Quantum scrambling of observable algebras
Paolo Zanardi
Abstract
In this paper we describe an algebraic/geometrical approach to quantum scrambling. Generalized quantum subsystems are described by an hermitian-closed unital subalgebra <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi></mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi></mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi></mml:math>. 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi></mml:math> and the full system of Hamiltonian eigenstates. Finally, a notion of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi></mml:mrow></mml:math>-chaoticity is suggested.