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arxiv: 2606.02505 · v1 · pith:SMYERC7Bnew · submitted 2026-06-01 · 🪐 quant-ph · cond-mat.stat-mech

Spatial and particle-particle entanglement in 1D quantum walks of two distinguishable or indistinguishable bosonic particles

Christopher Mastandrea , Chih-Chun Chien This is my paper

Pith reviewed 2026-06-28 14:24 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mech
keywords quantum walksentanglement measuresHubbard Hamiltonianbosonic particlesindistinguishable particlesspatial entanglementparticle-particle entanglementone-dimensional lattice
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The pith

In two-particle 1D quantum walks, long-time entanglement limits vary non-monotonically with onsite repulsion for both distinguishable and indistinguishable bosons.

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

This paper examines the development of entanglement in one-dimensional quantum walks involving two particles that may be distinguishable or indistinguishable bosons. It uses the Hubbard Hamiltonian to model their interactions and computes two types of entanglement: one between the left and right halves of the lattice based on particle counts, and another between the particles themselves based on the singular values of their Fock state representation. The analysis covers various initial states including separable, entangled, and doubly occupied configurations. A key observation is that the entanglement behaviors for indistinguishable particles closely track those for distinguishable particles under comparable starting conditions, yet the saturation values at long times depend on the onsite repulsion in a non-monotonic fashion. Understanding this dependence could inform how interactions influence entanglement in more complex quantum systems.

Core claim

The paper establishes that entanglement measures in the indistinguishable cases resemble those of the distinguishable cases when the initial states are comparable, but the long-time limits of the entanglement measures are typically non-monotonic as the onsite repulsion increases. This holds across different initial states and is derived from the time evolution under the Hubbard Hamiltonian using the specified entropy calculations for left-right spatial entanglement and particle-particle entanglement.

What carries the argument

Left-right entanglement from the entropy of coarse-grained states that count particles on each lattice half, together with particle-particle entanglement from the entropy of singular values of the time-evolved two-particle Fock state.

If this is right

  • Entanglement measures for indistinguishable particles can be compared directly to those for distinguishable particles when initial states are matched.
  • Long-time entanglement saturation values can be tuned by varying onsite repulsion without a simple increasing or decreasing trend.
  • The observed resemblance and non-monotonicity apply to continuous-time walks on a lattice and may guide studies of entanglement growth from separable, entangled, or doubly occupied starts.
  • These measures offer a route to track multi-particle quantum dynamics in future work.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The non-monotonic dependence might persist when the lattice is divided into more than two regions or when particle number exceeds two.
  • Experiments in optical lattices could directly measure the long-time limits to test the predicted variation with repulsion strength.
  • The close resemblance between distinguishable and indistinguishable cases may break for initial states that differ strongly in symmetry properties.

Load-bearing premise

The chosen coarse-graining into left and right particle counts together with singular-value entropy of the Fock state capture the physically relevant entanglement for the system's dynamics.

What would settle it

A direct calculation showing that long-time entanglement measures increase or decrease monotonically rather than non-monotonically as onsite repulsion is varied would contradict the reported behavior.

Figures

Figures reproduced from arXiv: 2606.02505 by Chih-Chun Chien, Christopher Mastandrea.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Left-right and (b) [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) Left-right and (b) [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) Left-right and (b) [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. (a) Left-right and (b) particle-particle correlation [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. (a) Left-right and (b) particle-particle correlation [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Alternative left-right entanglement entropy of Eq. [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Occupation numbers [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: shows the time evolution of occupation numbers of two distinguishable walkers with U/J = 10. Panels (a) and (b) show the results for the adjacent, singly oc￾cupied and doubly occupied initial states given in Eq. (11), and Eq. (12), respectively. Once again, we trun￾cate the time evolution of each plot to t/t0 = 35 as to not overcrowd the figures. The doubly-occupied initial state shows a slower spreading … view at source ↗
read the original abstract

