The JLMS formula in a large code with approximate error correction

As a fastidious and diligent researcher, I have meticulously analyzed the provided excerpts from this arXiv paper concerning Gauge/Gravity Duality as a quantum error-correcting code, specifically focusing on the JLMS (Jafferis-Lewkowycz-Maldacena-Suh) formula in the context of large codes.

Here is a detailed and comprehensive summary synthesizing the key technical contributions and arguments presented:

Detailed Research Summary: JLMS Formula in Large Codes with Approximate Error Correction

This research paper addresses a fundamental limitation in applying the Gauge/Gravity duality as an exact quantum error-correcting code, specifically concerning the JLMS relation. The authors argue that while exact quantum error correction with complementary recovery (and thus entanglement wedge reconstruction) is only achievable in the strict limit G to 0, their work constructs a single, large code designed to accommodate more general semiclassical bulk states and modular flows, even when G is not infinitesimally small.

Core Problem and Motivation

The central goal of the work is to return to the JLMS relation (Equation 1.1) and rigorously bound the associated error terms for a broader class of semiclassical bulk states than previously considered. The motivation stems from the observation that small codes only describe perturbative excitations near a fixed classical solution, precluding discussions of modular flows that alter the classical background. The paper seeks to overcome this by constructing a framework where modular flow effects are incorporated into a single structure.

Key Technical Innovations: The Large Code Framework

The authors introduce a sophisticated architecture based on assembling smaller codes:

1. Hierarchical Structure: The large code is not monolithic but is assembled from many small codes. Each constituent small code satisfies an approximate Faulkner-Lewkowycz-Maldacena (FLM) formula.

2. Accommodation of Flows: This large structure is specifically designed to accommodate modular flows and, crucially, in the G to 0 limit, it includes superpositions of states associated with distinct classical backgrounds.

3. Error Control Mechanisms: To manage the inherent approximations arising from this construction and the underlying physics, the authors introduce new concepts:

• 'Log-stability': A condition imposed on bulk states to control certain error terms.

• 'Approximate subsystem orthogonality': A condition used to bound errors related to state overlap.

Derivation of Approximate JLMS Relations

The paper rigorously derives approximate versions of the JLMS relation in this large code setting, focusing on two main forms: projected and exponentiated.

1. Approximate Projected JLMS Relation (Theorem 3 & Theorem 5)

Theorem 3 establishes the approximate projected JLMS formula for log-aligned states within a large code. The required conditions are stringent:

• The large code must be defined by a collection of Hamiltonians H alpha that satisfy an approximate FLM formula and an approximate isometry V alpha, controlled by small parameters epsilon FLM and epsilon iso.

• The log version of approximate subsystem orthogonality must hold.

• The states (rho, sigma) must satisfy enhanced log-stability with respect to each other (controlled by epsilon tail and epsilon enhanced) and be log-aligned (controlled by epsilon l-align).

Theorem 5 generalizes this to the operator-norm version for states that are log-smooth in a specific sense, providing an operator-norm error bound of the form:

V K rho V - V V A 4 G + K rho r times epsilon pJLMS + epsilon iso + epsilon enhanced + epsilon sub-log + epsilon l-smooth

2. Approximate Exponentiated JLMS Relation (Theorem 6)

Theorem 6 addresses the exponentiated version, which is particularly relevant for spectral analysis:

• It requires an approximate R´enyi FLM formula for n = 1 + i s.

• It depends on the exponentiated version of approximate subsystem orthogonality.

• The state rho must satisfy log-stability and be exponentiated-smooth (satisfying condition (4.76)).

