Modular Krylov Complexity in Quantum Field Theory

• Modular Krylov complexity is a diagnostic that quantifies operator spread under modular Hamiltonian evolution, revealing fine-grained entanglement dynamics.
• It employs the Lanczos algorithm to construct an orthonormal Krylov basis from modular flow, enabling boundary extraction of bulk area operators and holographic features.
• The framework aids in detecting island formation and Page transitions in evaporating black holes by analyzing time-dependent Lanczos coefficients and complexity growth.

Modular Krylov complexity is a computationally tractable diagnostic that quantifies the spread of operators or states under modular Hamiltonian evolution in quantum field theories and holographic systems. It captures fine-grained features of the entanglement spectrum by analyzing the modular flow through a Krylov basis constructed via the Lanczos algorithm. This framework provides a boundary-based method to extract the area operator of quantum extremal surfaces and diagnose island formation as well as the Page transition in evaporating black holes, offering insight into emergent spacetime geometry without explicit reference to bulk extremization (Vardian, 2 Feb 2026, Caputa et al., 2023).

1. Modular Hamiltonians and Operator-Algebraic Structure

For a pure code-subspace state $|\Psi\rangle$ in the CFT Hilbert space $\mathcal{H}_{\text{CFT}}$ and a region $A$, the reduced density matrix is $$\rho_A = \mathrm{Tr}_{\bar{A}} |\Psi\rangle\langle\Psi|$$, defining the modular Hamiltonian $K_A = -\log \rho_A$. In the context of AdS/CFT, the JLMS relation and the operator-algebra quantum error-correction (OAQEC) structure lead to the decomposition

$$K_A \otimes I_{\bar{A}} = K|_{\mathcal{A}} + \mathcal{L}_A,$$

where $\mathcal{A}$ is the von Neumann algebra of boundary operators reconstructing the entanglement wedge bulk algebra; $K|_{\mathcal{A}} = -\log (\rho_A|_\mathcal{A})$ is the noncentral “bulk-fields” part, and the central contribution $\mathcal{L}_A$ is the area operator of the quantum extremal surface, satisfying $\mathcal{L}_A = \frac{\hat{A}_X}{4G_N}$ and commuting with all operators in $\mathcal{A}$ (Vardian, 2 Feb 2026).

2. Construction of the Modular Krylov Basis

The Krylov basis in operator Hilbert–Schmidt space is constructed by repeated nested commutators under modular flow. For any $O \in \mathcal{A}$, define the modular evolution:

$$O(s) = e^{is(K_A \otimes I)}(O \otimes I)e^{-is(K_A \otimes I)} = \sum_{n=0}^{\infty} \frac{(is)^n}{n!} \mathcal{L}_S^n O,$$

where $\mathcal{L}_S(O) = [K_A \otimes I, O]$. By vectorizing $O \mapsto |O\rangle$ in $\mathcal{H}\otimes \mathcal{H}$, the Lanczos recursion builds an orthonormal chain:

$$| \phi_0 \rangle \propto | O \rangle, \qquad | \phi_{-1} \rangle \equiv 0,$$

$$\mathcal{L}_S | \phi_{n-1} \rangle = b_n | \phi_n \rangle + a_{n-1} | \phi_{n-1} \rangle + b_{n-1} | \phi_{n-2} \rangle,$$

where $\{a_n, b_n\}$ are the real-valued Lanczos coefficients (Vardian, 2 Feb 2026, Caputa et al., 2023).

3. Definition and Properties of Modular Krylov Complexity

Modular Krylov complexity for operators is defined via the expansion

$$|O(s)\rangle = \sum_{n=0}^{N-1} i^n \psi_n(s) | \phi_n \rangle, \qquad \sum |\psi_n(s)|^2 = 1,$$

with

$C_K(s) = \sum_{n=0}^{N-1} n |\psi_n(s)|^2,$

measuring the average “distance” traversed along the Krylov chain under modular evolution. This framework parallels the Krylov complexity applied to states (Caputa et al., 2023), and is sensitive not merely to the overall entanglement entropy but to higher moments of the entanglement spectrum, encoded in the full set of Lanczos coefficients. For generic systems and especially in many-body chaotic or holographic regimes, $C_K(s)$ exhibits early-time quadratic growth, exponential scrambling governed by a modular Lyapunov exponent $\lambda_L^{\text{mod}}=2\pi$, and eventual saturation at a plateau corresponding to the Page curve (Caputa et al., 2023).

