量子纠错与全息Rényi熵：最大纠缠与新型网络解释

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本文探讨了量子错误校正中的一个关键概念——Rényi熵，特别是其在量子纠错码中的表现，以及它如何与AdS/CFT（阿贝尔-达斯/康斯坦特-弗雷-西蒙公式）中的宇宙布兰公式进行对比。Rényi熵Sn被定义为$$S_n = \frac{n-1}{n}\cdot n^2 \frac{\partial}{\partial n}S_n$$，这是一个衡量量子系统混合程度的重要量，在量子信息理论和量子引力中的全息原理中有广泛应用。作者 Chris Akers 和 Pratik Rath 分析了通用算符代数代码中的Rényi熵行为，发现了一个更为普遍的计算规则。他们发现，为了使AdS/CFT模型中的Rényi熵与特定的宇宙布兰公式相符，纠错码在区域算子的本征空间内的纠缠度必须达到最大。这揭示了一个新的理解，即区域算子的定义可以通过最大化纠缠度得到改进，从而加强了Ryu-Takayanagi面积公式（描述在引力理论中量子信息的边界效应）与晶格规范理论中边缘模式之间的关系。此外，文章提出了一种全新的视角看待现有的全息张量网络，即它们可以被解释为区域本征态，而非传统的光滑几何结构。这种解释解决了过去张量网络在模拟全息CFT的Rényi熵谱时遇到的难题，为构建出具有正确频谱特征的全息网络提供了一种方法。这种转变不仅提升了对量子信息在全息理论中作用的理解，也为量子计算和量子引力的理论框架带来了新的洞察。本文的研究不仅深化了量子错误校正与Rényi熵之间的理论联系，而且对全息理论、量子信息在引力背景下的应用以及张量网络技术的发展都产生了深远的影响。通过这些发现，科学家们能够更好地理解量子系统的信息性质，并可能推动相关领域的实际应用和技术进步。

JHEP05(2019)052

for some state eρ

(n)

in the code subspace. We show that this is true if and only if χ

is maximally mixed. Second, when interpreted in the context of the AdS/CFT code, eρ

(n)

needs to manifest as the state eρ

with an inserted cosmic brane of exactly the right tension.

One of our primary focuses is to demonstrate that CFT states with geometric duals indeed

admit such an interpretation, as long as χ

is maximally mixed. Formulating this argument

requires that we carefully interpret eq. (1.4) in gravity. For example, we must understand

that each α-block corresponds to a particular geometry so that we can interpret some

α-blocks as geometries with cosmic branes. We must also understand that CFT states

with geometric duals have non-vanishing support p

on α-blocks corresponding to many

diﬀerent classical geometries. This way, eρ

(n)

can have its support predominantly on a

diﬀerent classical geometry than eρ

does. We provide these interpretations in section 3,

and we explicitly show how they manifest as a cosmic brane prescription within quantum

error-correction in section 4.

Also in section 4, we emphasize the fact that a maximally-mixed χ

for each α implies

both properties needed for a code to match the cosmic brane prescription. This leads us

to conclude that χ

is maximally-mixed in AdS/CFT. This has a number of interesting

implications, such as an improved deﬁnition of the area operator

= ⊕

log dim(χ

¯a

. (1.13)

In section 5, we discuss the implications of these results for tensor network models of

AdS/CFT. While tensor networks tend to nicely satisfy the RT formula [14–16], historically

they have struggled to have a non-ﬂat spectrum of Renyi entropies. Our results suggest

that there is a natural way to construct a holographic tensor network that not only has

the correct Renyi entropy spectrum, but also computes the Renyi entropies via a method

qualitatively similar to the cosmic brane prescription.

Finally, in section 6 we conclude with a discussion of implications, future directions

and related work. Note that this paper was released jointly with [17] where similar ideas

are discussed.

2 Operator-algebra quantum error correction

We start by reviewing the framework of operator-algebra quantum error correction (OQEC)

as discussed in [9]. Consider a ﬁnite dimensional “physical” Hilbert space H = H

⊗ H

and a “logical” code subspace H

code

⊆ H.

In the context of holography, one can think of

H as the boundary Hilbert space and H

code

as the Hilbert space of the bulk eﬀective ﬁeld

theory (EFT).

Let L(H

code

) be the algebra of all operators acting on H

code

and M ⊆ L(H

code

) be a

subalgebra. In particular, we require that M be a von Neumann algebra, i.e. it is closed

under addition, multiplication, hermitian conjugation and contains all scalar multiples of

the identity operator.

More generally the physical Hilbert space need not factorize, e.g. if the boundary theory has gauge

constraints. We expect the qualitative features of our result to be unchanged in that case.

– 4 –

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