Holographic Cavalieri Principle as a Universal relation between Holographic Complexity and Holographic Entanglement Entropy

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by  Davood Momeni

Holographic Cavalieri Principle as a Universal
relation between Holographic Complexity and
Holographic Entanglement Entropy
Davood Momeni1, Mir Faizal2,3, Ratbay Myrzakulov1
7
1
0 1 Eurasian International Center for Theoretical Physics
2
and Department of General Theoretical Physics,

n
Eurasian National University, Astana 010008, Kazakhstan
a
J 2 University of British Columbia - Okanagan

1 Kelowna, British Columbia V1V 1V7, Canada
3 3 Department of Physics and Astronomy, University of Lethbridge,

] Lethbridge, Alberta, T1K 3M4, Canada
h
t
-
p
Abstract
e
h
In this paper, we will propose a universal relation between the holo-
[
graphiccomplexity(dualtoavolumeinAdS)andtheholographicentan-

1 glemententropy(dualtoanareainAdS).Wewillexplicitlydemonstrate
v that our conjuncture hold for all a metric asymptotic to AdS3, and then
7 argue that such a relation should hold in general due to the AdS version
3
of the Cavalieri principle. Wewill demonstrate that it hold for Janus so-
3
lution,which havebeen recently beenobtained in typeIIBstringtheory.
1
Wewillalsoshowthatthisconjectureholdsforacirculardisk. Thiscon-
0
jecturewill beusedtoshowthattheproposalthatthecomplexityequals
.
3 action, and the proposal that the complexity equal volume can repre-
0 sent the same physics. Thus, using this conjecture, we will show that
7
theblack holesarefastest computers,usingtheproposal thatcomplexity
1
equals volume.
:
v
i
X 1 Introduction
r
a
Various studies done in different areas of physics have indicated that the laws
of physics can be represented in terms of the ability of an observer to process
relevant information [1, 2]. Entropy measures the amount of information that
is lost in a process, and hence, it is thought to be one of the most important
quantitiesassociatedwithanysuchinformationtheoreticalprocess. Theentropy
has been used to model physical phenomena from condensed matter physics to
gravitational physics. Even the geometry of spacetime can be viewed as an
emergent structure, which emerges due to an information theoretical process.
ThisisbecauseintheJacobsonformalism,theEinsteinequationcanbederived
from thermodynamics by assuming a certain scaling behavior of the entropy
[3,4]. Thisscalingbehaviorofentropyisthatthemaximumentropyofaregion
1
of space scales with its area, and this observation has been obtained using the
physics of black holes. This observation has also led to the development of the
holographic principle [5, 6], and the AdS/CFT correspondence is one of the
most important realizations of the holographic principle [7].
Theblackholeinformationparadoxoccursduetotheobservationthatblack
holes are maximum entropy objects and they evaporate due to Hawking radi-
ation. It is interesting to note that quantum entanglement has been used to
discuss the microscopic nature of black hole entropy, with the hope that it
may resolve the black hole information paradox [8, 9]. The AdS/CFT corre-
spondence makes it possible to quantify quantum entanglement in terms of the
holographic entanglement entropy [10, 11, 12]. The entanglement entropy has
beenusedinvariousbranchesofphysicsfromquantumcomputingtocondensed
matter physics, and AdS/CFT correspondence make it possible to calculate it
holographically. The holographic entanglement entropy of a CFT is dual to
the area of a minimal surface defined in the bulk of an asymptotically AdS
spacetime. So, for subsystem A with its complement, it is possible to write an
expression for the holographic entanglement entropy as
(γ )
A
S = A (1)
A
4G
d+1
where G is the gravitational constant in the AdS spacetime, γ is the (d 1)-
A
−
minimalsurface extendedinto the AdS bulk with the boundary ∂A,and (γ )
A
A
is the area of this minimal surface. It may be noted that usually there are
UV divergence in holographic entanglement entropy, and so we need to use
a regularization method to remove these divergences. Thus, for a deformed
geometry, we define the area in this paper as,
(γ )= (γ ) (γ ), (2)
A D A AdS A
A A −A
where (γ ) is the defined in deformed geometry (for example the geometry
D A
A
of a black hole), and (γ ) is defined in the background AdS spacetime.
