Relating Out-of-Time-Order Correlations to Entanglement via Multiple-Quantum Coherences

Martin Gärttner, Philipp Hauke, and Ana Maria Rey

Phys. Rev. Lett. 120, 040402 – Published 24 January 2018

Abstract

Out-of-time-order correlations (OTOCs) characterize the scrambling, or delocalization, of quantum information over all the degrees of freedom of a system and thus have been proposed as a proxy for chaos in quantum systems. Recent experimental progress in measuring OTOCs calls for a more thorough understanding of how these quantities characterize complex quantum systems, most importantly in terms of the buildup of entanglement. Although a connection between OTOCs and entanglement entropy has been derived, the latter only quantifies entanglement in pure systems and is hard to access experimentally. In this work, we formally demonstrate that the multiple-quantum coherence spectra, a specific family of OTOCs well known in NMR, can be used as an entanglement witness and as a direct probe of multiparticle entanglement. Our results open a path to experimentally testing the fascinating idea that entanglement is the underlying glue that links thermodynamics, statistical mechanics, and quantum gravity.

Received 7 June 2017
Revised 14 September 2017

DOI:https://doi.org/10.1103/PhysRevLett.120.040402

Physics Subject Headings (PhySH)

Entanglement measures Long-range interactions Many-body localization Nonlocality Quantum coherence & coherence measures Quantum entanglement Quantum simulation

Nuclear magnetic resonance

General PhysicsAtomic, Molecular & OpticalQuantum Information, Science & Technology

Authors & Affiliations

Martin Gärttner^1,2,*, Philipp Hauke^2,3,4, and Ana Maria Rey¹

¹JILA, NIST, and the University of Colorado, Boulder, Colorado 80309, USA
²Kirchhoff-Institut für Physik, Universität Heidelberg, Im Neuenheimer Feld 227, 69120 Heidelberg, Germany
³Institute for Theoretical Physics, University of Innsbruck, 6020 Innsbruck, Austria
⁴Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, 6020 Innsbruck, Austria

^*Corresponding author. martin.gaerttner@kip.uni-heidelberg.de

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Vol. 120, Iss. 4 — 26 January 2018

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Images

Figure 1
(a) Illustration of the scheme for measuring the coherences using time reversal. The state of interest ${\hat{ρ}}_{t}$ is reached after the first evolution period. The rotation then imprints a phase $m ϕ$ on each sector ${\hat{ρ}}_{m}$ of the density matrix (see text). Evolving backward and measuring the overlap with the initial state as a function of $ϕ$ , the coherences $I_{m}$ of ${\hat{ρ}}_{t}$ are retrieved as the Fourier components of this signal. (b) An example for the fidelity signal obtained from time evolution under the Ising Hamiltonian [Eq. (5) with $Ω = 0$ ] and rotations about the $z$ axis of the spin ( $\hat{A} = {\hat{S}}_{z}$ ). By Fourier transforming this signal, one obtains the intensities $I_{m}$ , which quantify the magnitude of the $m$ th order coherences of ${\hat{ρ}}_{t}$ .
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Figure 2
MQC spectra for evolution under the Ising (a) and transverse-field Ising (b) Hamiltonian as a function of the evolution time for $N = 48$ spins. The QFI per particle is shown on top of the density plot as a solid line. ( $k + 1$ )-particle entanglement is detected if $F_{Q} / N > b_{k} / N$ and, in the pure case, $F_{Q} = F_{I}$ . The direction of the rotation axis $n$ is optimized for each $t$ . The pixels corresponding to those $I_{m}$ that violate the bound for separable states are marked with a dot. At late times reflection at the boundary of the MQC spectrum at $m = N$ leads to self-interference and fragmentation of the coherence spectrum. (Right) The coherence spectrum (red solid) and entanglement bounds (black dashed) at specific times, indicated by the dashed lines (left). The gray shading shows for which $m$ the bounds are violated.
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Figure 3
(a),(b) Optimal QFI (black) and the lower bound $F_{I} (t)$ (red dashed) as a function of the total decoherence rate $Γ = (Γ_{u d} + Γ_{d u} + Γ_{e l}) / 2$ , scaled by $J / N$ , and pure Ising dynamics ( $Ω = 0$ ). The relative size of the individual decoherence rates for spontaneous emission and elastic scattering have been chosen $Γ_{u d} : Γ_{d u} : Γ_{e l} = 1 : 1 : 10$ . In the pure case ( $Γ = 0$ ), the bound coincides with the actual QFI. $F_{I} ({\hat{ρ}}_{t}, \hat{A})$ decays as $\exp [- N Γ t]$ , much faster than the QFI. The parameter choices are motivated by the parameters of Ref. [39], which corresponds to typical values of (a) $J = 2.9 kHz$ and $t = 0.6 ms$ and (b) $J = 5.8 kHz$ and $t = 1.2 ms$ . $N = 48$ spins have been used. (c),(d) Coherences $I_{m}$ for two different dephasing rates in each case. Increasing the incoherent processes by a factor of 2 [comparing (c) with (d)], the coherences globally decrease, but a violation of the entanglement bounds (dashed) is still found at large $m$ . For all values of $Γ$ , the QFI is calculated with respect to the rotation axis $n$ that is optimal for $Γ = 0$ .
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