黑洞α位：揭示AdS/CFT中的重构与熵关联

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在《学习黑洞的Alpha位》这篇论文中，作者Patrick Hayden和Geoffrey Penington探讨了在AdS/CFT背景下，黑洞与边界理论之间的深层次联系。他们聚焦于AdS空间中的整体几何结构，特别是当其中包含黑洞时的情况。AdS/CFT对偶理论表明，黑洞的微观状态对于理解边界上的局部区域如何重构特定的 bulk (整体) 操作至关重要。在传统的观点中，对于所有黑洞微状态，虽然存在重构现象，但没有一种通用方法可以适用于所有情况。论文的核心贡献在于细化了这一二元性，提出了一种改进的方法，即存在一个常数分数α，这个分数代表黑洞熵的一部分，对应的边界子区域能够重构大部分（即熵的α部分）的黑洞微状态下的特定算子。这个“α位”概念来源于量子信息科学，它在薛定谔图景中描述了边界子区域如何编码包含黑洞的体区域的量子信息，特别是由极表面（例如事件视界）所界定的部分。这些发现对AdS/CFT理论和量子信息论具有深远意义。首先，它们揭示了整体重构并非全局精确，而是存在非摄动下的误差界限，这有助于我们在处理这类问题时设定合理的近似范围。其次，论文提供了一个关于纠缠楔（Wedge Subregion Holography）的限制条件，即尽管它在不同状态下有所变化，但变化是受到严格控制的，这对于理解和控制量子纠缠在网络模型中的行为非常关键。论文进一步探讨了Papadodimas-Raju提出的观点，即黑洞视野背后的算子与普通量子错误纠正过程中的状态依赖性有关联。作者通过AdS/CFT的张量网络模型展示了如何通过操作（如“推入”）来在理论层面上产生状态依赖性。最后，他们强调了黑洞作为实现容量的α位代码的首个“显式”例子，这挑战了传统的霍金辐射观念，即认为辐射揭示了黑洞的α位信息。在技术细节上，论文于2019年12月发表在JHEP上，经历了从接收、修订到最终接受的过程。作者的研究不仅深化了对AdS/CFT的理解，而且对量子信息科学中黑洞信息悖论的解决提供了新的视角，推动了理论物理学和量子计算的前沿研究。

JHEP12(2019)007

say that the complementary channel N

is approximately k-forgetful. Conversely if the

complementary channel N

is approximately k-forgetful with uncertainty δ, then there

exists a decoding channel for any subspace of dimension at most k with error ε ≤ 2

√

2δ.

How important is the inclusion of a reference system with dimension k in (2.1)?

By writing out Schmidt decompositions and using triangle inequality (see for example

lemma 23 of [17]), one can easily show that, for any k,

max

ρ∈S(H

⊗H

)

k(N

⊗ Id

)ρ

− σ

⊗ ρ

≤ k max

ρ∈S(H

)

(ρ) − σ

. (2.2)

In other words, δ grows at most linearly with the reference dimension k. Hence, as claimed

above, exact universal subspace error correction is indeed equivalent to exact quantum

error correction; if ε = 0 for k = 2, then we also have δ = 0 for k = 2, and hence δ = ε = 0

for all k.

However there are examples of quantum channels that saturate the bound (2.2) [17].

This means that, when the dimension of the complete space H

is very large, the for-

getfulness δ and reconstruction error ε can end up being much larger for large subspace

dimensions k than for small subspace dimensions. In other words, it may be possible to re-

construct any suﬃciently small subspace with very high ﬁdelity, while still being completely

impossible to reconstruct the entire space.

To make precise statements without reference to epsilons and deltas, it is generally

necessary to consider a limit where the dimension d of the code space tends to inﬁnity, for

example the classical limit of the space of black hole microstates. In general, the question

of whether the error ε → 0 will then depend on how the subspace dimension k scales with

the dimension d of the complete space.

