Relevant OTOC operators: footprints of the classical dynamics

TL;DR

This paper investigates the relationship between out-of-time order correlators (OTOCs) and quantum entropy in coupled Arnold cat maps, showing that a small set of relevant operators can effectively reveal classical dynamical footprints and complexity.

Contribution

It demonstrates that summing over a small set of relevant operators accurately approximates entropy and uncovers classical dynamics, extending the OTOC-RE theorem to physically meaningful bases.

Findings

Small sets of operators suffice to approximate entropy.

Different operator bases reveal classical dynamical footprints.

Scaling of relevant operators indicates quantum complexity.

Abstract

The out-of-time order correlator (OTOC) has recently become relevant in different areas where it has been linked to scrambling of quantum information and entanglement. It has also been proposed as a good indicator of quantum complexity. In this sense, the OTOC-RE theorem relates the OTOCs summed over a complete base of operators to the second Renyi entropy. Here we have studied the OTOC-RE correspondence on physically meaningful bases like the ones constructed with the Pauli, reflection, and translation operators. The evolution is given by a paradigmatic bi-partite system consisting of two perturbed and coupled Arnold cat maps with different dynamics. We show that the sum over a small set of relevant operators, is enough in order to obtain a very good approximation for the entropy and hence to reveal the character of the dynamics, up to a time t 0 . In turn, this provides with an alternative natural indicator of complexity, i.e. the scaling of the number of relevant operators with time. When represented in phase space, each one of these sets reveals the classical dynamical footprints with different depth according to the chosen base.