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Cited by 13Quantum Computing Algorithms and ArchitectureQuantum Information and CryptographyQuantum Mechanics and Applications

The DOI matches its Crossref record; 60 of the 61 references it cites were found in a registry or an index; none of the 21 sources its claims cite is checked for support yet.

Maximal coin-walker entanglement in a ballistic quantum walk

Open access
Zhang, Rong, Yang, Ran, Guo, Jian, Sun, Chang-Wei, Duan, Jia-Chen, Zhou, Heng, Xie, Zhenda, Xu, Ping, Gong, Yan-Xiao, Zhu, Shi-Ning·2022·Physical Review A
DOI 10.1103/physreva.105.042216
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Abstract

We report that the position-inhomogeneous quantum walk (IQW) can be utilized to produce the maximal high-dimensional entanglement while maintaining the quadratic speedup spread of the wave function. Our calculations show that the maximal coin-walker entanglement can be generated in any odd steps or asymptotically in even steps, and the nearly maximal entanglement can be obtained in even steps after 2. We implement the IQW by a stable resource-saving time-bin optical network, in which a polarization Sagnac loop is employed to realize the precisely tunable phase shift. Our approach opens up an efficient way for high-dimensional entanglement engineering as well as promotes investigations on the role of coin-walker interactions in QW-based applications.

Overview

AI snapshot

Written from the paper’s full text.

TL;DR

The paper proposes a position-inhomogeneous quantum walk (IQW) that can generate maximal coin-walker entanglement while preserving ballistic quadratic speedup. It experimentally implements the IQW in a time-bin optical network and observes both maximal entanglement and quadratic speedup up to 11 steps.

Key findings

  • The IQW can produce maximal coin-walker entanglement \(E=1\) in any odd steps or asymptotically in even steps; nearly maximal entanglement \(E>0.99\) is obtained in even steps after step 2. For comparison, the Hadamard quantum walk (HQW) values are around 0.87.
  • Experimental von Neumann entropy for IQW: step 1 \(0.99 \pm 0.01\), step 2 \(0.86 \pm 0.01\), step 4 \(0.95 \pm 0.02\), step 11 \(0.99 \pm 0.06\); corresponding theoretical IQW values include step 1 \(1\), step 2 \(0.81128\), step 4 \(0.99967\), step 11 \(1\).
  • The IQW maintains ballistic spread and spreads faster than the HQW. At step 11, position variances are \(\nu_I(11)=46.91 \pm 2.99\), \(\nu_H(11)=37.46 \pm 2.76\), and \(\nu_C(11)=11\) for the classical random walk.
  • The trace distance between neighboring coin states scales as \(D(t)\sim t^{-1.90}\) up to 1000 steps, supporting asymptotic maximal entanglement in even steps. Experimental similarities are \(S_H=0.9805\pm0.0002\) and \(S_I=0.9673\pm0.0002\); fidelities are \(F_H=0.967\pm0.008\) and \(F_I=0.952\pm0.009\).

Methods

  • Theoretically, the IQW uses a position-dependent coin operation at \(x=0\) consisting of a Hadamard operation plus a phase shift \(\phi\), while other positions use the Hadamard operation; a state-dependent shift entangles coin and walker. Entanglement is quantified by the von Neumann entropy of the reduced coin density matrix, with trace distance and position variance also calculated.
  • Experimentally, the IQW is implemented with a time-bin encoded optical network using polarization states \(|H\rangle\) and \(|V\rangle\) as coin states. A 1550 nm pulsed laser with 70 ps pulse width and 1 MHz repetition rate is used; a polarization Sagnac loop provides tunable phase shifts, and an unbalanced polarization Mach-Zehnder interferometer implements the shift operation.
  • Measurement uses a single-photon detector and polarization projection, with all results averaged over more than \(10^4\) detected events. The one-loop time is \(T=72.9\) ns and the one-loop loss is measured to be 3.6 dB; experimental results are reported up to 11 steps.

Limitations

  • The paper states no explicit limitations section.
  • Evident from the text: the experimental demonstration is limited to 11 steps, while the asymptotic even-step behavior is shown theoretically up to 1000 steps.
  • The experiment uses a dimmed pulse laser to simulate single-photon dynamics based on the equivalence between coherent light in a linear optical network and single-photon evolution, rather than a true single-photon source. Statistical errors also grow with step number, e.g., IQW entropy uncertainty increases from \(\pm0.01\) at step 1 to \(\pm0.06\) at step 11.

