Entanglement Wedge Reconstruction

• Entanglement wedge reconstruction is a framework in gauge/gravity duality that maps local bulk operators to boundary regions using quantum error correction and extremal surface prescriptions.
• It employs geometric tools such as RT surfaces and quantum extremal surfaces to define recoverable bulk regions, accurately incorporating quantum corrections.
• Tensor-network and algebraic approaches reveal that state dependence and code subspace limitations pose challenges for maintaining bulk locality and resolving information paradoxes.

Entanglement wedge reconstruction is a central concept in gauge/gravity duality, particularly AdS/CFT, which formalizes the mapping between local bulk observables and boundary degrees of freedom. It asserts that given a boundary spatial region, one can identify a corresponding bulk region—the entanglement wedge—within which all bulk operators admit a boundary representation supported only on that region. This is realized via quantum error correction, modular flow, and the extremization of generalized entropy, with crucial implications for bulk locality, quantum gravity, and the resolution of information paradoxes.

1. Geometric Prescriptions: RT and Quantum Extremal Surfaces

The geometric foundation of entanglement wedge reconstruction relies on minimal surface prescriptions. For a boundary region $A$:

• Ryu–Takayanagi (RT) surface $X_0[A]$: In the classical (large-$N$, $G_N\to 0$) regime, the entropy of $A$ equals the area of the minimal bulk surface homologous to $A$, divided by $4G_N$:

$A_{\mathrm{RT}}[X] = \frac{\mathrm{Area}(X)}{4G_N}$

The classical entanglement wedge $EW_0[A]$ is the bulk region between $A$ and $X_0[A]$.

• Quantum Extremal Surface (QES): At finite $N$, quantum corrections appear through the von Neumann entropy of bulk fields:

$$S_{\mathrm{gen}}[X] = \frac{\mathrm{Area}(X)}{4G_N} + S_{\mathrm{bulk}}(\Sigma_X)$$

where $S_{\mathrm{bulk}}$ is the bulk entropy on any Cauchy slice $\Sigma_X$ bounded by $A\cup X$. Extremizing $S_{\mathrm{gen}}$ defines the QES $X_{\mathrm{QES}}[A]$, and $EW[A]$ is the domain between $A$ and $X_{\mathrm{QES}}[A]$ (Akers et al., 2019).

2. Reconstruction Wedge and State Dependence

While the entanglement wedge $EW[A]$ is state-dependent due to the bulk entropy term, exact quantum error correction implies that operators in $EW[A]$ are reconstructable from observables on $A$ only in specific states or subspaces. To formalize this:

• Reconstruction wedge $RW[A]$: Defined as the intersection of all entanglement wedges computed over every pure or mixed state in a code subspace $\mathcal{H}_{\mathrm{code}}$:

$$RW[A] = \bigcap_{\text{states}\in\mathcal{H}_{\mathrm{code}}} EW[A]_{\text{state}}$$

$RW[A]$ is operationally the region where all operators can be reconstructed in a state-independent fashion for the chosen code subspace. If a bulk point $p$ lies outside $RW[A]$, there exists a code subspace state in which $p$ is outside $EW[A]$, precluding its reconstruction from $A$ (Akers et al., 2019).

• Physical implication: At order $O(G_N)$ corrections, the ability to reconstruct operators in the deep wedge becomes subspace-dependent. For large code spaces (e.g., many microstates), $RW[A]$ can be macroscopically smaller than $EW[A]$, reflecting the breakdown of naïve locality in quantum gravity.

3. Microscopic Examples and Large Deviations

Two major families of explicit examples demonstrate macroscopically large separations between $RW[A]$ and $EW[A]$ at finite $G_N$:

Near-vacuum qubit models (AdS$_3$)

• Divide the AdS$_3$ boundary circle into four arcs; take $A=I \cup III$ and $\bar{A}=II \cup IV$.
• At the phase transition (degenerate RT surfaces), placing qubits in the central region modifies the bulk entropy, shifting the QES. For mixed states (maximal entropy), the QES excludes the central region for both $A$ and $\bar{A}$; thus, $RW[A]$ omits this region.
• Only when $A$'s entanglement can offset the bulk entropy, e.g., via added Bell pairs, does the reconstructable region enlarge (Akers et al., 2019).

Dustball in AdS$_3$

• A bulk region with $k$ dust particles, each with $s$ internal spins, gives entropy $S_{\mathrm{bulk}} = k\log s$.
• For maximally mixed spin states (large $S_{\mathrm{bulk}}$), the QES excludes the dustball, so $RW[A]$ does not include it, although pure spin states have $EW[A]$ containing the dustball.
• By tuning $k\sim O(G_N^{-1})$ and $R$ arbitrarily large, the excluded reconstruction region grows parametrically (Akers et al., 2019).

