Constraining gravity using entanglement in AdS/CFT

0.41 MB·

by  Shamik Banerjee

Constraining gravity using entanglement in
AdS/CFT
Shamik Banerjee∗, Arpan Bhattacharyya, Apratim Kaviraj, Kallol Sen
& Aninda Sinha
Centre for High Energy Physics, Indian Institute of Science,
4 C.V. Raman Avenue, Bangalore 560012, India.
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2

l
u
J

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Abstract

]
h We investigate constraints imposed by entanglement on gravity in the context of holography.
t First, by demanding that relative entropy is positive and using the Ryu-Takayanagi entropy func-
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p tional, we find certain constraints at a nonlinear level for the dual gravity. Second, by considering
e
h Gauss-Bonnet gravity, we show that for a class of small perturbations around the vacuum state,
[ the positivity of the two point function of the field theory stress tensor guarantees the positivity

4 of the relative entropy. Further, if we impose that the entangling surface closes off smoothly in the
v bulk interior, we find restrictions on the coupling constant in Gauss-Bonnet gravity. We also give
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an example of an anisotropic excited state in an unstable phase with broken conformal invariance
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0 which leads to a negative relative entropy.
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.
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0 Contents
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: 1 Introduction 2
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2 Relative entropy considerations 4
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3 Relative entropy in Gauss-Bonnet holography 11
3.1 Linear order calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2 Quadratic corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.3 Constant T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
µν
3.4 Shockwave background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.5 Correction from additional operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4 Relative entropy for an anisotropic plasma 20
5 Smoothness of entangling surface 24
∗On leave of absence from Kavli IPMU.
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6 Discussion 26
A Positivity of relative entropy 28
B Smoothness conditions arising from T = 0 29
tt
C Relative entropy for R2 theory in shockwave background 31
1 Introduction
In recent times, there has been a huge interest to see what quantum entanglement [1, 2, 3, 4, 5, 6, 7, 8,
9, 10] can teach us about gravity. Certain entanglement measures such as relative entropy [11], which
roughly speaking tells us how distinguishable two states are, need to be positive in a unitary theory.
The positivity of this quantity was studied in holographic field theories with two derivative gravity
duals in [12]–related work include [13]. In the context of quantum field theories with holographic dual
gravity descriptions, one can ask what this inequality translates into. Furthermore, the Ryu-Takayanagi
prescription [2] (and its extensions to more general gravity theories [14, 15, 16, 17, 18, 19] ) gives us a
way to compute how the entangling surface extends into the bulk. For the vacuum state in a conformal
field theory, one may expect that the surface closes off smoothly in the interior as can be checked by
explicit calculations in Einstein gravity. By demanding a smooth surface in the bulk, we can try to
see if a general theory of gravity gets constrained. These will be the main questions of interest in this
paper.
Let us begin by discussing relative entropy. Relative entropy between two states ρ and σ is defined
as
S(ρ|σ) = tr (ρlogρ)−tr (ρlogσ). (1)
As reviewed in appendix A, in quantum mechanics, this quantity is positive for a unitary theory. In
[12], relative entropy was discussed in the holographic context. The state σ was chosen to be the
reduced density matrix for a spherical entangling surface. In this case, σ ≡ e−H/tr e−H with H being
the modular hamiltonian. It can be easily shown that (see eg.[20, 12])
S(ρ|σ) = ∆H −∆S, (2)
where ∆H = (cid:104)H(cid:105) − (cid:104)H(cid:105) and ∆S = S(ρ) − S(σ) with S(ρ) = −tr ρlogρ being the von Neumann
1 0
entropy for ρ and is the entanglement entropy for a reduced density matrix ρ. Then the positivity of
S(ρ|σ) would require,
∆H ≥ ∆S. (3)
2
Now we can calculate the modular hamiltonian for the sphere [21], from the formula,
(cid:90) R2 −r2
H = 2π dd−1x T . (4)
00
2R
r<R
Here T is the d-dimensional field theory stress tensor and 00 is the time-time component. We know
µν
how to compute T in holography. The Ryu-Takayanagi prescription (and known generalizations)
