Boundary Modular Evolution

• Boundary Modular Evolution is the process where quantum states or operators evolve under a modular Hamiltonian, revealing fine-grained entanglement features.
• The method employs the Lanczos algorithm to construct a Krylov basis that quantifies modular Krylov complexity from boundary data.
• This approach uncovers phenomena such as entanglement islands, Page transitions, and holographic links between boundary and bulk geometries.

Boundary Modular Evolution refers to the quantum dynamical process in which states or operators living on a boundary quantum field theory (QFT), such as a conformal field theory (CFT), evolve under the action of a modular Hamiltonian associated with a spatial region. This evolution is studied via the Krylov basis generated by the Lanczos algorithm, giving rise to the notion of modular Krylov complexity—a rigorous metric quantifying the "spread" or growth of states and operators under modular flow. Boundary modular evolution provides a computationally tractable approach for probing entanglement structure, reconstructing fine-grained spectra, and diagnosing emergent geometrical features, including quantum extremal surfaces (QES), area operators, and entanglement islands, using only boundary data and without direct reference to bulk geometry (Caputa et al., 2023, Vardian, 2 Feb 2026).

1. Modular Hamiltonian and Krylov Basis Construction

The modular Hamiltonian $K_A$ for a region $A$ in a boundary QFT is defined by $K_A = -\log \rho_A$, where $\rho_A$ is the reduced density matrix in the code subspace. Modular evolution is implemented for states as $|\psi(s)\rangle = e^{-i K s} |\psi_0\rangle$ and for operators in the Heisenberg picture via the Liouvillian $\mathcal{L}_S(O) = [K_A \otimes I_{\bar A}, O]$ for $O \in \mathcal{B}(\mathcal{H}_A)$. The Krylov basis is constructed recursively using the Lanczos algorithm:

• States: For an initial vector $|\phi_0\rangle = |\psi_0\rangle$ and $|\phi_{-1}\rangle = 0$,

$$|A_{n+1}\rangle = (K - a_n^{\mathrm{mod}}) |\phi_n\rangle - b_n^{\mathrm{mod}} |\phi_{n-1}\rangle,$$

$$|\phi_{n+1}\rangle = |A_{n+1}\rangle / b_{n+1}^{\mathrm{mod}},$$

with $$a_n^{\mathrm{mod}} = \langle \phi_n | K | \phi_n \rangle$$ and $$b_{n+1}^{\mathrm{mod}} = \lVert |A_{n+1}\rangle \rVert$$.

• Operators: Repeated modular commutators span the operator Krylov subspace: $$\mathcal{K}_O = \mathrm{span}\{O, \mathcal{L}_S O, \mathcal{L}_S^2 O, \ldots\}$$. Orthonormalization yields a three-term recurrence for the Krylov vectors $|K_n\rangle$:

$$b_{n+1}|K_{n+1}\rangle = (\mathcal{L}_S - a_n)|K_n\rangle - b_n|K_{n-1}\rangle,$$

with $a_n = \langle K_n | \mathcal{L}_S | K_n \rangle$ and $$b_n = \langle K_n | \mathcal{L}_S | K_{n-1} \rangle$$.

This tridiagonalization protocol enables explicit computation of the modular Krylov complexity and recovery of the modular spectrum.

2. Definition and Interpretation of Modular Krylov Complexity

Given an evolved state or operator expanded as $$|\psi(s)\rangle = \sum_{n=0}^{N-1} U_n(s) |\phi_n\rangle$$ or $$O(s) = \sum_{n=0}^{d_O-1} i^n \varphi_n(s) |K_n\rangle$$, the modular Krylov complexity $C(s)$ quantifies how far evolution has spread in the Krylov chain. The definition is:

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

for states, or

$C(s) = \sum_{n=0}^{d_O-1} n\, |\varphi_n(s)|^2$

for operators.

This metric captures the fine-grained structure of entanglement dynamics. For instance, in a two-level qubit system where $$|\psi_0\rangle = \sqrt{p}\,|00\rangle + \sqrt{1-p}\,|11\rangle$$, $$C_{\mathrm{mod}}(s) = 4p(1-p) \sin^2[s \log((1-p)/p)]$$ with oscillatory behavior governed by the capacity of entanglement. In higher-dimensional cases, such as random modular Hamiltonians, the complexity grows and eventually saturates, mimicking Page-curve behavior (Caputa et al., 2023, Vardian, 2 Feb 2026).

3. Spectrum Reconstruction and Area Operator Extraction

Modular Krylov complexity provides a route to reconstruct the restricted spectrum of the modular Hamiltonian and isolate its central component, the area operator, entirely from boundary data. In the operator-algebra quantum error-correction (OAQEC) framework, the JLMS relation decomposes the boundary modular generator as $$K_A \otimes I_{\bar A} = K|_{\mathcal{A}} + \mathcal{L}_A$$, where $K|_{\mathcal{A}}$ acts non-centrally and $\mathcal{L}_A$ is the central area operator.

