Holography as Quantum Error Correction

Guiding question. Why can the same bulk operator have many different boundary representations without violating locality or the no-cloning theorem?

Main lesson. The low-energy bulk theory is not embedded into the boundary CFT as an ordinary tensor factor. It is embedded as a quantum error-correcting code. The bulk semiclassical Hilbert space is a code subspace,

$$V: \mathcal H_{\rm code}\longrightarrow \mathcal H_{\rm CFT}, \qquad V^\dagger V=I_{\rm code},$$

and bulk observables are logical operators. A logical operator can have different boundary representatives on different boundary regions, provided those representatives agree on the encoded subspace:

$$V^\dagger O_A V = V^\dagger O_B V = O_{\rm bulk}.$$

This is the quantum-information-theoretic form of bulk locality. It explains why subregion duality is redundant, why radial position is tied to error-correction strength, and why entanglement wedge reconstruction is possible.

The holographic QEC dictionary. The semiclassical bulk Hilbert space is a code subspace embedded into the exact CFT Hilbert space by an isometry $V$. Bulk fields are logical operators. Boundary operators acting on different physical regions can represent the same logical bulk observable, but only inside the code subspace.

1. The puzzle: too many ways to describe one bulk point

Suppose a scalar field excitation sits at a bulk point $x$. In global HKLL reconstruction, one represents it by an operator smeared over the entire boundary:

$$\phi(x)=\int_{\partial {\rm AdS}} K(x|X)\mathcal O(X)+\cdots.$$

In AdS-Rindler reconstruction, the same excitation can sometimes be represented using only a boundary subregion $A$. Entanglement wedge reconstruction goes further: if

$$x\in \mathcal E_A,$$

then there exists an operator $O_A$ supported on $A$ that represents the bulk operator at $x$. If $x$ also lies in the entanglement wedge of another region $B$, there is another representative $O_B$.

Naively this sounds impossible. How can the same bulk degree of freedom live in many different boundary places? If $O_A$ and $O_B$ are both versions of $O_x$, did the CFT clone a quantum degree of freedom?

The resolution is that the equalities are logical equalities, not microscopic equalities. The operators $O_A$ and $O_B$ need not be equal on the full CFT Hilbert space. They agree only after projecting to the image of the code subspace:

$$P_{\rm code}^{\rm CFT}O_A P_{\rm code}^{\rm CFT} = P_{\rm code}^{\rm CFT}O_B P_{\rm code}^{\rm CFT},$$

where

$$P_{\rm code}^{\rm CFT}=VV^\dagger.$$

Equivalently,

$$V^\dagger (O_A-O_B)V=0.$$

Thus $O_A$ and $O_B$ are not two independent copies of an unknown quantum system. They are two physical implementations of the same logical observable, exactly as in a quantum error-correcting code.

This is the conceptual leap introduced by the quantum-error-correcting interpretation of AdS/CFT: bulk locality is not microscopic boundary locality. Bulk locality is a property of the encoded low-energy sector.

2. Code subspaces in AdS/CFT

The full CFT Hilbert space $\mathcal H_{\rm CFT}$ contains black holes of many masses, highly stringy states, multi-particle states with large backreaction, and states with no simple semiclassical description. A local bulk effective field theory on a chosen background describes only a tiny sector. This sector is the code subspace.

A simple code subspace might be generated by acting on a semiclassical state $|\Psi_0\rangle$ with a controlled number of low-energy bulk operators:

$$\mathcal H_{\rm code} = {\rm span}\left\{ \phi_{i_1}(x_1)\cdots \phi_{i_k}(x_k)|\Psi_0\rangle: 0\leq k\leq k_{\rm max} \right\}.$$

The cutoff $k_{\rm max}$ is not just a technical nuisance. It is physical. If the code subspace includes too much energy or entropy, the geometry changes, the relevant HRT/QES surfaces move, and the reconstruction problem becomes a different problem.

There are several levels of code subspace:

• Small code subspace: a fixed classical geometry plus perturbative bulk fields.
• Medium code subspace: nearby geometries whose extremal surface remains stable.
• Large code subspace: states with enough entropy or backreaction to compete with area terms, where a fixed reconstruction can fail.

