2021/07/31 by Paolo Zanardi
Computer Science · Mathematics · Physics and Astronomy · #Algebra over a field #Algorithm #Bipartite graph #Centralizer and normalizer #Discrete mathematics #Eigenvalues and eigenvectors #Hamiltonian (control theory) #Hermitian matrix #Mathematics #Observable #Physics #Pure mathematics #Quantum #Quantum Computing Algorithms and Architecture #Quantum Information and Cryptography #Quantum entanglement #Quantum many-body systems #Quantum mechanics #Scrambling #Statistical physics #Subalgebra #quant-ph
paper · pdf · doi:10.22331/q-2022-03-11-666
published as Quantum 6, 666 (2022) · 6+3 pages. accepted version, to appear in Quantum
arxiv created 2022/03/08 · openalex publication_date 2022/03/11 · arxiv updated 2022/03/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
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.