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arXiv:2203.07692 · Physical review. E

Effect of chaos on information gain in quantum tomography

Abinash Sahu, Sreeram PG, Vaibhav Madhok

Published Aug 22, 2022• 8 citations
arXivHTMLPDF

Abstract

Does chaos in the dynamics enable or impede information gain in quantum tomography? We address this question by considering continuous measurement tomography in which the measurement record is obtained as a sequence of expectation values of a Hermitian observable evolving under the repeated application of the Floquet map of the quantum kicked top. For a given dynamics and Hermitian observables, we observe completely opposite behavior in the tomography of well-localized spin coherent states compared to random states. As the chaos in the dynamics increases, the reconstruction fidelity of spin coherent states decreases. This contrasts with the previous results connecting information gain in tomography of random states with the degree of chaos in the dynamics that drives the system. The rate of information gain and hence the fidelity obtained in tomography depends not only on the degree of chaos in the dynamics and to what extent it causes the initial observable to spread in various directions of the operator space, but, more importantly, how well these directions are aligned with the density matrix to be estimated. Our study also gives an operational interpretation for operator spreading in terms of fidelity gain in an actual quantum information tomography protocol.

DOI
10.1103/physreve.106.024209

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Does chaos in the dynamics enable or impede information gain in quantum tomography? We address this question by considering continuous measurement tomography in which the measurement record is obtained as a sequence of expectation values of a Hermitian observable evolving under the repeated application of the Floquet map of the quantum kicked top. For a given dynamics and Hermitian observables, we observe completely opposite behavior in the tomography of well-localized spin coherent states compared to random states. As the chaos in the dynamics increases, the reconstruction fidelity of spin coherent states decreases. This contrasts with the previous results connecting information gain in tomography of random states with the degree of chaos in the dynamics that drives the system. The rate of information gain and hence the fidelity obtained in tomography depends not only on the degree of chaos in the dynamics and to what extent it causes the initial observable to spread in various directions of the operator space, but, more importantly, how well these directions are aligned with the density matrix to be estimated. Our study also gives an operational interpretation for operator spreading in terms of fidelity gain in an actual quantum information tomography protocol.
We find quantum signatures of chaos in various metrics of information gain in quantum tomography. We employ a quantum state estimator based on weak collective measurements of an ensemble of identically prepared systems. The tomographic measurement record consists of a sequence of expectation values of a Hermitian operator that evolves under repeated application of the Floquet map of the quantum kicked top. We find an increase in information gain and, hence, higher fidelities in the reconstruction algorithm when the chaoticity parameter map increases. The results are well predicted by random matrix theory.
Abstract How does quantum chaos lead to rapid scrambling of information as well as errors across a system when one introduces perturbations in the dynamics? What are its consequences for the reliability of quantum simulations and quantum information processing? We employ continuous measurement quantum tomography as a paradigm to study these questions. The measurement record is generated as a sequence of expectation values of a Hermitian observable evolving under repeated application of the Floquet map of the quantum kicked top. Interestingly, we find that the reconstruction fidelity initially increases regardless of the degree of chaos or the strength of perturbations in the dynamics. For random states, when the measurement record is obtained from a random initial observable, the subsequent drop in the fidelity obtained is inversely correlated to the degree of chaos in the dynamics. More importantly, this also gives us an operational interpretation of Loschmidt echo for operators by connecting it to the performance of quantum tomography. We define a quantity to capture the scrambling of errors, an out-of-time-ordered correlator (OTOC) between two operators under perturbed and unperturbed system dynamics that serves as a signature of chaos and quantifies the spread of errors. Our results demonstrate not only a fundamental link between Loschmidt echo and scrambling of errors, as captured by OTOCs, but that such a link can have operational consequences in quantum information processing.