The encoding map

$$V:\mathcal H_{\rm code}\to \mathcal H_{\rm CFT}$$

identifies a bulk effective state $|\psi\rangle$ with the exact CFT state $V|\psi\rangle$. When we write an operator equality such as

$$O_{\rm CFT}\sim O_{\rm bulk},$$

the precise version is usually

$$V^\dagger O_{\rm CFT}V=O_{\rm bulk}.$$

This single equation already contains much of the subtlety of bulk emergence. It says that the CFT operator reproduces the matrix elements of the bulk operator among the states where semiclassical bulk effective theory is valid.

3. Ordinary quantum error correction

In ordinary quantum error correction one has:

• a logical Hilbert space $\mathcal H_L$;
• a larger physical Hilbert space $\mathcal H_P$;
• an encoding isometry $V:\mathcal H_L\to \mathcal H_P$;
• a noise channel $\mathcal N$;
• a recovery channel $\mathcal R$.

The code corrects the noise $\mathcal N$ if

$$\mathcal R\circ \mathcal N\circ \mathcal V = \mathcal V,$$

where

$$\mathcal V(\rho)=V\rho V^\dagger.$$

For an erasure error, the noise is especially simple. If the physical system factorizes as

$$\mathcal H_P=\mathcal H_A\otimes \mathcal H_{\bar A},$$

and the region $\bar A$ is lost, then

$$\mathcal N_A(\rho) = {\rm Tr}_{\bar A}\left(V\rho V^\dagger\right).$$

The erasure of $\bar A$ is correctable if there exists a recovery map $\mathcal R_A$ such that

$$\mathcal R_A\!\left({\rm Tr}_{\bar A}(V\rho V^\dagger)\right)=\rho$$

for every logical state $\rho$.

In the Heisenberg picture, this means every logical operator $O_L$ can be represented on the surviving physical region $A$:

$$V^\dagger (O_A\otimes I_{\bar A})V=O_L.$$

Thus erasure correction and operator reconstruction are two sides of the same statement.

Erasure correction in holography. Losing the boundary region $\bar A$ is correctable for bulk observables in $\mathcal E_A$. The surviving boundary region $A$ contains enough physical degrees of freedom to reconstruct the logical algebra of the corresponding entanglement wedge.

3.1 Knill–Laflamme condition

For a finite-dimensional subspace code with projector $P$ and error operators $E_i$, exact correctability is characterized by the Knill–Laflamme condition

$$P E_i^\dagger E_j P=c_{ij}P.$$

This means that the errors can change the syndrome but cannot reveal information about the encoded state. If the environment learned which logical state was encoded, no recovery operation could restore an unknown quantum state.

For erasure of $\bar A$, a useful equivalent statement is the decoupling condition:

$${\rm Tr}_A\left(V\rho V^\dagger\right)$$

is independent of the logical state $\rho$. In words, the erased region $\bar A$ must contain no information about the logical state. If the erased region knows nothing, the complement can know everything.

This logic will reappear in the Page curve and island story. After the Page time, the radiation region can contain logical information about an interior island because the complementary black hole degrees of freedom no longer contain an independent copy of the relevant logical algebra.

4. The holographic dictionary as a code dictionary

The quantum-error-correcting dictionary is:

Quantum error correction	Holography
logical Hilbert space	bulk effective Hilbert space $\mathcal H_{\rm code}$
physical Hilbert space	CFT Hilbert space $\mathcal H_{\rm CFT}$
encoding isometry	AdS/CFT map $V$
logical operator	bulk operator in the code subspace
physical operator	boundary CFT operator
erased physical qubits	missing boundary region
recoverable logical subsystem	bulk region in the entanglement wedge
code distance	size of boundary erasure that can be tolerated
syndrome	UV/boundary data not interpreted as bulk low-energy physics

This dictionary is not merely an analogy. It is a structural statement about subregion duality. The boundary representation of a bulk operator is redundant because the code is designed to tolerate erasures.

