Entanglement witnesses in the $XY$ chain: Thermal equilibrium and postquench nonequilibrium states

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Entanglement witnesses in the $X Y$ chain: Thermal equilibrium and postquench nonequilibrium states

Ferenc Iglói and Géza Tóth

Phys. Rev. Research 5, 013158 – Published 1 March 2023

Abstract

We use entanglement witnesses to detect entanglement in the $X Y$ chain in thermal equilibrium and determine the temperature bound below which the state is detected as entangled. We consider the entanglement witness based on the Hamiltonian. Such a witness detects a state as entangled if its energy is smaller than the energy of separable states. We also consider a family of entanglement witnesses related to the entanglement negativity of the state. We test the witnesses in infinite and finite systems. We study how the temperature bounds obtained are influenced by a quantum phase transition or a disorder line in the ground state. Very strong finite-size corrections are observed in the ordered phase due to the presence of a quasidegenerate excitation. We also study the postquench states in the thermodynamic limit after a quench when the parameters of the Hamiltonian are changed suddenly. In the case of the Ising model, we find that the mixed postquench state is detected as entangled by the two methods if the parameters of the Hamiltonian before and after the quench are close to each other. We find that the two witnesses give qualitatively similar results, showing that energy-based entanglement witnesses are efficient in detecting the nearest-neighbor entanglement in spin chains in various circumstances. For other XY models, we find that the negativity-based witnesses also detect states in some parameter regions where the energy-based witness does not, in particular, if the quench is performed from the paramagnetic phase to the ferromagnetic phase and vice versa. The domains in parameter space corresponding to postquench states detected as entangled by the energy-based witness have been determined analytically, which stresses further the utility of our method.

Received 6 December 2022
Accepted 3 February 2023
Corrected 7 March 2024

DOI:https://doi.org/10.1103/PhysRevResearch.5.013158

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Critical phenomena Eigenstate thermalization Magnetic phase transitions Magnetism Quantum algorithms & computation Quantum correlations in quantum information Quantum correlations, foundations & formalism XY model

Quantum Information, Science & TechnologyStatistical Physics & ThermodynamicsCondensed Matter, Materials & Applied Physics

Corrections

7 March 2024

Correction: The first sentence of Sec. V B contained an error in wording and has been fixed. A grant number in the Acknowledgments contained an error and has been fixed.

Authors & Affiliations

Ferenc Iglói ^1,2,* and Géza Tóth ^{3,4,5,6,1,†}

¹Wigner Research Centre for Physics, Institute for Solid State Physics and Optics, H-1525 Budapest, Hungary
²Institute of Theoretical Physics, University of Szeged, H-6720 Szeged, Hungary
³Department of Theoretical Physics, University of the Basque Country UPV/EHU, E-48080 Bilbao, Spain
⁴EHU Quantum Center, University of the Basque Country UPV/EHU, 48940 Leioa, Biscay, Spain
⁵Donostia International Physics Center (DIPC), E-20080 San Sebastián, Spain
⁶IKERBASQUE, Basque Foundation for Science, E-48013 Bilbao, Spain

^*igloi.ferenc@wigner.hu
^†toth@alumni.nd.edu; http://www.gtoth.eu

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Vol. 5, Iss. 1 — March - May 2023

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Images

Figure 1
Phase diagram of the $X Y$ chain with the Hamiltonian Eq. (1). The equation of the disorder line is given by Eq. (6). For a qualitative description of the regions on the two sides of the disorder line, see Sec. 2b1.
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Figure 2
Temperature bound of entangled equilibrium thermal states of an $X Y$ chain in the thermodynamic limit calculated using the energy-based entanglement witness given in Eq. (44). The Hamiltonian $H$ is given in Eq. (1). The witness detects a state entangled if $〈 H 〉$ is smaller than the minimal energy for separable states, given in Eq. (41). The value $γ = 1$ corresponds to the transverse Ising model. At the disorder point, $h$ is given by Eq. (7).
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Figure 3
Temperature bound of entangled equilibrium thermal states of an $X Y$ chain for finite systems calculated using the energy-based entanglement witness given in Eq. (44). Main panel: (solid) Temperature bound of entangled equilibrium thermal states in finite transverse Ising chains for (from top to bottom) $L = 4, 6, 8, 10, 12 .$ (Dashed) The $L \to \infty$ case, which is the same as the $γ = 1$ line in Fig. 2. Insets: finite-size corrections in a lin-log scale at different values of $h$ . Upper inset: corrections in the ordered phase for (from top to bottom) $h = 0.25, 0.5$ and 0.75. Lower inset: (top) at the critical point, $h = 1.0$ and (bottom) in the paramagnetic phase, $h = 1.25 .$ Note the different scales in the $x$ axis.
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Figure 4
Temperature bound of entangled equilibrium thermal states of an $X Y$ chain for finite systems calculated using the negativity-based entanglement witnesses. (Solid) (from top to bottom) $L = 4, 6, 8, 10, 12 .$ (Dashed) $L \to \infty$ case. Main panel: $γ = 1$ , quantum Ising chain. Upper inset: $γ = 0.8 .$ Lower inset: $γ = 0.6 .$
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Figure 5
Postquench states after a sudden quench protocol $(h_{0}, γ) \to (h, γ)$ in the $X Y$ chain. (Yellow) Entanglement is detected by the energy-based witness. (Blue) Entanglement is detected in the postquench state by the negativity-based method using $μ_{\min}^{(1)}$ in Eq. (63). Equivalently, the entangled states can be detected by the family of negativity-based entanglement witnesses given in Eq. (76). (Gray) Entanglement is detected in the postquench state by the negativity-based witness in Eq. (66). The blue and the gray areas touch each other at the disorder point, where $h_{0} = h_{d}$ and $h = h_{d},$ and $h_{d}$ is given in Eq. (7). The blue and the gray filled areas overlap with the yellow area such that all yellow-filled area is part of the blue- and the gray-filled areas. Thus, all postquench states detected by the energy-based witness are also detected by the negativity-based witness. In panel (d) the limit $γ \to 0^{+}$ is considered, which is different from the case when $γ = 0$ , see text.
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