Connecting Fisher information to bulk entanglement in holography

0.53 MB·

by  Souvik Banerjee

UUITP-03/17
LMU-ASC 01/17
Connecting Fisher information to bulk entanglement in holography
Souvik Banerjee∗
Department of Physics and Astronomy, Uppsala University,
SE-751 08 Uppsala, Sweden
Johanna Erdmenger†
Institut fu¨r Theoretische Physik und Astrophysik,
Julius-Maximilians-Universita¨t Wu¨rzburg,
Am Hubland, 97074 Wu¨rzburg, Germany.
and
Max-Planck-Institut fu¨r Physik (Werner-Heisenberg-Institut),
Fo¨hringer Ring 6, 80805 Mu¨nchen, Germany.
7
1 Debajyoti Sarkar‡
0
Arnold Sommerfeld Center, Ludwig-Maximilians-University,
2
Theresienstr. 37, 80333 Mu¨nchen, Germany.

r and
a
Max-Planck-Institut fu¨r Physik (Werner-Heisenberg-Institut),
M
Fo¨hringer Ring 6, 80805 Mu¨nchen, Germany.

1

In the context of relating AdS/CFT to quantum information theory, we propose a holographic
]
h dualoftheFisherinformationmetricformixedstatesintheboundaryfieldtheory. Thisamountsto
t aholographicmeasureforthedistancebetweentwomixedquantumstates. Forasphericalsubregion
-
in the boundary we show that this is related to a particularly regularized volume enclosed by the
p
e Ryu-Takayanagi surface. We further argue that the quantum correction to the proposed Fisher
h informationmetricisrelatedtothequantumcorrectiontotheboundaryentanglemententropy. We
[ discuss consequences of this connection.

2
v
I. INTRODUCTION to the boundary entangling region. In two dimensions,
9
this holographic computation of boundary entanglement
1
3 The theory of quantum information and measurement entropymatchesexactlywiththedirectreplicatrickcom-
2 [1] has been a key ingredient in the understanding of the putation in conformal field theory [6]. Subsequently, a
0 underlying quantum structure of the universe. There- direct holographic justification of the RT formula was
1. fore for the last two decades significant efforts have been provided in [7]. Moreover, the holographic realization
0 devoted to realizing connections ofquantum information of quantum entanglement is extremely useful in under-
7 theory to other branches of theoretical physics. Within standing the structure of the dual bulk spacetime. For
1 string theory, the first realization of such a connection is eternalblackholes[8],quantumentanglementwasfound
v: the calculation of black hole entropy through the count- to have a direct bulk interpretation in terms of regular-
i ing of black hole microstates for supersymmetric black ity of the horizon [9] which turns out to be an extremely
X holes [2]. useful input in understanding the black hole information
r These connections were put on an even stronger foot- paradox.
a
ingwiththeadventofAdS/CFTcorrespondence[3]. One
A further concept from quantum information theory
of the most important and crucial steps forward in this
recently studied in the context of holography is the dis-
direction is the holographic realization of entanglement
tance between two quantum states. There exist two well-
entropy [4, 5]. The entanglement entropy for an entan-
acceptedwaystodefinethedistancebetweentwogeneric
gling region in a conformal field theory living on the
quantumstates,namely(i)theFisherinformationmetric
boundary of an asymptotically AdS spacetime is pro-
and (ii) the Bures metric or fidelity susceptibility [1, 10].
posed to be given by a minimal codimension-two area
in the bulk. The associated bulk hypersurface, also re-
In order to define these, let us consider a generic den-
ferredtoasRyu-Takayanagi(RT)surface,ishomologous
sity matrix σ and perturb it by some parameter λ which
parametrizes the quantum state. Here for simplicity we
assume a one-parameter family of states which however
∗ [email protected] can be generalized for arbitrary number of parameters.
† [email protected] If ρ is the new density matrix corresponding to the fluc-
‡ [email protected] tuation λ → λ+δλ, then for nearby states, the density
2
matrix ρ can be expanded as Here V is the spatial (d−1) dimensional volume at
d−1
the boundary, G is d+1-dimensional Newton’s constant
1
ρ=σ+(δλ)ρ1+ 2(δλ)2ρ2+··· , vaniedwn,dna/nGOi(s1)sicmonpslytapnrto.2poFrrtoiomnathletofietlhdetcheenotrryalpcohinatrgoef
d
where for simplicity we have chosen the parametrization CT of the CFT considered.