For a boundary region $A$, define the restriction channel

$$\mathcal N_A(\rho)=\operatorname{Tr}_{\bar A}(V\rho V^\dagger).$$

If the logical bulk subsystem $a$ lies in the entanglement wedge $\mathcal E_A$, then $\mathcal N_A$ preserves the information in $a$. There exists a recovery map $\mathcal R_A$ whose adjoint gives boundary representatives:

$$O_a\longmapsto O_A=\mathcal R_A^*(O_a).$$

The statement is local in the emergent bulk but nonlocal in the microscopic boundary degrees of freedom. This is why a bulk field can look local in the interior while being encoded redundantly in boundary CFT variables.

5. Secret sharing as the simplest model

A useful toy model is quantum secret sharing. Imagine a logical qutrit encoded into three physical qutrits so that any two parties can recover the secret, but any one party alone has no information.

One standard three-qutrit code is

$$|0_L\rangle=\frac{1}{\sqrt 3}\left(|000\rangle+|111\rangle+|222\rangle\right),$$ $$|1_L\rangle=\frac{1}{\sqrt 3}\left(|012\rangle+|120\rangle+|201\rangle\right),$$ $$|2_L\rangle=\frac{1}{\sqrt 3}\left(|021\rangle+|102\rangle+|210\rangle\right).$$

For an arbitrary logical state

$$|\psi_L\rangle=\alpha|0_L\rangle+\beta|1_L\rangle+\gamma|2_L\rangle,$$

any single physical qutrit is maximally mixed:

$$\rho_1=\rho_2=\rho_3=\frac{I_3}{3}.$$

Therefore one share contains no information about $\alpha,\beta,\gamma$. But any two shares determine the third by linear constraints, so any two shares can recover the logical qutrit.

This captures the slogan:

$$\text{one region sees nothing, but two regions recover everything.}$$

Holographic subregion duality is a vastly more sophisticated, approximate, continuum version of this structure. A bulk logical operator can be reconstructed from different sufficiently large boundary regions, while too small a boundary region has no access to that operator.

Quantum secret sharing gives the simplest cartoon of holographic redundancy. Any two boundary shares can reconstruct the central logical degree of freedom, but one share alone contains no information about it. Holographic codes replace this discrete toy model by a geometrical pattern governed by entanglement wedges.

6. Redundant representations and no-cloning

Suppose a bulk operator $O_x$ lies in the entanglement wedge of both $A$ and $B$. Then there are boundary operators $O_A$ and $O_B$ satisfying

$$V^\dagger O_A V=O_x, \qquad V^\dagger O_B V=O_x.$$

This redundancy is not cloning. The no-cloning theorem forbids a unitary operation that maps

$$|\psi\rangle\longrightarrow |\psi\rangle|\psi\rangle$$

for every unknown state $|\psi\rangle$. Holographic reconstruction does not do this. It gives multiple representations of one logical operator on one encoded state, not multiple independent physical copies of the state.

A precise way to say this is:

$$V^\dagger(O_A-O_B)V=0.$$

Thus $O_A$ and $O_B$ have the same matrix elements between encoded states, but they can differ wildly outside the code subspace. The difference operator

$$O_A-O_B$$

is a null operator on the code subspace. It is invisible to low-energy bulk effective theory.

This is exactly how ordinary quantum error-correcting codes work. A logical Pauli operator may be represented by many different physical Pauli strings, related by stabilizers. The physical strings are different operators, but they induce the same logical transformation.

The bulk version is subtler because the equivalence is approximate, state-dependent through the code subspace, and constrained by gravity. But the logical structure is the same.

7. Radial locality and code distance

One of the original motivations for the QEC interpretation is radial locality. A bulk point close to the boundary is usually reconstructible from a relatively small boundary region. A bulk point deep in the interior requires a larger boundary region. A point near the center of global AdS may require more than half of the boundary.

This is what one expects from error correction. The deeper a logical degree of freedom sits in the bulk, the more protected it is against erasure of boundary regions. In code language, deeper bulk degrees of freedom have larger effective code distance.

Radial locality as error correction. A near-boundary excitation can be reconstructed from a relatively small boundary interval, while a deeper excitation requires a larger boundary region. The radial direction is tied to the amount of boundary erasure the logical degree of freedom can tolerate.