Model-written summary. Check it against the paper before relying on it.

Integrity signals

Registry

Crossref · 10.1103/physreva.105.042216

Cites

60 of 60 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Verification

Claims the paper makes

0 of 21 cited sources checked · 0 supported · 0 flagged

  • Findingown data

    The IQW spreads faster than the HQW.

    “Moreover, the IQW spreads even faster than the HQW.”
    • Also stated in: § III. EXPERIMENTAL DEMONSTRATIONS · plain
      “and moreover, the IQW spreads even faster than the HQW.”
    Quote in the paperNot a substantial claim
    • Rests on an assumption: The nonlocal shift operation leads to coin-walker entanglement.
  • Findingown data

    In the IQW, maximal coin-walker entanglement is achieved asymptotically in even steps.

    “asymptotically in even steps”
    • Also stated in: § II. THE MAXIMAL ENTANGLEMENT WITH CONSTRUCTIVE INTERFERENCE · plain
      “for the even steps, the maximal entanglement is achieved asymptotically”
    • Also stated in: § IV. CONCLUSION · plain
      “asymptotically in even steps”
    Quote in the paperSubstantial claim
    • Rests on an assumption: The nonlocal shift operation leads to coin-walker entanglement.
  • Findingown data

    The trace distance between coin states in adjacent steps for the IQW scales as D(t) ~ t^-1.90.

    “get a nonlinear fitting function of D(t) ∼ t -1.90 .”
    Quote in the paperSubstantial claim
    • Rests on an assumption: The nonlocal shift operation leads to coin-walker entanglement.
  • Findingown data

    In the HQW, the von Neumann entropy of coin-walker entanglement is around 0.87 for the steps considered.

    “the von Neumann entropy values for the HQW in Tab. 1 (Column E theo HQW ), which are all around 0.87.”
    Quote in the paperSubstantial claim
  • Premisereasoning

    Up to now, the simultaneous realization of maximal entanglement and quadratic speedup in the QW has remained a challenge.

    “Up to now, the simultaneous realization of maximal entanglement and quadratic speedup in the QW has remained a challenge.”
    Quote in the paperSubstantial claim
  • It may state two separate claims as one. 12
  • The quote only partly states this claim. 3
  • Probably not a substantial claim. 6

  • It may state two separate claims as one.

    Findingown data

    The fidelities of reconstructed coin density matrices were higher than F_H = 0.967 ± 0.008 for HQW and F_I = 0.952 ± 0.009 for IQW.

    “The fidelities, calculated by F = Tr ρ are all higher than F H = 0.967 ± 0.008 and F I = 0.952 ± 0.009 for HQW and IQW, respectively.”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one.

    Findingown data

    The similarities between experimental and theoretical position distributions were S_H = 0.9805 ± 0.0002 for HQW and S_I = 0.9673 ± 0.0002 for IQW.

    “The similarities for HQW and IQW are calculated as S H = 0.9805 ± 0.0002, and S I = 0.9673 ± 0.0002, respectively.”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one.

    Findingnone given

    The constructive interference is preserved in the IQW, leading to ballistic spread of the wavefunction in the high-dimensional Hilbert space.

    “Meanwhile, the constructive interference is preserved leading to the ballistic spread of the wavefunction in the high-dimensional Hilbert space.”
    Quote in the paperSubstantial claim
    • Rests on an assumption: The nonlocal shift operation leads to coin-walker entanglement.

  • It may state two separate claims as one.

    Findingown data

    At step 11, the position variances were ν_I(11) = 46.91 ± 2.99, ν_H(11) = 37.46 ± 2.76, and ν_C(11) = 11.

    “Taking step 11 as an example, the position variances of the IQW, HQW and CRW are ν I (11) = 46.91 ± 2.99, ν H (11) = 37.46 ± 2.76, and ν C (11) = 11, respectively.”
    Quote in the paperNot a substantial claim

  • The quote only partly states this claim.

    Findingown data

    In the IQW, nearly maximal coin-walker entanglement (E > 0.99) is obtained in even steps after step 2.