AdS A
A
Thus,wedefine the holographicentanglemententropyforadeformedgeometry
by subtracting the contribution coming from the background AdS spacetime.
This removes the divergent part and we are only left with a finite part. We
will use this finite part in this paper, and call it the holographic entanglement
entropy.
However,therecentstudieshaveindicatedthatitisnotenoughtoknowwhat
part of the information can be obtained by an observer from a system, but it
is also important to know how difficult is it to obtain that information. As the
entropyquantifiestheabstractnotionofthe lossofinformation,the complexity
quantifies the abstract notion of the difficulty to obtain the information (even
if it is present in the system). The complexity (like entropy) has been used to
study physical systems from black holes to condensedmatter physics, and even
quantumcomputing. Infact,recentlyithasbeenproposedthattheinformation
maynotbeideallylostinablackhole,butitmaybelostforallpracticepurposes
as it would be impossible to reconstructit fromthe Hawking radiation[13]. As
complexityhasonlybeenrecentlyusedtostudy variousphysicalsystems,there
are different proposals to define the complexity for a CFT. However, recently
motivated by holographic entanglement entropy, holographic complexity has
beenholographicallydefinedasaquantity dualto avolume ofcodimensionone
2
time slice in anti-de Sitter (AdS) [14]-[19]. Furthermore, it is possible to use a
subsystem A with its complement, and define this volume as V = V(γ ), i.e.,
A
the volume enclosed by the same minimal surface which was used to calculate
the holographic entanglement entropy [20],
V(γ )
A
= , (3)
A
C 8πRG
d+1
where R and V(γ ) are the radius of the curvature and the volume in the AdS
A
bulk. So, we will use this definition of the holographic complexity, and investi-
gatearelationbetweenthisdefinitionofholographiccomplexityandholographic
entanglemententropy. Itmaybenotedthatjustliketheminimalarea,thisvol-
ume also contains UV divergences, and so we need to regularize this volume.
So, for a deformed geometry, we define the volume as
V(γ )=V (γ ) V (γ ), (4)
A D A AdS A
−
whereV (γ )isthe volumeindeformedgeometry,andV (γ )isthe volume
D A AdS A
in the background AdS spacetime. So, we regularize the volume in a deformed
geometry by subtracting the contribution coming from the background AdS
spacetime. This again removes the divergent part and we are again left with
a finite part. In this paper, we will use this finite part of the holographic
complexity.
Now as both the holographic complexity and holographic entanglement en-
tropyarecalculatedusingthe sameminimalsurface,weexpectthatauniversal
relation to exist between them due to an AdS version of Cavalieri principle. In
this letter, we will explicitly demonstrate this to be the case, and also find the
explicit form of this universal relation. This can be used as a new holographic
dictionary to calculate the holographic complexity from holographic entangle-
ment entropy for different asymptotic AdS spacetimes. As both holographic
entanglement entropy and complexity are used in various different branches of
physics ranging from black hole physics to condensed matter physics, this gen-
eralconjecturecanhavealotofapplicationsinthosebranches. This is because
it is easier to calculate the entanglement entropy than complexity for various
complex systems, and if this conjecture holds in general, this can be used as
a holographic dictionary to obtain such quantities. Furthermore, as it is diffi-
cult to define complexity for the boundary theory, and a precise definition for
complexity does not exist for the boundary theory, we will use this relation to
obtain a definition for complexity of the boundary theory. This is because the
complexity will be defined in terms of quantities whose boundary dual is well
understood.
2 Excited States in Bulk Geometry
In this section, we will motivate a universal relation between the holographic
complexity andholographic entanglemententropy. Now we consider anexcited
state in a d+1 dimensional conformal field theory (CFT), and assume it to be
almost static and translational invariant. We want to analyse its gravity dual,
3
so we write the metric on AdS as [21]
d+2
R2 dz2 d
ds2 = f(z)dt2+ + (dx )2 . (5)
z2 "− h(z) i #
i=1
X
Near the boundary z 0, we can assume h(z) f(z) 1 z d+1, where z
→ ≃ ≃ − z0 0
is constant.
