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Entanglement switching via mobility edges in a quasiperiodic chain

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Pith's one-line read Tuning a mobility edge across the Fermi level switches long-range entanglement between distant spin pairs in a quasiperiodic chain.

desk verdict A clean numerical proposal for mobility-edge-controlled entanglement, but the switching claim rests on an unexamined RKKY validity check in exactly the regime where it is most likely to fail. read the letter →

arxiv 2507.06305 v1 pith:HMEOGZKM submitted 2025-07-08 cond-mat.str-el quant-ph

YouYoung Joung , Junmo Jeon , SungBin Lee This is my paper
classification cond-mat.str-elquant-ph
keywords generalizedAubry-Andrémodelmobilityedgequasiperiodicchainlong-rangeentanglementlogarithmicnegativityRKKYinteractioncriticalstatesswitching
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper aims to show that a one-dimensional quasiperiodic chain with a tunable mobility edge, the generalized Aubry-André model, can serve as a platform for controlling long-range entanglement between distant impurity spins. The central mechanism is that states near the mobility edge are critical and carry unusually strong indirect interactions between sites with similar local potential patterns, regardless of how far apart the sites are. By shifting the mobility edge across the Fermi level through the parameter $\alpha$, the paper demonstrates both a smooth transfer of entanglement from one distant pair to another and an abrupt switching of which pair is entangled. This matters because it offers a way to reroute nonlocal quantum correlations without changing the filling fraction, using parameters that are experimentally accessible in optical lattices.

What carries the argument

The central object is the generalized Aubry-André potential $V_i = \lambda \cos(2\pi q i + \phi)/(1-\alpha\cos(2\pi q i+\phi))$, whose spectrum has an exact mobility edge $\alpha E_{\mathrm{ME}} = \operatorname{sgn}(\lambda)(2t-|\lambda|)$. The argument works through an effective spin-exchange Hamiltonian $H_{\mathrm{eff}} = \sum_{i\neq j} J_{ij}\,\vec{S}_i\cdot\vec{S}_j$, with $J_{ij}$ computed from Eq. (4) as a second-order perturbative sum over eigenstates, the quasiperiodic analogue of RKKY exchange. The paper uses the inverse participation ratio to identify which states are extended, localized, or critical, and it shows that when the Fermi level sits at the mobility edge, the mediating critical states concentrate on sites whose local potentials are nearly equal, producing strong $J_{ij}$ between pattern-matched sites regardless of real-space distance. Logarithmic negativity, a standard entanglement measure, is used to quantify the resulting spin-pair entanglement.

What would settle it

Recompute the entanglement of the three impurity spins directly from the full Hamiltonian of Eq. (1), without relying on the second-order perturbation, at $J_K/t=0.05$, $T/t=10^{-4}$, and $\lambda/t=-1.88$, and check whether the logarithmic negativities $E^{AB}_{\mathcal N}$ and $E^{AC}_{\mathcal N}$ still jump at $\alpha_c\approx0.501$. If the direct calculation shows no sharp switch, the perturbative reduction used for $J_{ij}$ fails at the critical point.

Watch

Extended reading notes

Core claim

At half filling in the generalized Aubry-André model with $\lambda/t=-1.88$, the paper finds a critical point $\alpha_c \approx 0.501$ where the Fermi level coincides with the mobility edge. For three impurity spins placed at sites $(646, 879, 1938)$, as $\alpha$ increases toward $\alpha_c$, the logarithmic negativity of the distant pair $AC$ rises smoothly from $0$ to $1$ while the nearby pair $AB$ falls from $1$ to $0$; once $\alpha$ exceeds $\alpha_c$, the two values suddenly swap, with $AB$ jumping to $1$ and $AC$ dropping to $0$. The paper interprets this as a generic effect for any $\lambda$ with $|\lambda|\neq 2t$: near the mobility edge, critical states mediate pattern-selective long-range interactions, and the localization transition at the mobility edge acts as the switch. The abrupt switching is attributed to the faster exponential decay of the distant coupling once the Fermi level enters the localized regime.

