Quantum scrambling of observable algebras

Abstract

In this paper we describe an algebraic/geometrical approach to quantum scrambling. Generalized quantum subsystems are described by an hermitian-closed unital subalgebra $\mathcal{A}$ 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 $\mathcal{A}$ 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 $\mathcal{A}$. 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 $\mathcal{A}$ and the full system of Hamiltonian eigenstates. Finally, a notion of $\mathcal{A}$-chaoticity is suggested.

6+3 pages. accepted version, to appear in Quantum

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• Scrambling and operator entanglement in local non-Hermitian quantum systems
• Operational Quantum Mereology and Minimal Scrambling
• Operator Space Entangling Power of Quantum Dynamics and Local Operator Entanglement Growth in Dual-Unitary Circuits
• Coherence generation, symmetry algebras and Hilbert space fragmentation
• An operational definition of quantum information scrambling
• Long-time Quantum Scrambling and Generalized Tensor Product Structures
• Mutual averaged non-commutativity of quantum operator algebras
• Frobenius light cone and the shift unitary
• Quantum speed limit for the OTOC from an open systems perspective
• Tensor Product Structure Geometry under Unitary Channels