We present entanglement measures between spatially separated regions and between two distinguishable or indistinguishable particles in one-dimensional two-particle continuous-time quantum walks governed by the Hubbard Hamiltonian. The left-right entanglement checks the entropy of coarse-grained states counting the numbers of particles on the left and right halves of the lattice while the particle-particle entanglement is based on the entropy of the singular values of the time-evolved Fock state. With separable, entangled, and doubly occupied initial states, we examine initial entanglement and the following growth in different entanglement measures. While the entanglement measures of the indistinguishable cases resemble those of the distinguishable cases when the initial states are comparable, the long-time limits of the entanglement measures are typically non-monotonic as the onsite repulsion increases. We also discuss possible implications for future research of entanglement in multi-particle quantum dynamics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript numerically investigates spatial entanglement via left-right particle-number entropy and particle-particle entanglement via singular-value entropy of the Fock state in one-dimensional two-particle continuous-time quantum walks under the Hubbard model. It compares distinguishable and indistinguishable bosonic particles for separable, entangled, and doubly occupied initial states, reporting that the entanglement measures for indistinguishable cases resemble those for distinguishable cases with comparable initial states, and that the long-time limits are typically non-monotonic as a function of the onsite repulsion strength U.

Significance. If the reported numerical observations hold, the work supplies concrete descriptive data on entanglement growth and saturation in interacting lattice quantum walks, with the similarity between distinguishable and indistinguishable cases under matched initials and the non-monotonic U-dependence constituting the main findings. These could inform studies of multi-particle dynamics, though the paper presents the results as direct consequences of the defined measures and Hamiltonian without parameter-free derivations or machine-checked proofs.

minor comments (2)
  1. [Abstract] The abstract and discussion mention possible implications for multi-particle entanglement but do not outline any concrete next steps or testable predictions; adding a short paragraph would clarify the scope.
  2. Numerical details such as lattice size, time-stepping method, or convergence checks with respect to system size are not referenced in the provided summary; including these would allow readers to assess the robustness of the long-time limits.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive assessment that led to the recommendation of minor revision. The referee's summary correctly identifies the numerical focus on left-right and particle-particle entanglement measures, the comparison between distinguishable and indistinguishable bosons, and the non-monotonic long-time behavior with respect to onsite repulsion U. No major comments were listed in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper reports numerical observations of two explicitly defined entanglement measures (left-right coarse-grained particle-number entropy and singular-value entropy of the Fock state) under the Hubbard Hamiltonian for scanned values of onsite repulsion U. The central claims about resemblance between distinguishable and indistinguishable cases and non-monotonic long-time limits follow directly from these simulations with no fitted parameters, no self-citations invoked as load-bearing premises, and no reduction of outputs to inputs by construction. The derivation chain is self-contained computational evaluation of the stated Hamiltonian and measures.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central observations rest on standard quantum mechanics and the definition of the Hubbard Hamiltonian; no new entities are postulated and the repulsion parameter is scanned rather than fitted.

axioms (2)
  • domain assumption Time evolution is generated by the two-particle Hubbard Hamiltonian on a 1D lattice
    Invoked in the abstract as the governing dynamics for the continuous-time quantum walks.
  • domain assumption Entanglement is quantified by von Neumann entropy of the reduced density matrix obtained from left/right particle counts or from singular values of the Fock-state coefficients
    Stated as the definition of the two entanglement measures examined.

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Reference graph

Works this paper leans on

76 extracted references · 5 canonical work pages

  1. [1]

    Following the decomposition laid out in Ref

    Left-Right Entanglement Measure We first consider splitting the system in real space to a left and right half, and consider the correlations that might be created over the splitting partition. Following the decomposition laid out in Ref. [44], the Fock space states can be grouped by the occupation numbers on ei- ther side of the partition. These coarse-gr...

  2. [2]

    II A, we see that it is possible to consider the state to be a product state of the form|ψ⟩= P i,j ci,j| ↑, i⟩ ⊗ | ↓, j⟩

    Spin Up-Down Entanglement Measure From the basis construction shown in Section. II A, we see that it is possible to consider the state to be a product state of the form|ψ⟩= P i,j ci,j| ↑, i⟩ ⊗ | ↓, j⟩. The coeffi- cient matrixc i,j contains correlations between the| ↑, i⟩ and| ↓, j⟩states over the whole lattice. If we perform the singular value decomposit...