Load-bearing premise

The load-bearing premise is that the effective Heisenberg interaction $J_{ij}$ computed in second-order perturbation theory in $J_K$ faithfully describes the impurity-spin entanglement even at the mobility edge, where the mediating states are critical and the perturbative expansion is not explicitly validated.

Editorial extensions

If this is right

  • For any $|\lambda|\neq 2t$, the mobility edge can be moved across the Fermi level by tuning $\alpha$ alone, so entanglement routing is achieved without doping or changing the filling fraction.
  • Near $\alpha_c\approx0.501$, the distant pair's logarithmic negativity reaches the maximum value of $1$, and crossing $\alpha_c$ swaps it with the nearby pair, so the chain acts as an on-off switch for long-range entanglement.
  • The pattern-selective interaction is claimed to be a universal feature of critical states in generic incommensurate systems, not a special property of the GAA model.
  • At finite temperature $T/t=10^{-4}$ and $J_K/t=0.05$, the entanglement effect survives in the extended and critical regimes and vanishes only in the fully localized regime.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Inference beyond the paper: because $\alpha$ is externally controllable, a slow ramp of $\alpha$ could serve as a dynamic protocol for transferring or swapping entanglement between selected pairs in real time, but the paper does not explicitly simulate such a time-dependent protocol.
  • Inference beyond the paper: the same mobility-edge mechanism should appear in other one-dimensional incommensurate models with exact mobility edges, such as bichromatic potentials, so the switching could be tested in those settings as well.
  • Inference beyond the paper: the abrupt jump at $\alpha_c$ suggests the effective three-spin model undergoes a sharp qualitative change there; a direct exact-diagonalization study of the full Hamiltonian at larger $J_K$ would show how robust the switch is when the second-order perturbation used for $J_{ij}$ becomes questionable.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The manuscript studies the generalized Aubry–André (GAA) model with dilute magnetic impurities. Using the second-order RKKY expression for the exchange coupling J_{ij} (Eq. (4)), the authors compute indirect interactions between impurity spins and show that the interaction changes from power-law to exponential decay as the mobility edge crosses the Fermi level, with an intermediate regime of strong 'pattern-selective' couplings between sites that have nearly equal on-site potentials (Figs. 1 and 2). They then place three impurity spins at sites (646, 879, 1938), form the thermal state of the effective three-spin Heisenberg model at T/t = 10^{-4}, and compute logarithmic negativity as a function of α. The paper reports smooth transfer of entanglement from the short-range pair (646, 879) to the long-range pair (646, 1938) for α < α_c ≈ 0.501 and an abrupt switching at α_c, and argues that tuning the mobility edge provides a versatile experimental control knob for long-distance entanglement.

Significance. The proposed mechanism is interesting and timely: using an exactly solvable mobility edge to switch the character of RKKY interactions between power-law and exponential decay is a clean, falsifiable idea, and the pattern-selective enhancement by critical states is clearly illustrated in Fig. 2. The numerical procedure is transparent and the central qualitative observation—that J_{ij} follows the localization properties of states near E_F—is well supported by known GAA physics. If the effective Heisenberg description is validated in the relevant parameter regime, the paper would provide a concrete route to tunable long-range entanglement in quasiperiodic systems. However, the paper's strongest claims depend on an unchecked perturbative reduction and on terminology ('adiabatic transport') that is not backed by a dynamical calculation.

