi hbxyj5
=
=
1
X
R bulk
1
1
X
X
R bulk
2
Tr 4XR
2
Tr 4XR
x6=y
2
0
Tr 4XR
x6=y
Tr h aR bRi DjR\Bj VB
Y jbxyi haxyj jaxyi hbxyj5 D jR\Bj
with
= DV 1 + D2(V 1) Vb DV + D2V VB . To simplify this expression we can write
XR = XRXtot with Xtot the swap of all bulk vertices. Xtot will simply permute jbxyi and
jaxyi. Relabel R by R we obtain
jCabj2 =
Y jaxyi haxyj jbxyi hbxyj5 DjR\Bj VB
3
3
3
3
13
(5.1)
(5.2)
(5.3)
(5.4)
Here aR is the reduced density matrix of Qxy jaxyi haxyj in region R, and similarly for
R
b . If we consider the term with R the entire bulk, Tr
a b
R R
= j haj bi j2 =
ab is the
innerproject of the two bulk states. Roughly speaking, we can consider all other terms as
corrections to the overlap induced by the bulkboundary map that is not injective.
The overlap Tr
a b
R R is nonzero only if axy = bxy for all x; y 2 R. Denote the set
of R that satisfy this property as C. To obtain an upper bound of the overlap, we use
the inequality
Tr h a b i
R R
q
Tr
R
a 2 Tr
R
b 2 = e 21 Sa(2)(R)+Sb(2)(R)
where Sa(2;b)(R) are the second Renyi entropy of states jaxyi and jbxyi in region R. Therefore
jCabj
2
1 X e 21 Sa(2)(R)+Sb(2)(R) log D(VB jR\Bj)
R2C
To understand the physical meaning of eq. (5.6), we evaluate it in several situations.
1. The diagonal element. If axy = bxy 8x; y, R can be any subset of the bulk, and
the dominant term in the sum is given by R = entire bulk. Also in this case, the
inequality takes the equal sign. If we take the classical geometry discussed in this
(5.5)
(5.6)
HJEP08(217)6
section, with J1; log D
obtain Ca2a '
1. Therefore
Ca2a
Caa
! 1, we can ignore the contribution of other terms, and
2 ' 1 + D1 V
Vb 1 + D V
VB
' e VbD1 V VBD V
(5.7)
The ratio is close to 1 in the limit of large volume since VbD1 V and VBD V are
much smaller than 1. In other words, the uctuation of the norm of state j [faxyg]i
is exponentially suppressed, which justi es the computation of jCabj2 without rst
normalizing the two states.
2. Completely distinct states. If we consider two completely distinct states such that
q
axy 6= bxy 8x; y, then the only contribution comes from R = ;, and jCabj
1D VB =
Ca2aCb2bD VB . In other words, the overlap between these states,
after normalization, is the inverse of boundary Hilbert space dimension DVB . This is
equal to the average overlap between two completely random states in the boundary
2 =
Hilbert space dimension.5
3. Two states di erent in IR. Now we study a nontrivial example. In holography all
geometries considered are asymptotically antide Sitter space in UV (the region near
5Apparently, when the bulk volume V is large enough so that the basis j [axy]i is very overcomplete,
some of them will have a signi cant overlap. This fact, however, does not appear in the calculation of
averaged overlap jCabj2. The higher moments jCabj
2k shall be able to reveal the e ect of extremely large
V , which we postpone to future works. We would like to thank Lenny Susskind for helpful discussion on
this problem.
(orange) and between the two regions. The overlap of these two states are upper bounded by
the boundary) and are generically di erent in IR. For example we may consider two
geometries, one with a black hole in IR and one without black hole. As a toy model
of this situation, we can consider two geometries that are identical in a UV region
Rm bounding the boundary, and distinct in the IR region, as is illustrated in gure 8.