  3. [3]

    Each of these states are constructed by incoherently aggregating the Fock states belonging to each category

    Left-Right Entanglement Measure Similar to the case with distinguishable particles, the coarse-grained Fock states|n L, nR⟩also apply to indistin- guishable particles. Each of these states are constructed by incoherently aggregating the Fock states belonging to each category. For the time-evolved state described by |ψ(t)⟩= P n cn|n⟩, wherenlabels the Fock...

  4. [4]

    [47, 48], the Fock state|Ψ B⟩of two identical bosons can be cast in a second- quantization form as|Ψ b⟩= PL i,j=1 ωi,ja† i a† j|0⟩in a 1D lattice of sizeL

    Particle-Particle Entaglement Measure Following the discussion given in Refs. [47, 48], the Fock state|Ψ B⟩of two identical bosons can be cast in a second- quantization form as|Ψ b⟩= PL i,j=1 ωi,ja† i a† j|0⟩in a 1D lattice of sizeL. Here|0⟩is the vacuum state. The coefficient matrixω i,j is aL×L, symmetric matrix which can be decomposed with an appropria...

  5. [5]

    Explicitly, |ψ0⟩sep =|0,· · ·,0,↑ L/2−1,↓ L/2,0,· · ·,0⟩.(8) The spatial left-right entanglement measure given in Eq

    Separable initial state For two distinguishable particles, we turn our attention first to the case of an un-entangled, separable initial state, placing the particles on adjacent sites in the middle two locations of the lattice. Explicitly, |ψ0⟩sep =|0,· · ·,0,↑ L/2−1,↓ L/2,0,· · ·,0⟩.(8) The spatial left-right entanglement measure given in Eq. (3) with th...

  6. [6]

    Explicitly, |ψ0⟩ent = 1√ 2(|0,· · ·,↑ L/2−1,↓ L/2,· · ·,0⟩+ |0,· · ·,↓ L/2−1,↑ L/2,· · ·,0⟩).(9) We start with the left-right entanglement measure shown in Fig

    Entangled initial state Next, we look at the same spatial left-right and↑,↓en- tanglement measures from an entangled initial state of the two distinguishable particles across the middle two sites of the lattice. Explicitly, |ψ0⟩ent = 1√ 2(|0,· · ·,↑ L/2−1,↓ L/2,· · ·,0⟩+ |0,· · ·,↓ L/2−1,↑ L/2,· · ·,0⟩).(9) We start with the left-right entanglement measur...

  7. [7]

    Here we start the QW with both particles placed on the same lattice site near the middle

    Doubly occupied initial state In the previous sections, the initial states for both the separable and entangled initial states were created with particles placed such that the initial lattice states were only singly occupied. Here we start the QW with both particles placed on the same lattice site near the middle. Explicitly, |ψ0⟩d,d =|0,· · ·,0,↑ L/2,↓ L...

  8. [8]

    We start with the adjacent, singly occupied initial state |ψ0⟩s,i =|0,· · ·,0,1 L/2−1,1 L/2,0,· · ·0⟩,(11) which is a symmetrized bosonic Fock state [16]

    Singly occupied, adjacent initial state We now look at the entanglement measures defined for the indistinguishable two-particle walk. We start with the adjacent, singly occupied initial state |ψ0⟩s,i =|0,· · ·,0,1 L/2−1,1 L/2,0,· · ·0⟩,(11) which is a symmetrized bosonic Fock state [16]. For iden- tical particles, this is the only possible state for distr...

  9. [9]

    5(a) shows the left-right entanglement entropy for different values of the onsite repulsion

    Doubly occupied initial state We also analyze the entanglement measures with a dou- bly occupied initial state for indistinguishable particles given by |ψ0⟩d,i =|0,· · ·,0,2 L/2,0,· · ·0⟩.(12) 9 Fig. 5(a) shows the left-right entanglement entropy for different values of the onsite repulsion. For low values of onsite repulsion (U/J= 0,1), we see that the l...

  10. [10]

    The initial state is taken as (a) separable, (b) entangled, and (c) double occupancy initial state given in Eq. (8), Eq. (9), and Eq. (10) respectively. The red dashed vertical line indicates the time when the particles reach the boundary and reflect off. from the left to the right. The density matrix can then be constructed asρ tot =|ψ tot⟩⟨ψtot|, which ...