    “the nearly maximal entanglement can be obtained in even steps after 2”
    • Also stated in: § II. THE MAXIMAL ENTANGLEMENT WITH CONSTRUCTIVE INTERFERENCE · plain
      “nearly maximal entanglement with E > 0.99 has been achieved since step 2”
    • Also stated in: § IV. CONCLUSION · plain
      “approximately maximal entanglement can be generated in even steps after 2 as well”
    Quote in the paperSubstantial claim
    • Rests on an assumption: The nonlocal shift operation leads to coin-walker entanglement.

  • It may state two separate claims as one.

    Findingown data

    In the experiment, the IQW and HQW exhibit quadratic speedup spread, and the IQW spreads faster than the HQW.

    “We can see that in contrast to the diffusive spread in CRW, the IQW as well as the HQW can behave the quadratic speedup spread, and moreover, the IQW spreads even faster than the HQW.”
    Quote in the paperSubstantial claim

  • The quote only partly states this claim.

    Findingown data

    In the IQW, maximal coin-walker entanglement (E = 1) is obtained at any odd step.

    “the maximal coin-walker entanglement can be generated in any odd steps”
    • Also stated in: § II. THE MAXIMAL ENTANGLEMENT WITH CONSTRUCTIVE INTERFERENCE · plain
      “maximal entanglement E = 1 can be obtained”
    • Also stated in: § IV. CONCLUSION · plain
      “maximal coin-walker entanglement can be created in any odd steps”
    Quote in the paperSubstantial claim
    • Rests on an assumption: The nonlocal shift operation leads to coin-walker entanglement.

  • The quote only partly states this claim.

    Findingown data

    The experimental von Neumann entropy values for IQW and HQW agree with theoretical values (Table I).

    “The calculated values of the von Neumman entropy from the experimental results are listed in Tab. 1, which all agree well”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one.

    Findingown data

    The experiment simultaneously observed quadratic speedup transport and maximal entanglement entropy in the IQW.

    “We simultaneously observe the quadratic speedup transport and maximal entanglement entropy through the time-multiplexing measurement and the coin state tomography.”
    • Also stated in: § IV. CONCLUSION · boosted
      “both the maximal entanglement and quadratic speedup in the IQW have been confirmed experimentally.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim. The paper’s conclusions may not depend on it.

    Findingown data

    The one-loop loss of the optical network was measured to be 3.6 dB.

    “The one-loop loss is measured to be 3.6 dB.”
    Quote in the paperNot a substantial claim

  • It may state two separate claims as one.

    Findingown data

    The IQW and HQW both exhibit ballistic transport due to maintained constructive interference, in contrast to the diffusive spread of CRW.

    “One can see that the IQW and HQW can both exhibit the ballistic transport due to the maintained constructive interference, in contrast to the diffusive spread of CRW.”
    Quote in the paperSubstantial claim
    • Rests on an assumption: The nonlocal shift operation leads to coin-walker entanglement.

  • It may state two separate claims as one.

    Study design and conductreasoning

    The evolved coin-walker state remains pure due to unitary evolution and pure initial state, so the von Neumann entropy of the reduced coin state quantifies the coin-walker entanglement.

    “The evolved coin-walker state remains pure as a result of the unitary evolution together with the pure initial state, so the entanglement of |Ψ(t) can be quantified through the von Neumann entropy.”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one.

    Premisecitation

    The Hadamard quantum walk exhibits ballistic transport but cannot reach maximal coin-walker entanglement.

    “The Hadamard QW, though exhibits the ballistic transport, namely, the quadratic speedup, cannot reach the maximal coin-walker entanglement [19,20].”
    Quote in the paperSubstantial claim0 of 2 sources checked
    • cites [19]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites [20]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • Probably not a substantial claim.

    Premisecitation

    Maximal entanglement can be created asymptotically by introducing dynamical disorder.

    “the maximal entanglement can be created asymptotically by introducing dynamical disorder [23,25]”
    Quote in the paperSubstantial claim0 of 2 sources checked
    • cites [23]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites [25]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.

    Study design and conductreasoning

    The SPD and laser are both triggered by electrical pulses from the AWG, so the state evolution of each step can be accurately monitored.