major comments (2)
  1. [§Entanglement transfer and switching, Eq. (4), Fig. 2(b)] The switching and transfer curves in Fig. 3 are computed entirely from a three-spin Heisenberg Hamiltonian whose couplings J_{ij} are obtained from the second-order RKKY expression, Eq. (4), which is a perturbative expansion in J_K. For this reduction to describe the original Hamiltonian (1) at T = 10^{-4} t, the dimensionless expansion parameter J_K ρ_i(E_F) must be small at each impurity site and the single-impurity Kondo temperature must be far below T. At α = α_c the Fermi energy sits at the mobility edge, and Fig. 2(b) shows that the LDOS at E_F is sharply concentrated on sites with V_i ≈ 0. The selected impurity sites (646, 879, 1938) are chosen because their on-site potentials are nearly equal and small, placing them precisely in this high-LDOS region. Critical or multifractal states at N = 2500 can enhance the local DOS at such sites by one to two orders of magnitude relative to the average, which would make J_K ρ_i ≈ 0.1–0.3 for J_K/t = 0.05 (using an average ρ ≈ 0.3/t). In that range the Schrieffer–Wolff/RKKY expansion is uncontrolled and the Kondo temperature T_K ≈ D sqrt(J_K ρ_i) exp(−1/(J_K ρ_i)) can exceed T, so the impurity moments are not free local moments and the effective Heisenberg model is not a reliable representation of Eq. (1). The manuscript does not report ρ_i(E_F) at the impurity sites, does not estimate T_K, and does not check that fourth-order corrections are subleading. This missing check is load-bearing because the central entanglement-switching claim rests on the Heisenberg reduction. I request either a demonstration that J_K ρ_i remains small (and T_K < T) for the chosen sites, or a modified parameter set that satisfies both J_{ij} > T and J_K ρ_i << 1, or a treatment that accounts for the strong-coupling corrections.
  2. [Abstract and §Entanglement transfer and switching] The abstract and the main text claim 'adiabatic transport' of entanglement as a function of α. What is actually computed is the equilibrium logarithmic negativity E_N(α) of a thermal state of the static Heisenberg model (Fig. 3(a)), not a time-dependent process. No Schrödinger or Lindblad evolution under a ramp of α is presented, and no adiabaticity condition or transfer fidelity is given. The static redistribution of pairwise entanglement is an interesting result, but calling it 'adiabatic transport' overstates the demonstration. Please either add a slow-ramp dynamical simulation for the three-spin (or full) model and verify that the state follows the equilibrium curve, or revise the terminology to 'smooth transfer' or 'quasi-static switching' throughout.
minor comments (5)
  1. [Fig. 1 caption and text after Eq. (4)] The text assumes |J_K| << 1 when deriving Eq. (4), but the caption of Fig. 1 states J_K/t = 1. Please clarify whether the plotted quantity is the J_K^2-scaled RKKY kernel, or use a small J_K in the figure to avoid an apparent inconsistency.
  2. [§Pattern-selective interaction] The local density of states is defined through δ(E_F − E_n) without specifying the broadening or the finite-size regularization used for the numerical delta function. At N = 2500 and T = 0 it is important to state how states near E_F are collected (e.g., a small Lorentzian width or a window of eigenstates), since Fig. 2(a)-(b) otherwise depends on an unspecified convention.
  3. [Throughout] The boundary conditions (open versus periodic) for the N = 2500 chains are not stated. Because quasiperiodic systems have strong finite-size and boundary effects, please specify the boundary conditions and provide at least a brief finite-size check that α_c and the switching behavior are stable with N.
  4. [Fig. 3(c)] The text claims the mechanism applies to 'arbitrary λ', but Fig. 3(c) shows only a limited region of the (α, λ) plane. Please state the sampled range of λ and comment on whether the effect survives for both signs of λ and for λ near the |λ| = 2t limit.
  5. [§Entanglement transfer and switching] There are typos such as 'non of the spin pairs' (should be 'none of the spin pairs') and 'suprisingly' in Fig. 1(b) text; a careful proofreading pass is recommended.

Circularity Check

0 steps flagged · score 2.0 of 10

No derivation step reduces to its input; the entanglement-switching result is computed from the exact eigenstates via the standard RKKY expression, with only non-load-bearing self-citations in the motivation.

full rationale

The paper's central derivation is self-contained. The generalized Aubry-Andre Hamiltonian (Eqs. 1-2) and the mobility-edge formula (Eq. 3) are imported from independent literature [42], and the RKKY coupling expression (Eq. 4) is the standard second-order formula from [49,50]; no parameter is fitted to the entanglement data. The J_ij values are computed from numerically solved eigenstates of the tight-binding model, and the logarithmic negativity is then evaluated for the resulting three-spin Heisenberg model. The switching between the AB and AC pairs follows from the computed alpha-dependence of J_ij together with entanglement monogamy, not from any quantity that was defined in terms of the final entanglement. The self-citations [37,56] are used mainly as motivation that critical states can mediate long-range interactions and that pattern similarity matters; the present work recomputes these effects for the GAA model and does not rely on an unverified assertion from those papers as a load-bearing input. The possible breakdown of second-order RKKY theory at the mobility edge due to enhanced local density of states is a regime-validity concern, not a circularity, and is therefore outside the circularity score.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard results taken from the cited literature (the GAA mobility edge formula and the second-order RKKY expression), on the unverified validity of the effective-spin reduction near the mobility edge, and on hand-chosen parameters JK, T, and φ that set the regime of the demonstration.