  11. [11]

    Kempe, Quantum random walks: An introduc- tory overview, Contemporary Physics44, 307 (2003), https://doi.org/10.1080/00107151031000110776

    J. Kempe, Quantum random walks: An introduc- tory overview, Contemporary Physics44, 307 (2003), https://doi.org/10.1080/00107151031000110776

    work page doi:10.1080/00107151031000110776 2003

  12. [12]

    Wang and K

    J. Wang and K. Manouchehri,Physical Implementations of Quantum Walks(Springer, Berlin, Germany, 2014)

    2014

  13. [13]

    S. E. Venegas-Andraca, Quantum walks: a comprehen- sive review, Quantum Information Processing11, 1015 (2012)

    2012

  14. [14]

    Kadian, S

    K. Kadian, S. Garhwal, and A. Kumar, Quantum walk and its application domains: A systematic review, Com- put. Sci. Rev41, 100419 (2021)

    2021

  15. [15]

    Ambainis, Quantum walks and their algorithmic applications, Int

    A. Ambainis, Quantum walks and their algorithmic applications, Int. J. Quantum Inf.01, 507 (2003), https://doi.org/10.1142/S0219749903000383

    work page doi:10.1142/s0219749903000383 2003

  16. [16]

    Apers, S

    S. Apers, S. Chakraborty, L. Novo, and J. Roland, Quadratic speedup for spatial search by continuous-time quantum walk, Phys. Rev. Lett.129, 160502 (2022)

    2022

  17. [17]

    Solenov and L

    D. Solenov and L. Fedichkin, Continuous-time quantum walks on a cycle graph, Phys. Rev. A73, 012313 (2006)

    2006

  18. [18]

    Herrman and T

    R. Herrman and T. S. Humble, Continuous-time quan- tum walks on dynamic graphs, Phys. Rev. A100, 012306 (2019)

    2019

  19. [19]

    Prerana and S

    P. Prerana and S. Wald, Entanglement capacity of complex networks from quantum walks (2026), arXiv:2605.00772 [quant-ph]

    Pith/arXiv arXiv 2026

  20. [20]

    Coleman,Introduction to many-body physics(Cam- bridge University Press, 2015)

    P. Coleman,Introduction to many-body physics(Cam- bridge University Press, 2015)

    2015

  21. [21]

    Amico, D

    L. Amico, D. Anderson, M. Boshier, J.-P. Brantut, L.-C. Kwek, A. Minguzzi, and W. von Klitzing, Colloquium: Atomtronic circuits: From many-body physics to quan- tum technologies, Rev. Mod. Phys.94, 041001 (2022)

    2022

  22. [22]

    Browaeys and T

    A. Browaeys and T. Lahaye, Many-body physics with in- dividually controlled rydberg atoms, Nature Physics16, 132 (2020)

    2020

  23. [23]

    Eisert, M

    J. Eisert, M. Friesdorf, and C. Gogolin, Quantum many- body systems out of equilibrium, Nature Physics11, 124 (2015)

    2015

  24. [24]

    Fehske, R

    H. Fehske, R. Schneider, and A. Weisse,Computational many-particle physics(Springer, 2007)

    2007

  25. [25]

    Avella and F

    A. Avella and F. Mancini, eds.,Strongly correlated sys- tems: Theoretical methods(Springer Berlin, Heidelberg, Germany, 2012)

    2012

  26. [26]

    J. W. Negele,Quantum many-particle systems(CRC Press, London, England, 2019)

    2019

  27. [27]

    Ostahie, D

    B. Ostahie, D. Sticlet, C. u. u. u. u. P. m. c. Moca, B. D´ ora, M. A. Werner, J. K. Asb´ oth, and G. Zar´ and, Multiparticle quantum walk: A dynamical probe of topo- logical many-body excitations, Phys. Rev. B108, 035126 (2023)

    2023

  28. [28]

    M. G. de Andrade, W. Dai, S. Guha, and D. Towsley, A quantum walk control plane for distributed quantum computing in quantum networks, in2021 IEEE Interna- tional Conference on Quantum Computing and Engineer- ing (QCE)(2021) pp. 313–323

    2021

  29. [29]