inawaysuchthattheinitialstateσcoincideswithλ=0. In the first part of the present work, we will propose
The coefficients ρ and ρ are of first and second order a holographic dual to the Fisher information metric cor-
1 2
in δρ, respectively, with δρ a small deviation from σ. In respondingtoasphericalsubregionR intheholographic
this case, a distance metric may be defined by CFT. We consider a mixed state in a bipartite system
where σ and ρ denote the reduced density ma-
(cid:18) (cid:19) R,λ R,λ+δλ
G =(cid:104)δρδρ(cid:105)(σ) = 1tr δρ d (σ+δλδρ)| . (1) trices for the subregion, R corresponding to a decompo-
F,λλ λλ 2 dλ δλ=0 sition of the full Hilbert space H = H ⊗H . We
full R R(c)
will show that a natural holographic dual of Fisher in-
This is known as the Fisher information metric in the
formation for this mixed state is a particular regularized
literature.
volumeenclosedbytheRTsurfaceintheperturbedbulk
On the other hand, a second notion of the quantum
spacetime.
distance between the same two states is given by the
Furthermore, in analogy to (3), the reduced fidelity is
fidelity susceptibility, which is defined by
defined by
G =∂2F , (2) (cid:113)√ √
λλ λ F =Tr σ ρ σ .
R R(c) R,λ R,λ+δλ R,λ
where F is the quantum fidelity defined in terms of the
Intheabove,Tr denotespartialtracingoverthecom-
initial and final density matrices σ and ρ, R(c)
plementaryregionR(c). Accordingly,thereducedfidelity
(cid:113)√ √ susceptibility is given by
F =Tr σ ρ σ . (3)
λ λ+δλ λ
G =∂2F . (5)
For classical states when the density matrices com- R,λλ λ R
mute, (1) and (2) become equivalent up to an overall So long as the reduced density matrices in the vacuum
numerical factor. Hence, for classical states, the defini- and in the excited state are commuting i.e. simultane-
tion for the distance between quantum states is unique. ously diagonalizable, (5) is the same as the Fisher in-
The first holographic computation of the Fisher infor- formationmetriccorrespondingtothosereduceddensity
mationmetric(1)wasperformedin[11]usingitsrelation matrices. This is automatically true when we deal with
to the second order variation of relative entropy. On the classical states. Hence for this restricted class of states,
otherhand,theholographicdualof(3)wasfirstproposed our proposal also serves as holographic dual for the re-
in [12], but only for pure states when (3) reduces to an duced fidelity susceptibility.
inner product between nearby states, In [14], in the context of proposing a gravity dual for
complexity, Alishahiha proposed a holographic dual of
|(cid:104)ψ (x)|ψ (x)(cid:105)|=1−G(pure)(δλ)2+... . the reduced fidelity susceptibility (5) in terms of the vol-
λ λ+δλ λλ
ume enclosed by the RT surface γ(R). However, this
Here, G(pure) now refers to the fidelity susceptibility for quantity is UV divergent, while the Fisher information
λλ
thepurestate|ψ (cid:105). TheseauthorsconsidertheCFTvac- metric for a general mixed state must be finite. Hence,
λ
uumstate|ψ (cid:105)dualtopureAdSanddeformitbyanex- at least for the class of states where one can identify the
λ
actly marginal perturbation to obtain the state |ψ (cid:105). two notions of information metric introduced above, this
λ+δλ
In the dual gravity picture, this corresponds to a Janus proposal yields contradiction. However, our holographic
solution, where the pure AdS is deformed by a dilaton Fisher information proposal predicts a manifestly finite
[13]. Fisher information metric. In addition, at least for the
For a holographic CFT on Rd with marginal defor- above-mentionedrestrictedclassofstates,itgivesriseto
d
mation of dimension ∆ = d + 1, in [12] it was shown a UV-finite reduced fidelity susceptibility.
that1
n Vol(V )
G(pure) = d d−1 . (4)
λλ G (cid:15)d−1 2 HerewehavewrittenthefactorofNewton’sconstantexplicitly.
In the boundary field theory computation, it comes from the
leading-order term of the boundary two-point function (in the
1/N expansion of a large N CFT) via Ld−1 ∝ N2 (L is AdS
G
1 Note that [12] only considers marginal deformations of the radius). The matching of bulk and boundary computations of
ground state and not general massive deformations which we fidelitysusceptibilityin[12]thusonlyinvolvestheleading-order
shallbeconsideringinwhatfollows. Therefore,theholographic contributioninNewton’sconstantonbothsides. Inotherwords,
dual of information metric or reduced fidelity susceptibility for thefidelitysusceptibilitycomputedthereintermsofdualgravity
mixedstateswhichwewillproposewillnotsimplyreduceto(4) quantities,isonlytheleadingordersemiclassicaltermofthefull
inthepurestatelimit. boundaryfidelitysusceptibility.