free parameters (3)
  • Exchange coupling JK = 0.05 t
    Chosen as a compromise: small enough to justify the second-order RKKY expansion in Eq. (4), large enough to produce visible entanglement at T = 10^{-4} t. The entanglement curves in Fig. 3 depend on this value.
  • Temperature T = 10^{-4} t
    Chosen small so the switching and transfer features are sharp; the thermal state of the three-spin model determines the logarithmic negativity.
  • Phase shift φ = 0
    Set 'without loss of generality', but for a finite chain the phase determines which sites have similar Vi; the results are shown for a single realization with this phase, so the generic applicability of the pattern-selective rule is not tested over φ.
assumptions (5)
  • domain assumption The generalized Aubry-André mobility edge formula, αE_ME = sgn(λ)(2|t| - |λ|), from Eq. (3), taken from Ganeshan et al. [42].
    This standard result determines where the mobility edge sits as a function of α and λ, and the entire tuning mechanism relies on it.
  • domain assumption The second-order RKKY indirect exchange formula, Eq. (4), from Refs. [49,50], correctly gives the spin-spin interaction J_ij for the quasiperiodic chain.
    The paper derives the effective spin Hamiltonian from this formula without checking its validity near the mobility edge where critical states are present.
  • domain assumption The impurity spins are governed by the effective Heisenberg model H = Σ J_ij S_i·S_j with the computed J_ij.
    The entanglement switching in Fig. 3 is computed from this effective spin model, not from the full fermion-plus-impurity Hamiltonian of Eq. (1).
  • domain assumption The impurity spins are in a thermal state at temperature T = 10^{-4} t.
    Logarithmic negativity is evaluated on the thermal density matrix of the three-spin model; this assumes thermal equilibrium and neglects non-equilibrium or dynamical effects.
  • domain assumption The chain with q = (√5 - 1)/2 and N = 2500 is representative of the thermodynamic limit and the chosen sites are typical.
    No finite-size scaling or averaging over phases is provided, so the representativeness of this single realization is assumed.

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Cite this review

Pith. "Pith review of Entanglement switching via mobility edges in a quasiperiodic chain." pith.science (2026). https://pith.science/paper/HMEOGZKM

@misc{pith2026250706305,
  author       = {Pith},
  title        = {Pith review of: Entanglement switching via mobility edges in a quasiperiodic chain},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HMEOGZKM}},
  note         = {Machine review of arXiv:2507.06305}
}
read the original abstract

We propose quasiperiodic chains with tunable mobility edge physics, as a promising platform for engineering long-range quantum entanglement. Using the generalized Aubry-Andr\'e model, we show that the mobility edges play a key role in manipulating long-range indirect interactions in these systems. Near the mobility edge, critical states exhibit unexpectedly strong correlations between sites that share similar local structures, regardless of their spatial separation. Remarkably, by tuning the mobility edge across the Fermi level, one can induce both adiabatic transport and abrupt switching of entanglement between distant sites. These results highlight the potential of aperiodic structures for controlling nonlocal quantum correlations, opening new avenues for entanglement-based applications in quasiperiodic systems.

Figures

Figures reproduced from arXiv: 2507.06305 by the authors.

