Superadditivity of Krylov Complexity for Tensor Products

TL;DR

This paper proves that Krylov complexity is superadditive for tensor product systems, introduces a graph-based representation to analyze operator growth, and explains the geometric origin of excess complexity.

Contribution

It establishes superadditivity of Krylov complexity for tensor product Hamiltonians and introduces a Krylov graph framework for understanding operator growth.

Findings

Krylov complexity is superadditive under tensor products.

A positive operator quantifies excess complexity.

Diffusion on a lattice explains superadditivity.

Abstract

We study Krylov complexity for quantum systems whose Hamiltonians factorise as tensor products. We prove that complexity is superadditive under tensor products, $C_{12}\ge C_1+C_2$, and identify a positive operator that quantifies the resulting excess complexity. The underlying mechanism is made transparent by introducing a Krylov graph representation in which tensor products generate a higher-dimensional lattice whose diagonal shells encode operator growth and binomial path multiplicities. In the continuum limit, Krylov dynamics reduces to diffusion on this graph, with superadditivity arising from geometric broadening across shells. Explicit examples illustrate how deviations from synchronous evolution generate bounded, oscillatory excess complexity.