3
In the second part of this work, we will put forward a L is the radius of the AdS space-time.
conjecture which relates the leading 1/N quantum cor- In order to find the minimal RT surface in this per-
rection to the Fisher information metric and the corre- turbedAdSspacetimecorrespondingtoaball-shapeden-
sponding quantum correction to the boundary entangle- tangling region of radius R at the boundary, we proceed
mententropy. Accordingto[15,16],thisinturncoincides by parametrizing the RT surface as ρ = h(z). Then, on
withthebulkentanglemententropy. Thissecondconjec- the t=0 slice the RT area functional takes the form
ture is motivated by argueing that the reduced Fisher
information is related to the canonical energy as defined (cid:90) Rt dz (cid:113)
A=Ld−1Ω (h(z))d−2 f(z)+(h(cid:48)(z))2,
in [17]. Then, the connection between canonical energy d−2 zd−1
(cid:15)
and the bulk modular Hamiltonian [16] enables us to es- (7)
tablish our proposal. whereΩ isthevolumeoftheunit(d−2)sphere,given
d−2
Ourpaperisorganisedasfollows. InsectionIIandsec- by
tion III we establish the two proposals mentioned above,
namely (A) the holographic dual to the Fisher informa- πd−21
tion metric and (B) the connection between the leading Ω =2 .
d−2 Γ(cid:0)d−1(cid:1)
1/N quantum correction to the Fisher information met- 2
ric and the bulk entanglement entropy. We conclude in
R is the turning point of the bulk minimal surface.
section IV and discuss some of the consequences of our t
proposals, as well as a physical consistency check. We In order to find the minimal surface, we have to min-
also discuss directions for future work. imize the area functional (7) to solve for h(z). It is
hard to solve the equations of motion analytically. We
therefore aim at solving them perturbatively in orders of
II. PROPOSING A HOLOGRAPHIC DUAL FOR mRd << 1 and look for a solution of the form (up to
FISHER INFORMATION METRIC linear order; quadratic order to be done later)
Here we propose a candidate bulk dual for the Fisher h(z)=h0(z)+mh1(z). (8)
information metric as a regularized volume enclosed
by the RT surface in the bulk. For the holographic As shown in [18], this gives
dual,weconsideranasymptoticallyAdSspacetimeusing
(cid:113)
Fefferman-Graham coordinates. For the boundary CFT,
h = R2−z2
this amounts to considering states whose density matrix 0 t
deviates perturbatively from that of the vacuum state, 2Rd+2−zd(R2+z2)
h = t t . (9)
withthechangeintheboundarystresstensorplayingthe 1 (cid:112)
2(d+1) R2−z2
roleoftheperturbationparameter. Forthisexcitedstate, t
wecomputethevolumeundertheRTsurfacecorrespond-
With these ingredients, we now move on to compute the
ing to spherical entangling region at the boundary. This
volume under the RT minimal surface in the bulk. After
volume is generally UV divergent. However, subtracting
performing the integrations over the boundary coordi-
the RT volume for the same spherical subregion in the
nates ρ and Ω, this is given by
vacuum state yields a finite result. We propose that the
Fisher information metric is givern by the second order
variation of this regularized volume with respect to the V = LdΩd−2 (cid:90) Rt dz (h(z))d−1(cid:112)f(z). (10)
RT d−1 zd
perturbationparameter. Inwhatfollowswewillconsider (cid:15)
d>2.3
Our aim is now to compute the variation of this volume
Toelaborate,letusconsideraperturbationofAdS
d+1 order by order in the perturbation mRd.
given in Fefferman-Graham coordinates of the form
L2 (cid:20) 1 (cid:21)
ds2 = f(z)dz2+ dt2+dρ2+ρ2dΩ2 , (6)
z2 f(z) d−2 A. At linear order in stress-energy tensor
where
From fundamentals of AdS/CFT dualities, we know
f(z)=1+mzd. thatsuchFefferman-GrahamtypeexpansionofAdSmet-
ric conrresponds to the corresponsding expectation val-
ues of boundary stress tensor [19]. In order to find the
leading variation in the RT volume we first expand (8)
3 The d=2 case is special in the sense that the perturbative ex- up to leading order in mRd
pansion of the regularized volume does not contain a quadratic
tteicramll.yTahnedrtehaesroeniissntohactofnotrridbu=tio2n, hto2w(za)rdins (t8h)evmaninisihmeaslidareena- (cid:112) (cid:18) zd(R2+z2) (cid:19)
h(z)≈ R2−z2 1−m , (11)
surfaceatthesecondorderinmRd. 2(d+1)(R2−z2)
4
Inserting(11)into(10)andexpandingindividualterms readily solved to yield
in the integral again, we have
(cid:32)
h (z)= 1 (cid:0)160c R −11R8(cid:1)log(z−R )
V(m) ≈ LdΩd−2 (cid:90) Rtdz(R2−z2)d−21 2 320 1 t t t
RT (cid:18)d−1 (d(cid:15)−1)zd(R2+zdz2)(cid:19)(cid:18) mzd(cid:19) +(cid:0)59Rt8−160c1Rt(cid:1)log(Rt+z)+320c1z− R20R+t9z
× 1−m 1+ . t
2(d+1)(R2−z2) 2 9R2z6(cid:33)
−90R7z+34R6z2−30R5z3+22R4z4− t +c .