Figure 1
Figure 1. FIG. 1. Three distinct properties under the shift of the ME [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Pattern-selective interaction in the generalized [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Entanglement transfer and switching between the short range pair A, B [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

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Reference graph

Works this paper leans on

67 extracted references · 43 canonical work pages

  1. [1]

    Horodecki, P

    R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Quantum entanglement, Rev. Mod. Phys. 81, 865 (2009)

    2009

  2. [2]

    Audretsch,Entangled systems: new directions in quan- tum physics (John Wiley & Sons, 2007)

    J. Audretsch,Entangled systems: new directions in quan- tum physics (John Wiley & Sons, 2007)

    work page 2007

  3. [3]

    Aharonov, Quantum to classical phase transition in noisy quantum computers, Phys

    D. Aharonov, Quantum to classical phase transition in noisy quantum computers, Phys. Rev. A 62, 062311 (2000)

    work page 2000

  4. [4]

    Monroe, R

    C. Monroe, R. Raussendorf, A. Ruthven, K. R. Brown, P. Maunz, L.-M. Duan, and J. Kim, Large-scale mod- ular quantum-computer architecture with atomic mem- ory and photonic interconnects, Phys. Rev. A89, 022317 (2014)

    work page 2014

  5. [5]

    G. L. Long and X. S. Liu, Theoretically efficient high- capacity quantum-key-distribution scheme, Phys. Rev. A 65, 032302 (2002)

    2002

  6. [6]

    H. J. Kimble, The quantum internet, Nature453, 1023 (2008)

    2008

  7. [7]

    Gisin, G

    N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, Quan- tum cryptography, Rev. Mod. Phys.74, 145 (2002)

    2002

  8. [8]

    J. J. . Bollinger, W. M. Itano, D. J. Wineland, and D. J. Heinzen, Optimal frequency measurements with maxi- mally correlated states, Phys. Rev. A54, R4649 (1996)

    1996

Show all 67 references
  1. [9]

    Giovannetti, S

    V. Giovannetti, S. Lloyd, and L. Maccone, Quantum metrology, Phys. Rev. Lett.96, 010401 (2006)

    2006

  2. [10]

    R. J. Hughes, D. F. V. James, E. H. Knill, R. Laflamme, and A. G. Petschek, Decoherence bounds on quantum computationwithtrappedions,Phys.Rev.Lett. 77,3240 (1996)

    1996

  3. [11]

    J. M. Raimond, M. Brune, and S. Haroche, Manipulat- ing quantum entanglement with atoms and photons in a cavity, Rev. Mod. Phys.73, 565 (2001)

    2001

  4. [12]

    Haroche, Entanglement, decoherence and the quan- tum/classical boundary, Physics today51, 36 (1998)

    S. Haroche, Entanglement, decoherence and the quan- tum/classical boundary, Physics today51, 36 (1998)

    1998

  5. [13]

    Tantivasadakarn, R

    N. Tantivasadakarn, R. Thorngren, A. Vishwanath, and R. Verresen, Long-range entanglement from measuring symmetry-protected topological phases, Physical Review X 14, 021040 (2024)

    2024

  6. [14]

    Hetényi, A

    B. Hetényi, A. Mook, J. Klinovaja, and D. Loss, Long- distance coupling of spin qubits via topological magnons, Physical Review B106, 235409 (2022)

    2022

  7. [15]

    Z. Tang, T. Chen, X. Tang, and X. Zhang, Topolog- ically protected entanglement switching around excep- tional points, Light: Science & Applications 13, 167 (2024)

    2024

  8. [16]

    Y. Wang, J. Ren, W. Zhang, L. He, and X. Zhang, Topo- logically protected strong coupling and entanglement be- tween distant quantum emitters, Physical Review Ap- plied 14, 054007 (2020)

    2020

  9. [17]

    Mittal, E

    S. Mittal, E. A. Goldschmidt, and M. Hafezi, A topolog- ical source of quantum light, Nature561, 502 (2018)

    2018

  10. [18]

    Omran, H

    A. Omran, H. Levine, A. Keesling, G. Semeghini, T. T. Wang, S. Ebadi, H. Bernien, A. S. Zibrov, H. Pich- ler, S. Choi, et al. , Generation and manipulation of schrödinger cat states in rydberg atom arrays, Science 365, 570 (2019)

    2019

  11. [19]

    Ðorđević, P

    T. Ðorđević, P. Samutpraphoot, P. L. Ocola, H. Bernien, B. Grinkemeyer, I. Dimitrova, V. Vuletić, and M. D. Lukin, Entanglement transport and a nanophotonic in- terface for atoms in optical tweezers, Science373, 1511 (2021). 6