    A. M. Childs, D. Gosset, and Z. Webb, Universal computation by multiparticle quantum walk, Science339, 791 (2013), https://www.science.org/doi/pdf/10.1126/science.1229957

    work page doi:10.1126/science.1229957 2013

  30. [30]

    J. J. Sakurai and J. Napolitano,Modern Quantum Me- chanics, 2nd ed. (Addison Wesley Longman, Boston, MA, 2010)

    2010

  31. [31]

    H. A. Carteret, N. Linden, S. Popescu, and A. Sudbery, Multiparticle entanglement, Foundations of Physics29, 527 (1999)

    1999

  32. [32]

    Eisert and D

    J. Eisert and D. Gross, Multi-particle entanglement (2006), arXiv:quant-ph/0505149 [quant-ph]

    Pith/arXiv arXiv 2006

  33. [33]

    Wen, Choreographed entangle- ment dances: Topological states of quan- tum matter, Science363, eaal3099 (2019), https://www.science.org/doi/pdf/10.1126/science.aal3099

    X.-G. Wen, Choreographed entangle- ment dances: Topological states of quan- tum matter, Science363, eaal3099 (2019), https://www.science.org/doi/pdf/10.1126/science.aal3099

    work page doi:10.1126/science.aal3099 2019

  34. [34]

    Mastandrea and C.-C

    C. Mastandrea and C.-C. Chien, Robustness and classi- cal proxy of entanglement in variants of quantum walks, Phys. Rev. E110, 064124 (2024)

    2024

  35. [35]

    Yamagishi, N

    M. Yamagishi, N. Hatano, K. Kawabata, C. Kiumi, A. Nishino, F. Nori, and H. Obuse, Quantum walk with a local spin interaction (2026), arXiv:2603.24444 [quant- 14 ph]

    arXiv 2026

  36. [36]

    S. K. Goyal and C. M. Chandrashekar, Spatial entan- glement using a quantum walk on a many-body system, Journal of Physics A: Mathematical and Theoretical43, 235303 (2010)

    2010

  37. [37]

    Camacho, J

    G. Camacho, J. Meinecke, and J. Wolters, Quantum walk on a square lattice with identical particles, Phys. Rev. A 111, 052416 (2025)

    2025

  38. [38]

    Zhuang, Z

    Q. Zhuang, Z. Zhang, and J. H. Shapiro, Distributed quantum sensing using continuous-variable multipartite entanglement, Phys. Rev. A97, 032329 (2018)

    2018

  39. [39]

    Liu, Y.-B

    Y.-C. Liu, Y.-B. Cheng, X.-B. Pan, Z.-Z. Sun, D. Pan, and G.-L. Long, Quantum integrated sensing and com- munication via entanglement, Phys. Rev. Appl.22, 034051 (2024)

    2024

  40. [40]

    Zhang, S

    Z. Zhang, S. Mouradian, F. N. C. Wong, and J. H. Shapiro, Entanglement-enhanced sensing in a lossy and noisy environment, Phys. Rev. Lett.114, 110506 (2015)

    2015

  41. [41]

    Jozsa and N

    R. Jozsa and N. Linden, On the role of en- tanglement in quantum-computational speed-up, Proceedings of the Royal Society A: Mathemati- cal, Physical and Engineering Sciences459, 2011 (2003), https://royalsocietypublishing.org/rspa/article- pdf/459/2036/2011/640791/rspa.2002.1097.pdf

    arXiv 2011

  42. [42]

    Dadras, A

    S. Dadras, A. Gresch, C. Groiseau, S. Wimberger, and G. S. Summy, Quantum walk in momentum space with a bose-einstein condensate, Phys. Rev. Lett.121, 070402 (2018)

    2018

  43. [43]

    B. C. Travaglione and G. J. Milburn, Implementing the quantum random walk, Phys. Rev. A65, 032310 (2002)

    2002

  44. [44]

    Zhou, X.-W

    W.-H. Zhou, X.-W. Wang, R.-J. Ren, Y.-X. Fu, Y.-J. Chang, X.-Y. Xu, H. Tang, and X.-M. Jin, Multi-particle quantum walks on 3D integrated photonic chip, Light Sci. Appl.13, 296 (2024)

    2024

  45. [45]

