We investigate circuit complexity to characterize chaos in multiparticle quantum systems. In the process, we take a stride to analyze open quantum systems by using complexity. We propose a new diagnostic of quantum chaos from complexity based on the reduced density matrix by exploring different types of quantum circuits. Through explicit calculations on a toy model of two coupled harmonic oscillators, where one or both of the oscillators are inverted, we demonstrate that the evolution of complexity is a possible diagnostic of chaos.
We study the connections between three quantities that can be used as diagnostics for quantum chaos, i.e., the out-of-time-order correlator (OTOC), Loschmidt echo (LE), and complexity. We generalize the connection between OTOC and LE for infinite dimensions and extend it for higher-order OTOCs and multi-fold LEs. Novel applications of this intrinsic relation are proposed. We also propose a relationship between a specific circuit complexity and LE by using the inverted oscillator model. These relationships signal a deeper connection between these three probes of quantum chaos.
Quantum Chaos Diagnostics for non-Hermitian Systems from Bi-Lanczos Krylov Dynamics
Shanghai Jiao Tong University（上海交通大学）；Shanghai Research Center for Quantum Sciences（上海量子科学研究中心）；Instituto de Física Teórica UAM/CSIC（UAM/CSIC理论物理研究所）；Asia Pacific Center for Theoretical Physics（亚太理论物理中心）；Pohang University of Science and Technology（韩国科学技术院）；Technical University of Munich（慕尼黑工业大学）；Ludwig-Maximilians-Universität München（慕尼黑大学）；Gwangju Institute of Science and Technology（光州科学技术研究院）
We study three prominent diagnostics of chaos and scrambling in the context of two-dimensional conformal field theory: the spectral form factor, out-of-time-ordered correlators, and unitary operator entanglement. With the observation that all three quantities may be obtained by different analytic continuations of the torus partition function, we address the connections and distinctions between the information that each quantity provides us. In this process, we study the emergence of irrationality from "large-N" limits of rational conformal field theories (RCFTs) as well as the explicit breakdo…
We propose a new diagnostic for quantum chaos. We show that time evolution of complexity for a particular type of target state can provide equivalent information about the classical Lyapunov exponent and scrambling time as out-of-time-order correlators. Moreover, for systems that can be switched from a regular to unstable (chaotic) regime by a tuning of the coupling constant of the interaction Hamiltonian, we find that the complexity defines a new time scale. We interpret this time scale as recording when the system makes the transition from regular to chaotic behaviour.
Topological Control of Quantum Chaos Diagnostics: OTOCs, Spectral Statistics, and Information Scrambling in Ising Model
School of Quantum Physics and Matter Institute for Research in Fundamental Sciences (IPM)（量子物理与物质学院，基础科学研究院（IPM））；Ershad Damavand, Institute of Higher Education (EDI)（厄尔沙德达马万德高等教育研究所（EDI））；Department of Mathematics and Computer Science, Amirkabir University of Technology（阿米尔卡比尔理工大学数学与计算机科学系）；Department of Mathematics, Faculty of Mathematical Sciences, Alzahra University（阿尔扎哈拉大学数学科学学院数学系）；Department of Physics, CT.C, Islamic Azad University（伊斯兰阿扎德大学CT.C物理学系）；Institute of Biosocial and Quantum Science and Technologies, CT.C, Islamic Azad University（伊斯兰阿扎德大学CT.C生物社会与量子科学与技术研究所）
We establish a rigorous connection between quantum coherence and quantum chaos by employing coherence measures originating from the resource theory framework as a diagnostic tool for quantum chaos. We quantify this connection at two different levels: quantum states and quantum channels. At the level of states, we show how several well-studied quantifiers of chaos are, in fact, quantum coherence measures in disguise (or closely related to them). We further this connection for all quantum coherence measures by using tools from majorization theory. Then, we numerically study the coherence of chao…
It is believed that thermalization drives the reduced density matrix of a subsystem to approach a short-range entangled operator. If the initial state is also short-range entangled, it is possible that the reduced density matrix remains low-entangled throughout thermalization; or there could exist a barrier with high operator entanglement between the initial and thermalized reduced density matrix. In this paper, we study such dynamics in three classes of models: the rational CFTs, the random unitary circuit, and the holographic CFTs, representing systems of increasing quantum chaoticity. We sh…