(cid:0) (cid:1)
Let us consider an entangling region (subsystem A) in the shape of a strip
definedby0<x <l, L/2<x <L/2,whereListakentobe infinite.
1 2,3,...,d
−
So, we can parameterize the minimal surface γ by x = x(z), and write its
A 1
area as
dz 1
(γ )=2Ld 1Rd +x (z)2. (6)
A A − zdsh(z) ′1
Z
wherethederivativewithrespecttothez isdenotesby . Wecandeterminethe
′
shape of x(z) by minimizing this area function. We note that the Lagrangian
in Eq. (6) is independent of x (z), and so the first integral associated with
1
Euler-Lagrangeequation can be expressed as
1 ( z )d+1
x (z)= z∗ . (7)
′1 h(z) 1 ( z )2(d+1)
− z∗
q
here we have assumed that x (z ) = . The total entangled length l, the
′1 ∗ ∞
entanglement area functional (γ ) and volume of codimension one time slice
A
A
V(γ ) of the metric (5) are given by
A
z∗ z d 1
l =2 dz (8)
Zǫ (cid:0)z∗(cid:1) sh(z)(1− zz∗ 2d)
Ld 1Rd 1 dξ (cid:0) (cid:1) 1
−
(γ )= (9)
A A (z∗)d Zzǫ∗ ξd+1sh(ξ)(1−ξ2(d+1))
Ld 1Rd 1 x (ξ)dξ
− 1
V(γ )= . (10)
A (z∗)d−1 Zzǫ∗ ξd+1 h(ξ)
where ξ = z . Now at the tip point z = z =pz , these Eqs. (8,9,10), can be
z∗ ∗ 0
written as
l =2z b(d,z ) (11)
∗ ∗
Ld 1Rd
−
(γ )=a(d,z ) (12)
A A ∗ (z )d
∗
Ld 1Rd
−
V(γ )=ν(d,z ) . (13)
A ∗ (z )d 1
∗ −
Now using Eqs. (11)-13), along with the expression for holographic entangle-
ment entropy given by Eq. (1), and the expression for holographic complexity
given by Eq. (3), we obtain
l
=n S (14)
A d A
C R
(cid:16) (cid:17)
4
where n = ν(d,z∗) is a constant. It may be noted that the effective
d 4πa(d,z∗)b(d,z∗)
temperature(‘entanglementtemperature’)T is proportionaltothe inverseof
ent
l [21],
T =c l 1. (15)
ent −
·
Thus, we obtain the following Universal relation
T c n
ent A d
C = · . (16)
S R
A
This is a universal relation between the holographic complexity and holo-
graphicentanglemententropy. ItexistsduetoanAdSversionoftheholographic
cavalieriprinciple, and so we can call this universalrelationas the Holographic
CavalieriPrinciple Conjuncture, andexplicitly state it as following: Let us as-
sumethat tworegions existbetween two parallel AdSslice, and these tworegions
in codimension two slice of an asymptotically AdS space have equal areas, and
the CFT duals to these two regions have equal entangled temperature, then they
will have equal holographic complexity. .
Thisconjecturecanhavealotofapplicationsasitcanalsobeusedtoobtain
a definition of complexity for the boundary theory. There is no agreed defini-
tion of the complexity for the boundary theory, however,using this conjecture,
we can define the complexity of the boundary theory. Thus, we can obtain the
complexityfortheboundarytheoryusing =c n S /RT ,becausethethe
A d A ent
C ·
boundary dualofallthese quantities exceptcomplexity is welldefined. So, this
relation can be used as a definition for the complexity of the boundary theory.
It is also expected that the holographic complexity would be directly propor-
tional to the holographic entanglement entropy, as the difficulty of obtaining
the information from a system will increase as the amount of information lost
from a system will increase. Thus, this result is something we would expect on
physical grounds for the boundary theory.
