Out-of-time-ordered correlators (OTOCs) have been extensively studied in recent years as a diagnostic of quantum information scrambling. In this paper, we study quantum information-theoretic aspects of the regularized finite-temperature OTOC. We introduce analytical results for the bipartite regularized OTOC (BROTOC): the regularized OTOC averaged over random unitaries supported over a bipartition. We show that the BROTOC has several interesting properties, for example, it quantifies the purity of the associated thermofield double state and the operator purity of the analytically continued time-evolution operator. At infinite-temperature, it reduces to one minus the operator entanglement of the time-evolution operator. In the zero-temperature limit and for nondegenerate Hamiltonians, the BROTOC probes the groundstate entanglement. By computing long-time averages, we show that the equilibration value of the BROTOC is intimately related to eigenstate entanglement. Finally, we numerically study the equilibration value of the BROTOC for various physically relevant Hamiltonian models and comment on its ability to distinguish integrable and chaotic dynamics.Out-of-time-ordered correlators (OTOCs) have been extensively studied in recent years as a diagnostic of quantum information scrambling. In this paper, we study quantum information-theoretic aspects of the regularized finite-temperature OTOC. We introduce analytical results for the bipartite regularized OTOC (BROTOC): the regularized OTOC averaged over random unitaries supported over a bipartition. We show that the BROTOC has several interesting properties, for example, it quantifies the purity of the associated thermofield double state and the operator purity of the analytically continued time-evolution operator. At infinite-temperature, it reduces to one minus the operator entanglement of the time-evolution operator. In the zero-temperature limit and for nondegenerate Hamiltonians, the BROTOC probes the groundstate entanglement. By computing long-time averages, we show that the equilibration value of the BROTOC is intimately related to eigenstate entanglement. Finally, we numerically study the equilibration value of the BROTOC for various physically relevant Hamiltonian models and comment on its ability to distinguish integrable and chaotic dynamics.

arXiv:2111.07086QuantumJun 27, 202213 citations

Information Scrambling over Bipartitions: Equilibration, Entropy Production, and Typicality

Georgios Styliaris, Namit Anand, Paolo Zanardi A

In recent years, the out-of-time-order correlator (OTOC) has emerged as a diagnostic tool for information scrambling in quantum many-body systems. Here, we present exact analytical results for the OTOC for a typical pair of random local operators supported over two regions of a bipartition. Quite remarkably, we show that this "bipartite OTOC" is equal to the operator entanglement of the evolution, and we determine its interplay with entangling power. Furthermore, we compute long-time averages of the OTOC and reveal their connection with eigenstate entanglement. For Hamiltonian systems, we uncover a hierarchy of constraints over the structure of the spectrum and elucidate how this affects the equilibration value of the OTOC. Finally, we provide operational significance to this bipartite OTOC by unraveling intimate connections with average entropy production and scrambling of information at the level of quantum channels.In recent years, the out-of-time-order correlator (OTOC) has emerged as a diagnostic tool for information scrambling in quantum many-body systems. Here, we present exact analytical results for the OTOC for a typical pair of random local operators supported over two regions of a bipartition. Quite remarkably, we show that this "bipartite OTOC" is equal to the operator entanglement of the evolution, and we determine its interplay with entangling power. Furthermore, we compute long-time averages of the OTOC and reveal their connection with eigenstate entanglement. For Hamiltonian systems, we uncover a hierarchy of constraints over the structure of the spectrum and elucidate how this affects the equilibration value of the OTOC. Finally, we provide operational significance to this bipartite OTOC by unraveling intimate connections with average entropy production and scrambling of information at the level of quantum channels.

arXiv:2007.08570Physical Review LettersJan 20, 202177 citations

Proposal for measuring out-of-time-ordered correlators at finite temperature with coupled spin chains

Bhuvanesh Sundar, Andreas Elben, Lata Kh Joshi +1 more

Abstract Information scrambling, which is the spread of local information through a system’s many-body degrees of freedom, is an intrinsic feature of many-body dynamics. In quantum systems, the out-of-time-ordered correlator (OTOC) quantifies information scrambling. Motivated by experiments that have measured the OTOC at infinite temperature and a theory proposal to measure the OTOC at finite temperature using the thermofield double state, we describe a protocol to measure the OTOC in a finite temperature spin chain that is realized approximately as one half of the ground state of two moderately-sized coupled spin chains. We consider a spin Hamiltonian with particle–hole symmetry, for which we show that the OTOC can be measured without needing sign-reversal of the Hamiltonian. We describe a protocol to mitigate errors in the estimated OTOC, arising from the finite approximation of the system to the thermofield double state. We show that our protocol is also robust to main sources of decoherence in experiments.Abstract Information scrambling, which is the spread of local information through a system’s many-body degrees of freedom, is an intrinsic feature of many-body dynamics. In quantum systems, the out-of-time-ordered correlator (OTOC) quantifies information scrambling. Motivated by experiments that have measured the OTOC at infinite temperature and a theory proposal to measure the OTOC at finite temperature using the thermofield double state, we describe a protocol to measure the OTOC in a finite temperature spin chain that is realized approximately as one half of the ground state of two moderately-sized coupled spin chains. We consider a spin Hamiltonian with particle–hole symmetry, for which we show that the OTOC can be measured without needing sign-reversal of the Hamiltonian. We describe a protocol to mitigate errors in the estimated OTOC, arising from the finite approximation of the system to the thermofield double state. We show that our protocol is also robust to main sources of decoherence in experiments.