    Gr¨ afe, R

    M. Gr¨ afe, R. Heilmann, M. Lebugle, D. Guzman-Silva, A. Perez-Leija, and A. Szameit, Integrated photonic quantum walks, J. Opt.18, 103002 (2016)

    2016

  46. [46]

    Zhou, X.-W

    W.-H. Zhou, X.-W. Wang, R.-J. Ren, Y.-X. Fu, Y.-J. Chang, X.-Y. Xu, H. Tang, and X.-M. Jin, Multi-particle quantum walks on 3d integrated photonic chip, Light: Science & Applications13, 296 (2024)

    2024

  47. [47]

    M. A. Nielsen and I. L. Chuang,Quantum Computation and Quantum Information: 10th Anniversary Edition (Cambridge University Press, Cambridge, UK, 2010)

    2010

  48. [48]

    H. M. Wiseman and J. A. Vaccaro, Entanglement of indistinguishable particles shared between two parties, Phys. Rev. Lett.91, 097902 (2003)

    2003

  49. [49]

    Benatti, R

    F. Benatti, R. Floreanini, F. Franchini, and U. Mar- zolino, Entanglement in indistinguishable particle sys- tems, Physics Reports878, 1 (2020)

    2020

  50. [50]

    Y. S. Li, B. Zeng, X. S. Liu, and G. L. Long, Entangle- ment in a two-identical-particle system, Phys. Rev. A64, 054302 (2001)

    2001

  51. [51]

    B. J. Dalton, J. Goold, B. M. Garraway, and M. D. Reid, Quantum entanglement for systems of identical bosons: I. general features, Phys. Scr.92, 023004 (2017)

    2017

  52. [52]

    Morris, B

    B. Morris, B. Yadin, M. Fadel, T. Zibold, P. Treutlein, and G. Adesso, Entanglement between identical parti- cles is a useful and consistent resource, Phys. Rev. X10, 041012 (2020)

    2020

  53. [53]

    Lambert, E

    N. Lambert, E. Gigu‘ere, P. Menczel, B. Li, P. Hopf, G. Su’arez, M. Gali, J. Lishman, R. Gadhvi, R. Agarwal, A. Galicia, N. Shammah, P. Nation, J. R. Johansson, S. Ahmed, S. Cross, A. Pitchford, and F. Nori, Qutip 5: The quantum toolbox in Python, Physics Reports1153, 1 (2026)

    2026

  54. [54]

    Salomon, G

    C. Salomon, G. V. Shlyapnikov, and L. F. Cugliandolo, Many-Body Physics with Ultracold Gases: Lecture Notes of the Les Houches Summer School: Volume 94, July 2010(Oxford University Press, 2012) Chap. 4

    2010

  55. [55]

    Strang,Linear Algebra and Its Applications(Thom- son, Brooks/Cole, Belmont, CA, 2006)

    G. Strang,Linear Algebra and Its Applications(Thom- son, Brooks/Cole, Belmont, CA, 2006)

    2006

  56. [56]

    Klema and A

    V. Klema and A. Laub, The singular value decompo- sition: Its computation and some applications, IEEE Transactions on Automatic Control25, 164 (1980)

    1980

  57. [57]

    Paˇ skauskas and L

    R. Paˇ skauskas and L. You, Quantum correlations in two- boson wave functions, Phys. Rev. A64, 042310

  58. [58]

    Wang and B

    X. Wang and B. C. Sanders, Canonical entanglement for two indistinguishable particles, Journal of Physics A: Mathematical and General38, L67 (2005)

    2005

  59. [59]

    Naudts and T

    J. Naudts and T. Verhulst, Ensemble-averaged entangle- ment of two-particle states in fock space, Phys. Rev. A 75, 062104 (2007)

    2007

  60. [60]

    Kapcia and S

    K. Kapcia and S. Robaszkiewicz, The effects of the next-nearest-neighbour density–density interaction in the atomic limit of the extended hubbard model, Journal of Physics: Condensed Matter23, 105601 (2011)

    2011

  61. [61]

    Hofmann and M

    F. Hofmann and M. Potthoff, Doublon dynamics in the extended fermi-hubbard model, Phys. Rev. B85, 205127 (2012)

    2012

  62. [62]