This is only a heuristic unless the entanglement wedge is specified. The precise condition is:

$$x\in \mathcal E_A \quad\Longleftrightarrow\quad \text{operators near }x\text{ can be reconstructed on }A.$$

For nested boundary regions,

$$A\subset B,$$

entanglement wedge nesting gives

$$\mathcal E_A\subseteq \mathcal E_B.$$

Thus if $x$ is reconstructible from $A$, it is also reconstructible from the larger region $B$. In error-correction language, adding physical qubits cannot make the logical information harder to recover.

8. Complementary recovery

A particularly important QEC structure is complementary recovery. Suppose a bulk Cauchy slice is divided by an RT/HRT/QES surface into two wedge regions $a$ and $\bar a$, dual to boundary regions $A$ and $\bar A$:

$$\mathcal E_A\leftrightarrow a, \qquad \mathcal E_{\bar A}\leftrightarrow \bar a.$$

Then operators in $a$ should be reconstructible on $A$, while operators in $\bar a$ should be reconstructible on $\bar A$:

$$O_a\longleftrightarrow O_A, \qquad O_{\bar a}\longleftrightarrow O_{\bar A}.$$

This is stronger than merely saying that $A$ can recover $a$. It says that the two complementary boundary regions recover complementary bulk algebras. Complementary recovery is one of the key inputs behind the quantum-error-correcting derivation of the RT formula.

At leading order, the boundary entropy of $A$ takes the schematic form

$$S(A)=\frac{{\rm Area}(X_A)}{4G_N}+S_{\rm bulk}(a)+\cdots.$$

From the QEC viewpoint, the first term behaves like a universal entropy associated with the encoding itself, while $S_{\rm bulk}(a)$ is the logical entropy of the bulk degrees of freedom in the wedge. In the more precise operator-algebra formulation, the area term is related to a central element of the logical algebra. That refinement is important enough to deserve its own later page.

9. QEC and the RT formula

The RT formula originally looked like a purely geometric statement:

$$S(A)=\frac{{\rm Area}(\gamma_A)}{4G_N}.$$

But from the code perspective, this formula has an information-theoretic meaning. The entropy of a physical boundary region has two contributions:

1. an encoding entropy, associated with the number of microscopic boundary states compatible with the same low-energy bulk data;
2. a logical entropy, associated with the bulk degrees of freedom reconstructed in the wedge.

The quantum-corrected version is therefore natural:

$$S(A)=\left\langle \frac{\widehat{\rm Area}(X_A)}{4G_N}\right\rangle+S_{\rm bulk}(a)+\cdots.$$

In ordinary subspace QEC with a fixed area term, one can model this as

$$S(A)_\rho=S_0+S(a)_\rho,$$

where $S_0$ is independent of the logical state. In holography the area term is generally an operator, and the correct treatment requires operator-algebra QEC:

$$S(A)_\rho =\operatorname{Tr}(\rho\mathcal L_A)+S_{\rm alg}(\rho_a).$$

Here $\mathcal L_A$ is the abstract area operator and $S_{\rm alg}$ is the entropy of the bulk algebra in the entanglement wedge. This is not an optional mathematical decoration. It is forced by gauge constraints and by the fact that gravitational Hilbert spaces do not factorize cleanly across surfaces.

The basic moral remains simple:

$$\text{RT/QES is the entropy formula of a gravitational quantum error-correcting code.}$$

10. Approximate codes and finite-$N$ corrections

Textbook QEC often studies exact finite-dimensional codes. Holographic QEC is not exact in that sense. Several approximations are layered together:

10.1 Large-$N$ approximation

Classical bulk locality is sharp only in the large-$N$ limit. At finite $N$, bulk locality is approximate, and the code has finite accuracy.

10.2 Semiclassical expansion

The code subspace is defined around a semiclassical background. If states have enough energy or entropy to move the extremal surface, the original reconstruction can fail.

10.3 Gauge constraints

Gauge-invariant bulk operators must be dressed. In gravity the dressing often reaches the boundary, so the naive tensor-factor language must be replaced by algebraic language.

