一环阶全息代码的通用性与量子纠缠的奥秘

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全息代码的一环通用性研究最近取得重要进展，特别是在量子信息领域。论文《One-loop universality of holographic codes》揭示了全息纠错码在大规范群G中的O(1/G)级简单通用特性。这类代码中固定 Ryu-Takayanagi (RT) 区域的状态与一个平坦纠缠谱紧密相关，这种纠缠反映了适当子空间之间的最大纠缠度。作者们通过对主体的有效作用进行高阶校正控制，以及处理低于截止值的动态量子涨落，将这一发现扩展到了一环阶（即O(1)校正），从而进一步提升了对全息理论的理解。这一成果加深了对体积路径积分与量子编码之间关系的认识，它支持了以下几点关键见解： 1. **超越主导级次的简单张量网络模型**：这一工作证实了简单的张量网络模型在描述全息现象时，不仅限于G中的主导行为，而是展现出更深层次的统一性，与全息CFT的行为相匹配。 2. **体积与边界模态的关系**：论文指出，Jafferis、Lewkowycz、Maldacena和Suh提出的关于体积与边界模哈密顿量的关联不仅仅适用于代码子空间的期望值，而且在适当的模块化流概念下，这个关系保持不变。这意味着它们在量子编码的全局结构中扮演着核心角色。 3. **代码子空间的不变性**：在全息量子码的整体重归一化群流中，出现了有趣的消减现象，这可能暗示着代码子空间在特定的流变换下保持不变，这对于理解编码的稳定性和有效性至关重要。中间技术成果包括将RT熵的Lewkowycz-Maldacena计算形式化为边界条件作用下的Hamilton-Jacobi变分问题，以及对高阶导数修正的贡献，同时提供了一般化的形式来处理具有有限圆锥角的RT表面。这些数学工具和技术的提升，为全息理论与量子信息的交叉研究提供了坚实的基础。这篇论文在全息代码的理论框架内，探索了一环阶的通用性，加深了我们对全息理论物理性质的理解，特别是其与边界理论的相互作用以及量子编码的深层次结构。这些发现对于进一步发展全息理论和量子信息科学都具有重要意义。

JHEP03(2020)191

for some unitaries U

, U

on H

and H

, bases {|α, ii}, {|α, ji} of H

, H

, and some

set of states |χ

i ∈ H

⊗H

. Ref. [5] called such codes “operator algebra quantum error-

correcting codes with complementary recovery”, as (2.8) is equivalent to the requirement

that the action of any operator in M

on a state in H

can be reproduced by acting on

that state with an operator in R, and correspondingly for M

, R.

Since arbitrary unitaries on H

, H

can be absorbed into U

, U

, the

only independent structure in (2.8) comes from the coeﬃcients in the Schmidt decom-

position of |χ

, or equivalently the spectrum of eigenvalues of the density matrix

≡ Tr

|χ

ihχ

| (which is also the spectrum of χ

≡ Tr

|χ

ihχ

|). This spectrum

is thus the essence of any code, and it is this spectrum that was shown in [21] to be ﬂat at

O(1/G) in states of ﬁxed RT-area; see also [20].

Tracing (2.8) over R yields



⊗ χ



†

, (2.9)

where ρ

is the image of ρ

under the isomorphism between H

and H

. Using (2.6),

the von Neumann entropy of (2.9) immediately takes the form (2.1) with the identiﬁcation

S(χ

. (2.10)

As described in [20, 21] the entropies of the normalized density matrices ρ

/(Tr ρ

) take

a similar form, though they will satisfy (2.1) with the same identiﬁcation (2.10) if and

only if each χ

satisﬁes χ

∝ χ

; i.e., if each such density matrix is proportional to a

projection operator.