µν
gives us a way to compute ∆S. Thus we can check if and how the inequality ∆H ≥ ∆S is satisfied. In
[12], many examples were considered and in each case it was shown that this inequality is respected in
Einstein gravity. If we consider a small excitation around the vacuum state then to linear order in the
perturbation ∆H = ∆S. This can be shown to be equivalent with the linearized Einstein equations
[22, 23]. This equality has been recently shown to hold for a general higher derivative theory of gravity
[24]. Itisthusveryinterestingtoaskwhatconstraintswegetatthenonlinearorder. Wewilladdressthis
question for the special case of a constant stress tensor for the case where the holographic entanglement
entropy is given by the Ryu-Takayanagi prescription—in other words, we will ask if even at non-linear
level we get Einstein gravity. We find that the constraints arising from relative entropy give us a larger
class of models than just Einstein gravity. However, we show that there exists matter stress tensor
for which the bulk null energy condition is violated everywhere except at the Einstein point. This in
turn implies that relative entropy can continue to be positive although the bulk null energy condition
is violated. In fact we can ask the question the other way round: are there examples where the relative
entropy is negative but the bulk null energy condition still holds? We will give an example where this
happens. Thus the connection between energy conditions and the positivity of relative entropy, which in
some sense is reminiscent of the connection between energy conditions and the laws of thermodynamics,
appears to be less direct than what one would have expected.
In order to get some intuition about what feature of gravity ensures the positivity of relative entropy,
we extend the calculations in [12] to higher derivative theories. In particular we focus on Gauss-Bonnet
gravity in 5 bulk dimensions [25, 26, 27] since in this context there is a derivation [16, 18] of the
corresponding entropy functional [28, 14, 15]. We find that in all examples that we consider, the
positivity of the two point function of the stress tensor guarantees that the relative entropy is positive.
In particular we show this for a constant field theory stress tensor as well as for a disturbance that is
far from the entangling surface.
The inequality for relative entropy can only be explicitly checked when the modular hamiltonian is
known. Unfortunately, currently this is not known for cases when the entangling region is a cylinder
or a slab. From explicit calculations in the context of Einstein gravity, it is known that the entangling
surfaces for sphere, cylinder and the slab in the bulk corresponding to the vacuum state close off
smoothly. Theentanglingsurfaceequationgetsmodifiedinthepresenceofhigherderivativecorrections.
The smoothness of the surface imposes constraints on the higher derivative coupling constants. For the
slab, this question was addressed in [29]. We will extend this calculation to the other two cases and find
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boundsontheGauss-Bonnetcouplingconstant. Wewillfindthatthissimplecriterionleadstosomewhat
weaker constraints on the coupling as compared to what arises from micro-causality considerations
[30, 31]. However, quite curiously, the bounds are in good agreement with the a/c bounds [31] for a
non-supersymmetric field theory.
This paper is organized as follows. In section 2, we consider constraints arising from the positivity of
relative entropy in a holographic set up where the entanglement entropy is given by the Ryu-Takayanagi
entropy functional. These constraints arise at a quadratic order in a perturbation with a constant field
theory stress tensor. In section 3, we turn to the study of relative entropy in Gauss-Bonnet holography.
In section 4, we investigate the relative entropy for an anisotropic plasma which breaks conformal
invariance. We find that the relative entropy in this case is negative and we suggest some possible
explanations for this. In section 5, we derive constraints on the Gauss-Bonnet coupling by demanding
thattheentanglingsurfaceextendingintothebulkclosesoffsmoothly. Weconcludewithopenproblems
in section 6. The appendices contain further calculations relevant for the rest of the paper. We will use
capital latin letters to indicate bulk indices and greek letters to indicate boundary indices. Lower case
latin letters will indicate an index pertaining to the co-dimension 2 entangling surface.
Note added: The paper by Erdmenger et al [50] which appeared on the same day in the arXiv, deals
with a related idea of looking for pathological surfaces in certain higher derivative theories of gravity.