10.4 Nonperturbative effects

Exact quantum gravity may not contain an exact operator corresponding to a sharply localized bulk point. The low-energy local operator is an effective observable, valid within a controlled code subspace.

10.5 Complexity

The existence of a recovery map does not imply an efficient recovery map. This distinction becomes crucial for black hole interiors, Hayden–Preskill recovery, Python’s lunch, and the practical decoding of Hawking radiation.

A useful hierarchy is therefore:

$$\text{information-theoretically reconstructible} \not\Rightarrow \text{efficiently reconstructible} \not\Rightarrow \text{locally visible}.$$

Much confusion about black hole information comes from collapsing these three notions into one.

11. Relation to black hole information

The QEC viewpoint changes the way one reads the black hole information paradox.

In Hawking’s semiclassical description, an outgoing mode $B$ is paired with an interior partner $A_{\rm in}$. In a naive tensor-factor picture, if $B$ is also entangled with early radiation $R$, monogamy appears to produce the firewall paradox.

The QEC lesson is that the interior partner need not be an independent microscopic tensor factor. It may be a logical degree of freedom with different reconstructions in different boundary or radiation regions.

This does not make the paradox disappear by a slogan. It reframes the question:

$$\text{Which algebra is reconstructible from which subsystem, in which code subspace?}$$

After the Page time, the island formula says that the entanglement wedge of the radiation contains an island $\mathcal I$:

$$\mathcal E_R=R\cup \mathcal I.$$

Entanglement wedge reconstruction then says that operators in $\mathcal I$ have representatives on the radiation region $R$. This is a QEC statement:

$$O_{\mathcal I}\longleftrightarrow O_R.$$

It does not mean the interior information is visible as a simple local pattern in the Hawking quanta. It means the radiation subsystem contains the logical algebra of the island, possibly in a highly scrambled and computationally difficult form.

12. What the QEC picture does not say

The QEC interpretation is powerful, but it should not be oversold.

It does not say that the bulk is an ordinary stabilizer code. Stabilizer codes are useful toy models. The actual CFT encoding is dynamical, approximate, continuum, gauge constrained, and strongly interacting.

It does not by itself derive the CFT. QEC explains how bulk locality can be compatible with a boundary description once the duality exists. It is not a complete microscopic construction of every holographic CFT.

It does not make reconstruction unique. Redundancy is the point. Many physical operators can represent the same logical observable.

It does not eliminate gravitational dressing. Bulk operators must be gauge invariant, and gravitational dressing is part of the reconstruction problem.

It does not guarantee efficient decoding. A recovery channel can exist while being exponentially complex to implement.

It does not allow superluminal signaling. Entanglement wedge inclusion is not causal accessibility. The causal wedge and entanglement wedge answer different questions.

13. Common pitfalls

Pitfall 1: “A bulk point is stored at a boundary point.”

No. A bulk point is encoded nonlocally in boundary degrees of freedom. In a good code, local boundary regions may contain no direct information about deep bulk degrees of freedom.

Pitfall 2: “Multiple reconstructions mean multiple copies.”

No. Multiple boundary representatives agree only inside the code subspace. They are not independent clones.

Pitfall 3: “The code subspace is the whole CFT Hilbert space.”

No. The code subspace is tiny compared with the full CFT Hilbert space. It is the regime where semiclassical bulk effective theory is valid.

Pitfall 4: “QEC is just a metaphor.”

No. The reconstruction conditions, relative-entropy equalities, and entropy formulas have precise QEC formulations. The toy models are metaphors; the code-subspace structure is mathematical.

Pitfall 5: “A recovery map gives a practical decoding algorithm.”

No. Existence and efficiency are different. Black hole decoding can be information-theoretically possible but computationally prohibitive.