We assume axy and bxy are completely distinct if x or y are outside region Rm, so
that all regions contributing to the overlap are Rm or its subsets. In this case the
dominant contribution to eq. (5.6) is given by the R
Rm that has minimal averaged
entropy 12 Sa(2)(R) + Sb(2)(R) . If both geometries are classical geometries with all
connected links axy = a0, the entropies satisfy area law Sa(2;b)(R) = s0 j@Rja;b with s0
the entropy contributed by each link state ja0i. j@Rja;b denotes the area (number
of links crossing the boundary of R) in graphs of a; b respectively. In summary we
obtain for two classical geometries a; b
q
jCabj
2
2
Caa Cbb
2
(5.8)
where R is chosen to minimize the averaged area. For example if we consider two
geometries with and without a black hole, and assume that the geometry to be
identical in UV until a certain distance to the horizon, then j@Rja;b > ABH is bounded
by the area of black hole horizon, so that the overlap is upper bounded by e SBH .
More generally, the overlap is bounded by the entropy of the minimal area surface
that enclose the region where the two geometries are (macroscopically) distinct.
From our de nition of code subspace, it's clear that if two classical geometries are
distinct at a link xy, the small uctuations axy + axy are still distinct from bxy + bxy.
Therefore the overlap upper bound for jCabj2 between two classical geometries a; b also
applies to any pair of states from the code subspaces of a and b. Consequently, if we choose
a set of macroscopically distinct geometries axny, the code subspaces HCn of each of them are
almost orthogonal subspaces of the boundary Hilbert space. One can de ne a bigger code
subspace HC =
nHCn such that the bulkboundary isometry is still wellde ned in the
bigger code subspace. In the bigger code subspace HC , operators that can be reconstructed
on a boundary region A form an algebra with nontrivial center, a structure that has been
investigated in ref. [28]. More speci cally, for example one can de ne an area operator
LA =
snIn which takes eigenvalue sn in each code subspace HCn. sn is the number of
connected links between EA and its complement. Similar structure has been explicitly
constructed in the perfect tensor networks [21] by introducing link degrees of freedom [26].
We would like to comment a bit more on the mapping between bulk and boundary
operators. A generic bulk operator in this code subspace has the form
Pn the projection operator onto nth code subspace HCn, and n an operator acting only in
that subspace. If we denote the linera map from boundary to bulk as M , a local operator
n in the code subspace of geometry axny is mapped to a boundary operator M yPn nPnM .
Although the bulkboundary mapping is linear and isometric, one can consider PnM as
the linear map restricted to a code subspace, which is \statedependent"[29]. Locality in
the bulk can only be de ned in a code subspace around a given classical geometry, and the
local operators in a code subspace (such as an operator xy that only slightly changes axy
value for one link) is actually an operator Pn xyPn in the large bulk Hilbert space. The
\state dependence" of operator correspondence in each code subspace is encoded in the
support of the operator in the bulk Hilbert space, speci ed by Pn.
= P
n Pn nPn, with
Conclusion and discussions
In conclusion, we have shown that the random tensor network states on all graphs form an
overcomplete basis of the boundary Hilbert space, which we name as holographic coherent
states. A generic boundary state is mapped to a superposition of geometries. The
semiclassical geometries are de ned as small uctuations around reference classical geometries
with strongly entangled edges. We show that small uctuations around a classical geometry
form a code subspace, the states in which are mapped to the boundary isometrically, with
local reconstruction properties. Furthermore, we show that states in the code subspaces
of two di erent classical geometries are almost orthgonal to each other, with their overlap
decaying exponentially as a function of the minimal area surface that covers the bulk region
in which the two bulk geometries are distinct.
The holographic coherent state basis has a lot of similarity to the coherent state basis
of a boson
eld. If we consider a complex boson
Ac + 12 2
A
coherent state basis j (x)i is an overcomplete basis of the system, with which one can
write a path integral representation of the partition function. The action of the system
may have multiple local minima, for example con gurations with and without vortices.