t t t t 2 2
Here the superscript m denotes that this corresponds to
the volume under the RT surface corresponding to per-
turbedgeometry. Nextwesubtractfromitthesamevol-
c and c are integration constants which should be
1 2
ume for the unperturbed background, i.e for pure AdS
suitably chosen in order to extract the physical solu-
obtained by setting m=0 in (6). This yields
tion. We note from the solution that in order to ensure
VR(mT)−VR(0T) ≈m(d−Ld1Ω)(dd−+2 1) hathn2e(dzt)uc/2r(cid:112)n=iRngt26p−140ozi(cid:0)n21t1→a3lRs0ot8ar−secz9e6→ivRet8sRlato,gnw(e2ewRmct)uo(cid:1)ns.ttrhiCbavuoentiscoe1nq=uaetn1t11th6Rl0yit7s,
(cid:90) R order of perturbation theory and is given by
× dz(R2−z2)d−23 (cid:0)R2−dz2(cid:1)
(cid:15) 3 mR4
R = m2R7(29+32log(2))− +R.
=0, t 640 4
Now expanding (10) up to quadratic order, we find
where in the first line, we have only kept terms up to
oprladceerdmRRdbyinRthseincinetethgreatnedrm. FliunretahrerinmmorRe,dwine Rhavgeivrees- V(m2)−V(0) ≈ 21R6m2.
t t RT RT 128
a quadratic correction to the volume. This shows that
the leading correction to the volume under RT surface In general dimensions, this result generalizes to
vanishesidenticallyasclaimedin[14,20]. Thevanishing
of this linear term in m is also crucial for our porposal V(m2)−V(0) ≈ LdΩd−2A m2R2d,
RT RT d−1 d
(A) to work, as will be seen in the next section.
with A a dimension-dependent constant.
d
This is the first central result of our paper. We see
that we have arrived at a UV-finite notion of a regular-
B. At quadratic order in stress-energy tensor
ized volume under the RT surface. It is of second order
in m. We will exploit this fact for proposing it as the
At quadratic order, we have holographic dual of Fisher information. We emphasize
thatthefinitenessoftheregularizedvolumedefinedhere
1 is critical for this proposal. In particular, this ensures a
f(z)=1+mzd+ m2z2d,
4 meaningful gravity dual for mixed states.4
We are thus lead to propose that the holographic dual
where coefficients of individual terms in the expansion is of the Fisher information metric is given by
fixed by comparing with the FG expansion of AdS black
hole. Now in order to compute the quadratic correction GF,mm =∂m2 F, (13)
to the RT surface in the bulk, we start with an ansatz
with
h(z)=h0(z)+mh1(z)+ (cid:112)Rm22−z2h2(z), F = G2dπ+321(dd(d+−1)1Γ)(cid:0)Γd(+d−3(cid:1)1L)A (cid:16)VR(mT2)−VR(0T)(cid:17) .
t 2 d
with h and h as given in (9). (14)
0 1
Inserting (14) back into (13) yields
The equation of minimal surface for h is again ob-
2
tained by extremizing the area functional (7) ,
G =∂2 F = π32dΩd−2Γ(d−1) Ld−1R2d.
F,mm m G2d+1(d+1)Γ(cid:0)d+ 3(cid:1)
(d−1)R2 2
h(cid:48)(cid:48)(z)+ t h(cid:48)(z)+C (z)=0, (12) (15)
2 z(z2−R2) 2 d
t
with C (z) being a complicated function of z. It is very
d
hard to solve (12) for general dimensions. However, as 4 See also [21] and [22] for some recent suggestions on possible
an illustration, we will consider d = 3 when (12) can be regularizedquantitieswhichcouldberelatedtocomplexity.
5
The choice of prefactors in (14) will become clear in the All quantities on the right-hand side of (20) belong to
next section. As mentioned before, mRd plays the role the gravity side of the correspondence. E is the classical
of perturbation parameter in the dual bulk picture, in canonical energy for the unperturbed vacuum state and
agreement with the holographic dictionary. canbeexpressedasanintegralofboundarystress-energy
Although so far, (13) applies only to ball-shaped re- tensor [17],
gions in the CFT, one may also wish to consider general
(cid:90)
entanglingregions,e.g.strips. InsuchcasestheO(mRd) E = ξµT vν, (21)
µν
terms do not necessarily vanish [20]. However, in prin-
Σ
ciple, one can still define the holographic dual to Fisher
where Σ is any Cauchy slice in the entanglement wedge
information metric as the second order variation of the
correspondingtotheball-shapedentanglingregioninthe
regularizedRTvolumewithrespecttothemassparame-
boundary. ξ is the conformal Killing vector, v is the
ter, with this parameter playing the role of perturbation
volume form defined as (for a D-dimensional spacetime)
parameter in the dual bulk theory.