We use results on Virasoro conformal blocks to study chaotic dynamics in CFT$_2$ at large central charge c. The Lyapunov exponent $λ_L$, which is a diagnostic for the early onset of chaos, receives $1/c$ corrections that may be interpreted as $λ_L = \frac{2 π}β \left( 1 + \frac{12}{c} \right)$. However, out of time order correlators receive other equally important $1/c$ suppressed contributions that do not have such a simple interpretation. We revisit the proof of a bound on $λ_L$ that emerges at large $c$, focusing on CFT$_2$ and explaining why our results do not conflict with the analysis le…
We study chaos and scrambling in unitary channels by considering their entanglement properties as states. Using out-of-time-order correlation functions to diagnose chaos, we characterize the ability of a channel to process quantum information. We show that the generic decay of such correlators implies that any input subsystem must have near vanishing mutual information with almost all partitions of the output. Additionally, we propose the negativity of the tripartite information of the channel as a general diagnostic of scrambling. This measures the delocalization of information and is closely…
We study the Eigenstate Thermalization Hypothesis (ETH) in chaotic conformal field theories (CFTs) of arbitrary dimensions. Assuming local ETH, we compute the reduced density matrix of a ball-shaped subsystem of finite size in the infinite volume limit when the full system is an energy eigenstate. This reduced density matrix is close in trace distance to a density matrix, to which we refer as the ETH density matrix, that is independent of all the details of an eigenstate except its energy and charges under global symmetries. In two dimensions, the ETH density matrix is universal for all theori…
The spectral form factor (SFF), characterizing statistics of energy eigenvalues, is a key diagnostic of many-body quantum chaos. In addition, partial spectral form factors (PSFFs) can be defined which refer to subsystems of the many-body system. They provide unique insights into energy eigenstate statistics of many-body systems, as we show in an analysis on the basis of random matrix theory and of the eigenstate thermalization hypothesis. We propose a protocol that allows the measurement of the SFF and PSFFs in quantum many-body spin models, within the framework of randomized measurements. Aim…
The exponential growth or decay with time of the out-of-time-order commutator (OTOC) is one widely used diagnostic of many-body chaos in spatially-extended systems. In studies of many-body classical chaos, it has been noted that one can define a velocity-dependent Lyapunov exponent, $λ({\bf v})$, which is the growth or decay rate along "rays" at that velocity. We examine the behavior of $λ({\bf v})$ for a variety of many-body systems, both chaotic and integrable. The so-called light cone for the spreading of operators is defined by $λ({\bf \hat n}v_B({\bf \hat n}))=0$, with a generally directi…
We apply different integrability analysis procedures to a reduced (spatially homogeneous) mechanical system derived from an off-shell non-minimally coupled N=2 Maxwell-Chern-Simons-Higgs model that presents BPS topological vortex excitations, numerically obtained with an ansatz adopted in a special - critical coupling - parametric regime. As a counterpart of the regularity associated to the static soliton-like solution, we investigate the possibility of chaotic dynamics in the evolution of the spatially homogeneous reduced system, descendant from the full N=2 model under consideration. The ori…
We examine the $5d$ Yang-Mills matrix model in $0+1$-dimensions with $U(4N)$ gauge symmetry and a mass deformation term. We determine the explicit $SU(4)\approx SO(6)$ equivariant parametrizations of the gauge field and the fluctuations about the classical four concentric fuzzy four sphere configuration and obtain the low energy reduced actions(LEAs) by tracing over the $S_F^4$s for the first five lowest matrix levels. The LEA's so obtained have potentials bounded from below indicating that the equivariant fluctuations about the $S_F^4$ do not lead to any instabilities. These reduced systems e…
The harmonic oscillator is the paragon of physical models; conceptually and computationally simple, yet rich enough to teach us about physics on scales that span classical mechanics to quantum field theory. This multifaceted nature extends also to its inverted counterpart, in which the oscillator frequency is analytically continued to pure imaginary values. In this article we probe the inverted harmonic oscillator (IHO) with recently developed quantum chaos diagnostics such as the out-of-time-order correlator (OTOC) and the circuit complexity. In particular, we study the OTOC for the displacem…
