对数黑洞熵修正与全息Rényi熵的双曲空间映射

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本文主要探讨了在具有重力对偶的CFT（Conformal Field Theory）背景下，对球形缠结表面的纠缠和Rényi熵的计算方法。作者Subhash Mahapatra从理论上提出了一个新颖的视角，即首先将熵从CFT中的纠缠和Rényi熵映射到双曲空间上的热熵，这是一种基于AdS/CFT（Anti-de Sitter/Conformal Field Theory）对应关系的策略。这种对应关系通常用于理解量子引力理论中的黑洞熵。传统的Wald熵是描述拓扑黑洞的熵量，但作者在此基础上引入了对Wald熵的修正。他们采用了一种基于地平线对称性和渐近Cardy公式的计算方法，发现这些修正值与黑洞事件视界的面积对数成正比。这意味着，随着纠缠表面面积的增加，Wald熵的修正表现出一种对数增长，这在理论物理学中具有重要意义。通过对黑洞熵的校正表达式进行分析，作者进而推导出了Rényi熵的校正形式。他们具体应用到了爱因斯坦和高斯-贝内特引力对偶的模型中，验证了即使考虑这些校正，熵的相关性质如熵c函数依然保持不变，且满足Rényi熵的不等式。这项研究不仅深化了我们对CFT和引力理论之间关系的理解，而且为纠缠熵和黑洞信息悖论提供了新的洞察。通过对数校正的引入，研究者们揭示了一个可能的量子效应，即在黑洞熵的计算中，对数因素可能反映了量子力学的深层次特性。这篇论文在量子引力、信息理论以及CFT等领域都有着重要的理论贡献。

23 Page 4 of 17 Eur. Phys. J. C (2018) 78 :23

Consider a d-dimensional CFT on Minkowski space and

choose a spherical entangling surface of radius R as a subsys-

tem. The computation of entanglement entropy of this sub-

system with the rest of the system can be performed by calcu-

lating the reduced density matrix ρ

. However, the authors in

[8], using the conformal structure of the theory, mapped this

problem of the entanglement entropy to the thermal entropy

on a hyperbolic space R×H

d−1

. They showed that the causal

development of the ball enclosed by the spherical entangling

surface can be mapped to a hyperbolic space R ×H

d−1

; with

curvature of H

d−1

space given by the radius of the spherical

entangling region R. An important point of this mapping was

that the vacuum of the original CFT mapped to a thermal bath

with temperature

2π R

(5)

on the hyperbolic space. Now relating the density matrix

therm

in the new spacetime R × H

d−1

to the old spacetime

by a unitary transformation, ρ

= U

−1

therm

U , we get

= U

−1

[−H / T

]

Z(T

)

U, (6)

where Z(T

) = Tr[e

−H / T

]. For the Rényi entropy we also

need the qth power of ρ

. From the above equation, we get

= U

−1

[−qH/T

]

Z(T

)

U. (7)

Taking the trace of both sides of Eq. (7), we get

Tr[ρ

Z(T

/q)

Z(T

)

, (8)

as U and its inverse cancel each other upon taking trace. Now,

using the deﬁnition of Rényi entropy as in Eq. (2), we arrive

1 − q



ln Z(T

/q) − q ln Z(T

)



. (9)

The above expression for the Rényi entropy can also be writ-

ten in terms of the free energy F (T ) =−T ln Z (T ):

1 − q



F(T

) − F(T

/q)



, (10)

and further, using the thermodynamic relation S

therm

−∂ F/∂ T , we can rewrite the above expression as

q − 1



dTS

therm

(T ); (11)

here, just to clarify again, S

therm

is the thermal entropy of a

d-dimensional CFT on R × H

d−1

, while S

is the desired

Rényi entropy. Equation (11) was the main result of [8,10],

which relates Rényi (and hence entanglement) entropy of

a spherical entangling region in d-dimensional CFT to the

thermal entropy on a hyperbolic space. As pointed out in

[8], the above analysis just mapped one difﬁcult problem to

another equally difﬁcult problem and is not particularly use-

ful for practical purposes. However, its true usefulness can be

realized via the AdS/CFT correspondence. In the AdS/CFT

correspondence, the thermal state of the boundary CFT cor-

responds to an appropriate non-extremal black hole in the

bulk AdS spacetime, with thermal entropy corresponding

to black hole entropy. Therefore, using the AdS/CFT cor-

respondence, we can relate S

therm

appearing in Eq. (11)to

that of the black hole entropy, which is relatively easy to com-

pute. Since on the boundary side our CFT is on R × H

d−1

its dual gravity theory will be described by a topological

black hole with hyperbolic event horizon. In any event, in

this AdS/CFT approach, the Rényi entropy is now given by

the horizon entropy of the corresponding hyperbolic black

hole, which can easily be computed using Ward’s standard

formula. However, since we are interested in calculating the

effects of the corrections of the entropy of the black hole on

the Rényi entropy, here we will use an approach based on

symmetry arguments on the horizon to calculate the black

hole entropy, instead of Wald’s formula. This is the topic of

the discussion of the next section.

3 Logarithmic corrections to the entropy of the black

hole from the Cardy formula

In this section, we will describe the necessary steps to calcu-

late the asymptotic form of density of states from a two-

dimensional conformal algebra.

This form of density of

states will be used in a later section to calculate the black hole

entropy and, further, to compute logarithmic corrections. In

this section, we will mostly follow the notations used in [35]

and refer the reader to [35] for a detailed discussion.

We start with a standard Virasoro algebra of the two con-

formal ﬁeld theory with central charges c, ¯c:



, L



= (m − n)L

m+n

m(m

− 1)δ

m+n,0





= (m − n)

m+n

¯c

m(m

− 1)δ

m+n,0





= 0; (12)

here L

and

are the generators of holomorphic and

antiholomorphic diffeomorphisms. If ρ(,

) denotes the

degeneracy of states carrying L

=  and

 eigen-

values, then one can deﬁne the partition function on the two-

torus of modulus τ = τ

+iτ

See [42], for generalization of the Cardy formula in higher-

dimensional CFT.

Extension of the Cardy formula beyond the subleading order was

performed in [43].

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