Out-of-time-order correlations (OTOCs) characterize the scrambling, or delocalization, of quantum information over all the degrees of freedom of a system and thus have been proposed as a proxy for chaos in quantum systems. Recent experimental progress in measuring OTOCs calls for a more thorough understanding of how these quantities characterize complex quantum systems, most importantly in terms of the buildup of entanglement. Although a connection between OTOCs and entanglement entropy has been derived, the latter only quantifies entanglement in pure systems and is hard to access experimentally. In this work, we formally demonstrate that the multiple-quantum coherence spectra, a specific family of OTOCs well known in NMR, can be used as an entanglement witness and as a direct probe of multiparticle entanglement. Our results open a path to experimentally testing the fascinating idea that entanglement is the underlying glue that links thermodynamics, statistical mechanics, and quantum gravity.
Out-of-time-order correlations (OTOCs) characterize the scrambling, or delocalization, of quantum information over all the degrees of freedom of a system and thus have been proposed as a proxy for chaos in quantum systems. Recent experimental progress in measuring OTOCs calls for a more thorough understanding of how these quantities characterize complex quantum systems, most importantly in terms of the buildup of entanglement. Although a connection between OTOCs and entanglement entropy has been derived, the latter only quantifies entanglement in pure systems and is hard to access experimentally. In this work, we formally demonstrate that the multiple-quantum coherence spectra, a specific family of OTOCs well known in NMR, can be used as an entanglement witness and as a direct probe of multiparticle entanglement. Our results open a path to experimentally testing the fascinating idea that entanglement is the underlying glue that links thermodynamics, statistical mechanics, and quantum gravity.
Out-of-time-ordered correlation functions (OTOCs) play a crucial role in the study of thermalization, entanglement, and quantum chaos, as they quantify the scrambling of quantum information due to complex interactions. As a consequence of their out-of-time-ordered nature, OTOCs are difficult to measure experimentally. Here we propose an OTOC measurement protocol that does not rely on the reversal of time evolution and is easy to implement in a range of experimental settings. The protocol accounts for both pure and mixed initial states, and is applicable to systems that interact with environmental degrees of freedom. We demonstrate the application of our protocol by the characterization of scrambling in a periodically driven spin that exhibits quantum chaos.
The idea of the out-of-time-order correlator (OTOC) has recently emerged in the study of both condensed matter systems and gravitational systems.It not only plays a key role in investigating the holographic duality between a strongly interacting quantum system and a gravitational system, it also diagnoses the chaotic behavior of many-body quantum systems and characterizes information scrambling.Based on OTOCs, three different concepts-quantum chaos, holographic duality, and information scrambling-are found to be intimately related to each other.Despite its theoretical importance, the experimental measurement of the OTOC is quite challenging, and thus far there is no experimental measurement of the OTOC for local operators.Here, we report the measurement of OTOCs of local operators for an Ising spin chain on a nuclear magnetic resonance quantum simulator.We observe that the OTOC behaves differently in the integrable and nonintegrable cases.Based on the recent discovered relationship between OTOCs and the growth of entanglement entropy in the many-body system, we extract the entanglement entropy from the measured OTOCs, which clearly shows that the information entropy oscillates in time for integrable models and scrambles for nonintgrable models.With the measured OTOCs, we also obtain the experimental result of the butterfly velocity, which measures the speed of correlation propagation.Our experiment paves a way for experimentally studying quantum chaos, holographic duality, and information scrambling in many-body quantum systems with quantum simulators.
Out-of-time-order correlators (OTOCs) serve as a proxy for quantum information scrambling, which refers to the process where information stored locally disperses across the many-body degrees of freedom in a quantum system, rendering it inaccessible to local probes. Most experimental implementations of OTOCs to probe information scrambling rely on indirect measurements based on global observables, using techniques such as Loschmidt echoes andmultiple quantum coherences, via time-reversal evolutions. In this article, we establish a direct connection between OTOCs with global and local observables in the context of NMR experiments, where the observable is the total magnetization of the system. We conduct a numerical analysis to quantify the differences in the evolution of both magnitudes, evaluating the excitation dynamics in spin ring systems with 8 to 16 spins, using a many-body Hamiltonian and long-range interactions. Our analysis decomposes the global echo into a sum of local echoes and cross-contributions, leading to local and global OTOCs. The results indicate that, after an initial transient period, local OTOCs determine the global ones. We observe that the difference between the average of local OTOCs and the global one, as well as their fluctuations, becomes negligible as the system size increases. Thus, for large homogeneous systems, global and local OTOCs become equivalent. This behavior aligns with that observed in highly interacting or chaotic systems in several experiments.
The idea of the out-of-time-order correlator (OTOC) has recently emerged in the study of both condensed matter systems and gravitational systems.It not only plays a key role in investigating the holographic duality between a strongly interacting quantum system and a gravitational system, it also diagnoses the chaotic behavior of many-body quantum systems and characterizes information scrambling.Based on OTOCs, three different concepts-quantum chaos, holographic duality, and information scrambling-are found to be intimately related to each other.Despite its theoretical importance, the experimental measurement of the OTOC is quite challenging, and thus far there is no experimental measurement of the OTOC for local operators.Here, we report the measurement of OTOCs of local operators for an Ising spin chain on a nuclear magnetic resonance quantum simulator.We observe that the OTOC behaves differently in the integrable and nonintegrable cases.Based on the recent discovered relationship between OTOCs and the growth of entanglement entropy in the many-body system, we extract the entanglement entropy from the measured OTOCs, which clearly shows that the information entropy oscillates in time for integrable models and scrambles for nonintgrable models.With the measured OTOCs, we also obtain the experimental result of the butterfly velocity, which measures the speed of correlation propagation.Our experiment paves a way for experimentally studying quantum chaos, holographic duality, and information scrambling in many-body quantum systems with quantum simulators.
Out-of-time-ordered correlators (OTOCs) provide a powerful probe of quantum chaos and scrambling. While the Lindblad formalism is often used to describe OTOCs in open systems, microscopic mechanisms that give rise to the Lindblad equation at the level of time-ordered correlation functions do not necessarily justify this approach at the level of OTOCs. Starting from a microscopic model of a subsystem coupled to a random bath, we show that the resulting OTOC dynamics contains an additional contribution compared to the standard Lindblad approach. This contribution, induced by correlations between different replicas, allows key properties of the underlying unitary dynamics such as long-time freeness to be retained and information scrambling to be captured more faithfully. As an application, we show that this extended formalism can be used to numerically study OTOCs in large closed systems by replacing part of the system with a dissipative boundary, offering a way to study operator spreading and higher-order correlations beyond directly accessible system sizes.
Quantum many-body systems in equilibrium can be effectively characterized using the framework of quantum statistical mechanics. However, there still exist a lot of questions regarding how to understand the nonequilibrium dynamical behavior of quantum many-body systems, which are not accessible with the thermodynamic description. Experiments in quantum simulators are opening up a route toward the generation of quantum states beyond the equilibrium paradigm. As an example, in closed quantum many-body systems, dynamical quantum phase transitions act as phase transitions in time, with physical quantities becoming nonanalytic at a critical time, extending important principles such as universality to the nonequilibrium realm. Here, in a solid-state quantum simulator, we report the experimental detection of out-of-time-order correlators in the presence of nonequilibrium phase transitions with the transverse field Ising model, which are a central concept to quantify quantum information scrambling and quantum chaos. Through measuring the multiple quantum spectra, we eventually observe the buildup of quantum correlation. Further applications of this protocol could potentially enable studies of other exotic phenomena such as many-body localization and tests of the holographic duality between quantum and gravitational systems.