    2021

  12. [20]

    A. L. Shaw, P. Scholl, R. Finkelstein, R. B.-S. Tsai, J. Choi, and M. Endres, Erasure cooling, control, and hyperentanglement of motion in optical tweezers, Science 388, 845 (2025)

    2025

  13. [21]

    Hartung, M

    L. Hartung, M. Seubert, S. Welte, E. Distante, and G. Rempe, A quantum-network register assembled with optical tweezers in an optical cavity, Science 385, 179 (2024)

    2024

  14. [22]

    Iqbal, N

    M. Iqbal, N. Tantivasadakarn, T. M. Gatterman, J. A. Gerber, K. Gilmore, D. Gresh, A. Hankin, N. Hewitt, C. V. Horst, M. Matheny,et al., Topological order from measurements and feed-forward on a trapped ion quan- tum computer, Communications Physics7, 205 (2024)

    2024

  15. [23]

    Bäumer and S

    E. Bäumer and S. Woerner, Measurement-based long- range entangling gates in constant depth, Physical Re- view Research 7, 023120 (2025)

    2025

  16. [24]

    Bäumer, V

    E. Bäumer, V. Tripathi, D. S. Wang, P. Rall, E. H. Chen, S. Majumder, A. Seif, and Z. K. Minev, Efficient long- range entanglement using dynamic circuits, PRX Quan- tum 5, 030339 (2024)

    2024

  17. [25]

    T.-C. Lu, L. A. Lessa, I. H. Kim, and T. H. Hsieh, Mea- surement as a shortcut to long-range entangled quantum matter, PRX Quantum3, 040337 (2022)

    2022

  18. [26]

    Sangouard, C

    N. Sangouard, C. Simon, H. De Riedmatten, and N. Gisin, Quantum repeaters based on atomic ensem- bles and linear optics, Reviews of Modern Physics83, 33 (2011)

    2011

  19. [27]

    Agustí, Y

    J. Agustí, Y. Minoguchi, J. M. Fink, and P. Rabl, Long- distance distribution of qubit-qubit entanglement using gaussian-correlated photonic beams, Phys. Rev. A105, 062454 (2022)

    2022

  20. [28]

    Krutyanskiy, M

    V. Krutyanskiy, M. Galli, V. Krcmarsky, S. Baier, D. Fioretto, Y. Pu, A. Mazloom, P. Sekatski, M. Can- teri, M. Teller,et al., Entanglement of trapped-ion qubits separated by 230 meters, Physical Review Letters130, 050803 (2023)

    2023

  21. [29]

    H. Qiu, Y. You, X.-H. Guan, X.-J. Du, J. He, and Z.-J. Yang, Efficient entanglement between two long-distance quantum emitters mediated by dark-gap-plasmon waveg- uides, Physical Review B110, 075432 (2024)

    2024

  22. [30]

    Burkard, T

    G. Burkard, T. D. Ladd, A. Pan, J. M. Nichol, and J. R. Petta, Semiconductor spin qubits, Reviews of Modern Physics 95, 025003 (2023)

    2023

  23. [31]

    Y. Lei, F. Kimiaee Asadi, T. Zhong, A. Kuzmich, C. Si- mon, and M. Hosseini, Quantum optical memory for en- tanglement distribution, Optica10, 1511 (2023)

    2023

  24. [32]

    S.-H. Wei, B. Jing, X.-Y. Zhang, J.-Y. Liao, C.-Z. Yuan, B.-Y. Fan, C. Lyu, D.-L. Zhou, Y. Wang, G.-W. Deng, et al., Towards real-world quantum networks: a review, Laser & Photonics Reviews16, 2100219 (2022)

    2022

  25. [33]

    Hu, C.-X

    X.-M. Hu, C.-X. Huang, Y.-B. Sheng, L. Zhou, B.-H. Liu, Y. Guo, C. Zhang, W.-B. Xing, Y.-F. Huang, C.-F. Li, and G.-C. Guo, Long-distance entanglement purifica- tion for quantum communication, Phys. Rev. Lett.126, 010503 (2021)

