ResearchPod Summary
The "complexity = volume" conjecture suggests that the volume of a black hole interior is dual to the quantum state complexity of the boundary theory. Recent proposals have identified Krylov complexity as a potential candidate for this volume operator. However, a significant theoretical tension exists: Krylov complexity is typically defined using the entire energy spectrum (the UV Hilbert space), whereas the Heisenberg time—the point at which complexity should saturate—depends on the entropy of the specific low-energy state. This paper investigates whether this global definition of Krylov complexity can correctly reproduce the expected saturation behavior at the energy-dependent Heisenberg time.
The authors analyze the dynamics of Krylov state complexity using the theory of orthogonal polynomials on discrete energy spectra. By treating the Krylov basis as a set of orthonormal polynomials, they examine how these polynomials behave near the edges of the spectrum. They derive a rigorous bound on the number of Krylov basis states required to represent a low-energy state, effectively showing that the basis "crowds out" or saturates once the Krylov index exceeds the number of states available at that specific energy level.
The study proves that for any low-energy state, the Krylov complexity stops growing shortly after the Heisenberg time, regardless of the fact that the operator is sensitive to the entire UV spectrum. This is achieved by demonstrating that the Krylov polynomials are effectively localized in energy space; they do not "see" the high-energy states when the system is in a low-energy configuration. In the context of the DSSYK model, this confirms that the Krylov-based length operator correctly saturates at the expected scale, providing a robust link between non-perturbative quantum gravity and quantum information theory.
This work resolves a major conceptual hurdle in the "complexity = volume" program. By proving that a simple, global definition of Krylov complexity naturally respects the energy-dependent saturation time, the authors provide a stronger theoretical foundation for using Krylov complexity as a proxy for geometric observables in quantum gravity. It suggests that the holographic duality between gravity and quantum complexity is more robust than previously thought, as it does not require fine-tuned, state-dependent definitions of the complexity operator to match semiclassical expectations.
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