1 (cid:112)
v = g˜(cid:15) dxµ1 ∧dxµ2···∧dxµD,
ν D! ν,µ1µ2...µD
C. Connection to canonical energy and boundary
relative entropy and (cid:15) being the usual Levi-Civita tensor. Eg denotes
gravitational equations of motion with proper cosmolog-
In a related development, [11] connects the quantum ical constant, e.g, for pure gravity in AdS
Fisherinformationcorrespondingtoperturbationsofthe
1 √ (cid:18) 1 1 (cid:19)
CFT vacuum density matrix of a ball-shaped region to Eg = −g R − Rg + Λg . (22)
thecanonicalenergyforperturbationsinthecorrespond- µν 16πG µν 2 µν 2 µν
ing Rindler wedge of the dual AdS space-time. This is
For the perturbed AdS space-time as in (6), and
obtained by using the definition of boundary relative en- d+1
for the case when the entangling region is a sphere,
tropy
the canonical energy as on the right-hand side of (20)
S(bdy)(ρ ||σ )=Tr(ρlogρ)−Tr(ρlogσ) maybecomputedexplicitly. Onecanalsoindependently
rel λ(cid:48) λ compute the second order variation of relative entropy,
=(cid:104)logρ(cid:105)ρ−(cid:104)logσ(cid:105)ρ, ∂2 S (ρ ||ρ ). Both calculations were done for d = 2
∂λ2 rel λ 0
which gives in [11] and were shown to match explicitly. Again, in
the holographic setup λ is identified with the boundary
S(bdy)(ρ ||σ )=∆(cid:104)H(σ)(cid:105)−∆S . (17) energy, mRd where m appears as a mass parameter in
rel λ(cid:48) λ R EE (6).
The first term on the right-hand side denotes the change Wenowgeneralizethisresulttogeneraldimensions. It
intheexpectationvalueofthemodularHamiltonianH reads
R
corresponding to the change in the reduced density ma-
trix. The modular Hamiltonian corresponding to a re- ∂2 S (ρ ||ρ )= π23dΩd−2Γ(d−1) Ld−1R2d
duced density matrix σ is defined through ∂λ2 rel λ 0 G2d+1(d+1)Γ(cid:0)d+ 3(cid:1)
λ 2
=G , (23)
F,λλ
e−HR,λ
σ = . (18)
R,λ Tr(e−HR,λ) where in the last line we have used the definition (19).
Thebasicingredientsinthiscomputationis(17)andthe
Here, thesecondtermrepresentsthechangeinentangle- fact that in this particular case of spherical entangling
ment entropies in the two mentioned states. When the region in CFT, the modular Hamiltonian has a local ex-
twostatesinquestionareperturbativelyclosetoonean- pression in terms of the boundary stress energy tensor
other, expanding the density matrix ρλ(cid:48) around λ=0 in as
(17) gives
(cid:90) R2−|x|2
H = dd−1x T
∂2 R 2R 00
G =(cid:104)δρδρ(cid:105)(σ) = S (ρ ||ρ ), (19) |x|<R
F,λλ λλ ∂λ2 rel λ 0
T being the temporal component of the stress-energy
00
where the left-hand side denotes the Fisher information tensor in the boundary CFT. Hence one can vary both
metric as defined in (1). ρ = σ is identified with the the terms in the right hand side of (17) up to second
0
CFT vacuum. order in mRd which yields (23).
Furthermore,theright-handsideof(19)isequaltothe Hence we find our proposal (15) fully consistent with
classical canonicalenergyingravityasdefinedin[17],i.e, the expression for Fisher information metric obtained in
(23). Thisalsojustifiesthechoiceofprefactorsasin(14).
∂2 (cid:12) (cid:90) ∂2Eg As discussed before, it is worth mentioning again at
S (ρ ||ρ )(cid:12) =E −2 ξµ µνvν. (20)
∂λ2 rel λ 0 (cid:12)λ=0 Σ ∂λ2 this point that for a restricted class of states when the
6
reduced density matrices in the vacuum and in the ex-
citedstatearesimultaneouslydiagonalizable-orinother
words, when the subregions are maximally entangled
even after perturbation, one should expect, analogous to
(19),
∂2
G = S (ρ ||ρ ), (24)
R,λλ ∂λ2 rel λ 0 A
GR,λλ being the reduced fidelity susceptibility defined Rc Rcb Rb R
in (5). For these states, our proposal (15) also serves
B
as a holographic dual to reduced fidelity susceptibility
while (14) can be interpreted as the holographic dual to
reduced fidelity.
Let us then briefly summarize our results of this sec-
tion. Weshowthatthereisawelldefined,finitenotionof
regularized volume which serves as the holographic dual
toFisherinformationfortwoperturbativelyclosestates.