Computing Lanczos coefficients for operator evolution and diagonalizing the resulting tridiagonal matrix yields the non-central spectrum $\{ \lambda_i \}$ of $K|_{\mathcal{A}}$. The area operator is then extracted as the difference between the diagonal elements of the full generator and these reconstructed eigenvalues:

$$\mathcal{L}_A = (K_A \otimes I)_{\mathrm{diag}} - \mathrm{diag}(\lambda_1, \ldots, \lambda_{d_O}).$$

This process is robust for code-subspace operators with full support and enables boundary-only identification of quantum extremal surfaces (QES) and their geometric area (Vardian, 2 Feb 2026).

4. Entanglement Islands and Page Transition Diagnostics

Boundary modular Krylov evolution is an effective diagnostic for entanglement island formation and Page transitions in evaporating black hole systems. By monitoring the spectrum and central component (area operator) as a function of real time, abrupt emergence of a nonzero central $\mathcal{L}_A$ signals the growth of an entanglement island at the Page time. This is accompanied by a marked change in the pattern of Lanczos coefficients and a kink or saturation in the complexity $C(s)$.

Two diagnostic signatures are:

• Sudden jump in $\mathcal{L}_A$ as a function of boundary time $T$ (identifying the QES area contribution onset).
• Transition from linear Lanczos growth in the trivial/no-island phase to a plateau in the island phase.

Explicit calculations in evaporating brane and CFT+qubit bath models validate the capacity of boundary-only Krylov reconstruction to capture the Page transition without bulk extremization (Vardian, 2 Feb 2026).

5. Relation to Entanglement Spectrum and Quantum Information

The modular Krylov spectrum encodes all moments of the modular Hamiltonian, thus being in one-to-one correspondence (up to degeneracies) with the entanglement spectrum. For example, in quantum mechanics and two-dimensional CFTs, the Renyi entropies $S_n = (1-n)^{-1} \log \operatorname{Tr} \rho_A^n$ and their analytic continuations provide the full modular evolution data. Moments $\mu_k = (-i)^k \partial^k_s S(s)|_{s=0}$ are polynomials in Lanczos coefficients and reconstruct both the entanglement entropy ($a_0^{\mathrm{mod}} = \mu_1$) and variance (capacity of entanglement, $b_1^{\mathrm{mod}}$).

Spread complexity thus acts as a fine-grained probe, surpassing the descriptive power of entropy alone, revealing the underlying quantum information-not algebraic structure (Caputa et al., 2023).

6. Universal Growth, Chaos Bounds, and Holographic Connections

At late modular times, operator Krylov complexity in 2D CFTs exhibits exponential growth governed by a universal modular Lyapunov exponent $\lambda_L^{\mathrm{mod}} = 2\pi$. The scrambling time $s_\ast$ is proportional to $\log(1/\epsilon B(u))$, dictated by the local inverse temperature profile $B(u)$ in the modular Hamiltonian $H_{\mathrm{mod}} = \int B(u) T(u) du$. Inserting the initial operator deeper into the bulk reduces the scrambling time, enhancing complexity growth rates.

In holographic contexts, the JLMS correspondence equates boundary and bulk modular Hamiltonian Krylov subspaces, suggesting that boundary modular Krylov complexity computes bulk geometric quantities, such as interior volumes of entanglement wedges: a new complexity-geometry duality. This enables investigation of black hole interiors and fine-grained emergent spacetime geometry without bulk extremization through purely boundary dynamics (Vardian, 2 Feb 2026, Caputa et al., 2023).

7. Limitations, Caveats, and Open Problems

Several technical caveats are noted:

• Code-subspace leakage can be avoided by projecting the modular Hamiltonian into the effective field theory code subspace, suppressing $O(1/N)$ errors for low-energy boundary excitations.
• The inverse Lanczos (turnpike) problem is non-unique if the Krylov chain is short or dominated by noisy moment data, complicating practical spectrum reconstruction.
• Operator Krylov complexity is directly holographic; state Krylov complexity is sensitive to central phase effects.
• Extensions include boundary modular flows combined with physical Hamiltonian evolution, tensor-network realizations, and refined bulk complexity proposals.

A key open question is the boundary-only reconstruction of local bulk fields behind the horizon by choosing different operators in the Krylov algebra, potentially allowing resolution of fine-grained interior geometry directly from Lanczos data. This suggests a powerful link between boundary modular evolution and the entanglement structure of spacetime (Vardian, 2 Feb 2026).