Around each local minimum one can expand the action in small uctuations, A [ c +
. The quantization of such uctuations are low energy quasiparticles such
as super uid phonons. The Hilbert space of such quasiparticle excitations is a \low energy
subspace" of the entire Hilbert space. Di erent classical minima j c1(x)i ; j c2(x)i are not
exactly orthogonal, but the overlap of macroscopically di erent states are exponentially
eld described by a j j4 theory, the
suppressed. Therefore one can view the low energy excitations associated with each of
them as physically independent subspaces.6 There are two key di erences between the
holographic coherent states we consider and the boson coherent states. Firstly, the overlap
in the former case is suppressed by exponential of the minimal area covering the distinct
region, while that in the latter case is suppressed by exponential of the volume of the
distinct region, which can be viewed as a manifestation of holographic principle. Secondly,
in the gravity case, locality in the bulk is only de ned in the code subspaces, which can
be seen in the fact that the log of Hilbert space dimension log(dim(HC )) is proportional to
the volume of the bulk, while that of the total Hilbert space log(dim(H)) is proportional to
the boundary. On comparison, in ordinary boson coherent state case both quantities are
proportional to the volume of the system.
unitary transformations Q
There are a lot of open questions along this direction. For a given boundary
Hamiltonian, a natural problem is to use the holographic coherent states as variational
wavefunctions. The geometry described by axy can be used as a \mean eld order parameter" that
is optimized by minimizing the energy. The di culty of this approach is the random
average, which introduces the ambiguity of a local unitary transformation and therefore mixes
states with very di erent energy. In principle, this problem can be solved in the following
procedure. For each given geometrical state jai
j [faxyg]i, one can consider all local
X2B uX jai, with uX 2 SU (D), and variationally determine uX
by minimizing energy. Denote the minimal energy in this class of states as E [a], we can
then minimize energy to determine the optimal bulk geometry axy. It is not clear whether
such a variational procedure is technically feasible. We will reserve that to future works.
Another natural question is how to obtain the bulk equation of motion  the analog
of Einstein's equation. By writing the boundary dynamics into a path integral in the
geometrical basis, one can in principle obtain a bulk action. Is the Einstein equation or
its analog the saddle point equation if the bulk action? Will such saddle point equation
be related to previous entanglement approaches to Einstein equation [31{36]. Yet another
interesting question is whether a similar arealaw bound of state inner product exists in
general relativity, where the inner product between two states is de ned by a path integral
with these states as boundary conditions [37, 38]. It is interesting to compare our results
with other recent discussions about the overcompleteness of the geometry basis. [39{41]
Acknowledgments
We would like to acknowledge helpful discussions with Ahmed Almheiri, Patrick Hayden,
Aitor Lewkowycz, Don Marolf, Sepehr Nezami, Leonard Susskind and Michael Walter.
This work is supported by the National Science Foundation through the grant No.
DMR1151786 (ZY), and the David and Lucile Packard Foundation (XLQ).
6It is interesting to note that the 4 theory example appeared in a related discussion in ref. [30] about
statedependent operators (see section 5).
Fluctuations and higher Renyi entropies
In section 3, we make the following approximation in the calculation of the second Renyi
entropy.
where h1, h0 denote the boundary eld con guration for the calculation of Tr[ 2B], Tr[ B]2.
This calculation is valid if the uctuation around the minimum is small [23]. Formally,
the following conditions should be satis ed
by requiring
Tr h
Here we have used that Tr[ 2B]
e A(m2i)n[h1], since at
nite temperature the partition
k
B
function receives contributions from all spin con gurations, not just the minimal energy
con guration. Similarly for the calculation of Tr [ B]2 one can require
1. Thus the calculation of the
uctuation requires the random average over four copies
of the density matrix. Similarly, when calculating the kth Renyi entropy, we need to
calculate Tr
which involves k copies of the density matrix. For example Tr
Tr[ B]4
e A(m4i)n[h0]
1
B
2 2 are both average of 4 copies of density matrices, with di erent boundary conditions
which specify the contraction of indices. More explicitly they can be written as Tr
i
4h(B1234) and Tr
B
2 2 = Tr h
i
4h(B12)(34)
with h(B1234) the cyclic permutation acting
on 4copies of B, and h(B12)(34) the permutation of 12 and 34 acting on the same region.
