A universal approach to Krylov State and Operator complexities

TL;DR

The work presents a unified framework that places Krylov state and operator complexities on the same footing by formulating complexity as a trace ${ m Tr}(oldsymbol{ extell} ho)$ of a density matrix with a Lanczos-derived label operator. It extends Krylov complexity to operator growth via a channel-state mapping onto a doubled Hilbert space, connects to Liouvillian dynamics, and provides a clear route to subregion and mixed-state complexities. The formalism naturally reproduces late-time linear growth and saturation, clarifies the holographic interpretation through doubled-state constructions and Reeh–Schlieder-type entanglement, and offers practical extensions to open systems, mutual complexity, and transition-matrix related observables. Overall, the paper establishes a versatile, density-matrix-centered approach that unifies Krylov, operator, and holographic notions of complexity, with potential replicas and wormhole interpretations guiding future investigations.

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.