全息双工编码与亚AdS局域性

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"双向全息代码和次AdS位置 - JHEP01(2016)175 - Open Access" 这篇论文深入探讨了量子信息理论与弦理论中的一个关键概念，即全息原理（Holographic Principle）及其在AdS/CFT对应关系中的应用。AdS/CFT对应是一种强大的工具，它建立了反德西特空间（Anti-de Sitter space, AdS）的引力理论与边界上的共形场论（Conformal Field Theory, CFT）之间的等价性。全息原理提出，在一个高维空间的边界上可以完全描述该空间内部的所有信息。文章指出，利用实现量子纠错码的张量网络可以构建全息二元性的玩具模型。这些模型能够重现Ryu-Takayanagi公式，该公式是计算边界间隔熵的一种方法，揭示了边界理论与体理论之间的信息关联。此外，通过这些模型，可以将大量的体理论算子以冗余的方式映射到边界理论中，这有助于理解体-边界对应的关系。然而，早期的模型存在一些局限性，例如仅能对“代码”子空间中的特定边界状态提供全息解释，无法涵盖所有边界状态。为了解决这个问题，论文提出了新的张量网络模型，这些模型不仅包含之前的进展，还引入了更多全息对偶性的特性： 1. 对所有边界状态提供全息解释，不再局限于“代码”子空间。 2. 一组体态模拟“经典几何”，再现了边界间隔的Ryu-Takayanagi公式，使得几何结构的描述更为完整。 3. 类似于重力理论中的微分不变性，即体规对称性，这在理解和模拟引力效应时至关重要。 4. 能够产生足够稀疏的体局部激发，这反映了体理论中的局部性质，与真实物理系统更接近。 5. 允许在低于AdS分辨率甚至平坦空间的描述下理解几何，这意味着模型具有更广泛的适用性和灵活性。通过这些改进，新的张量网络模型为理解和构建全息对偶性提供了更加全面和精确的框架，进一步深化了我们对AdS/CFT对应的理解。这不仅有助于量子引力理论的研究，也可能对量子信息处理、量子计算等领域产生重要影响。

JHEP01(2016)175

of the perfect tensors used as building blocks by PYHP, but must satisfy additional, more

stringent conditions, hence their name. To be concrete, throughout this work we take

n = 2, although generalization to other n is straightforward. For n = 2 the tensor can

be written as T

αβγδ

, as drawn in ﬁgure 1. The four in-plane indices α, β, γ, δ, which run

from 1, 2, . . . , D, will be contracted to form the tensor network. The perpendicular index

I = 1, 2, . . . , D

will represent a bulk degree of freedom. A pluperfect tensor is required to

satisfy the following three conditions:

1. T is a unitary mapping from indices αβγδ to the bulk index I. As an equation, this

means that

αβγδ

J∗

αβγδ

= δ

. (2.1)

2. Among the D

values of I, there exists a subset I = 1, 2, . . . , D

, for which T

αβγδ

is a

four-leg perfect tensor when I is ﬁxed. In other words, T

αβγδ

for ﬁxed I = 1, 2, . . . , D

deﬁnes a unitary mapping from any two of the four indices αβγδ to the other two

(up to a normalization factor). As an equation, this reads

αβγδ

I∗

µνγδ

αµ

βν

, for I = 1, 2, . . . , D

(2.2)

and other equations obtained by permutation of the four in-plane indices.

3. For any one of the four in-plane indices, say α, T

αβγδ

for I = 1, 2, . . . , D

is a unitary

mapping from Iα to βγδ, again up to normalization. As an equation,

αβγδ

J∗

µβγδ

αµ

, for I, J = 1, 2, . . . , D

. (2.3)

We will refer to these as the pluperfection conditions.

Each condition will serve a speciﬁc role in the construction of the BHC. Condition 1

facilitates the deﬁnition of a unitary holographic mapping applicable to all states on the

boundary. Condition 2 deﬁnes the particular set of “classical geometry” states satisfying the

RT formula. Condition 3 guarantees that certain sets of operators acting on the classical

geometry states are mapped to the boundary isometrically. These statements will be

explained in detail as we proceed. Before doing that, however, we would like to give an

explicit example to demonstrate that pluperfect tensors do exist, and also discuss why it

is natural to consider them.

2.2 Explicit construction of a pluperfect tensor

To demonstrate that pluperfect tensors do exist, we present an explicit example. We start

from a particular [403]

stabilizer code , which is a four-leg tensor R

αβγδ

with each leg

having dimension D = 3 [19]. We can view this as a quantum state |Ψi = R

αβγδ

|αi|βi|γi|δi.

This state is deﬁned by the equation S

|ψi = |ψi, for i = 1, 2, 3, 4, with the stabilizer

– 4 –

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