Therefore in general we can evaluate the random average of k copies of density matrix
with an arbitrary boundary condition, and study how to control its deviation from the
contribution of the dominant con guration. The k copy quantity with most general boundary
condition can be expressed as
Z(k)
Tr
"
k Y
X2B
Y Tr h
hX
#
=
X
gxi2Sk xy
xy gxgy
k i j i Y Tr h
EkP RgxihX
i
x2B
1, which can be achieved
(A.1)
(A.2)
4
B
and
4
B
=
(A.3)
(A.4)
(A.5)
with boundary permutations hX 2 Sk de ning the boundary conditions. We label the
permutation group elements as gxi; i = 0; 1; 2; : : : ; k! 1, with gx0 = Ix the identity operator.
The averaged entanglement quantity is mapped to a partition function of a Sk statistical
mechanical model de ned on the complete graph.
In the following we will prove that the
uctuation of such quantities with general
boundary conditions is bounded if the following su cient conditions are satis ed:
(V
Tr( xykgxigyi)Tr( xykgxjgyj ) > jTr( xykgxigyj )j2; 8; i 6= j
In section A.1, we bound the uctuations based on conditions (A.4) and eq. (A.5). In
section A.2, we propose a stronger condition of the density matrix that implies eq. (A.5).
In section A.3, we construct the an explicit example in spin system and show that eq. (A.4)
and eq. (A.5) are satis ed.
A.1
General results
In this section, we prove that eq. (A.4) and eq. (A.5) are su cient to bound the uctuations
and to guarantee that higher Renyi entropies are close to the maximum.
First we rewrite eq. (A.5) as
HJEP08(217)6
where
Lii(k) + Ljj (k)
2Lij (k) > 0
1
2
Lij (k) =
log Tr( xykgxiIy) + log Tr( xy Ixgyj )
k
k i j
log Tr( xy gxgy)
Next, it is straightforward to show that eq. (A.5) implies Lii(k) > 0, i 6= 0, a condition
we will use to bound the uctuation. If we take j = 0, eq. (A.5) means
Lii(k) + L00(k)
2Li0(k) > 0
Since xy is normalized. L00(k) = Li0(k) = 0. Thus Lii(k) > 0.
Now we calculate the partition function in (k > 2) replica with an arbitrary boundary
condition. Using permutation symmetry between vertices in the complete graph, eq. (A.3)
e A(ni;mi)
Vb!
VB!
n0!n1!
nk! 1! m0!m1!
mk! 1!
A(ni; mi) =
X J ij (ni + mi) (nj + mj ) + X J ii
(ni + mi)(ni + mi
i
2
1) + X Bimi
i
J ij =
log tr
xykgxigyj
Bi =
log tr
EPkRgxih
Lij = (J i0 + J 0j
J ij )=2
can be rewritten as
1
Z(k) =
X
fnig;fmig
i>j
with
X ni = Vb
i
X
i>j 1
+ X J ii
i 1
2
where ni(mi) is the number of bulk(boundary) points occupied by the group element gi;
Vb(VB) is the total number of bulk(boundary) points; hX = h xes the boundary condition.
Then we replace n0 = Vb
i 1 ni, m0 = VB
P
i 1 mi. Since J 00 = 0, we have
A(ni; mi) =
J ij (ni + mi) (nj + mj ) + X J i0 (ni + mi) @Vb + VB
0
1
X(nj + mj )A
j 1
X mi = VB
i
P
(ni + mi)(ni + mi
i 1
1) + X Bimi
i
(A.6)
(A.7)
(A.8)
(A.9)
=
+ X
i 1
(ni + mi)Lij(nj + mj) + X
(Vb + VB)Lii + (Vb + VB
1)
J ii
2
(Vb + VB)Lii + (Vb + VB
+ Bi
B0
mi + B0VB
1)
2
ni
(A.10)
In the large Vb; VB limit, we treat ni and mi as continuous variables to decide where
F (ni; mi)
A(ni; mi)
Pi log ni!