14. Summary

Holography realizes bulk effective field theory as a quantum error-correcting code:

$$\boxed{ V:\mathcal H_{\rm code}\hookrightarrow \mathcal H_{\rm CFT}. }$$

The central reconstruction statement is

$$\boxed{ O_a\in {\rm Alg}(\mathcal E_A) \quad\Longrightarrow\quad \exists\,O_A\in {\rm Alg}(A): V^\dagger O_A V=O_a. }$$

The QEC interpretation explains:

• why bulk operators can have many boundary representatives;
• why no-cloning is not violated;
• why radial locality is related to erasure protection;
• why entanglement wedge reconstruction is the natural form of subregion duality;
• why the RT/QES formula resembles an entropy formula for a code;
• why island reconstruction is a statement about logical algebras, not local signals.

The next pages sharpen this picture. The HaPPY code provides an exactly solvable tensor-network model of holographic QEC, while operator-algebra QEC explains why area terms, centers, and gauge constraints are not optional details but part of the exact grammar of gravitational subregion duality.

Exercises

Exercise 1. Matrix elements of a logical representative

Let $V:\mathcal H_L\to \mathcal H_P$ be an isometric encoding. Suppose $O_A$ is a physical operator satisfying

$$V^\dagger O_A V=O_L.$$

Show that for all logical states $|\psi\rangle,|\chi\rangle\in\mathcal H_L$,

$$\langle V\psi|O_A|V\chi\rangle = \langle \psi|O_L|\chi\rangle.$$

Why does this not imply that $O_A$ is unique?

Solution

Using $V^\dagger V=I_L$,

$$\langle V\psi|O_A|V\chi\rangle = \langle \psi|V^\dagger O_A V|\chi\rangle = \langle \psi|O_L|\chi\rangle.$$

The equality only fixes matrix elements inside the image of the code subspace. If $N$ is any physical operator satisfying

$$V^\dagger N V=0,$$

then $O_A+N$ represents the same logical operator. Thus logical representatives are equivalence classes of physical operators modulo operators that vanish on the code subspace.

Exercise 2. Decoupling criterion for erasure

Consider an exact erasure-correcting code with physical Hilbert space $\mathcal H_A\otimes\mathcal H_{\bar A}$. Show that if the erased region $\bar A$ has reduced density matrix independent of the logical input state,

$${\rm Tr}_A(V\rho V^\dagger)=\sigma_{\bar A}$$

for all logical states $\rho$, then $\bar A$ contains no information about the logical state. Explain why this is necessary for recovery from $A$.

Solution

The reduced state on $\bar A$ is exactly the same state $\sigma_{\bar A}$ for every logical input $\rho$. Therefore no measurement on $\bar A$ can distinguish two logical states. In information-theoretic language, the channel to $\bar A$ is constant.

If recovery from $A$ is possible, then losing $\bar A$ cannot destroy the logical state. If $\bar A$ nevertheless contained information about the logical state, the encoded state would distribute information into both $A$ and $\bar A$ in a way that would conflict with the no-cloning/no-broadcasting constraints for unknown quantum states. Exact QEC makes this precise: correctability of erasure on $\bar A$ is equivalent to decoupling of $\bar A$ from the logical reference system.

Exercise 3. One share of the three-qutrit secret-sharing code

For the three-qutrit code

$$|0_L\rangle=\frac{1}{\sqrt 3}(|000\rangle+|111\rangle+|222\rangle),$$ $$|1_L\rangle=\frac{1}{\sqrt 3}(|012\rangle+|120\rangle+|201\rangle),$$ $$|2_L\rangle=\frac{1}{\sqrt 3}(|021\rangle+|102\rangle+|210\rangle),$$

show that for any logical basis state $|i_L\rangle$, the reduced density matrix of the first qutrit is $I_3/3$. Why does the same conclusion hold for an arbitrary logical superposition?

Solution

For each logical basis state, the first qutrit takes the values $0,1,2$ exactly once with equal amplitude. For example,

$$|1_L\rangle=\frac{1}{\sqrt 3}(|012\rangle+|120\rangle+|201\rangle).$$

Tracing out qutrits 2 and 3 kills cross terms because the corresponding states of qutrits 2 and 3 are orthogonal. Hence

\rho_1=rac{1}{3}\left(|0\rangle\langle0|+|1\rangle\langle1|+|2\rangle\langle2|\right)=\frac{I_3}{3}.

