Complexity growth and the Krylov-Wigner function

• DOI: 10.1007/jhep05(2024)264
• Publication Date: 2024-05-23T19:01:42Z
• Authors: Ritam Basu; Anirban Ganguly; Souparna Nath; Onkar Parrikar
• Keywords: High Energy Physics - Theory; Quantum Physics; Conformal Field Models in String Theory; Nuclear and particle physics. Atomic energy. Radioactivity; 0103 physical sciences; Models of Quantum Gravity; QC770-798; Lattice Models of Gravity; 01 natural sciences; Holography and Condensed Matter Physics (AdS/CMT)

Abstract

For any state in a D-dimensional Hilbert space with a choice of basis, one can define a discrete version of the Wigner function — a quasi-probability distribution which represents the state on a discrete phase space. The Wigner function can, in general, take on negative values, and the amount of negativity in the Wigner function has an operational meaning as a resource for quantum computation. In this note, we study the growth of Wigner negativity for a generic initial state under time evolution with chaotic Hamiltonians. We introduce the Krylov-Wigner function, i.e., the Wigner function defined with respect to the Krylov basis (with appropriate phases), and show that this choice of basis minimizes the early time growth of Wigner negativity in the large D limit. We take this as evidence that the Krylov basis (with appropriate phases) is ideally suited for a dual, semi-classical description of chaotic quantum dynamics at large D. We also numerically study the time evolution of the Krylov-Wigner function and its negativity in random matrix theory for an initial pure state. We observe that the negativity broadly shows three phases: it rises gradually for a time of

$$ O\left(\sqrt{D}\right) $$

O

D

, then hits a sharp ramp and finally saturates close to its upper bound of

$$ \sqrt{D} $$

D

.