Pi log mi! reaches its maximum. We use Stirling
formula and calculate the second derivatives of this function
M =
M1ij =
M2ij =
M3ij =
"
M1 M2#
M2 M3
2
2
2
F (ni; mj) = 2Lij
F (ni; mj) = 2Lij
F (ni; mj) = 2Lij
ij
ni
ij
mi
1
n0
1
m0
(A.11)
Now we show that F does not have local minimum away from the corners of the parameter
space. A local minimum requires M to be a negative de nite matrix, so to prove that F does
not have local minimum one just needs to show that M is not negative de nite anywhere
away from the corners. A corner of the parameter space (labeled by ni=Vb; mi=VB) is
de ned by having one ni = Vb; mj = VB and all other numbers vanishing. Therefore for
any point away from these corners, there are either two numbers ni; nj of order Vb, or two
numbers mi; mj of order VB. Let's assume there are ni; nj of order Vb since the discussion
with mi; mj is exactly in parallel. This includes the following two cases:
If n0 is of O(
1
), then there are two ni, nj with i; j > 0 of order O(Vb). De ne a vector
~v whose ith element is 1, jth element is
1 and all others are 0. Obviously,
vT M v = 2 Lii + Ljj
2Lij
1
ni
1
nj
Since Lii + Ljj
2Lij > 0, and ni, nj are O(Vb), vT M v > 0. So M is not negative
de nite and there is no local maximum in this case away from the corners.
If n0 is of O(Vb), then there is at least another ni being O(Vb). We choose ~v whose
only nonzero element is 1 at the ith element. Thus
vT M v = 2Lii
1
ni
1
n0
(A.12)
(A.13)
Since Lii > 0 is O(
1
) and n0, ni is O(Vb), vT M v > 0.
Therefore we conclude that eq. (A.5) is the su cient condition that guarantees F (ni; mi)
does not have local minimum away from the corners.
The next step is to compare the value of F (ni; mi) of each corner solution and bound
the near corner solutions. The corner solutions are categorized as
J iiVb
2
J jj VB
2
J iiVb + J jj VB
2
(A.14)
(A.15)
(A.16)
(A.17)
(A.18)
(g 1h),
HJEP08(217)6
Sn0;m0 : n0 = Vb, m0 = VB,
Sni;m0 : ni = Vb, i
1, m0 = VB,
Sn0;mj : n0 = Vb, mj = VB, j
1,
Sni;mj : ni = Vb, mj = VB, i; j
1,
F (Sni;mj ) =
Bj VB
VbVB Lii + Ljj
2Lij
(V
1)
F (Sn0;m0 ) =
B0VB
F (Sni;m0 ) =
B0VB
VbVBLii
(V
1)
F (Sn0;mj ) =
Bj VB
VbVBLjj
(V
1)
Firstly, we notice that F (Sn0;m0 )
F (Sni;m0 ) is always true, because Lii > 0 is
assumed and J ii = log DL k
(gi) > 0, where (g) denotes the number of cycles in a
permutation g.
Secondly,
F (Sn0;m0 )
F (Sn0;mj )
= VB log D k
((gj ) 1h)
(h)) + VBVbLii + (V
VB log D k
((gj ) 1) + (V
2
1) VB log DL k
1)
J jj VB
2
((gj ) 1)
(V
In the inequality, we use Lii > 0, and the triangle inequality of d(g; h)
k
which is equal to the minimal number of transpositions (i.e., permutations that exchange
only two indices) required to write a permutation g 1h. d(g; h) de nes a distance on
Sk, which satis es the triangle inequality d(g; I) + d(I; h)
d(g; h) [23]. Thus eq. (A.4)
2 log D is a su cient condition for F (Sn0;m0 )
F (Sn0;mj ).
F (Sn0;m0 )
F (Sni;mj )
= VB log D k
+ (V
1)
> VB log D k
J iiVb + J jj VB
2
(k
(k
VB log D k
((gj ) 1) + (V
((gj ) 1)
((gj ) 1)
(A.19)
where in the rst inequality, we use J ii > 0 and Lii + Ljj
2Lij > 0. In the second
inequality, we use the triangle inequality of k
we also have F (Sn0;m0 )
F (Sni;mj ).
((gi) 1gj ) again. Thus if eq. (A.4) holds,
In fact, we can make tighter bounds in F (Sn0;m0 ) F (Sni;mj ) and F (Sn0;m0 ) F (Sn0;mj )
if we do not simply discard Lii or Lii + Ljj
2Lij . However, using condition eq. (A.4) has
the advantage that it does not depend on k and the details of the link state.
i
Finally, we can bound Z(k) by analyzing the con gurations near the corners.
