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arXiv:2304.00564

Metrological detection of multipartite entanglement through dynamical symmetries

Yu-Ran Zhang, Franco Nori

Published Apr 2, 2023• 0 citations
arXivHTMLPDF

Abstract

Multipartite entanglement, characterized by the quantum Fisher information (QFI), plays a central role in quantum-enhanced metrology and understanding quantum many-body physics. With a dynamical generalization of the Mazur-Suzuki relations, we provide a rigorous lower bound on the QFI for the thermal Gibbs states in terms of dynamical symmetries, i.e., operators with periodic time dependence. We demonstrate that this bound can be saturated when considering a complete set of dynamical symmetries. Moreover, this lower bound with dynamical symmetries can be generalized to the QFI matrix and to the QFI for the thermal pure states, predicted by the eigenstate thermalization hypothesis. Our results reveal a new perspective to detect multipartite entanglement and other generalized variances in an equilibrium system, from its nonstationary dynamical properties, and is promising for studying emergent nonequilibrium many-body physics.

DOI
10.48550/arxiv.2304.00564

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Multipartite entanglement, characterized by the quantum Fisher information (QFI), plays a central role in quantum-enhanced metrology and understanding quantum many-body physics. With a dynamical generalization of the Mazur-Suzuki relations, we provide a rigorous lower bound on the QFI for the thermal Gibbs states in terms of dynamical symmetries, i.e., operators with periodic time dependence. We demonstrate that this bound can be saturated when considering a complete set of dynamical symmetries. Moreover, this lower bound with dynamical symmetries can be generalized to the QFI matrix and to the QFI for the thermal pure states, predicted by the eigenstate thermalization hypothesis. Our results reveal a new perspective to detect multipartite entanglement and other generalized variances in an equilibrium system, from its nonstationary dynamical properties, and is promising for studying emergent nonequilibrium many-body physics.
The characterization of entanglement is a central problem for the study of quantum many-body dynamics. Here, we propose the quantum Fisher information (QFI) as a useful tool for the study of multipartite-entanglement dynamics in many-body systems. We illustrate this by considering the regular-to-ergodic transition in the Dicke model---a fully connected spin model showing quantum thermalization above a critical interaction strength. We show that the QFI has a rich dynamical behavior which drastically changes across the transition. In particular, the asymptotic value of the QFI as well as its characteristic timescales witness the transition both through their dependence on the interaction strength and through the scaling with the system size. Since the QFI also sets the ultimate bound for the precision of parameter estimation, it provides a metrological perspective on the characterization of entanglement dynamics in many-body systems. Here, we show that quantum ergodic dynamics allows for a much faster production of metrologically useful states.
We study the quantum Fisher information (QFI) and, thus, the multipartite entanglement structure of thermal pure states in the context of the eigenstate thermalization hypothesis (ETH). In both the canonical ensemble and the ETH, the quantum Fisher information may be explicitly calculated from the response functions. In the case of the ETH, we find that the expression of the QFI bounds the corresponding canonical expression from above. This implies that although average values and fluctuations of local observables are indistinguishable from their canonical counterpart, the entanglement structure of the state is starkly different; with the difference amplified, e.g., in the proximity of a thermal phase transition. We also provide a state-of-the-art numerical example of a situation where the quantum Fisher information in a quantum many-body system is extensive while the corresponding quantity in the canonical ensemble vanishes. Our findings have direct relevance for the entanglement structure in the asymptotic states of quenched many-body dynamics.
We derive a universal upper bound on the speed of multipartite entanglement generation. For arbitrary differentiable pure multipartite states undergoing parameter-dependent evolution, we prove that the rate of change of generalized concurrence is bounded by the square root of the quantum Fisher information (QFI). The proof follows directly from the symmetric logarithmic derivative formalism, revealing the bound as a consequence of quantum estimation theory, while a complementary Schmidt-spectrum analysis identifies the conditions for saturation and yields a tighter bound for rank-two bipartitions. We illustrate these results in interacting many-body spin models and investigate their behavior across the quantum phase transition of the transverse-field Ising model. At the critical point, we find that the QFI can be strongly enhanced while the generalized concurrence is nearly stationary and its sensitivity to the coupling is suppressed. This demonstrates that the enhanced parameter sensitivity associated with quantum criticality need not arise from rapid changes in the magnitude of multipartite entanglement.
In a recent study concerning the multipartite entanglement structure in states satisfying the eigenstate thermalization hypothesis (ETH), a measure known as the quantum Fisher information (QFI) was observed to detect differences in the entanglement content between thermal states of the same temperature; thus establishing a new, distinct way to distinguish between thermal states of different types (pure or mixed). In this work, we try to extend the previous analysis to the class of integrable models–i.e., systems which do not thermalize in the conventional way. We start this investigation with the simplest case, that of noninteracting systems. We derive exact analytic expressions of the QFI for a state described by the generalized Gibbs ensemble (GGE) and for a quenched pure state. We then show through various examples that this hierarchy of entanglement content persists
Many-body entanglement in condensed matter systems can be diagnosed from equilibrium response functions through the use of entanglement witnesses and operator-specific quantum bounds. Here, we investigate the applicability of this approach for detecting entangled states in quantum systems driven out of equilibrium. We use a multipartite entanglement witness, the quantum Fisher information, to study the dynamics of a paradigmatic fermion chain undergoing a time-dependent change of the Coulomb interaction. Our results show that the quantum Fisher information is able to witness distinct signatures of multipartite entanglement both near and far from equilibrium that are robust against decoherence. We discuss implications of these findings for probing entanglement in light-driven quantum materials with time-resolved optical and x-ray scattering methods.
Multipartite entanglement is a crucial resource for advancing quantum technologies, with considerable research efforts directed toward achieving its rapid and scalable generation. In this work, we derive an analytical expression for the time-averaged quantum Fisher information (QFI), enabling the detection of scalable multipartite entanglement dynamically generated by all quantum chaotic systems governed by Dyson’s ensembles. Our approach integrates concepts of randomness and quantum chaos, demonstrating that the QFI is universally determined by the structure and dimension of the Krylov space that confines the chaotic dynamics. In particular, the QFI ranges from N^2/3 N 2 / 3 for N N qubits in the permutation-symmetric subspace (e.g. for chaotic kicked top models with long-range interactions), to N N when the dynamics extend over the full Hilbert space with or without bit reversal symmetry or parity symmetry (e.g. in chaotic models with short-range Ising-like interactions). In the former case, the QFI reveals multipartite entanglement among N/3 N / 3 qubits and highlights the power of chaotic collective spin systems in generating scalable multipartite entanglement. Interestingly this result can be related to isotropic substructures in the Wigner distribution of chaotic states and demonstrates the efficacy of quantum chaos for Heisenberg-scaling quantum metrology. Finally, our general expression for the QFI agrees with that obtained for random states and, differently from out-of-time-order-correlators, it can also distinguish chaotic from integrable unstable spin dynamics.