3 Circular Disk
In this section, we will demonstrate that this conjecture holds for a circular
disk. The holographic entanglement entropy [11] and holographic complexity
[20] for sucha geometry has alreadybeen analyzed. We wouldlike to pointout
thatduetoourdefinitionoftheareaandvolumeoftheminimalsurface,wewill
onlyusethefinitepartoftheholographicentanglemententropyandholographic
complexity. This is because these quantities are regularized by subtracting the
contributionscomingfromthebackgroundAdS.NowusingthebulkofAdS ,
d+2
it is possible to define a sphere of radius l. So, we parametrizesuch a metric as
R2
ds2 = dt2+dr2+dρ2+ρ2dΩ . (17)
r2 − d−1
(cid:16) (cid:17)
The entanglingregionis representedby t=0,r l ,wherel is the radiusofa
{ ≤ }
circulardisk. Theareaandvolumefunctionalsfortheparametrizationρ=ρ(r),
can be written as
l ρ(r)d 1 dρ(r)
(γ )=Ω Rd dr − 1+( )2, (18)
A A d−1 Zǫ rd r dr
Ω Rd+1 l ρ(r)d
d 1
V(γA)= −d drrd+1 . (19)
Zǫ
5
Now we can write the solution of equation of motion describing this system
as ρ(r) = √l2 r2. Using this solution, we can obtain an expression for Eqs.
−
(18,19),
l (l2 r2)d/2 1
(γ )=Ω Rd dr − − , (20)
A A d−1 Zǫ rd
Ω Rd+1 l (l2 r2)d/2
d 1
V(γA)= −d dr −rd+1 . (21)
Zǫ
Nowweexpandtheaboveintegralsinseries,andusethesuitableregularization
for them. So, we are only left with the finite part of the volume and area
functionals,
(γ )=Ω Rd ∞ (1−d/2)n , (22)
A d 1
A − n!(2n d+1)
n=0 −
X
Ωd 1Rdl ∞ ( d/2)n
V(γA)= − − . (23)
d n!(2n d)
n=0 −
X
Now using (1,3), we obtain
Ωd 1Rd ∞ (1 d/2)n
SA = − − (24)
4G n!(2n d+1)
n=0 −
X
Ωd 1Rdl ∞ ( d/2)n
A = − − (25)
C 8πRGd n!(2n d)
n=0 −
X
So, we can write the ratio of such terms as
c l
A d
C = . (26)
S R
A
Thus, we observe that even for this geometry, the holographic entanglement
entropy is propotional to the holographic complexity. This is the expression
we expected from our conjecture. Thus, this conjecture holds for the such a
geometry.
4 Asymptotically AdS Spacetime
We can try to argue that such a conjecture is justified for a general asymptoti-
callyAdS . TheappropriateformofthemetricwritteninFefferman-Graham
d+1
coordinates is given by
R2
ds2 = dr2+g dxµdxν (27)
d+1 r2 µν
(cid:16) (cid:17)
Wechooseanentangledstripparametrizedby t=0,x =x(r) [ l/2,l/2],x
1 i
{ ∈ − ∈
[0,L],i=2,..,d 1, and write the area functional for this metric as
−
d 2 g(r)(1+G(r)x2)
(γ )=Rd 1 − xdr ′ (28)
A A − rd 1
Z p −
6
where g g ,G(r) = g g1igj1. A conserved charge can be constructed
≡ | ij| 11 − gij
using this area functional (for general x(r))
R g(r)G(r)x R
( )d 1 ′ =( )d 1 g(r )G(r ) (29)
r − gp(r)(1+G(r)x′2) r∗ − ∗ ∗
p
Thus, we obtain the follpowing result,
(R)d 1 g(r )G(r )
x(r)= dr r∗ − ∗ ∗ (30)
Z G(r) g(r)G(r)(R)2(dp1) g(r )G(r )(R)2(d 1)
r − − ∗ ∗ r∗ −
r
(cid:16) (cid:17)
Total entangled length l and the total volume V(γ ), can be written as
A
r∗ (R)d 1 g(r )G(r )
l = 2 r∗ − ∗ ∗ .(31)
Z0 G(r) g(r)G(r)(R)2(dp1) g(r )G(r )(R)2(d 1)
r − − ∗ ∗ r∗ −
r
r∗(cid:16) R g g (cid:17)
V(γ ) = dd 2x x(r)( )d g 1+ 1i j1dr (32)
A − ij
r | | g
Z Z0 q r ij
Asitisnotpossibletoexplicitlycalculatetheholographicentanglemententropy
for a general metric, we will simplify our analysis to the general form of the
metriconthe AdS . ThemetricofanyasymptoticallyAdS canbe represented
3 3
by Eq. (27) when g = h rd, where h is a uniform metric. Thus, for such
µν µν µν
a metric, we can easily integrate Eq. (29), and obtain
1
x(r)= (33)
√Hcosh( r )
r∗
where H = h h1ihj1. Now for AdS , we can obtain the area as Area =
11 − hij 3
2R h , and so the entanglement entropy can be written as
µν
| |
p R h
µν
S = | | (34)
A
2G
p 3
Similarly,wecancalculationthevolume,andexpressitasV(γ )=R2 h
A µν
| |N
where = 1 dξ 19 ln(ǫ)+ (ǫ6) 1, and so the holographic
N ǫ ξcosh(ξ) ≈ −96 − O ≫ p
complexity can be written as
R
R h
µν
= | |N. (35)
A
C 8πG
p 3
Furthermore, the total length is given by
2
l= (36)
√HB
where = 1 dξ = π iln ǫ + (ǫ6) 1. Note that √H 1 .
