In this paper we will attempt to answer the following question: what are the natural quantum subsystems which emerge out of a system's dynamical laws? To answer this question we first define generalized tensor product structures (gTPS) in terms of observables, as dual pairs of an operator subalgebra A and its commutant. Second, we propose an operational criterion of minimal information scrambling at short time scales to dynamically select gTPS. In this way the emergent subsystems are those which maintain the longest informational identity. This strategy is made quantitative by defining a Gaussian scrambling rate in terms of the short-time expansion of an algebraic version of the Out of Time Order Correlation (OTOC) function i.e., the A -OTOC. The Gaussian scrambling rate is computed analytically for physically important cases of general division into subsystems, and is shown to have an intuitive and compelling physical interpretation in terms of minimizing the interaction strength between subsystems.
In this paper we will attempt to answer the following question: what are the natural quantum subsystems which emerge out of a system's dynamical laws? To answer this question we first define generalized tensor product structures (gTPS) in terms of observables, as dual pairs of an operator subalgebra A and its commutant. Second, we propose an operational criterion of minimal information scrambling at short time scales to dynamically select gTPS. In this way the emergent subsystems are those which maintain the longest informational identity. This strategy is made quantitative by defining a Gaussian scrambling rate in terms of the short-time expansion of an algebraic version of the Out of Time Order Correlation (OTOC) function i.e., the A -OTOC. The Gaussian scrambling rate is computed analytically for physically important cases of general division into subsystems, and is shown to have an intuitive and compelling physical interpretation in terms of minimizing the interaction strength between subsystems.
Much recent work has been devoted to the study of information scrambling in quantum systems. In this paper, we study the long-time properties of the algebraic out-of-time-order-correlator (``$\mathcal{A}\text{\ensuremath{-}}\mathrm{OTOC}$'') and derive an analytical expression for its long-time average under the nonresonance condition. The $\mathcal{A}\text{\ensuremath{-}}\mathrm{OTOC}$ quantifies quantum scrambling with respect to degrees of freedom described by an operator subalgebra A, which is associated with a partitioning of the corresponding system into a generalized tensor product structure. Recently, the short-time growth of the $\mathcal{A}\text{\ensuremath{-}}\mathrm{OTOC}$ was proposed as a criterion to determine which partition arises naturally from the system's unitary dynamics. In this paper, we extend this program to the long-time regime where the long-time average of the $\mathcal{A}\text{\ensuremath{-}}\mathrm{OTOC}$ serves as the metric of subsystem emergence. Under this framework, natural system partitions are characterized by the tendency to minimally scramble information over long timescales. We consider several physical examples, ranging from quantum many-body systems and stabilizer codes to quantum reference frames, and perform the minimization of the $\mathcal{A}\text{\ensuremath{-}}\mathrm{OTOC}$ long-time average both analytically and numerically over relevant families of algebras. For simple cases subject to the nonresonant condition, minimal $\mathcal{A}\text{\ensuremath{-}}\mathrm{OTOC}$ long-time average is shown to be related to minimal entanglement of the Hamiltonian eigenstates across the emergent system partition. Finally, we conjecture and provide evidence for a general structure of the algebra that minimizes the average for nonresonant Hamiltonians.
In this paper we describe an algebraic/geometrical approach to quantum scrambling. Generalized quantum subsystems are described by an hermitian-closed unital subalgebra 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 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 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 A and the full system of Hamiltonian eigenstates. Finally, a notion of A -chaoticity is suggested.
We assess the possibilities offered by Hilbert space fundamentalism, an attitude towards quantum physics according to which all physical structures (e.g. subsystems, locality, spacetime, preferred observables) should emerge from minimal quantum ingredients (typically a Hilbert space, Hamiltonian, and state). As a case study, we first mainly focus on the specific question of whether the Hamiltonian can uniquely determine a tensor product structure, a crucial challenge in the growing field of quantum mereology. The present paper reviews, clarifies, and critically examines two apparently conflicting theorems by Cotler et al. and Stoica. We resolve the tension, show how the former has been widely misinterpreted and why the latter is correct only in some weaker version. We then propose a correct mathematical way to address the general problem of preferred structures in quantum theory, relative to the characterization of emergent objects by unitary-invariant properties. Finally, we apply this formalism in the particular case we started with, and show that a Hamiltonian and a state are enough structure to uniquely select a preferred tensor product structure.
In quantum many-body systems, complex dynamics delocalize the physical degrees of freedom. This spreading of information throughout the system has been extensively studied in relation to quantum thermalization, scrambling, and chaos. Locality is typically defined with respect to a tensor product structure (TPS) which identifies the local subsystems of the quantum system. In this paper, we investigate a simple geometric measure of operator spreading by quantifying the distance of the space of local operators from itself evolved under a unitary channel. We show that this TPS distance is related to the scrambling properties of the dynamics between the local subsystems and coincides with the entangling power of the dynamics in the case of a symmetric bipartition. Additionally, we provide sufficient conditions for the maximization of the TPS distance and show that the class of 2-unitaries provides examples of dynamics that achieve this maximal value. For Hamiltonian evolutions at short times, the characteristic timescale of the TPS distance depends on scrambling rates determined by the strength of interactions between the local subsystems. Beyond this short-time regime, the behavior of the TPS distance is explored through numerical simulations of prototypical models exhibiting distinct ergodic properties, ranging from quantum chaos and integrability to Hilbert space fragmentation and localization.
This tutorial article introduces the physics of quantum information scrambling in quantum many-body systems. The goals are to understand how to precisely quantify the spreading of quantum information and how causality emerges in complex quantum systems. We introduce a general framework to study the dynamics of quantum information, including detection and decoding. We show that the dynamics of quantum information is closely related to operator dynamics in the Heisenberg picture, and, under certain circumstances, can be precisely quantified by the so-called out-of-time-ordered correlator (OTOC). The general behavior of the OTOC is discussed based on several toy models, including the Sachdev-Ye-Kitaev model, random circuit models, and Brownian models, in which the OTOC is analytically tractable. We introduce numerical methods, including exact diagonalization and tensor network methods, to calculate the OTOC for generic quantum many-body systems. We also survey current experimental schemes for measuring the OTOC in various quantum simulators. Published by the American Physical Society 2024
In this work, we develop a quantum metrological framework for quantum chaos by showing that local subsystems of information scrambling systems naturally function as quantum stopwatches. The reduced quantum state of a subsystem encodes the passage of time through its growing distinguishability from the initial preparation. Treating time as the estimation parameter, we then derive a generalized quantum Cramer-Rao bound that directly relates the precision of time estimation to the decay of out-of-time ordered correlators (OTOCs) and subsystem quantum Fisher information (QFI). As a main result, we obtain continuity bounds for quantum Lyapunov exponent in terms of the subsystem QFI in quantumly chaotic dynamics. Furthermore, using a scaling analysis based on imaginary-time correlators, we show that the subsystem QFI exhibits universal critical amplification near quantum phase transitions. Our results are demonstrated and verified by a numerical analysis of the dynamics of a chaotic Ising chain.