A universal approach to Krylov State and Operator complexities

TL;DR

This paper introduces a unified framework for Krylov state and operator complexities using density matrices, enabling extensions to subregion and mixed state complexities, and connecting with holographic complexity concepts.

Contribution

It provides a general formalism that unifies Krylov state and operator complexities and extends them to new contexts like subregions and mixed states.

Findings

Unified framework for Krylov complexities

Extension to subregion and mixed states

Connection with holographic complexity

Abstract

We present a general framework in which both Krylov state and operator complexities can be put on the same footing. In our formalism, the Krylov complexity is defined in terms of the density matrix of the associated state which, for the operator complexity, lives on a doubled Hilbert space obtained through the channel-state map. This unified definition of complexity in terms of the density matrices enables us to extend the notion of Krylov complexity, to subregion or mixed state complexities and also naturally to the Krylov mutual complexity. We show that this framework also encompasses nicely the holographic notions of complexity.