arXiv:gr-qc/9907090v1  27 Jul 1999
Entanglement/Brick-wall entropies correspondence
Shinji Mukohyama
Department of Physics and Astronomy, University of Victoria, Victoria, BC, Canada V8W 3P6
Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan
Abstract.
There have been many attempts to understand the statistical origin of black-hole entropy.
Among them, entanglement entropy and the brick wall model are strong candidates. In this paper we show
a relation between entanglement entropy and the brick wall model: the brick wall model seeks the maximal
value of the entanglement entropy. In other words, the entanglement approach reduces to the brick wall
model when we seek the maximal entanglement entropy .
I
INTRODUCTION
Black hole entropy is given by a mysterious formula called the Bekenstein-Hawking formula [1,2]:
SBH = A
4l2
pl
,
(1)
where A is area of the horizon. There have been many attempts to understand the statistical origin of the
black-hole entropy.
Entanglement entropy [3,4] is one of the strongest candidates of the origin of black hole entropy.
It is
originated from a direct-sum structure of a Hilbert space of a quantum system: for an element |ψ⟩of the
Hilbert space F of the form
F = FI ¯⊗FII,
(2)
the entanglement entropy Sent is deﬁned by
Sent = −TrI[ρI ln ρI],
ρI = TrII|ψ⟩⟨ψ|.
(3)
Here ¯⊗denotes a tensor product followed by a suitable completion and TrI,II denotes a partial trace over
FI,II, respectively.
On the other hand, there is another strong candidate for the origin of black hole entropy: the brick wall model
introduced by ’tHooft [5]. In this model, thermal atmosphere in equilibrium with a black hole is considered.
In this situation, we encounter with two kinds of divergences in physical quantities. The ﬁrst is due to inﬁnite
volume of the system and the second is due to inﬁnite blue shift near the horizon. We are not interested in the
ﬁrst since it represents contribution from matter in the far distance. Hence we introduce an outer boundary
in order to make our system ﬁnite. It is the second divergence that we would like to associate with black hole
entropy. Namely, it can be shown by introducing a Planck scale cutoﬀthat entropy of the thermal atmosphere
near the horizon is proportional to the area of the horizon in Planck units.
In this paper, we show that the brick wall model seeks the maximal value of the entanglement entropy.
II
MODEL DESCRIPTION
For simplicity, we consider a minimally coupled, real scalar ﬁeld described by the action

 
I
II
1
1
r = r
r = r
t = const.
FIGURE 1. The Kruskal-like extension of the static, spherically symmetric black-hole spacetime. We consider only
the regions satisfying r > r1 (the shaded regions I and II ).
S = −1
2
Z
d4x√−g

gµν∂µφ∂νφ + m2
φφ2
,
(4)
in the spherically symmetric, static black-hole spacetime
ds2 = −f(r)dt2 + dr2
f(r) + r2dΩ2.
(5)
We denote the area radius of the horizon by r0 and the surface gravity by κ0 (̸= 0):
f(r0) = 0,
κ0 = 1
2f ′(r0).
(6)
We quantize the system of the scalar ﬁeld with respect to the Killing time t in a Kruskal-like extension of the
black hole spacetime. The corresponding ground state is called the Boulware state and its energy density is
known to diverge near the horizon. Although we shall only consider states with bounded energy density, it
is convenient to express these states as excited states above the Boulware ground state for technical reasons.
Hence, we would like to introduce an ultraviolet cutoﬀα with dimension of length to control the divergence.
The cutoﬀparameter α is implemented so that we only consider two regions satisfying r > r1 (shaded regions
I and II in Figure 1), where r1 (> r0) is determined by
α =
Z r1
r0
dr
p
f(r)
.
(7)
[Evidently, the limit α →0 corresponds to the limit r1 →r0. Thus, in this limit, the whole region in which
∂/∂t is timelike is considered.] Strictly speaking, we also have to introduce outer boundaries, say at r = L
(≫r0), to control the inﬁnite volume of the constant-t surface. However, even if there are outer boundaries,
the following arguments still hold.
In this situation, there is a natural choice for division of the system of the scalar ﬁeld: let HI be the space
of mode functions with supports in the region I and HII be the space of mode functions with supports in the
region II. Thence, the space F of all states are of the form (2), where FI and FII are deﬁned as symmetric
Fock spaces constructed from HI and HII, respectively:
FI ≡C ⊕HI ⊕(HI ¯⊗HI)sym ⊕· · · ,
FII ≡C ⊕HII ⊕(HII ¯⊗HII)sym ⊕· · · .
(8)
Here (· · ·)sym denotes the symmetrization.
III
SMALL BACKREACTION CONDITION
Let us investigate what kind of condition should be imposed for our arguments to be self-consistent. A clear
condition is that the backreaction of the scalar ﬁeld to the background geometry should be ﬁnite. For the brick