arXiv:2107.02196New Journal of PhysicsJan 28, 202217 citations

Information scrambling and chaos in open quantum systems

Paolo Zanardi A, Namit Anand

Out-of-time-ordered correlators (OTOCs) have been extensively used over the past few years to study information scrambling and quantum chaos in many-body systems. In this paper, we extend the formalism of the averaged bipartite OTOC of Styliaris et al. [Phys. Rev. Lett. 126, 030601 (2021)] to the case of open quantum systems. The dynamics is no longer unitary but it is described by more general quantum channels (trace preserving, completely positive maps). This ``open bipartite OTOC'' can be treated in an exact analytical fashion and is shown to amount to a distance between two quantum channels. Moreover, our analytical form unveils competing entropic contributions from information scrambling and environmental decoherence such that the latter can obfuscate the former. To elucidate this subtle interplay, we analytically study special classes of quantum channels, namely, dephasing channels, entanglement-breaking channels, and others. Finally, as a physical application we numerically study dissipative many-body spin chains and show how the competing entropic effects can be used to differentiate between integrable and chaotic regimes.Out-of-time-ordered correlators (OTOCs) have been extensively used over the past few years to study information scrambling and quantum chaos in many-body systems. In this paper, we extend the formalism of the averaged bipartite OTOC of Styliaris et al. [Phys. Rev. Lett. 126, 030601 (2021)] to the case of open quantum systems. The dynamics is no longer unitary but it is described by more general quantum channels (trace preserving, completely positive maps). This ``open bipartite OTOC'' can be treated in an exact analytical fashion and is shown to amount to a distance between two quantum channels. Moreover, our analytical form unveils competing entropic contributions from information scrambling and environmental decoherence such that the latter can obfuscate the former. To elucidate this subtle interplay, we analytically study special classes of quantum channels, namely, dephasing channels, entanglement-breaking channels, and others. Finally, as a physical application we numerically study dissipative many-body spin chains and show how the competing entropic effects can be used to differentiate between integrable and chaotic regimes.

arXiv:2012.13172Physical Review AJun 15, 202175 citations

Experimental Measurement of Out-of-Time-Ordered Correlators at Finite Temperature

Alaina Green, Andreas Elben, C. Huerta Alderete +6 more

Out-of-time-ordered correlators (OTOCs) are a key observable in a wide range of interconnected fields including many-body physics, quantum information science, and quantum gravity. Measuring OTOCs using near-term quantum simulators will extend our ability to explore fundamental aspects of these fields and the subtle connections between them. Here, we demonstrate an experimental method to measure OTOCs at finite temperatures and use the method to study their temperature dependence. These measurements are performed on a digital quantum computer running a simulation of the transverse field Ising model. Our flexible method, based on the creation of a thermofield double state, can be extended to other models and enables us to probe the OTOC's temperature-dependent decay rate. Measuring this decay rate opens up the possibility of testing the fundamental temperature-dependent bounds on quantum information scrambling.Out-of-time-ordered correlators (OTOCs) are a key observable in a wide range of interconnected fields including many-body physics, quantum information science, and quantum gravity. Measuring OTOCs using near-term quantum simulators will extend our ability to explore fundamental aspects of these fields and the subtle connections between them. Here, we demonstrate an experimental method to measure OTOCs at finite temperatures and use the method to study their temperature dependence. These measurements are performed on a digital quantum computer running a simulation of the transverse field Ising model. Our flexible method, based on the creation of a thermofield double state, can be extended to other models and enables us to probe the OTOC's temperature-dependent decay rate. Measuring this decay rate opens up the possibility of testing the fundamental temperature-dependent bounds on quantum information scrambling.

arXiv:2112.02068Physical Review LettersApr 6, 202250 citations

Long-time quantum scrambling and generalized tensor product structures

Faidon Andreadakis, Emanuel Dallas, Paolo Zanardi A

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 $\mathcal{A}$ , 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.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 $\mathcal{A}$, 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.

arXiv:2312.13386Physical Review AMay 15, 20246 citations

\mathcal{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.