Quantum scrambling of observable algebras

TL;DR

This paper introduces an algebraic and geometric framework for understanding quantum scrambling, unifying various measures like OTOC and operator entanglement through a new algebraic measure called GAAC.

Contribution

It develops a novel algebraic/geometrical approach to quantum scrambling, defining the GAAC measure and connecting it to existing concepts, with explicit calculations for random unitaries.

Findings

GAAC characterizes quantum scrambling and unifies multiple measures.

Explicit formulas for GAAC in random unitaries and energy spectra.

Upper bounds and saturation conditions for GAAC values.

Abstract

In this paper we describe an algebraic/geometrical approach to quantum scrambling. Generalized quantum subsystems are described by an hermitian-closed unital subalgebra $\cal A$ of operators evolving through a unitary channel. Qualitatively, quantum scrambling is defined by how the associated physical degrees of freedom get mixed up with others by the dynamics. Quantitatively, this is accomplished by introducing a measure, the geometric algebra anti-correlator (GAAC), of the self-orthogonalization of the commutant of $\cal A$ induced by the dynamics. This approach extends and unifies averaged bipartite OTOC, operator entanglement, coherence generating power and Loschmidt echo. Each of these concepts is indeed recovered by a special choice of $\cal A$. We compute typical values of GAAC for random unitaries, we prove upper bounds and characterize their saturation. For generic energy spectrum we find explicit expressions for the infinite-time average of the GAAC which encode the relation between $\cal A$ and the full system of Hamiltonian eigenstates. Finally, a notion of ${\cal A}$-chaoticity is suggested.