Note that eigenstates of the superselection parameter α are also eigenstates of L

. In

holography, it is an interesting question whether α is deﬁned completely by the eigenvalue of

or whether it contains additional information, but in either case let us simply consider

an eigenstate of α. In such a state, if we for the moment ignore information within a

distance 1/Λ (set by the cutoﬀ scale Λ) away from the bulk entangling surface γ

, the

remaining information about bulk quantum ﬂuctuations below the cutoﬀ Λ in a state |ψi

appears to be captured by the amplitudes hψ|α, iji and the details of the state factors

|α, ii

⊗ |α, ji

; indeed, these ingredients suﬃce to determine the correlation functions

of operators in W ∪ W . The state |χ

must thus be associated with bulk degrees of

freedom with energies above the cutoﬀ Λ. Together with the sources at the AdS boundary,

such high energy degrees of freedom determine a natural classical background on which

dynamical quantum ﬂuctuations propagate through the condition that the background be

a stationary point of the eﬀective action that arises from integrating them out.

It is thus

natural to conjecture that many properties of |χ

can be computed by using the

cutoﬀ-scale eﬀective action as a corresponding classical variational principle. Indeed, the

identiﬁcation (2.10) shows that its von Neumann entanglement entropy can be calculated

Note that the existence of a preferred classical background determined by a variational principle does

not necessarily imply that quantum ﬂuctuations around this background are small.

– 6 –

JHEP03(2020)191

using the classical Lewkowycz-Maldacena procedure. It is thus reasonable to expect this

to extend to Renyi entropies S

(χ

). Below, we refer to this idea as the classical eﬀective

action conjecture for the quantum code.

We give a deﬁnitive, though somewhat indirect, argument for this result in section 4

below. The rough sketch of the idea is to consider a state |ψi

in an appropriate code

subspace H

such that |ψi

is an eigenstate of L

with eigenvalue σ. We use this state to

build a new state

|ψ



⊗ 1



|ψi

, (2.11)

where ρ

is the density matrix deﬁned on R by tracing |ψi

over R. The new state is

labelled with a subscript 3 because Tr

|ψ

σσ

hψ

| = (ρ

)

and thus

hψ

|ψ

= Tr





. (2.12)

We use a bulk calculation to argue that |ψ

also lies in the same code subspace H

Thus both states deﬁne the same density matrix χ

≡ Tr

|χ

ihχ

| on H

for each

superselection sector α consistent with the ﬁxed geometric entropy σ (and on which |ψi

has non-zero projection). The relation (2.11) then requires





∝ χ

. (2.13)

Since eigenvalues of density matrices are real and non-negative, the relation (2.13) allows

to have only the eigenvalues 0 and 1 up to an overall normalization. Thus the density

matrix χ

is proportional to a projector onto a subspace of dimension dictated by its

entropy, which is in turn dictated by the associated eigenvalue of L

. Using the decompo-

sition (2.8), one can then more generally show that multiplication by



⊗ 1



preserves

the given code subspace H

We note that this result provides evidence supporting the above-mentioned classical

eﬀective action conjecture. In particular, the bulk calculation deriving (2.13) relies on

properties of variational principles for higher-derivative actions that we will establish in

section 3 below. These properties imply that with ﬁxed geometric entropy, a purely classical

(saddle-point) calculation of Renyi entropies would again give a ﬂat entanglement spectrum

for the |χ

state, and thus that the associated density matrix on R

would be a

projector onto a subspace of dimension set by the associated saddle-point von Neumann

entropy (i.e., by the geometric entropy). This is thus the prediction of the classical eﬀective

action conjecture, and we see that it agrees precisely with the results for the spectrum of

|χ

described above.

3 Higher derivative saddle-point Renyi entropies and states of ﬁxed ge-

ometric entropy

Before proceeding to the main argument in section 4, we must ﬁrst develop certain tech-

niques for studying higher derivative actions and ﬁxing the associated geometric entropy.

We emphasize that our main results (derived in section 4) do not rely on this conjecture. Rather, the

conjecture is supported by — and provides an intuitive way of understanding — our main results.

– 7 –

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一环阶全息代码的通用性与量子纠缠的奥秘

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