    “The SPD and the laser are both triggered by the electrical pulses from the AWG.Therefore, the state evolution of each step can be accurately monitored in our setup.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim.

    Premisecitation

    Maximal entanglement can be created after 10-step evolution via an optimal set of coin operations.

    “or after 10-step evolution via an optimal set of coin operations [26,27]”
    Quote in the paperSubstantial claim0 of 2 sources checked
    • cites [26]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites [27]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • Probably not a substantial claim. The paper’s conclusions may not depend on it.

    Study design and conductnone given

    All experimental results were averaged over more than 10^4 detected events.

    “All experimental results are averaged over more than 10 4 detected events.”
    Quote in the paperNot a substantial claim

  • It may state two separate claims as one.

    Interpretationown data

    The experimental setup maintained nearly perfect coherence, as confirmed by non-Gaussian distributions after 11-step evolution.

    “Therefore, nearly perfect coherence in our setup is confirmed by the non-Gaussian distributions after 11-step evolution.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim. The paper’s conclusions may not depend on it.

    Interpretationcitation

    The IQW approach is applicable to a variety of physical systems such as atoms, trapped ions, and photons.

    “which is applicable in a variety of different physical systems such as atoms [44], trapped ions [45,46] and photons [47,48].”
    Quote in the paperNot a substantial claim0 of 5 sources checked
    • cites [44]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites [45]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites [46]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites [47]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites [48]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • It may state two separate claims as one.

    Interpretationown data

    The experimental results demonstrate that coherence was maintained perfectly and confirm that the coin and walker were in a coherent superposition state during the whole evolution.

    “The above experimental results, including both the non-Gaussian distribution and quadratic speedup, have demonstrated that the coherence are maintained perfectly and further confirm that the coin-walker are in the coherent superposition state during the whole evolution [52] with the corresponding theoretical values.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [52]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

20 of 20

  • The paper relies on this without support.

    Applications of quantum walks usually require an efficient way for maximal entanglement generation.

    “All these applications usually require an efficient way for maximal entanglement generation.”

  • The paper relies on this without support.

    The nonlocal shift operation leads to coin-walker entanglement.

    “The nonlocal operation Ŝx leads to the coin-walker entanglement.”
2 of 2

Self-check

Percentages, sums and reported differences re-computed from the paper’s own numbers.

Each contradiction between two passages listed here was confirmed by an independent check.

  • Step-2 IQW entropy contradicts the claimed threshold

    Minor

    The text says the IQW achieves E > 0.99 from step 2, but Table I gives a theoretical step-2 entropy of 0.81128.

1 of 1

Citations

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Metadata

Authors
Zhang, R., Yang, R., Guo, J., Sun, C.-W., Duan, J.-C., Zhou, H., Xie, Z., Xu, P., Gong, Y.-X., & Zhu, S.-N.
Affiliations
  • Nanjing University of Posts and Telecommunications (CN)
  • Collaborative Innovation Center of Advanced Microstructures (CN)
  • University of Electronic Science and Technology of China (CN)
  • National University of Defense Technology (CN)
ORCID
Venue
Physical Review A
Publisher
American Physical Society (APS)
Published
2022-04-26
Volume
105
Issue
4
ISSN
  • 0556-2791 (ISSN-L)
  • 2469-9926
  • 2469-9934
DOI
10.1103/physreva.105.042216
OpenAlex
W4221154396
Semantic Scholar
f29e529dd8f7074727fbdeb8247118533aaaa6ad
Cited by
13
References
61
Citations by year
2026: 1 · 2025: 4 · 2024: 4 · 2023: 3 · 2022: 1
FWCI
0.80
Type
journal article
Language
en
Open access
Licence
link.aps.org/licenses/aps-default-license (vor)
Locations
  • Physical Review A (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • arXiv (Cornell University)
Funders
  • National Basic Research Program of China (2019YFA0705000, 2019YFA0308700)
  • National Natural Science Foundation of China (51890861, 11547031, 11705096, 11674169, 11627810, 61705033, 11690031, 11974178)
  • Key R&D Program of Guangdong Province (2018B030329001)
Topics
  • Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Information and Cryptography
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
Keywords
quantum walks, high-dimensional entanglement, entanglement engineering, time-bin entanglement, tunable phase shift, quantum advantage

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