wall model this condition is satisﬁed. Namely, in Ref. [6], it was shown that the total mass of the thermal
atmosphere of quantum ﬁelds is actually bounded. Thus, also for our system, we would like to impose the
condition that the contribution ∆M of the subsystem FI to the mass of the background geometry should be
bounded in the limit α →0.
It is easily shown that ∆M is given by
∆M ≡−
Z
x∈I
T t
t 4πr2dr = HI,
(9)
where HI is the Hamiltonian of the subsystem FI with respect to the Killing time t. Hence, the expectation
value of ∆M with respect to a state |ψ⟩of the scalar ﬁeld is decomposed into the contribution of excitations
and the contribution from the zero-point energy:
⟨ψ|∆M|ψ⟩= Eent + ∆MB,
(10)
where Eent is entanglement energy deﬁned by
Eent ≡⟨ψ| : HI : |ψ⟩,
(11)
and ∆MB is the zero-point energy of the Boulware state. Here, the colons denote the usual normal ordering.
[This deﬁnition of entanglement energy corresponds to E(I′)
ent in Ref. [7] and ⟨: H2 :⟩in Ref. [8].]
Since the Boulware energy ∆MB diverges as ∆MB ∼−ATHα−2 in the limit α →0 [6], we should impose
the condition
Eent ≃|∆MB|,
(12)
where A = 4πr2
0 is the area of the horizon, TH = κ0/2π is the Hawking temperature. We would like to call this
condition the small backreaction condition (SBC). Note that the right hand side of SBC (12) is independent of
the state |ψ⟩.
IV
MAXIMAL ENTANGLEMENT ENTROPY
Now, we shall show that the Hartle-Hawking state is a maximum of the entanglement entropy in the space
of quantum states satisfying SBC. For this purpose, we prove a more general statement for a quantum system
with a state-space of the form (2): a state of the form
|ψ⟩= N
X
n
e−En/2T |n⟩I ⊗|n⟩II
(13)
is a maximum of the entanglement entropy in the space of states with ﬁxed expectation value of the operator
EI deﬁned by
EI =
 X
n
En|n⟩I · I⟨n|
!
⊗
 X
m
|m⟩II · II⟨m|
!
,
(14)
provided that the real constant T is determined so that the expectation value of EI is actually the ﬁxed value.
Here, {|n⟩I} and {|n⟩II} (n = 1, 2, · · ·) are bases of the subspaces FI and FII, respectively, and En are
assumed to be real and non-negative. Note that this statement is almost the same as the following statement
in statistical mechanics: a canonical state is a maximum of statistical entropy in the space of states with ﬁxed
energy, provided that the temperature of the canonical state is determined so that the energy is actually the
ﬁxed value.
Note that the expectation value of EI is equal to the entanglement energy (11), providing that |n⟩I and En
are an eigenstate and an eigenvalue of the normal-ordered Hamiltonian : HI : of the subsystem FI. Hence,
for the system of the scalar ﬁeld, the above general statement insists that the state (13) is a maximum of
the entanglement entropy in the space of states satisfying SBC, which corresponds to ﬁxing the entanglement
entropy. Oﬀcourse, in this case, the constant T should be determined so that SBC (12) is satisﬁed.

