全息关联：Fisher信息与量子纠缠的边界与核心体积

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"将Fisher信息与全息中的整体纠缠联系起来" 这篇研究论文探讨了量子信息理论与AdS/CFT对应（Anti-de Sitter/Conformal Field Theory correspondence）之间的关系，特别是在混合态Fisher信息度量的全息对偶方面。Fisher信息度量是一种衡量概率分布差异的数学工具，在统计推断和信息理论中有广泛应用。在量子领域，它被用来度量两个量子态之间的差异，包括混合态，即非纯态的量子系统。 AdS/CFT对应是一个强大的理论工具，它建立了高维引力理论（例如AdS空间）与低维边界上的量子场论之间的等价性。通过这个对应，物理学家能够利用量子场论的方法来研究引力问题，反之亦然。论文中提出，Fisher信息度量的全息对应在边界场论中可以用来衡量混合量子态之间的距离。具体来说，当边界上有球形子区域时，这个距离与Ryu-Takayanagi（RT）表面所包围的特定规则体积有关。RT公式是AdS/CFT对应中的一个关键结果，它关联了边界上的纠缠熵与引力背景中的最小曲面面积。作者进一步研究了Fisher信息度量的量子修正与边界纠缠熵量子修正之间的关系。纠缠熵是量子系统中子系统间信息丢失的一种度量，它在量子信息处理和黑洞物理学中扮演着核心角色。如果Fisher信息的量子修正与纠缠熵的量子修正有关联，那么这将提供一个理解量子引力效应如何影响信息处理的新途径。论文还讨论了这种联系的潜在后果。这可能对理解和计算高维引力系统中的信息处理能力有重大意义，尤其是在黑洞物理和宇宙学的背景下。此外，它也可能有助于解决量子引力中的信息丢失问题，这是理论物理学中的一个长期挑战。这项工作深化了我们对AdS/CFT对应的理解，并为量子信息理论与引力理论的交叉提供了新的洞察。通过揭示Fisher信息与整体纠缠之间的联系，它为探索量子纠缠的全息描述和量子引力的性质开辟了新的研究方向。

JHEP08(2018)001

zeroth order, i.e. for AdS space itself. The ﬂuctuation considered is dual to the energy-

momentum tensor on the ﬁeld-theory side. The proposal (1.9) modiﬁes the pure-state

volume expression (1.6) in a natural way such as to obtain a ﬁnite expression. C

is a

dimensionless constant which cannot be ﬁxed from ﬁrst principles on the gravity side.

will determine C

by comparison with results for the relative entropy [15, 16]. For metric

and marginal perturbations, this coeﬃcient depends only on the spacetime dimension, while

for relevant scalar perturbations also the operator dimension enters.

We suggest that the holographic reduced ﬁdelity susceptibility is obtained by taking

the second order variation of F with respect to m,

R,mm

= ∂

F. (1.10)

The deﬁnition (1.10), along with the proposal (1.9), ensures that the holographic ﬁdelity

susceptibility is ﬁnite, as required for mixed states. G

R,mm

scales as G

R,mm

∝ R

, with

R the radius of the spherical entangling region in the dual ﬁeld theory in d spacetime

dimensions. This scaling behaviour is expected, as we discuss below. Moreover, the volume

diﬀerence in (1.9) entirely encodes the dependence on the shape of the entangling region.

The particular scaling behaviour G

R,mm

∝ R

also follows from the alternative deﬁ-

nition for Fisher information proposed in [11] in terms of the relative entropy ∆S, which

is a measure of entropic distance between two states. An example is the relative entropy

measuring this distance between a perturbed state and the ground state. The Fisher infor-

mation metric proposed in [11] is given by the second order variation of ∆S measuring the

entropic distance between the ground state of the boundary CFT and the state obtained

by perturbing this ground state by injecting energy. As holographically shown in [15] and

later conﬁrmed in [17, 18] by a direct ﬁeld theory computation, the relative entropy for this

perturbation scales precisely as R

for a spherical subregion of radius R, at quadratic order

in energy ﬂuctuations. In fact as discussed in [18], when taking into account a calculational

issue, both the expressions of relative entropy obtained holographically in [15] and from

the ﬁeld theory computation of [18] match exactly including the prefactor. Consequently,

the Fisher information metric also scales as R

. Now for the restricted class of states

introduced above, the reduced ﬁdelity susceptibility coincides with the Fisher information

metric. Therefore, for this class of states, the reduced ﬁdelity susceptibility G

R,mm

also

scales with R

Our proposal of identifying the expression (1.10) with the holographic dual of the

ﬁdelity susceptibility for mixed states, using the renormalized volume proposal (1.9), thus

provides a ﬁnite expression with the correct scaling behaviour R

. This expression encodes

all information about the shape of the entangling region.

As a further example, we also consider the ﬂuctuations caused by the insertion of

a scalar in AdS and compute the corresponding subtracted volume at the quadratic or-

der in the perturbation parameter. We obtain a scaling behaviour of the corresponding

contribution to the Fisher information metric of the form R

2∆

, where ∆ is the scaling

dimension of the operator dual to the scalar bulk AdS ﬁeld. This behaviour is again con-

sistent with the quadratic variation of the relative entropy for such ﬂuctuations [15], and

A similar undetermined coeﬃcient is also present in the pure state proposal of [12], as is seen from

equations (2), (9) and (18) in that paper.

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