We
have shown that when eq. (A.4) and eq. (A.5) are satis ed, all other corner solutions are
exponentially small compared with the dominating corner Sn0;m0 , and the exponent is
suppressed by
(gj) VB ((V
1) log DL
2 log D). Thus the next biggest con guration
is at the neighborhood of the corner solution Sn0;m0 . In fact we can bound all con gurations
that are nite distance away from Sn0;m0 by C exp
O(
1
) number. Thus we obtain that
Z(k)
i
e B0VB
1 + C(VBVb)k! 1 exp
(V
1) log DL
where (VBVb)k! 1 is the total number of con gurations of F (ni; mj ).
We conclude that if eq. (A.4), (A.5) are satis ed, the
uctuation is controlled and all
higher Renyi entropies are close to VB log D. Thus there is an isometry from the boundary
to the bulk.
A.2
A su
cient condition for eq. (A.5)
In this section, we provide a su cient condition that deduces eq. (A.5), which helps to
clarify what density matrices satisfy this equation. In a basis j xi = Qsk=1
kcopied Hilbert space, density operators and permutation operators are written as
xk of the
(A.20)
(A.21)
(A.22)
(A.23)
(A.24)
(A.25)
One can rearrage the indices and write
xyk =
k
xy
gj = (gj ) ; j ih j
; ; ; (j xi
j yi) (h xj
h yj)
Tr h
xykgxigyj i = (gi) ;
k
xy
; ; ;
~xyk
jgj i
k
xy
; ; ; (j xi
(gj ) ; j i
j i;
(gj ) ; = hgij ~xykjgj i
j xi) (h yj
h yj)
hgij
(gi) ; h j
h j
where in the last step, we have used the fact that the matrix elements gi ; in the product
basis are real. In this representation, Tr h
xykgxigyj i becomes an inner product between states
gi ; gj with metric ~xyk. Therefore eq. (A.5) follows from CauchySchwarz inequality if
~xyk is Hermitian and positive semide nite for all k. Thus we conclude that a su cient but
not necessary condition for eq. (A.5) is that ~xy is Hermitian and positive semide nite.
This condition is not necessary since eq. (A.5) is only required for permutation operators
and does not need to hold for general operators.
An explicit example of states jaxyi
In this section, we provide an explicit example of link states jaxyi and prove that condition
(eq. (3.17) is satis ed. We de ne the state jJ i as a SU(2) singlet formed by two spins each
carrying spin J representation:
jJ i
M
X ( )
J M
p2J + 1
jJ; M ; J;
M i
with J = 0; 1; : : : ; DL
1 labeling the link states. The Hilbert space of each site is a direct
sum of di erent representations Hx =
DL 1
J=0 HJ . States with di erent J obviously are
HJEP08(217)6
orthogonal. (A subtlety is that the entropy of state jJ i is log (2J + 1), so that we should
think the link variable a / log (2J + 1) if we still want a to label the entropy across the
link. This does not a ect any discussion here.) The density matrix xy is given by
1
DL
1
DL
X
J=1
jJ ihJ j
uy =
X(
1
)M jJ;
J;M
M i hJ; M j
If one directly obtains ~xy in eq. (A.24) for xy, the resulting xy is Hermitian but not
positive semide nite. However, we can prove xy satis es condition (A.5) by de ning a
unitary operator on the y site
The density matrix in the new basis is
uy xyuyy =
1
DL
DL 1
X
1
1 J=0 2J + 1
X
M;N
jJ; M ; J; M i hJ; N ; J; N j
Since uy k commutes with permutation operators gyi, we have Tr h
xykgxigyj i = Tr h
xykgxigyj i.
For xy, the corresponding operator ~xy de ned in eq. (A.24) is
~xy =
1
DL
DL 1
X
1
1 J=0 2J + 1
X
M;N
jJ; M ; J; N i hJ; M ; J; N j =
DL 1
J=0 (DL
1
1)(2J + 1) IJ
with IJ an identity matrix of the size (2J + 1)2
(2J + 1)2. Obviously ~xy is diagonal and
positive de nite, so that we prove xy and therefore xy satisfy eq. (A.5).
Open Access.
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