2 Relative entropy considerations
In this section we will use the results in [12] to derive certain constraints at nonlinear order that arise
due to the positivity of relative entropy. In Fefferman-Graham coordinates, the bulk metric can be
written as
L2
ds2 = dz2 +g dxµdxν . (5)
z2 µν
ForEinsteingravity,thebulkequationsofmotionallowustosystematicallysolveforg asanexpansion
µν
around the boundary z = 0 (see eg.[32]). The idea here is to see what mileage we get if we do not know
what the bulk theory is but we demand that the relative entropy calculated using the Ryu-Takayanagi
entropy functional is positive. We want to calculate the quadratic correction to the entanglement
entropy for the following form of boundary metric,
L2
(cid:2) (cid:3)
g = η +azdT +a2z2d(n T Tα +n η T Tαβ)+··· , (6)
µν z2 µν µν 1 µα ν 2 µν αβ
where a = 2 (cid:96)dP−1. This form is consistent with Lorentz invariance for a constant T . We will treat T
dLˆd−1 µν µν
as a small perturbation to the vacuum. At linearized order, it has been shown in [22, 23, 24] Einstein
equations arise from the condition ∆H = ∆S. We wish to investigate what happens at the next order.
We will keep n and n arbitrary and derive constraints on them arising from the inequality ∆H ≥ ∆S.
1 2
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Our analysis follows [12] very closely, the only change being that we will not specify n and n to be the
1 2
Einstein values. Since at linear order (the argument will be reviewed in the next section) we have the
equality ∆H = ∆S and since T from the holographic calculation is just given by the coefficient of the
00
zd term in the metric, the inequality implies ∆S ≤ 0 at quadratic order. Thus our task is to calculate
∆(2)S, the quadratic correction to ∆S, as a function of n ,n . The analysis below is valid for d > 2.
1 2
We start with the Ryu-Takayanagi prescription for calculating entanglement entropy in holography,
2π (cid:90) √
S = dd−1x h. (7)
(cid:96)d−1
P
√
From Taylor expansion one can show that the quadratic correction to h is,
√ 1√ 1√ 1√
δ(2) h = h(hijδh )2 + h δhijδh + h hijδ(2)h . (8)
ij ij ij
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The induced metric is,
L2
h = g + ∂ z∂ z. (9)
ij ij z2 i j
√
This is evaluated at the extremal surface z = z + (cid:15)z = R2 −r2 + (cid:15)z . Hence, at 0-th order, the
0 1 1
metric and its inverse are,
L2 (cid:18) x x (cid:19) z2 (cid:18) xixj(cid:19)
h = η + i j and hij = 0 ηij − . (10)
ij z2 ij z2 L2 R2
0 0
In ∆(2)S, we get 3 kinds of second order contributions. To be systematic, we write,
(cid:90) √
dd−1x δ(2) h = A +A +A , (11)
(2,0) (2,1) (2,2)
where schematically, these are the (δg)2, z δg and z2 contributions respectively. To calculate the first
1 1
term, we can set z = 0. Then
1
δ(2)h
δh = aL2zd−2T and ij = a2L2z2d−2(n T Tα +n η T Tαβ). (12)
ij ij 2 0 1 iα j 2 ij αβ
This gives,
(cid:90) (cid:18) (cid:18)n r2 (cid:19) (cid:18)n n r2(cid:19)
A = Ld−1a2 dd−1x Rzd T Ti0 1 +(d−1)n −n +(T )2 2(d−1)− 2
(2,0) 0 i0 2 2 2R2 00 2 2R2
(cid:18)n n n r2 1(cid:19) n (cid:18) 1 n (cid:19) 1 (cid:19)
+T Tij 1 + 2(d−1)− 2 − − 1 xixjT T0 +xixjT Tk − 1 + (cid:0)T2 −T2 −2TT (cid:1) ,
ij 2 2 2R2 4 2R2 i0 j ik j 2R2 2R2 8 x x