Returning to the subject, let us prove the general statement. (The following proof is the almost same as
that given in the Appendix of Ref. [9] for a slightly diﬀerent statement. However, for completeness, we shall
give the proof. )
First, we decompose an element |ψ⟩of F as
|ψ⟩=
X
n,m
Cnm|n⟩I ⊗|m⟩II,
(15)
where the coeﬃcients Cnm (n, m = 1, 2, · · ·) are complex numbers satisfying P
n,m |Cnm|2 = 1 and can be
considered as matrix elements of a matrix C.
Since C†C is a non-negative Hermitian matrix, it can be
diagonalized as
C†C = V †PV,
(16)
where P is a diagonal matrix with diagonal elements pn (≥0) and V is a unitary matrix. For this decomposition
and diagonalization, the entanglement entropy and the expectation value of the operator EI are written as
follows.
Sent = −
X
n
pn ln pn,
(17)
Eent =
X
n,m
Enpm|Vnm|2,
(18)
where Vnm is matrix elements of V . The constraints P
n,m |Cnm|2 = 1 and V †V = 1 are equivalent to
X
n
pn = 1,
X
l
V ∗
lnVlm = δnm.
(19)
Next, we shall show that these expressions are equivalent to those appearing in statistical mechanics in FI.
Let us consider a density operator ¯ρ on FI:
¯ρ =
X
n,m
˜Pnm|n⟩I · I⟨m|,
(20)
where ( ˜Pnm) is a non-negative Hermitian matrix with unit trace. By diagonalizing the matrix ˜P as
˜P = ¯V † ¯P ¯V ,
(21)
we obtain the following expressions for entropy S and an expectation value E of the operator ¯EI ≡P
n En|n⟩I ·
I⟨n|.
S = −
X
n
¯pn ln ¯pn,
E =
X
n
X
n,m
En¯pm|Vnm|2,
(22)
where ¯pn is the diagonal elements of ¯P. The constraints Tr¯ρ = 1 and ¯V † ¯V = 1 are restated as
X
n
¯pn = 1,
X
l
¯V ∗
ln ¯Vlm = δnm.
(23)
From these and those expressions, the following correspondence is easily seen:

Sent ↔S,
Eent ↔E,
C†C ↔˜P.
(24)
Hence, a maximum of S in the space of statistical states with a ﬁxed value of E gives a set of maxima of Sent
in the space of quantum states with a ﬁxed value of Eent. (All of them are related by unitary transformations
in the subspace FII.) Thus, since the thermal state ˜Pnm = e−En/T δnm is a maximum of S in the space of
statistical states with a ﬁxed value of E, Cnm = e−En/2T δnm is a maximum of Sent in the space of quantum
states with a ﬁxed value of Eent. Here the temperature (or the constant) T should be determined so that E
(or Eent) has the ﬁxed value. This completes the proof of the general statement.
Therefore, for the system of the scalar ﬁeld, a state of the form (13) is a maximum of the entanglement
entropy in the space of quantum states satisfying SBC, provided that the constant T is determined so that
SBC is satisﬁed. The value of T is easily determined as T = TH by using the well-known fact that the negative
divergence in the Boulware energy density can be canceled by thermal excitations if and only if temperature
with respect to the time t is equal to the Hawking temperature.
Finally, we obtain the statement that the Hartle-Hawking state [10] is a maximum of entanglement entropy
in the space of quantum states satisfying SBC since the Hartle-Hawking state is actually of the form (13) with
T = TH [11]. [Strictly speaking, in order to obtain the Hartle-Hawking state, we have to take the limit α →0
(and L →∞). However, the following arguments still hold for a ﬁnite value of α (and L).] The corresponding
reduced density matrix is the thermal state with temperature equal to the Hawking temperature. Therefore,
the maximal entanglement entropy is equal to the thermal entropy with the Hawking temperature, which is
sought in the brick wall model.
V
CONCLUSION
In summary the brick wall model seeks the maximal value of entanglement entropy. In other words, the
entanglement approach reduces to the brick wall model when we seek the maximal entanglement entropy .
Our arguments suggests strong connection among three kinds of thermodynamics: black hole thermodynam-
ics, statistical mechanics, and entanglement thermodynamics [9,7,8,12]. It will be interesting to investigate
close relations among them in detail.
ACKNOWLEDGMENTS
The author would like to thank Professors W. Israel and H. Kodama for their continuing encouragement.
This work was supported partially by the Grant-in-Aid for Scientiﬁc Research Fund (No. 9809228).
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