For an arbitrary superposition $\alpha|0_L\rangle+\beta|1_L\rangle+\gamma|2_L\rangle$, cross terms between different logical basis states also vanish after tracing out qutrits 2 and 3. This follows because, for fixed first-qutrit value, the remaining two-qutrit states associated with different logical labels are orthogonal. Therefore the first share is always maximally mixed and contains no information about the logical state.

Exercise 4. Redundant representatives and no-cloning

Suppose $O_A$ and $O_B$ are two physical representatives of a logical operator $O_L$:

$$V^\dagger O_A V=V^\dagger O_B V=O_L.$$

Show that $O_A-O_B$ has vanishing matrix elements between encoded states. Explain in one paragraph why this avoids the no-cloning paradox.

Solution

Subtracting the two equations gives

$$V^\dagger(O_A-O_B)V=0.$$

Therefore, for any logical states $|\psi\rangle,|\chi\rangle$,

$$\langle V\psi|(O_A-O_B)|V\chi\rangle=0.$$

The two operators are not two independent copies of the logical degree of freedom. They are two representatives with identical action on the encoded subspace. Outside the code subspace they may differ. The no-cloning theorem forbids copying an unknown quantum state into two independent systems; it does not forbid multiple physical representatives of one logical observable in a redundant encoding.

Exercise 5. Radial position and minimal boundary size

Consider a family of boundary regions $A(\theta)$ in a static holographic geometry, where $\theta$ measures the angular size of the region. Suppose a bulk point $x$ lies in $\mathcal E_{A(\theta)}$ only for $\theta\geq \theta_*(x)$. Interpret $\theta_*(x)$ as an error-correction diagnostic. What should happen to $\theta_*(x)$ as $x$ moves deeper into the bulk?

Solution

The condition $x\in\mathcal E_{A(\theta)}$ means operators near $x$ can be reconstructed on $A(\theta)$. Thus $\theta_*(x)$ is the minimal boundary region size needed to recover the logical information at $x$.

If $x$ moves deeper into the bulk, it should generally require a larger boundary region for reconstruction. Therefore $\theta_*(x)$ increases. In QEC language, deep bulk degrees of freedom are more protected against erasure of boundary regions: small boundary regions do not contain enough information to reconstruct them, while sufficiently large regions do.

Exercise 6. Code subspace size and area budget

Suppose a wedge reconstruction is valid for a code subspace whose bulk entropy in $a$ is much smaller than the area term:

$$S_{\rm bulk}(a)\ll \frac{{\rm Area}(X_A)}{4G_N}.$$

Explain why enlarging the code subspace until $S_{\rm bulk}(a)$ is comparable to the area term can invalidate the original reconstruction.

Solution

The entanglement wedge is determined by a generalized entropy competition. If the bulk entropy contribution is small compared with the area term, the extremal surface and wedge are stable under changes of the code state. A single reconstruction map can then work throughout the code subspace.

If the code subspace is enlarged so much that

$$S_{\rm bulk}(a)\sim \frac{{\rm Area}(X_A)}{4G_N},$$

then the generalized entropy of competing surfaces can change by an amount comparable to the classical area difference. The dominant QES may move or jump. The bulk region dual to $A$ is then not fixed across the enlarged code subspace, so the original reconstruction map need not remain valid.

Further reading

• A. Almheiri, X. Dong, and D. Harlow, “Bulk Locality and Quantum Error Correction in AdS/CFT,” arXiv:1411.7041.
• F. Pastawski, B. Yoshida, D. Harlow, and J. Preskill, “Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence,” arXiv:1503.06237.
• X. Dong, D. Harlow, and A. C. Wall, “Reconstruction of Bulk Operators within the Entanglement Wedge in Gauge-Gravity Duality,” arXiv:1601.05416.
• D. Harlow, “The Ryu–Takayanagi Formula from Quantum Error Correction,” arXiv:1607.03901.
• D. Harlow, “TASI Lectures on the Emergence of the Bulk in AdS/CFT,” arXiv:1802.01040.
• J. Cotler, P. Hayden, G. Penington, G. Salton, B. Swingle, and M. Walter, “Entanglement Wedge Reconstruction via Universal Recovery Channels,” arXiv:1704.05839.