A phase space approach to the wavefunction and operator spreading in the Krylov basis

TL;DR

This work develops a phase-space framework for Krylov-based complexity in quantum dynamics by constructing phase-space Krylov functions via the Weyl transform and representing Krylov-state growth as a phase-space average of the time-evolved Wigner function. It extends the formalism to operators through a double Weyl transform and double phase space, establishing a Lanczos-like recursion in Liouville space and deriving explicit double-phase-space expressions for both single- and two-operator objects. The authors separate classical and quantum contributions to the growth of Krylov state and operator complexities using the Moyal expansion, and connect these Krylov-based measures to harmonic-expansion complexities of the Wigner function. This phase-space unification provides a concrete route to compare quantum complexity growth with classical-chaos indicators, and offers a versatile toolkit for future explorations in discrete phase spaces and open systems. Overall, the paper formulates a comprehensive phase-space description of Krylov dynamics and complexity, linking Krylov coefficients, Wigner-function evolution, and double-phase-space operator representations to a common framework for time-evolution complexity.

Abstract

In the Wigner-Weyl phase space formulation of quantum mechanics, we analyse the problem of the spreading of an initial state or an initial operator under time evolution when described in terms of the Krylov basis. After constructing the phase space representations of the Krylov basis states generated by a Hamiltonian from a given initial state by using the Weyl transformation, we subsequently use them to cast the Krylov state complexity as an integral over the phase space in terms of the Wigner function of the time-evolved initial state, so that the contribution of the classical Liouville equation and higher-order quantum corrections to the Wigner function time evolution equation towards the Krylov state complexity can be identified. Next, we construct the double phase space functions associated with the Krylov basis for the operators by using a suitable generalisation of the Weyl transformation applicable for superoperators, and use them to rewrite the Krylov operator complexity as an integral over the double phase space in terms of a generalisation of the usual Wigner function. These results, in particular, show that the complexity measures based on the expansion of a time-evolved state (or an operator) in the Krylov basis can be thought to belong to a general class of complexity measures constructed from the expansion coefficients of the time-dependent Wigner function in an orthonormal basis in the phase space, and help us to connect these complexity measures with measures of complexity of time-evolved state based on harmonic expansion of the time-dependent Wigner function.