Both of them are in turn related to the classical canoni-
FIG.1. AttheboundaryCFT oftheglobalAdS cylinder,
cal energy in the subregion. This set of connections will d d+1
wehaveadiscshapedregionRdenotedbyAB(redline,color
play a crucial role for the next part of our paper where
online). The dashed (black) line γ represents the RT surface
we make statements regarding their quantum counter- whichdividesthebulkregionintotwosubregionsR andRc.
b b
part. Here we also noted that for the special class of The area of this minimal surface gives the leading semiclas-
states, all the above definitions coincide with the defini- sical term of the total boundary entanglement entropy S .
EE
tion of reduced fidelity susceptibility, thus modifying the The O(G0) term of bulk entanglement entropy of the region
previously existing proposal of [14]. R is a measure of the first-order quantum correction term
b
S of S .
EE,q EE
III. FISHER INFORMATION AND BULK
ENTANGLEMENT where the first term on the right-hand side of (25) scales
as 1/G (or equivalently is of order N2) and corresponds
totheminimalareasurfaceterm6,whilethesecondterm
We now turn to the second part of our proposal on
scales as G0 and corresponds to the first quantum cor-
relating the 1/N quantum correction to reduced fidelity
rection to the boundary entanglement entropy.
susceptibility with bulk entanglement entropy.
In [15], the quantum correction to the boundary en-
tanglement entropy S is given by
EE,q
A. Bulk entanglement entropy and quantum (cid:12)
canonical energy SEE,q =Sbulk =−∂n(logZn,q−nlogZ1,q)(cid:12)n→1,
where Z is the bulk partition function of the replicated
n
Our investiagtion bulk entanglement entropy is moti- geometry in the bulk.
vated by a recent study in [15]. There the authors argue Taking these results into account, we now proceed to
thatthe1/N quantumcorrectiontotheboundaryentan- stateourobservations. Inthepath-integrallanguage,the
glement entropy for a boundary subregion R is given by decomposition of (25) can be realized by writing the full
the bulk entanglement between two regions - the region bulk partition function Zbulk as
inside the corresponding RT surface in the bulk and its
complement. The relevant regions are depicted in figure Zbulk =Wbulk+Wbulk,
eff
1. This bulk entanglement entropy can be computed or-
der by order in Newton’s constant G, using the replica where Wbulk denotes the classical action. This gives the
trick in the bulk, [7, 15] as5 classical part of bulk entanglement entropy. It is essen-
tially the same minimal area surface term that appears
S (R)=S (R)+S (R), (25) in Wald’s treatment of the first law [23]. Wbulk is the
bulk bulk,cl bulk,q eff
one-loop effective bulk action which gives S (R).
bulk,q
5 ForthecaseswherewehaveaU(1)symmetryasforstaticblack
holes, it is easier to implement the replica trick for non-integer 6 Notethatthemotivationof[15]wastoconnectthebulkentan-
n, n being the number of replicated geometries. For more gen- glement entropy with the quantum correction of boundary en-
eral cases without any U(1) symmetry, one needs to define the tanglement entropy. So, these authors only studied the S
bulk,q
partitionfunctionfornon-integernseparately[15]. part,whichtheyrefertoasS .
bulk−ent
7
In the framework of replicated n-fold geometries gˆ , to (29), and inserting the expansion back into (20), we
n
the full density matrix ρˆ(cid:48) is given in terms of a bulk obtain
n
time dependent Hamiltonian Hτ,full which generates the ∂2 (cid:12) (cid:90)
t[7im].eTthraantsilsa,tionalongtheEuclideanizedtimeτ direction ∂λ2Srbedly(ρλ||ρ0)(cid:12)(cid:12)λ=0 =E − ΣξµTµgνrav,(2)(δg(0))vν
(cid:20)(cid:90)
− ξµTgrav,(2)(h)vν
ρˆ(cid:48)n =e−(cid:82)02πnHb,n,full (cid:90) Σ µν (cid:21)
=e−(cid:82)02πn(Hb,n,cl+Hb,n,q) =ρˆ(cid:48)n,cl·ρˆ(cid:48)n,q, (26) + ξµTµmνatter,(2)(g)vν
Σ
where the subscripts b, n, cl or q above respectively sug- +boundary terms. (30)
gestthattheassociatedHamiltonianH isinthebulk, in
This gives a clean separation of classical and quantum
n-deformed spacetime and it is either classical or quan-
contributions in the Fisher metric and also the classical
tum(orderbyorderintheGexpansion). Theseclassical
andquantumcontributionsintheleadingordercanonical
and quantum parts give rise to the classical and quan-
energyintheperturbedbackground. Thefirsttwoterms
tumpartsofthecorrespondingbulkentanglementS .
