Krylov's State Complexity and Information Geometry in Qubit Dynamics

TL;DR

This paper compares Krylov's state complexity with an information-geometric measure for qubit dynamics, revealing they capture different aspects of quantum evolution and are fundamentally complementary.

Contribution

It formulates Krylov complexity in geometric terms and contrasts it with IG complexity, highlighting their distinct interpretations in quantum state evolution.

Findings

Krylov complexity measures the directional spread of quantum states.

IG complexity reflects the volume of the trajectory on the Bloch sphere.

The two measures behave differently, emphasizing different dynamical features.

Abstract

We compare Krylov's state complexity with an information-geometric (IG) measure of complexity for the quantum evolution of two-level systems. Focusing on qubit dynamics on the Bloch sphere, we analyze evolutions generated by stationary and nonstationary Hamiltonians, corresponding to geodesic and nongeodesic trajectories. We formulate Krylov complexity in geometric terms, both instantaneously and in a time-averaged sense, and contrast it with an IG complexity of quantum evolutions characterized in terms of efficiency and curvature. We show that the two measures reflect fundamentally different aspects of quantum dynamics: Krylov's state complexity quantifies the directional spread of the evolving state relative to the initial state, whereas the IG complexity captures the effective volume explored along the trajectory on the Bloch sphere. This geometric distinction explains their inequivalent behavior and highlights the complementary nature of state-based and information-geometric notions of complexity in quantum systems.