    Strohmaier, D

    N. Strohmaier, D. Greif, R. J¨ ordens, L. Tarruell, H. Moritz, T. Esslinger, R. Sensarma, D. Pekker, E. Alt- man, and E. Demler, Observation of elastic doublon de- cay in the fermi-hubbard model, Phys. Rev. Lett.104, 080401 (2010)

    2010

  63. [63]

    Winkler, G

    K. Winkler, G. Thalhammer, F. Lang, R. Grimm, J. H. Denschlag, A. J. Daley, A. Kantian, H. P. B¨ uchler, and P. Zoller, Repulsively bound atom pairs in an optical lat- tice, Nature441, 853 (2006)

    2006

  64. [64]

    Chien, D

    C.-C. Chien, D. Gruss, M. D. Ventra, and M. Zwolak, Interaction-induced conducting–non-conducting transi- tion of ultra-cold atoms in one-dimensional optical lat- tices, New Journal of Physics15, 063026 (2013)

    2013

  65. [65]

    Acasiete, F

    F. Acasiete, F. P. Agostini, J. K. Moqadam, and R. Por- tugal, Implementation of quantum walks on ibm quan- tum computers, Quantum Information Processing19, 426 (2020)

    2020

  66. [66]

    Georgopoulos, C

    K. Georgopoulos, C. Emary, and P. Zuliani, Comparison of quantum-walk implementations on noisy intermediate- scale quantum computers, Phys. Rev. A103, 022408 (2021)

    2021

  67. [67]

    Tornow and K

    S. Tornow and K. Ziegler, Measurement-induced quan- tum walks on an ibm quantum computer, Phys. Rev. Res.5, 033089 (2023)

    2023

  68. [68]

    Cramer, M

    M. Cramer, M. B. Plenio, S. T. Flammia, R. Somma, D. Gross, S. D. Bartlett, O. Landon-Cardinal, D. Poulin, and Y.-K. Liu, Efficient quantum state tomography, Nat. Commun.1, 149 (2010)

    2010

  69. [69]

    Karski, L

    M. Karski, L. F¨ orster, J.-M. Choi, A. Steffen, W. Alt, D. Meschede, and A. Widera, Quantum walk in position space with single optically trapped atoms, Science325, 174 (2009)

    2009

  70. [70]

    T. J. Dunn, I. A. Walmsley, and S. Mukamel, Experimen- tal determination of the quantum-mechanical state of a molecular vibrational mode using fluorescence tomogra- phy, Phys. Rev. Lett.74, 884 (1995)

    1995

  71. [71]

    C.-K. Hu, C. Wei, C. Liu, L. Che, Y. Zhou, G. Xie, H. Qin, G. Hu, H. Yuan, R. Zhou, S. Liu, D. Tan, T. Xin, 15 and D. Yu, Experimental sample-efficient quantum state tomography via parallel measurements, Phys. Rev. Lett. 133, 160801 (2024)

    2024

  72. [72]

    Baumgratz, D

    T. Baumgratz, D. Gross, M. Cramer, and M. B. Plenio, Scalable reconstruction of density matrices, Phys. Rev. Lett.111, 020401 (2013)

    2013

  73. [73]

    Gigena and R

    N. Gigena and R. Rossignoli, Bipartite entanglement in fermion systems, Phys. Rev. A95, 062320 (2017)

    2017

  74. [74]

    Lo Franco and G

    R. Lo Franco and G. Compagno, Quantum entanglement of identical particles by standard information-theoretic notions, Scientific Reports6, 20603 (2016)

    2016

  75. [75]

    Tziperman, D

    O. Tziperman, D. Nabben, R. Ruimy, J. Holder, E. Nussinson, Y. Fang, A. Gorlach, D. Kazenwadel, A. Karnieli, I. Kaminer, and P. Baum, Two-electron quantum walks for probing entanglement and deco- herence in an electron microscope, Nature Physics 10.1038/s41567-026-03254-y (2026)

    work page doi:10.1038/s41567-026-03254-y 2026

  76. [76]

    M. K. Giri, S. Mondal, B. P. Das, and T. Mishra, Two component quantum walk in one-dimensional lattice with hopping imbalance, Scientific Reports11, 22056 (2021)

    2021