bulk ontheright-handsideof(30)areclassical(O(1/G))con-
Fromtheabove,itiseasytoseefordiagonaldensityma-
tributions, with the second term arising from (21) upon
trices that by inserting the expression (26) into the von the second order variation in δg(0). These two classical
Neumann bulk entanglement entropy, S divides into
bulk terms are the same as the right hand side of (20). The
classical and quantum parts as in (25), i.e.
superscript (2) signifies the fact that all the variations
areofsecondorderinδg(0) andh.8 Theremainingterms
S =−∂ (cid:2)logTr(ρˆ(cid:48) )−nlogTr(ρˆ(cid:48) )(cid:3)
bulk n n,cl 1,cl inthebracketarequantumcorrections.9 Furthermore,it
−∂ (cid:2)logTr(ρˆ(cid:48) )−nlogTr(ρˆ(cid:48) )(cid:3)+... wasshownthattheboundarytermscanbetakencareof
n n,q 1,q
through a suitable choice of gauge as pointed out in [24].
=S (R)+S (R)+... , (27)
bulk,cl bulk,q Now combining (19) and (30) enables us to schemati-
cally write
where the dots denote terms that are local integrals on
the RT surface. When only the background metric has a ∂2 (cid:12)
Sbdy(ρ ||ρ )(cid:12) =G
non-zero vacuum expectation value, there are two other ∂λ2 rel λ 0 (cid:12)λ=0 F,λλ
terms that in principle can contribute to the O(G0) cor-
=G +G , (31)
F,cl F,q
rection to the boundary entanglement entropy. These
come from a change in area due to the back-reaction on Thus from (19) and (30), we obtain an expansion of
the classical background and from general higher deriva- Fisher information metric at order by order in Newton’s
tive terms, respectively.7 For our present purpose we do constant and their respective connections with the clas-
not consider the higher derivative terms in the bulk ac- sical and quantum part of the canonical energy. Bearing
tion. in mind (24), for the special case of commuting density
matrices,thisalsocallsforadecompositionanalogousto
Our key observation in this section will be the term-
(31) for the reduced fidelity susceptibility, as
by-term matching of the expansions (25) or (27) to an
analogous expansion of the Fisher information metric, ∂2 (cid:12)
namely, ∂λ2Srbedly(ρλ||ρ0)(cid:12)(cid:12)λ=0 =GR,cl+GR,q. (32)
Inthenextsubsection,wefurtherdevelopthisconnec-
G =G +G . (28)
F,λλ F,cl F,q tion, where we relate G (and G for the restricted
F,q R,q
class of states corresponding to commuting density ma-
Once again, the subscripts cl and q denote the classical
trices) to the bulk modular Hamiltonian and hence to
and quantum parts.
bulk entanglement entropy.
To begin with, let us consider a perturbation of the
background metric g(0) of the form
B. Canonical energy and bulk modular
g =g(0)+δg(0)+h. (29)
Hamiltonian
Here we consider two different kinds of perturbations of
√
Here we complete our arguments by invoking the fact
the bulk metric. h is an O( G) quantum fluctuation,
while δg(0) takes into account the λ variation. Expand- that the quantum correction to canonical energy (the
ing the right-hand side of (22) in powers of λ according
8 Note that the first order variation in either case vanishes by
virtueoflinearizedequationofmotion.
9 Alsonotethatthematterpartofthestresstensoronlyappearsat
7 In addition, counterterms may be necessary to ensure finite en- thequantumlevelasclassicallyforemptyAdS,thecontribution
tanglement. isidenticallyzero.
8
bracketed term in the second and third lines of (30)) is least for this class of states we can compare our result
essentially the same as the bulk modular Hamiltonian for a previous proposal for the holographic dual of the
H bulk that appears as the first quantum correction to reduced fidelity susceptibility given in [14] in terms of
R
the boundary modular Hamiltonian [25], [16] holographic complexity, namely, the leading term in the
volume under the RT surface. However, the proposal
H = Area(γ) +H bulk+... . (33) given there suffers from the following shortcomings. As
R 4G R we mentioned before, for classical (or effectively classi-
cal) states, fidelity susceptibility is physically the same
This is the operator equivalent10 of the expansion of
as Fisher information, which is defined by the second
boundary entanglement entropy at order by order in G,
order variation of relative entropy. Now relative en-
namely
tropy for a mixed state is always UV-finite. Hence it
is hard to justify that holographic complexity, which is
Area(γ)
SEE = 4G +SEE,q+... . (34) UV-divergent, should be its bulk dual. UV-convergent
behaviour was also advocated from a purely field the-
Thustheresults(30),(31),(32),(33),(34)clearlysuggest ory computation in [26], at least for free theories and
that the quantum Fisher information metric and equiv- conformal field theories with large central charge. Fur-
alently the reduced fidelity susceptibility for the men- thermore, the second-order variation of relative entropy
tionedrestrictedclassofstatescanindeedbeunderstood was computed explicitly [11, 27], and its behaviour dif-
as sum of two terms as in (28), namely a leading semi- fers significantly from that of holographic complexity as
classical term and a subleading quantum term. In this proposed in [14]. On the other hand, as we have shown,
division we are simply keeping track of the orders G−1 our proposal for the holographic dual of reduced fidelity
and G0, respectively. susceptibility for those states meets both requirements
Forexample, ifwejustfocusonthequantumpart, i.e. by construction. Our proposal for the bulk dual is simi-
the O(G0) part, we see from (31) and (30) that lar in spirit to the recently proposed idea of complexity
of formation [28–30], which measures the relative com-
(cid:20)(cid:90) (cid:90) (cid:21)
G =− ξµTgrav,(2)(h)vν + ξµTmatter,(2)(g)vν plexity between two states. One natural question might
F,q µν µν
thenbewhetherthisisalsorelatedtothecomputational
Σ Σ
=S , complexityasdiscussedin[31–33]. Infact,itsconnection
EE,q
withtherelativeentropyisreminiscentofthedefinitions
where the last quantity arises from the quantum canon-
of complexities used in [34],[35].