B 0 ξ√ξ2−1 −2 − 2 O ≫ ∼ √|hij|
If we combinRe three equations and using the definition of the temperature, we
obtain
T
ent A
C = N . (37)
S 8π R
A
B
7
Even though we have explicitly calculated this for all asymptotically AdS , we
3
canfollowthe samealgorithmandcalculatethesequantitiesforanyasymptoti-
callyAdSmetric. However,suchcalculation,eventhoughconceptuallystraight-
forward, can become computational complicated. So, in the next section, we
will demonstrate this conjecture holds for an important asymptotically AdS.
5 Janus solution
Now we will explicitly test this conjuncture for Janus solution. First of all,
we will consider a AdS Janus solution [37]. The Janus solution interpolates
3
between two AdS spaces [29]. This bulk model for this solution is defined by
the following action,
1 2
S = dx3√g gab∂ φ∂ φ+ . (38)
−16πG R− a b R2
N Z (cid:18) (cid:19)
TheJanussolutionisathreedimensional(actuallythesimplestanalytic)mem-
ber of the generally AdS -sliced domain walls with the isometry group SO(d
d
−
1,2) [23, 24, 25, 26, 27, 28]. The metric for such a solution can be written as
ds2 =e2A(r)g (x)dxidxj +e2h(r)dr2, (39)
ij
where g (x) is a metric on AdS with scale R . It may be noted that such a
ij d d
domain wall has also been obtained as a solution in the type IIB supergravity
[29]. This solution has no r-dependent matter fields except a flowing dilaton
φ(r). This solution is regular, if parameters are chosen such that the rate of
variation of the dilaton is sufficiently slow. We use the radial coordinate r for
which h(r)=0, so we take φ =cexp( dA(r)), and by this the scalar equation
′
−
of is also satisfied. Now the wall profile equation can be written as
A′2 =(1/L2)[1 e−2A+be−2dA], (40)
−
The constant b is related to c by b = κ2 c2R2. We have set R = L for
d(d 1) d
simplicity. So, when b=0, the solution giv−es pure AdS ,
d+1
ds2 =cosh2(r/L)g (x)dxidxj +dr2. (41)
ij
However, for b = 0, it is not possible to obtain a simple solution for Eq. (40),
6
(unless d=2 and this is the Janus metric). The metric for AdS sliced domain
2
walls in AdS can be written as
3
ds2 =e2A(r)ds2 +dr2. (42)
AdS2
The explicit solution of the equations of motion is the Janus solution, is given
by the metric
ds2 =R2 dy2+f(y)ds2 , (43)
AdS2
1
f(y)= ((cid:0)1+ 1 2γ2cosh(cid:1)(2y)), (44)
2 −
p
and the dilaton is given by the function
y dy
φ(y)=γ +φ , (45)
1
f(y)
Z−∞
8
where γ ( 1 ) is the parameter of Janus deformation.