4. Code-Theoretic and Tensor-Network Perspectives

Quantitative investigations in tensor-network models (HaPPY code and variants) provide further insight:

• Wedge types:
  • Causal wedge: always too small, fails as a reconstruction criterion ($p_c=0$).
  • Greedy entanglement wedge: code-dependent reconstructability thresholds, sensitive to tiling and bulk-boundary ratios.
  • Minimum entanglement wedge: best state-independent proxy for geometric wedge, showing a sharp reconstruction threshold $p_c=0.5$ for all HaPPY-type codes (Linden, 2 Jan 2024).
• State-dependent extensions: Stabilizer-symmetry manipulations reveal operator reconstruction beyond the greedy wedge, implying subspace-specific extensions reminiscent of gravitational entanglement-wedge subtleties (Linden, 2 Jan 2024).
• Rejection of mutual-information wedges: In Monte-Carlo and entropy analyses, mutual information between boundary and bulk qubits does not reliably track geometric wedge inclusion, invalidating a "mutual-information wedge" as an operational reconstruction region (Linden, 2 Jan 2024).

5. Algebraic and Recovery-Channel Formulations

Rigorous operator-algebra approaches equate entanglement-wedge reconstruction to the equality of bulk and boundary relative entropies. This yields:

• JLMS formula:

$$S(\rho_A \Vert \sigma_A) = S(\rho_{EW(A)} \Vert \sigma_{EW(A)}) + O(G_N)$$

establishes the code-theoretic duality and modular flow intertwining, validated even in infinite-dimensional Hilbert spaces and for continuum von Neumann algebras (Kang et al., 2018, Kang et al., 2019).

• Recovery maps: The Petz channel, and more generally universal recovery channels (twirled Petz), provide explicit boundary representations of bulk operators. For code subspaces of fixed finite dimension, the Petz map gives non-perturbatively small error bounds without requiring averaging over modular flow (Bahiru et al., 2022, Chen et al., 2019, Cotler et al., 2017).
• Modular flow and reflection operator: Modern constructions relate relational bulk reconstruction between different code subspaces via modular theory flows, with connections to Connes cocycle limits and modular reflection operators (Parrikar et al., 4 Mar 2024).

6. Extensions, Covariant Definitions, and Physical Implications

The standard entanglement wedge (QES prescription) generalizes to:

• Covariant max and min entanglement wedges: In non-AdS or time-dependent spacetimes, one defines covariant max-EW (reconstructible region) and min-EW (influential region) via one-shot conditional entropy and quantum focusing conjectures. The max-EW is state-specific and operationally reconstructible up to errors controlled by generalized entropy gaps (Akers et al., 2023, Bousso et al., 2023).
• Modular chaos and ergodicity: Algebraic modular flows generate entanglement wedge operator algebras from causal wedge subalgebras, e.g., in JT gravity, through boost-like modular actions about the RT surface (Gao, 28 Feb 2024).
• Information paradox and firewall resolution: In evaporating AdS black holes, entanglement wedge phase transitions at Page time solve the Page curve and Hayden–Preskill decoding criterion, with the minimal state dependence required to avoid cloning paradoxes (Penington, 2019).

7. Limitations and Open Problems

Macroscopic deviations between $RW[A]$ and $EW[A]$ persist at finite $G_N$ for large code subspaces. State dependence, subalgebra selection, 1/$G_N$ corrections, and lack of long-range entanglement in certain code models signal the breakdown of naive subregion duality, demanding refinement of holographic algebraic structure and the scope of bulk operator reconstructability (Akers et al., 2019, Gesteau et al., 2020).

References (selected arXiv papers)

• "Large Breakdowns of Entanglement Wedge Reconstruction" (Akers et al., 2019)
• "To Wedge Or Not To Wedge, Wedges and operator reconstructability in toy models of AdS/CFT" (Linden, 2 Jan 2024)
• "Explicit reconstruction of the entanglement wedge via the Petz map" (Bahiru et al., 2022)
• "Reconstruction of Bulk Operators within the Entanglement Wedge in Gauge-Gravity Duality" (Dong et al., 2016)
• "Holographic Relative Entropy in Infinite-dimensional Hilbert Spaces" (Kang et al., 2018)
• "Microscopic Entanglement Wedges" (Koch, 2023)
• "Explicit reconstruction of the entanglement wedge" (Kim, 2016)
• "Entanglement Wedge Reconstruction via Universal Recovery Channels" (Cotler et al., 2017)
• "One-shot holography" (Akers et al., 2023)
• "Modular flow in JT gravity and entanglement wedge reconstruction" (Gao, 28 Feb 2024)
• "Relational bulk reconstruction from modular flow" (Parrikar et al., 4 Mar 2024)