4. Extraction of Lanczos Coefficients from Boundary Correlators

The coefficients $\{a_n, b_n\}$ are extracted from boundary correlators such as the modular return amplitude:

$$R(s) = \langle O | O(s) \rangle = \mathrm{Tr}[ \rho_A O^\dagger e^{is K_A} O e^{-is K_A} ].$$

The Taylor expansion

$R(s) = \sum_{k=0}^\infty \frac{(is)^k}{k!} M_k,$

with $M_k = \left. \frac{d^k R}{ds^k} \right|_{s=0}$, enables reconstruction of $\{a_n, b_n\}$ via the Viswanath–Petersen moment method. This entire procedure is boundary-centric, not requiring knowledge of the bulk geometry (Vardian, 2 Feb 2026).

Quantity	Boundary Measurement	Use
Modular return $R(s)$	Boundary correlator	Extract moments $M_k$
Moments $M_k$	Taylor coefficients of $R(s)$	Viswanath–Petersen recursion for Lanczos coefficients
Lanczos $\{a_n, b_n\}$	Algorithmic reconstruction	Krylov basis, complexity, area operator computation

5. Area Operator Reconstruction and Bulk Probes

The area operator $\hat{A}_X$ of a quantum extremal surface is reconstructed as

$$\hat{A}_X = 4G_N [ K_A \otimes I - P_O K|_{\mathcal{A}} P_O ],$$

where $P_O$ projects onto the Krylov subspace generated by the chosen operator $O$. The full modular Hamiltonian $K_A \otimes I$ is determined on the boundary, and subtraction of the noncentral component $K|_{\mathcal{A}}$ (isolated using the Krylov-Lanczos machinery) yields the area operator. This process bypasses bulk extremization and applies equally to superpositions of semiclassical geometries (Vardian, 2 Feb 2026).

6. Island Formation and the Page Transition via Modular Krylov Complexity

In black hole evaporation scenarios with a CFT plus bath, one constructs the modular Hamiltonian $K_{\text{CFT}}(T)$ for the evolving reduced boundary state at time $T$. Applying the Krylov algorithm to appropriate global operators yields time-dependent $\{a_n(T), b_n(T)\}$ and area operator $\mathcal{L}_A(T)$. The behavior is as follows:

• For $T < T_{\text{Page}}$, $\mathcal{L}_A$ remains trivial, indicating no island and zero area.
• For $T > T_{\text{Page}}$, a nonzero eigenvalue appears in $\mathcal{L}_A$, signifying the formation of an entanglement island inside the horizon and an area jump matching bulk predictions.

This probe utilizes only boundary modular evolution and Krylov data, avoiding bulk calculations. The growth and plateau of $C_K(s)$ across $T_{\text{Page}}$ coincide with the Page curve transition and the emergence of interior geometry (Vardian, 2 Feb 2026).

7. Relationship to Entanglement Spectrum and Holographic Applications

All Lanczos coefficients and thus modular Krylov complexity are determined by the full entanglement spectrum $\{\lambda_j\} = \text{Spec}(K_A)$, accessible via Rényi partition functions and their analytic continuations:

$$S(s) = \langle \Psi_0 | e^{-iK s} | \Psi_0 \rangle = \text{Tr}( \rho_A^{1 - is} ) = Z(1 - is ),$$

where $Z(n) = \text{Tr}(\rho_A^n)$. Lower-order coefficients recover the von Neumann entropy and capacity of entanglement, while higher orders encode further spectral data. This framework confirms that entanglement entropy alone is insufficient; only the entire spectrum suffices to determine complexity dynamics (Caputa et al., 2023). In holography, modular Krylov complexity provides a boundary-accessible probe of bulk entanglement structure, area operators, and emergent spacetime phenomena.

8. Summary and Future Directions

Modular Krylov complexity, combining operator-algebra quantum error-correction structures and boundary modular flow analysis, enables boundary reconstruction of central geometric quantities and dynamics previously considered accessible only via bulk geometrical extremization. Its sensitivity to the full entanglement spectrum marks it as a refined diagnostic for quantum information, many-body chaos, and holographic bulk reconstruction. Its application to time-dependent scenarios, such as black hole evaporation, establishes a concrete method for detecting island phases and Page transitions from boundary data alone (Vardian, 2 Feb 2026, Caputa et al., 2023). A plausible implication is a broader utility of modular Krylov diagnostics in characterizing boundary-to-bulk emergence and entanglement phase transitions.