≤ √2
The metric of AdS slice is given by
2
ds2 =(dz2+dx2)/z2, φ =φ( ), (46)
AdS2 1 −∞
and it is dual to the coupling constant of the exactly marginal deformation for
the groundstate Ω >. The value φ =φ( ) for the other groundstate Ω >
1 2 2
| ∞ |
is obtained by performing the integral give in Eq. (45). So, we have
1 1 2γ2
φ φ =√2arctan − − γ, (47)
2 1
− " p√2γ #≃
when γ 1. Now if the bulk extension of the surface is parameterized by
≪
x=x(r), then the corresponding area is given by
R2f(0)
(γ )= 1+x(z)2dz. (48)
A A z2 ′
Z
p
The minimal surface x(z) is the solution of the following equation,
x(z) 1
′
= , (49)
z2 1+x′(z)2 (z∗)2
with the auxiliary boundary conpdition, x′(z)z=z∗ = . Now the solution of
| ∞
this equation can be written as
x(z)=z∗ E(ξ,i) F(ξ,i) (50)
−
(cid:16) (cid:17)
where ξ = z , and E(z,k),F(z,k) are elliptic functions,
z∗
z √1 k2r2
E(z,k)= − dr, (51)
√1 r2
Z0 −
z dr
F(z,k)= . (52)
√1 k2r2√1 r2
Z0 − −
Thus, minimizing the area, we obtain
R2f(0) 1 dξ
(γ )= (53)
A
A z∗ Zzǫ∗ ξ2 1−ξ4
p
where z is turning point and ǫ is a UV cut off. Now for the solution Eq. (50),
∗
the total entangled length l, the finite part of the entanglement area functional
A(γ)andvolumeofcodimensiononetimesliceV(γ )ofthemetricareobtained
A
as follows,
l =2z E(1,i) F(1,i) (54)
∗
−
(cid:16) R2f(0) (cid:17)∞ (1/2)n
(γ )= ( 1+ ) (55)
A
A z − n!(4n 1)
∗ n=1 −
X
π
V(γ )=R2 +K(i) E(1,i) . (56)
A
4 −
(cid:0) (cid:1)
9
Here K(k) is another elliptic functions,
1 dr
K(k)= (57)
√1 k2r2√1 r2
Z0 − −
So, using Eqs. (54-56), it is possible to explicitly demonstrate that the holo-
graphic Cavalieri principle holds for this solution,
T
ent A
C = N. (58)
S R
A
where = cn1 is a numeric factor, and n ,n are given by
N πn2·f(0) 1 2
π +K(i) E(1,i)
n = 4 − ,
1
2 E(1,i) F(1,i)
−
(cid:16) (cid:17)
∞ (1/2)n
n = 1+ . (59)
2
− n!(4n 1)
n=1 −
X
Thus, we have been able to explicitly demonstrate that holographic Cavalier
principle hold for Janus solution. This can be used to analyse the holographic
complexity for the boundary theory dual to the Janus solution. It may be
notedthatJanussolutionisdualtoaninterestingfieldtheoreticalsystem. This
is because the field theoretical system dual to Janus solution is a boundary
spacetimedividedbyacodimensiononedefect[29]. AdifferentYang-Millscou-
pling exists in each of the two halves of this boundary spacetime. In fact, the
the string theoretical configurations for this solution have also been analysed
[29]. The conformalperturbationtheoryhas been usedto analysethe quantum
level conformal symmetry of the Janus solution [30]. The holographic entan-
glement entropy for Janus solution has been calculated, and it has been used
for analyzing the behavior of boundary theory dual to the Janus solution [31].
The holographic complexity can also be used to analyse the behavior of the
boundary theory dual to the Janus solution. So, it would be interesting to use
the resultsofthis paperto analysethe behaviorofthe boundarytheorydualto
Janus solution.
6 Application
In this paper,we have useda proposalfor holographiccomplexity, which states
that the holographic complexity of a system is equal to the volume enclosed
by a minimal surface. There is another recent proposal for the holographic
complexityandthisproposalstatesthattheholographiccomplexityofasystem
is equal to the bulk action, calculated on a Wheeler-DeWitt patch [32, 33]
A
= , (60)
C π¯h
where A is the bulk action evaluated on the Wheeler-deWitt patch with a suit-
able temporal boundary, and is this holographic complexity obtained using
C
this new proposal. It is possible to calculate the action on a Wheeler-DeWitt
patch for such geometries, using the null boundaries of the Wheeler-DeWitt
10