ical energy and is equal to the modular Hamiltonian in
One naive intuition to justify the relation to computa-
the bulk Hbulk [25], [16].
R tional complexity comes from the positivity of both re-
Thus following the separation in the classical and the
duced fidelity susceptibility and Fisher information. As
quantum parts in (30), we conclude that while the clas-
already mentioned in [36], the identification of Fisher
sicalpartoftheFisherinformationmetricG isgiven
F,λλ information with the Hollands-Wald canonical energy
by the classical part of canonical energy in agreement
[11],[17] implies the positive energy theorem for asymp-
with [11], the quantum part of it can be thought of as a
totically AdS spacetime. Our result hints at an alterna-
dualtothebulkmodularHamiltonian. Thesameconclu-
tivewaytoviewthederivationofthepositiveenergythe-
sion holds for the reduced fidelity susceptibility, however
oremforasymptoticallyAdSspacetimesintermsofpos-
only for the restricted class of states leading to commut-
itivity of reduced fidelity susceptibility for mixed states.
ing density matrices.
A suggestion is to interprete this positivity of canonical
energy (in our case, we consider the canonical energy as-
sociated to the bulk Rindler wedge corresponding to the
IV. CONCLUSIONS AND OUTLOOK
spherical boundary region R) as the transition from a
reference vacuum state to a more complex excited state.
In the first part of this work we have proposed a
In other words, the monotonically increasing nature of
holographic dual of Fisher information metric for mixed
reduced fidelity susceptibility mimics that of computa-
states. This is given by a regularized (i.e. finite) vol-
tional complexity. However, a full justification behind
ume contained under the RT surface in the bulk. This
such a connection is yet to be understood. Work in this
also serves as a holographic dual for the reduced fidelity
direction is in progress and we hope to report on this
susceptibility but for a restricted class of states, namely,
generalisation in near future. In particular, we expect a
whenthedensitymatricescommutebeforeandafterper-
calculationwithinquantumfieldtheorytoreproducethe
turbation,i.ewhenthestatesareeffectivelyclassical. At
scaling with R2d as in (15) and (23).
The second part of our proposal relates the quantum
contributiontoFisherinformationorreducedfidelitysus-
10 This is possible by noting the connection between the entan- ceptibilitytobulkentanglement. Thelatterhasbeenar-
glement entropy and the modular Hamiltonian via the density guedtobeinstrumentalinunderstandingthereconstruc-
matrixasin(18). tion of the bulk points inside an entanglement wedge in
9
terms of local operators in the boundary CFT through serve a closer look.
modular evolution [16]. We expect our proposed duality
mightaddanusefulcomponenttowardsaconcretestudy
in this direction.
Acknowledgements
Therearemanyotherimportantissuesthatneedtobe
investigatedinfutureforacompleteunderstandingofthe We would like to thank Dean Carmi, Dan Kabat, Nima
proposed connections. We already mentioned the quan- Lashkari for several helpful correspondence. We thank
tum information origin of holographic complexity itself, Mohsen Alishahiha and Tadashi Takayanagi for numer-
which is missing so far in the literature. It will also be oushelpfuldiscussionsandcommentsonthemanuscript.
interesting to understand how our construction changes We would like to thank Ayan Mukhopadhyay for com-
formorecomplicatedboundarystates,suchassubregions ments on an earlier version of the manuscript. The work
with arbitrary shapes or thermofield double geometries. of SB is supported by the Knut and Alice Wallenberg
A generalization to multi-dimensional parameter space Foundation under grant 113410212. The work of DS is
and a covariant generalization of our proposal also de- supported by the ERC grant ‘Self-completion’.
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