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where T = xixjTij and T = Ti. The last two terms in (11) are same as they appear in [12] . Quoting
x R2 i
the result,
(cid:90) R (cid:20) (cid:18) z2 (cid:19) T (cid:18) z2xixjxk∂ z (cid:19)(cid:21)
A = Ld−1a dd−1x T z − 0 xi∂ z + ij 2z2xi∂jz −z xixj − 0 k 1 , (13)
(2,1) 2z 1 R2 i 1 R2 0 1 1 R2
0
(cid:90) R (cid:20)d(d−1)z2 z2(∂z )2 z2(xi∂ z )2 (d−1)xi∂ z2(cid:21)
A = Ld−1 dd−1x 1 + 0 1 − 0 i 1 + i 1 . (14)
(2,2) zd 2z2 2R2 2R4 2R2
0 0
We can find z by minimizing A +A , which gives,
1 (2,1) (2,2)
aR2zd−1
z = − 0 (T +T ). (15)
1 x
2(d+1)
Plugging this and summing we get from eq.(11),
(cid:32) (cid:33)
(cid:90) √ (cid:90) xixjT Tk xixjT T0
dd−1xδ(2) h = Ld−1a2 dd−1x c T2 +c T2 +c T2 +c T Ti0 +c ik j +c i0 j +c TT ,
1 2 x 3 ij 4 i0 5 R2 6 R2 7 x
(16)
where unlike [12]1, the coefficients c ···c are dependent on n and n ,
1 7 1 2
(R2 −r2)(d−4)/2
(cid:0)
c = −4(1+d)2n (r2 −R2)2(r2 −(d−1)R2) (17)
1 8(1+d)2R 2
(cid:1)
+R2(2(d2 +2d−1)r4 +(1−5d2)r2R2 +(2d2 −d−1)R4) , (18)
(−r2 +R2)21(−4+d)((1−5d2)r2R3 +(−3+d(3+4d))R5)
c = , (19)
2 8(1+d)2
(−r2 +R2)d/2(−2n r2 +(−1+2n +2(−1+d)n )R2)
2 1 2
c = , (20)
3
4R
(−r2 +R2)d/2(n R2 −2n (r2 −(−1+d)R2))
1 2
c = , (21)
4
2R
(d2 −(1+d)2n )R(−r2 +R2)d/2
1
c = , (22)
5 2(1+d)2
n
c = − 1R(cid:0)−r2 +R2(cid:1)d/2 , (23)
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2
(−1+d)R3(−r2 +R2)21(−4+d)((1−3d)r2 +(1+2d)R2)
c = . (24)
7 4(1+d)2
Now we integrate the expression (16) over the (d−2)-sphere on the boundary. We use the trick,
(cid:90) (cid:90)
dd−1x f(r)xixjxkxl···n pairs = N(δ δ ···+permutations) dd−1x f(r)r2n, (25)
ij kl
1There appears to be an overall sign missing for c in [12].
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where N is some normalization constant. For n = 1, N = 1/(d−1); and for n = 2, N = 1/((d−1)2 +
2(d−1)). The final result comes out in the form 2,
(cid:90) √
(cid:0) (cid:1)
dd−1x h = a2Ld−1Ω C T2 +C T2 +C T2 , (26)
d−2 1 2 ij 3 i0
with √
2−3−dd(1+4(d2 −1)n ) πR2dΓ[d+1]
2
C = , (27)
1 (d2 −1)Γ(cid:2)3 +d(cid:3)
2
√
2−3−dd πR2dΓ[1+d]
(cid:0) (cid:0) (cid:1) (cid:1)
C = −1−2d+4(d+1)n +4 d2 −1 n , (28)
2 (d2 −1)Γ(cid:2)3 +d(cid:3) 1 2
2
√
2−1−dd(n +2(d−1)n ) πR2dΓ[1+d]
1 2
C = − . (29)
3 (d−1)Γ(cid:2)3 +d(cid:3)
2
Now we must demand that ∆(2)S ≤ 0. We can write ∆(2)S = VTMV with V being a (d−1)(d+2)/2
dimensional vector with the independent components of T as its components. Then demanding that
µν
the eigenvalues of M are ≤ 0 will ensure ∆(2)S ≤ 0. This leads to
n +2(d−1)n ≥ 0, (30)
1 2
2d+1−4(d+1)n −4(d2 −1)n ≥ 0, (31)
1 2
d+2−4(d+1)n −4d(d2 −1)n ≥ 0. (32)
1 2
We get the region indicated in fig.1 allowed by this set of inequalities. One interesting observation is
that when d → ∞, then the allowed region becomes the interval 0 ≤ n ≤ 1 with n = 0 coinciding
1 2
with the Einstein result. The area of the triangle is given by
d2
Area = . (33)
d 8(d+1)2(d−2)
Notice that the (extrapolated) Area is infinity. This makes sense since in d = 2 we expect constraints
d=2
on only 2 eigenvalues (since T2 and T2 are no longer independent) which will give us an unbounded
ij
region. Further Area → 0 which leads to a line interval for d → ∞ as shown in fig.1.