The resulting operator-norm formula is:

V e-is K rho V - e-is A 4 G + K rho r times epsilon(s)e JLMS+s epsilon iso + epsilon OD + epsilon sub-tr + epsilon(s) sub-exp + epsilon e-smooth

Technical Arguments and Error Analysis

The paper dedicates significant effort to justifying the approximations and bounding the errors:

• Justification of Log-Stability (Log-stability Condition 2): The authors use the gravitational path integral in AdS/CFT to show that Condition 2 of log-stability follows from Condition 1 when epsilon tail is not exponentially small in the G to 0 limit. This involves analyzing semiclassical versions of bulk traces (Tr R(rho alpha)) and showing that the required coincidence of backgrounds on a specific cut (r) is enforced by the normalization constraints, even if they differ elsewhere. The calculation demonstrates that the relevant action difference simplifies to A 4 G (n-1), leading to the desired result in the limit n to 1.

• Isometry Bounds: The approximate isometry condition (V V alpha - 1/alpha infinity = delta) is proven by showing that delta is bounded by 2 sqrt epsilon FLM, establishing the required relationship between epsilon iso and epsilon FLM.

• State Support: Lemma 8 is invoked to show that the support of sigma is contained within the support of rho, ensuring that the relevant Hilbert space for analysis can be restricted to H rho.

Conclusion and Significance

The ultimate achievement of this work is the construction of a large code where both projected and exponentiated JLMS relations hold to good accuracy, even when certain entanglement eigenvalues are exponentially small. The entire framework is built upon abstract, general codes, with AdS/CFT providing specific physical examples. The results demonstrate that the JLMS formula holds approximately with errors that vanish in the limit G to 0 under specific conditions related to log-stability and smoothness.

**In essence, this paper moves beyond the strict G to 0 requirement for exact error correction by developing a robust, hierarchical structure (the large code) that allows for the approximate recovery of entanglement wedge information across a wider class of semiclassical bulk states.

1. System can perform error correction by leveraging two-sided recovery in holographic codes, as described in Section 1, which is crucial for reconstructing bulk operators from boundary modular Hamiltonians even when exact quantum error correction is not achieved.

2. System can analyze modular flows generated by semiclassical bulk states at order-one modular parameters without being restricted to perturbative excitations near a single classical solution, enabling the consideration of modularity flows generated by semiclassical bulk states at order-one modular parameters as presented in the abstract.

3. System can construct and utilize a large code that accommodates superpositions of states associated with distinct classical backgrounds, allowing it to describe physics across multiple semiclassical geometries simultaneously in the G → 0 limit.

4. System can derive approximate JLMS relations for projected and exponentiated forms under various stability conditions, such as log-stability (Definition 4) or relative log-stability with respect to a specific test state (Definition 4b).

5. System can compute operator-norm bounds on the projected and exponentiated JLMS formulas using criteria like log-smoothness (Definition 12) for states, providing robust error estimates independent of the choice of parameters.

6. System can apply these derived relations to analyze systems where entanglement eigenvalues are exponentially small, as demonstrated by the analysis in Section 4.5 and Theorem 6, which yields error bounds dependent on ε(s) e-smooth.

Gauge/gravity duality is often described as a quantum error correcting code. However, as seen in the Jafferis-Lewkowycz-Maldacena-Suh (JLMS) formula, exact quantum error correction with complementary recovery (and thus entanglement wedge reconstruction) emerges only in the limit G to 0. As a result, precise arguments controlling error terms have focused on what we call `small' codes which, as G to 0, describe only perturbative excitations near a given classical solution. Such settings are quite restrictive and, in particular, they prohibit discussion of any modular flow that would change the classical background. As a result, they forbid consideration of modular flows generated by semiclassical bulk states at order-one modular parameters. In contrast, we present a single `large' code for the bulk theory that can accommodate such flows and, in particular, in the G to 0 limit includes superpositions of states associated with distinct classical backgrounds. This large code is assembled from small codes that each satisfy an approximate Faulkner-Lewkowycz-Maldacena formula. In this extended setting we clarify the meaning of the (approximate) JLMS relation between bulk and boundary modular Hamiltonians and quantify its validity in an appropriate class of states.