    2021

  26. [34]

    Ursin, F

    R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauen- steiner, T. Jennewein, J. Perdigues, P. Trojek, et al. , Entanglement-based quantum communication over 144 km, Nature physics3, 481 (2007)

    2007

  27. [35]

    Yuan, X.-H

    Z.-S. Yuan, X.-H. Bao, C.-Y. Lu, J. Zhang, C.-Z. Peng, and J.-W. Pan, Entangled photons and quantum com- munication, Physics Reports497, 1 (2010)

    2010

  28. [36]

    L.-M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, Long- distance quantum communication with atomic ensembles and linear optics, Nature414, 413 (2001)

    2001

  29. [37]

    Jeon and S

    J. Jeon and S. Lee, Hidden hyperspace geometry and long-distance quantum coupling, Phys. Rev. B 111, L020411 (2025)

    2025

  30. [38]

    Wilkinson, Critical properties of electron eigenstates in incommensurate systems, Proceedings of the Royal So- ciety of London

    M. Wilkinson, Critical properties of electron eigenstates in incommensurate systems, Proceedings of the Royal So- ciety of London. A. Mathematical and Physical Sciences 391, 305 (1984)

    1984

  31. [39]

    Kohmoto, Metal-insulator transition and scaling for incommensurate systems, Phys

    M. Kohmoto, Metal-insulator transition and scaling for incommensurate systems, Phys. Rev. Lett. 51, 1198 (1983)

    1983

  32. [40]

    Aubry and G

    S. Aubry and G. André, Analyticity breaking and ander- son localization in incommensurate lattices, Ann. Israel Phys. Soc 3, 18

  33. [41]

    B. I. Shklovskii and A. L. Efros,Electronic properties of doped semiconductors, Vol. 45 (Springer Science & Busi- ness Media, 2013)

    2013

  34. [42]

    Ganeshan, J

    S. Ganeshan, J. H. Pixley, and S. Das Sarma, Nearest neighbor tight binding models with an exact mobility edge in one dimension, Phys. Rev. Lett. 114, 146601 (2015)

    2015

  35. [43]

    Biddle and S

    J. Biddle and S. Das Sarma, Predicted mobility edges in one-dimensional incommensurate optical lattices: An ex- actly solvable model of anderson localization, Phys. Rev. Lett. 104, 070601 (2010)

    2010

  36. [44]

    F. A. An, K. Padavić, E. J. Meier, S. Hegde, S. Gane- shan, J. H. Pixley, S. Vishveshwara, and B. Gadway, In- teractions and mobility edges: Observing the generalized aubry-andré model, Phys. Rev. Lett.126, 040603 (2021)

    2021

  37. [45]

    Wang, J.-H

    Y. Wang, J.-H. Zhang, Y. Li, J. Wu, W. Liu, F. Mei, Y. Hu, L. Xiao, J. Ma, C. Chin, and S. Jia, Observation of interaction-induced mobility edge in an atomic aubry- andré wire, Phys. Rev. Lett.129, 103401 (2022)

    2022

  38. [46]

    X. Qiu, J. Zou, X. Qi, and X. Li, Precise programmable quantum simulations with optical lattices, npj Quantum Information 6, 87 (2020)

    2020

  39. [47]

    Lukin, M

    A. Lukin, M. Rispoli, R. Schittko, M. E. Tai, A. M. Kauf- man, S. Choi, V. Khemani, J. Léonard, and M. Greiner, Probing entanglement in a many-body–localized system, Science 364, 256 (2019)

    2019

  40. [48]

    Gauthier, I

    G. Gauthier, I. Lenton, N. McKay Parry, M. Baker, M. Davis, H. Rubinsztein-Dunlop, and T. Neely, Direct imaging of a digital-micromirror device for configurable microscopic optical potentials, Optica3, 1136 (2016)

    2016

  41. [49]

    M. A. Ruderman and C. Kittel, Indirect exchange cou- pling of nuclear magnetic moments by conduction elec- trons, Phys. Rev.96, 99 (1954)

    1954

  42. [50]