d→∞
At this stage, we have a wider class of theories that are allowed by the inequality than the Einstein
theory. The other theories need extra matter in addition to Einstein gravity to support them. As such
we could ask if the matter needed satisfies the null energy condition.
2The expression for C in [12] after substituting for n ,n is off by a factor of d/(d+2) although the overall sign is
3 1 2
correct. This appears to be related to the opposite sign used for c . We have cross-checked our results on mathematica
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for various cases and the notebook may be made available on request.
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Figure 1: (colour online) For d > 2 we get the allowed n ,n region to be the blue triangle above for a
1 2
generic stress tensor. The region above the blue solid line and below the blue dashed and dotted lines
are allowed from the relative entropy positivity. For d → ∞ the region collapses to a line 0 ≤ n ≤ 1
1
indicated in green. The Einstein value (n ,n ) = (1,− 1 ) is shown by the black dot. The region
1 2 2 8(d−1)
below the solid red line and above the dashed and dotted red lines are allowed by the null energy
condition. By turning on a generic component of the stress tensor only the Einstein value is picked out.
By switching off certain components of the stress tensor, various bands bounded by the solid, dashed
and dotted lines are picked out.
As an example consider turning on a constant T in d = 4. Then we find
01
1 12
R − g (R+ ) = Tbulk, (34)
AB 2 AB L2 AB
with Tbulk working to be
AB
(cid:34) (cid:35)
3 (cid:88)
Tbulk = 16z6T2 (δn +4δn )δzδz +(δn +6δn )δ0δ0 −(δn +6δn )δ1δ1 −2(δn +3δn ) δi δi .
AB 01 2 1 2 A B 1 2 A B 1 2 A B 1 2 A B
i=2,3
(35)
Hereδn = n −1/2andδn = n +1/24. UsingthiswefindthatthenullenergyconditionTbulkζAζB ≥ 0
1 1 2 2 AB
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leads to
Tbulk +Tbulk = −δn ≥ 0, (36)
00 22 1
5
Tbulk +Tbulk = δn +12δn ≥ 0, (37)
00 zz 2 1 2
with Tbulk + Tbulk = 0. These simplify to n ≤ 1/2 and n ≥ −1/24. Thus the region in fig.1 that
00 11 1 2
respects the null energy condition is smaller than that allowed by the positivity of relative entropy.
For a general constant stress tensor in general d we proceed as follows. We note that for a metric of
the form in eq.(5), with g a function of z only, we have [33]
µν
R = R(cid:48) −(z∂ K +KK −2K Kκ), (38)
µν µν z µν µν µκ ν
R = 0, (39)
µz
z2R = −gµνz∂ K +KµνK , (40)
zz z µν µν
R = R(cid:48) −(2zgµν∂ K +K2 −3KµνK ), (41)
z µν µν
where K = 1z∂ g . Here (cid:48) denotes a quantity computed with g . Using these it is straightforward
µν 2 z µν µν
(but tedious) to compute (setting L = 1 for convenience, defining S = n T Tα +n η T Tαβ and
µν 1 µα ν 2 µν αβ
aborbing the factors of a into T ; the raising and lowering of indices on T ,S are done with η .
