{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,15]],"date-time":"2026-06-15T10:14:44Z","timestamp":1781518484912,"version":"3.54.1"},"reference-count":42,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2022,3,11]],"date-time":"2022-03-11T00:00:00Z","timestamp":1646956800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Science Foundation","award":["PHY-1819189"],"award-info":[{"award-number":["PHY-1819189"]}]},{"DOI":"10.13039\/100000183","name":"Army Research Office","doi-asserted-by":"crossref","award":["W911NF-20-1-0075"],"award-info":[{"award-number":["W911NF-20-1-0075"]}],"id":[{"id":"10.13039\/100000183","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"short-container-title":["Quantum"],"abstract":"<jats:p>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.<\/jats:p>","DOI":"10.22331\/q-2022-03-11-666","type":"journal-article","created":{"date-parts":[[2022,3,11]],"date-time":"2022-03-11T09:59:48Z","timestamp":1646992788000},"page":"666","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":14,"title":["Quantum scrambling of observable algebras"],"prefix":"10.22331","volume":"6","author":[{"given":"Paolo","family":"Zanardi","sequence":"first","affiliation":[{"name":"Department of Physics and Astronomy, and Center for Quantum Information Science and Technology, University of Southern California, Los Angeles, California 90089-0484, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"9598","published-online":{"date-parts":[[2022,3,11]]},"reference":[{"key":"0","unstructured":"A. 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