To take a trivial example, universal subspace error correction for subspaces of dimen-

sion at most

k =

for some ﬁxed t is equivalent to conventional error correction (i.e. either both have errors

that tend to zero in some limit or neither does). However, if the subspace dimension grows

sublinearly with d, universal subspace error correction is inequivalent to ordinary error

correction. The most natural possiblility to consider is that the dimension of the subspaces

grows proportionally to d

for some 0 ≤ α ≤ 1. If universal subspace error correction is

possible with vanishing error for such a dimension we say that Bob has the α-bits of the

state sent by Alice.

The case α = 0, which we call zero-bits, corresponds to the ability to do universal

subspace error correction for any constant dimension k that is independent of d. Just as

we saw above when d/k was held ﬁxed, the exact value of k does not matter (we generally

take k = 2 for convenience) since, if ε → 0 for any k ≥ 2, it will also tend to zero for

all ﬁxed k, even though, for ﬁnite errors, the size of k will aﬀect the size of the error ε.

Formally, we deﬁne the α-bit decoding error based on subspaces of dimension k = 2

αn

+ 1,

since this formula gives k = 2 for zero-bits (k = 1 would be trivial) and scales as d

for

α > 0. However, as we have discussed, the exact dimension of the subspace is essentially

unimportant; we only care about how this dimension scales with n.

– 9 –

JHEP12(2019)007

3 Alpha-bits from the Hawking radiation

In this section we argue based on simple qubit toy models that an evaporating black hole

reveals its α-bits through its Hawking radiation as quickly as possible, saturating the α-bit

capacity of a noiseless quantum channel. This is in sharp contrast with the usual notion

that the Hawking radiation tries to hide information about the black hole state for as

long as possible, but we shall show that the two ideas are not merely reconcilable but in

fact equivalent. We generalise the arguments made by Hayden and Preskill in [14], which

can be interpreted as describing the special case where α = 0. This section is relatively

self-contained and is not necessary to understand the main claims of the paper, which are

developed in section 4; however, it is both of interest in its own right and features strong

similarities to the way the α-bits of black holes are encoded in AdS/CFT — suggesting

that the lessons from AdS/CFT may well be important in a signiﬁcantly broader context.

It is often incorrectly implied that the Hawking radiation contains no information

until the Page time, after which it begins to reveal the qubits of the black hole one by

one. In fact there is good reason to think that (at least in simple toy models) not a single

qubit of the black hole will be revealed, to an observer knowing nothing about the original

black hole state, until the black hole has almost entirely evaporated.

Instead, after the

Page time, the α-bits of the entire black hole will be revealed for increasing values of α,

until eventually all the qubits are revealed, essentially simultaneously, at the very end of

the evaporation process. No particular subsystem is revealed before any other subsystem;

however, increasingly large subspaces of the entire system become decodable.

Consider a large semiclassical black hole A in a pure microstate that is already known

by some observer Bob. Alice wants to hide her diary D, a small quantum state, from Bob

by dropping it into this black hole. After she has done so, Bob knows that the black hole

is in some particular small-dimensional subspace of the large Hilbert space of black hole

microstates — speciﬁcally the subspace of states that could have been created by the diary

falling in. In the semiclassical limit, the dimension of the space of black hole microstates

tends to inﬁnity, while the dimension of the small subspace remains ﬁxed.

The black hole is then allowed to evaporate into Hawking radiation. We assume Bob

has a perfect understanding of the microscopic dynamics of the black hole and the ability

to collect all the Hawking radiation that is emitted by it, as well as inﬁnite computational

power. However, even with these awesome powers, he has no ability to measure the internal

black hole degrees of freedom themselves. How long does Bob have to wait in order to

determine the original state of the diary with a high degree of conﬁdence?

This question was studied in detail in [14]. Since the dynamics of the black hole interior

are expected to be highly chaotic, Bob cannot hope for the small subspace of possible black

hole states that could have been created by the diary to be especially easy to decode from

the Hawking radiation. The problem is essentially equivalent to the question of whether

More precisely, there will no tensor product factorisation of the black hole H

∼

qubit

⊗ H

rest

such

that the reduced state on H

qubit

can be determined from the Hawking radiation before the black hole has

almost evaporated, even if the dimension of H

qubit

is only two.

– 10 –

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