µν µν µν µν
Also we have used Tµ = 0.)
µ
gµν = z2[ηµν −Tµνzd +(TµαTν −Sµν)z2d], (42)
α
1
K = − (2η −(d−2)zdT −2(d−1)z2dS ), (43)
µν 2z2 µν µν µν
1
Kν = − [2δν −dzdTν +dz2d(T Tαν −2Sν)], (44)
µ 2 µ µ µα µ
1
K = − [2d+dz2d(T Tαβ −2Sα)], (45)
2 αβ α
z2
Kµν = − [2ηµν −(d+2)zdTµν +2(d+1)z2d(TµTνα −Sµν)], (46)
2 α
1
zgµν∂ K = [4d+z2d(4d(d−2)Sα −d(d−4)T Tαβ)], (47)
z µν 2 α αβ
z2d
K Kµν = d+ [d(d+4)T Tαβ −8dSα], (48)
µν 4 αβ α
1
K Kκ = [4η −4(d−1)zdT +z2d(d2TκT −4(2d−1)S )]. (49)
µκ ν 4z2 µν µν µ κν µν
Using these we find
Tbulk = −d(d−1)z2d−2T Tαβ(δn +dδn ), (50)
zz αβ 1 2
(cid:2) (cid:3)
Tbulk = d2z2d−2 −δn T Tκ +η T Tαβ(δn +(d−1)δn ) . (51)
µν 1 µκ ν µν αβ 1 2
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Here δn = n − 1/2 and δn = n + 1/(8(d − 1)) i.e., the deviations from the Einstein values. Now
1 1 2 2
we are in a position to ask if the matter supporting this bulk stress tensor satisfies the null energy
conditions or not. First we note that Tbulk +Tbulk ≥ 0 immediately leads to
00 11
−d2(−T2 +T2)δn ≥ 0. (52)
00 ij 1
This leads to a definite sign for δn if and only if (−T2 +T2) has a definite sign. But in general, there
1 00 ij
is no reason for this combination to have a definite sign. So we are led to suspect that for a generic
stress tensor, δn = 0. To confirm this suspicion let us look at Tbulk +Tbulk.
1 zz 00
Tbulk +Tbulk =
zz 00
(cid:2) (cid:3)
−d (d−1)T2 (δn +2dδn )+T2[(2d−1)δn +2d(d−1)δn ]+T2[(2−3d)δn −4d(d−1)δn ] .
00 1 2 ij 1 2 0i 1 2
(53)
As in the relative entropy analysis, we write the RHS as VTMV where V is a (d−1)(d+2)/2 dimen-
sional vector whose non-zero independent components are the T ,T ,T ’s. Then we demand that the
00 ij 0i
eigenvalues of M are positive for the null energy condition to hold for a generic constant traceless stress
tensor T . This yields
µν
(3d−2)δn +4d(d−1)δn ≥ 0, (54)
1 2
(2d−1)δn +2d(d−1)δn ≤ 0, (55)
1 2
δn +2(d−1)δn ≤ 0. (56)
1 2
Only for δn = δn = 0 are these inequalities satisfied for d > 2. Thus the null energy condition picks
1 2
out the Einstein value if we ask if for a generic constant stress tensor the O(T2) terms are supported
by matter. Of course as we saw for d = 4 we can turn on T and set everything else to zero, there
0i
would be a region in the n ,n parameter space where the null energy condition and the positivity of
1 2
the relative entropy would hold (this corresponds to the region between the red and blue solid lines
in fig.1). For the generic case, only the Einstein value is picked out. To emphasise, that the Einstein
value was picked out for the generic case, relied only on the null energy condition analysis and did not
rely on the positivity of the relative entropy. To summarize, we found that there exists a larger class of
theories in the (n ,n ) parameter space than just the Einstein theory. However, except at the Einstein
1 2
point, we found that there always exists some matter stress tensor which violates the bulk null energy
condition.
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