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Scientia Insularum Revista de Ciencias Naturales en islas

An educated review of "Quantum work statistics, Loschmidt echo and information scrambling".

Eduardo González Padrón

Published Jan 1, 2020• 0 citations

Abstract

The formulation of quantum work statistics as a dynamical problem through the \nLoschmidt echo is at the heart of this work. An introduction to each of these \nconcepts is presented together with the notion of information scrambling, which \nextends the scope of this work to areas such as quantum chaos or even black hole \nphysics. Using the paper of A. Chenu et al. [1] as the guidelines, we first show \nthat the work statistics associated with an arbitrary driving protocol of an isolated quantum system in a generic initial state is equivalent to the Loschmidt echo \ndynamics of a purified density matrix in an enlarged Hilbert space. When the initial state is thermal, the purification leads to a thermofield double state, which is \nused to describe eternal black holes through the AdS/CFT correspondence, often \nargued to be the fastest information scramblers. The field of quantum chaotic \nsystems is shown to emerge naturally from the previous content, and a full description of it in terms of Random Matrix Theory is also presented. Numerical and \nanalytical results are finally obtained for the quantities introduced after imposing \ntime-reversal symmetry in our problem, hence selecting the Gaussian Orthogonal \nEnsemble as the framework within we shall take our averages.

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The formulation of quantum work statistics as a dynamical problem through the \nLoschmidt echo is at the heart of this work. An introduction to each of these \nconcepts is presented together with the notion of information scrambling, which \nextends the scope of this work to areas such as quantum chaos or even black hole \nphysics. Using the paper of A. Chenu et al. [1] as the guidelines, we first show \nthat the work statistics associated with an arbitrary driving protocol of an isolated quantum system in a generic initial state is equivalent to the Loschmidt echo \ndynamics of a purified density matrix in an enlarged Hilbert space. When the initial state is thermal, the purification leads to a thermofield double state, which is \nused to describe eternal black holes through the AdS/CFT correspondence, often \nargued to be the fastest information scramblers. The field of quantum chaotic \nsystems is shown to emerge naturally from the previous content, and a full description of it in terms of Random Matrix Theory is also presented. Numerical and \nanalytical results are finally obtained for the quantities introduced after imposing \ntime-reversal symmetry in our problem, hence selecting the Gaussian Orthogonal \nEnsemble as the framework within we shall take our averages.

A universal relation is established between the quantum work probability distribution of an isolated driven quantum system and the Loschmidt echo dynamics of a two-mode squeezed state. When the initial density matrix is canonical, the Loschmidt echo of the purified double thermofield state provides a direct measure of information scrambling and can be related to the analytic continuation of the partition function. Information scrambling is then described by the quantum work statistics associated with the time-reversal operation on a single copy, associated with the sudden negation of the system Hamiltonian.

A universal relation is established between the quantum work probability distribution of an isolated driven quantum system and the Loschmidt echo dynamics of a two-mode squeezed state. When the initial density matrix is canonical, the Loschmidt echo of the purified double thermofield state provides a direct measure of information scrambling and can be related to the analytic continuation of the partition function. Information scrambling is then described by the quantum work statistics associated with the time-reversal operation on a single copy, associated with the sudden negation of the system Hamiltonian.

Characterizing the work statistics of driven complex quantum systems is generally challenging because of the exponential growth with the system size of the number of transitions involved between different energy levels. We consider the quantum work distribution associated with the driving of chaotic quantum systems described by random matrix Hamiltonians and characterize exactly the work statistics associated with a sudden quench for arbitrary temperature and system size. Knowledge of the work statistics yields the Loschmidt echo dynamics of an entangled state between two copies of the system of interest, the thermofield double state. This echo dynamics is dictated by the spectral form factor. We discuss its relation to frame potentials and its use to assess information scrambling.

Information scrambling, the process by which quantum information spreads and becomes effectively inaccessible, is central to modern quantum statistical physics and quantum chaos. These lecture notes provide an introduction to information scrambling from both static and dynamical perspectives. The spectral properties of reduced density matrices arising from Haar-random states are developed through the geometry of the unitary group and the universal results of random matrix theory. This geometric framework yields universal, model-independent predictions for entanglement and spectral statistics, capturing generic features of quantum chaos without reference to microscopic details. Dynamical diagnostics such as the spectral form factor and out-of-time-ordered correlators further reveal the onset of chaos in time-dependent evolution. The notes are aimed at advanced undergraduate and graduate students in physics, mathematics, and computer science who are interested in the connections between quantum chaos, information dynamics, and quantum computing. Concepts such as entanglement growth, Haar randomness, random-matrix statistics, and unitary t-designs are introduced through their realization in random quantum circuits and circuit complexity. The same mathematical framework also underpins modern quantum-device benchmarking, where approximate unitary designs and random circuits translate geometric ideas into quantitative tools for assessing fidelity, noise, and scrambling efficiency in real quantum processors.

Quantum chaos entails an entropic and computational obstruction to describing a system and thus is intrinsically difficult to characterize. An understanding of quantum chaos is fundamentally related to the mechanism of thermalization in many-body systems and the quantum nature of black holes. In this thesis we adopt the view that quantum information theory provides a powerful framework in which to elucidate chaos in strongly-interacting quantum systems. We first push towards a more precise understanding of chaotic dynamics by relating different diagnostics of chaos, studying the time-evolution of random matrix Hamiltonians, and quantifying random matrix behavior in physical systems. We derive relations between out-of-time ordered correlation functions, spectral quantities, and frame potentials to relate the scrambling of quantum information, decay of correlators, and Haar-randomness. We give analytic expressions for these quantities in random matrix theory to explore universal aspects of late-time dynamics. Motivated by our random matrix results, we define k-invariance in order to capture the onset of random matrix behavior in physical systems. We then refine our diagnostics in order to study chaotic systems with symmetry by considering Haar-randomness with respect to quotients of the unitary group, and in doing so we generalize our quantum information machinery. We further consider extended random matrix ensembles in the context of strongly-interacting quantum systems dual to black holes. Lastly, we study operator growth in classes of random quantum circuits.