We adopt a continuous weak measurement tomography protocol to explore the signatures of chaos in the quantum system(s). We generate the measurement record as a series of expectation values of an observable evolving under the desired dynamics, which can show a transition from integrability to chaos. We find that the rate of information gain depends on the degree of chaos in the dynamics, the choice of initial observable, and how well the operator is aligned along the density matrix. The amount of operator spreading in the Krylov subspace, as quantified by the fidelity in quantum tomography and various other metrics of information gain, increases with the degree of chaos in the system. We study operator spreading in many-body quantum systems by its potential to generate an informationally complete measurement record. Our quantifiers for operator spreading are more consistent indicators of quantum chaos than Krylov complexity. Our study gives an operational interpretation for operator spreading in terms of fidelity gain in quantum tomography. Continuing in our journey of finding the footprints of chaos in the quantum domain, we explore the growth of errors in noisy tomography. For random states, when the measurement record is obtained from a random operator, the subsequent drop in the fidelity obtained is inversely correlated to the degree of chaos in the dynamics. This gives us an operational interpretation of Loschmidt echo for operators by connecting it to the performance of quantum tomography. We find a quantity to capture the scrambling of errors, an out-of-time-ordered correlator (OTOC) between two operators under perturbed and unperturbed dynamics that serves as a signature of chaos. Our results demonstrate a fundamental link between Loschmidt echo and scrambling of errors, as captured by OTOCs, with operational consequences in quantum information processing.
We study operator spreading in many-body quantum systems by its potential to generate an informationally complete measurement record in quantum tomography. We adopt continuous weak measurement tomography for this purpose. We generate the measurement record as a series of expectation values of an observable evolving under the desired dynamics, which can show a transition from integrability to complete chaos. We find that the amount of operator spreading, as quantified by the fidelity in quantum tomography, increases with the degree of chaos in the system. We also observe a remarkable increase in information gain when the dynamics transitions from integrable to nonintegrable. We find our approach in quantifying operator spreading is a more consistent indicator of quantum chaos than Krylov complexity as the latter may correlate/anticorrelate or show no explicit behavior with the level of chaos in the dynamics. We support our argument through various metrics of information gain for two models: the Ising spin chain with a tilted magnetic field and the Heisenberg XXZ spin chain with an integrability-breaking field. Our paper gives an operational interpretation for operator spreading in quantum chaos.
Quantum chaos is the study of quantum systems whose classical description is chaotic. How does chaos manifest itself in the quantum world? In this spirit, we study the dynamical generation of entanglement as a signature of chaos in a system of periodically kicked coupled-tops, where chaos and entanglement arise from the same physical mechanism. The long-time entanglement as a function of the position of an initially localized wave packet very closely correlates with the classical phase space surface of section - it is nearly uniform in the chaotic sea, and reproduces the detailed structure of the regular islands. The uniform value in the chaotic sea is explained by the random state conjecture. As classically chaotic dynamics take localized distributions in phase space to random distributions, quantized versions take localized coherent states to pseudo-random states in Hilbert space. Such random states are highly entangled, with an average value near that of the maximally entangled state. Continuing on our journey to find the footprints of chaos in the quantum world, we explore quantum signatures of classical chaos by studying the rate of information gain in quantum tomography. We find an increase in the rate of information gain and hence higher fidelities in the process when the Floquet maps employed increase in chaoticity. We make predictions for the information gain using random matrix theory in the fully chaotic regime and show a remarkable agreement between the two. The last part of this thesis is devoted to the study of the nature of quantum correlations themselves. We present an operational interpretation of quantum discord based on the quantum state merging protocol. Quantum discord is the markup in the cost of quantum communication in the process of quantum state merging, if one discards relevant prior information.
How does quantum chaos lead to rapid scrambling of information as well as systematic errors across a system when one introduces perturbations in the dynamics? What are its consequences for the reliability of quantum simulations and quantum information processing? We employ continuous measurement quantum tomography as a paradigm to study these questions. The measurement record is generated as a sequence of expectation values of a Hermitian observable evolving under repeated application of the Floquet map of the quantum kicked top. We construct a quantity to capture the scrambling of systematic errors, an out-of-time-ordered correlator (OTOC), which serves as a signature of chaos and quantifies the spread of errors. We show that the spread of errors, as quantified by the OTOC, is related to the operator Loschmidt echo, which is defined as the Hilbert-Schmidt inner product of the operators On, and O'n generated from repeated application of the Floquet map for ideal (unperturbed) dynamics and the true (perturbed) dynamics, respectively. This also gives us an operational interpretation of Loschmidt echo (LE) for operators by connecting it to the performance of quantum tomography. We show how our results demonstrate not only a link between LE and scrambling of errors different than previous studies, but also that such a link can have operational consequences in quantum information processing.