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arXiv:1809.01091 · New Journal of Physics

Multipartite-entanglement tomography of a quantum simulator

Marco Gabbrielli, Luca Lepori, Luca Pezzè

Published Jan 2, 2019• 37 citations
arXivHTMLPDF

Abstract

Multipartite-entanglement tomography, namely the quantum Fisher information (QFI) calculated with respect to different collective operators, allows to fully characterize the phase diagram of the quantum Ising chain in a transverse field with variable-range interaction. In particular, it recognizes the phase stemming from long-range (LR) antiferromagnetic interaction, a capability also shared by the spin squeezing. Furthermore, the QFI locates the quantum critical points, both with vanishing and nonvanishing mass gap. In this case, we also relate the finite-size power-law exponent of the QFI to the critical exponents of the model, finding a signal for the breakdown of conformal invariance in the deep LR regime. Finally, the effect of a finite temperature on the multipartite entanglement, and ultimately on the phase stability, is considered. In light of the current realizations of the model with trapped ions and of the potential measurability of the QFI, our approach yields a promising strategy to probe LR physics in controllable quantum systems.

DOI
10.1088/1367-2630/aafb8c

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Multipartite-entanglement tomography, namely the quantum Fisher information (QFI) calculated with respect to different collective operators, allows to fully characterize the phase diagram of the quantum Ising chain in a transverse field with variable-range interaction. In particular, it recognizes the phase stemming from long-range (LR) antiferromagnetic interaction, a capability also shared by the spin squeezing. Furthermore, the QFI locates the quantum critical points, both with vanishing and nonvanishing mass gap. In this case, we also relate the finite-size power-law exponent of the QFI to the critical exponents of the model, finding a signal for the breakdown of conformal invariance in the deep LR regime. Finally, the effect of a finite temperature on the multipartite entanglement, and ultimately on the phase stability, is considered. In light of the current realizations of the model with trapped ions and of the potential measurability of the QFI, our approach yields a promising strategy to probe LR physics in controllable quantum systems.
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Multipartite entanglement in the 1-D spin-12\frac{1}{2}21​ Heisenberg Antiferromagnet
Multipartite entanglement refers to the simultaneous entanglement between multiple subsystems of a many-body quantum system. While multipartite entanglement can be difficult to quantify analytically, it is known that it can be witnessed through the Quantum Fisher information (QFI), a quantity that can also be related to dynamical Kubo response functions. In this work, we first show that the finite temperature QFI can generally be expressed in terms of a static structure factor of the system, plus a correction that vanishes as T→0T\rightarrow 0T→0. We argue that this implies that the static structure factor witnesses multipartite entanglement near quantum critical points at temperatures below a characteristic energy scale that is determined by universal properties, up to a non-universal amplitude. Therefore, in systems with a known static structure factor, we can deduce finite temperature scaling of multipartite entanglement and low temperature entanglement depth without knowledge of the full dynamical response function of the system. This is particularly useful to study 1D quantum critical systems in which sub-power-law divergences can dominate entanglement growth, where the conventional scaling theory of the QFI breaks down. The 1D spin-12\frac{1}{2}21​ antiferromagnetic Heisenberg model is an important example of such a system, and we show that multipartite entanglement in the Heisenberg chain diverges non-trivially as ∼log⁡(1/T)3/2\sim \log(1/T)^{3/2}∼log(1/T)3/2. We verify these predictions with calculations of the QFI using conformal field theory and matrix product state simulations. Finally we discuss the implications of our results for experiments to probe entanglement in quantum materials, comparing to neutron scattering data in KCuF3_33​, a material well-described by the Heisenberg chain.
Quantum entanglement uncovers the essential principles of quantum matter, yet determining its structure in realistic many-body systems poses significant challenges. Here, we employ a protocol, dubbed entanglement microscopy, to reveal the multipartite entanglement encoded in the full reduced density matrix of the microscopic subregion in spin and fermionic many-body systems. We exemplify our method by studying the phase diagram near quantum critical points (QCP) in 2 spatial dimensions: the transverse field Ising model and a Gross-Neveu-Yukawa transition of Dirac fermions. Our main results are: i) the Ising QCP exhibits short-range entanglement with a finite sudden death of the LN both in space and temperature; ii) the Gross-Neveu QCP has a power-law decaying fermionic LN consistent with conformal field theory (CFT) exponents; iii) going beyond bipartite entanglement, we find no detectable 3-party entanglement with our two witnesses in a large parameter window near the Ising QCP in 2d, in contrast to 1d. We further establish the singular scaling of general multipartite entanglement measures at criticality and present an explicit analysis in the tripartite case. Quantum entanglement contains important information about quantum systems, but its calculation is challenging. Here the authors develop a quantum Monte Carlo technique for full tomography on microscopic subregions of a system, enabling extraction of multipartite quantum entanglement in large scale models.
Abstract We investigate the multipartite entanglement for a slow quantum quench crossing a critical point. We consider the quantum Ising model and the Lipkin-Meshkov-Glick model, which are local and full-connected quantum systems, respectively. The multipartite entanglement is quantified by quantum Fisher information with the generator defined as the operator of the ferromagnetic order parameter. The quench dynamics begins with a ground state in a paramagnetic phase, and then the transverse field is driven slowly to cross a quantum critical point, and ends with a zero transverse field. For the quantum Ising model, based on methods of matrix product states, we calculate the quantum Fisher information density of the final state. Numerical results of both linear and nonlinear quenches show that the quantum Fisher information density of the final state scales as a power law of the quench rate, which overall conforms to the prediction of the Kibble-Zurek mechanism with a small correction. We show that this correction results from the long-range behaviors. We also calculate the quantum Fisher information density in the Lipkin-Meshkov-Glick model. The results show that the scaling of quantum Fisher information in this full-connected system conforms to the Kibble-Zurek mechanism better, since the long-range physics cannot be defined in this nonlocal system. Our results reveal that the multipartite entanglement provides an alternative viewpoint to understand the dynamics of quantum phase transitions, specifically, the nontrivial long-range physics.
We witness multipartite entanglement in the ground state of the Kitaev chain-a benchmark model of a one dimensional topological superconductor-also with variable-range pairing, using the quantum Fisher information. Phases having a finite winding number, for both short- and long-range pairing, are characterized by a power-law diverging finite-size scaling of multipartite entanglement. Moreover, the occurring quantum phase transitions are sharply marked by the divergence of the derivative of the quantum Fisher information, even in the absence of a closing energy gap.
The interplay of quantum and thermal fluctuations in the vicinity of a quantum critical point characterizes the physics of strongly correlated systems. Here we investigate this interplay from a quantum information perspective presenting the universal phase diagram of the quantum Fisher information at a quantum phase transition. Different regions in the diagram are identified by characteristic scaling laws of the quantum Fisher information with respect to temperature. This feature has immediate consequences on the thermal robustness of quantum coherence and multipartite entanglement. We support the theoretical predictions with the analysis of paradigmatic spin systems showing symmetry-breaking quantum phase transitions and free-fermion models characterized by topological phases. In particular we show that topological systems are characterized by the survival of large multipartite entanglement, reaching the Heisenberg limit at finite temperature.