    Yafet, Ruderman-kittel-kasuya-yosida range function of a one-dimensional free-electron gas, Phys

    Y. Yafet, Ruderman-kittel-kasuya-yosida range function of a one-dimensional free-electron gas, Phys. Rev. B36, 3948 (1987)

    1987

  43. [51]

    Bauer, T.-M

    J. Bauer, T.-M. Chang, and J. Skinner, Correlation length and inverse-participation-ratio exponents and multifractal structure for anderson localization, Physical Review B 42, 8121 (1990)

    1990

  44. [52]

    Niimi, T

    Y. Niimi, T. Matsui, H. Kambara, K. Tagami, M. Tsukada, and H. Fukuyama, Scanning tunneling mi- croscopy and spectroscopy of the electronic local den- sity of states of graphite surfaces near monoatomic step edges, Physical Review B—Condensed Matter and Ma- terials Physics 73,...

    2006

  45. [53]

    Chicanne, T

    C. Chicanne, T. David, R. Quidant, J.-C. Weeber, Y. Lacroute, E. Bourillot, A. Dereux, G. C. Des Francs, 7 and C. Girard, Imaging the local density of states of op- tical corrals, Physical review letters88, 097402 (2002)

    2002

  46. [54]

    Gisin and R

    N. Gisin and R. Thew, Quantum communication, Nature photonics 1, 165 (2007)

    2007

  47. [55]

    A.Al-QasimiandD.F.V.James,Suddendeathofentan- glement at finite temperature, Phys. Rev. A77, 012117 (2008)

    2008

  48. [56]

    Jeon and S

    J. Jeon and S. Lee, Immortal quantum correlation in quasiperiodic quasi-one-dimensional systems, Physical Review B 111, L161117 (2025)

    2025

  49. [57]

    Vidal and R

    G. Vidal and R. F. Werner, Computable measure of en- tanglement, Phys. Rev. A65, 032314 (2002)

    2002

  50. [58]

    Horodecki, Entanglement measures., Quantum Inf

    M. Horodecki, Entanglement measures., Quantum Inf. Comput. 1, 3 (2001)

    2001

  51. [59]

    Koashi and A

    M. Koashi and A. Winter, Monogamy of quantum entan- glement and other correlations, Physical Review A69, 022309 (2004)

    2004

  52. [60]

    Z. V. Vardeny, A. Nahata, and A. Agrawal, Optics of photonic quasicrystals, Nature photonics7, 177 (2013)

    2013

  53. [61]

    Ozawa and H

    T. Ozawa and H. M. Price, Topological quantum matter in synthetic dimensions, Nature Reviews Physics1, 349 (2019)

    2019

  54. [62]

    X. Li, X. Li, and S. Das Sarma, Mobility edges in one-dimensional bichromatic incommensurate potentials, Physical Review B96, 085119 (2017)

    2017

  55. [63]

    Verresen, N

    R. Verresen, N. Tantivasadakarn, and A. Vishwanath, Efficiently preparing schr\" odinger’s cat, fractons and non-abelian topological order in quantum devices, arXiv preprint arXiv:2112.03061 (2021)

    2021 arXiv

  56. [64]

    Hogben, P

    H. Hogben, P. Hore, and I. Kuprov, Multiple decoherence-free states in multi-spin systems, Journal of Magnetic Resonance 211, 217 (2011)

    2011

  57. [65]

    Neschen, An efficient multi-spin coding algorithm for neuralnetworks,InternationalJournalofModernPhysics C 2, 623 (1991)

    M. Neschen, An efficient multi-spin coding algorithm for neuralnetworks,InternationalJournalofModernPhysics C 2, 623 (1991)

    1991

  58. [66]

    B. L. Higgins, D. W. Berry, S. D. Bartlett, H. M. Wise- man, and G. J. Pryde, Entanglement-free heisenberg- limited phase estimation, Nature450, 393 (2007)

    2007

  59. [67]

    Cialdi, D

    S. Cialdi, D. Brivio, E. Tesio, and M. G. Paris, Pro- grammable entanglement oscillations in a non-markovian channel, Physical Review A—Atomic, Molecular, and Optical Physics 83, 042308 (2011)

    2011

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