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

TL;DR

This paper introduces a phase space framework using Wigner functions to analyze the spreading of quantum states and operators in the Krylov basis, linking complexity measures to phase space representations.

Contribution

It develops a phase space approach to express Krylov state and operator complexity as integrals over Wigner functions, connecting quantum complexity with classical and quantum phase space dynamics.

Findings

Krylov complexity can be represented as an integral over phase space.

The approach connects quantum complexity measures with harmonic expansions of Wigner functions.

The method incorporates classical Liouville dynamics and quantum corrections.

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 functions corresponding to 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 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.