Abstract

How to reveal topological phases and their boundaries is an intriguing issue in various systems. Entanglement, which plays a fundamental role in quantum information, has been found profoundly related to topological phases. However, experimentally exploring this relation is precluded by the limited ability to obtain entanglement in many-body systems. In this work, we propose and experimentally demonstrate that the robustness of entanglement, quantified by the von Neumann entropy, can be used to reveal the topological phase with winding number $ {\cal W} = 1 $ and topological phase with $ {\cal W} = 0 $ in quantum walks. With the different robustness of entanglement against perturbations of a parameter, the phase boundaries between the distinct topological phases can be further determined. As a result, our work not only offers a new perspective for quantum walks but also exhibits the deep connection between entanglement and topological physics.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2019 (3)

B. Sephton, A. Dudley, G. Ruffato, F. Romanato, L. Marrucci, M. Padgett, S. Goyal, F. Roux, T. Konrad, and A. Forbes, “A versatile quantum walk resonator with bright classical light,” PLoS One 14, e0214891 (2019).
[Crossref]

Y. Wang, Y.-H. Lu, F. Mei, J. Gao, Z.-M. Li, H. Tang, S.-L. Zhu, S. Jia, and X.-M. Jin, “Direct observation of topology from single-photon dynamics,” Phys. Rev. Lett. 122, 193903 (2019).
[Crossref]

F. Nejadsattari, Y. Zhang, F. Bouchard, H. Larocque, A. Sit, E. Cohen, R. Fickler, and E. Karimi, “Experimental realization of wave-packet dynamics in cyclic quantum walks,” Optica 6, 174–180 (2019).
[Crossref]

2018 (8)

Q.-Q. Wang, X.-Y. Xu, W.-W. Pan, K. Sun, J.-S. Xu, G. Chen, Y.-J. Han, C.-F. Li, and G.-C. Guo, “Dynamic-disorder-induced enhancement of entanglement in photonic quantum walks,” Optica 5, 1136–1140 (2018).
[Crossref]

A. D. Verga and R. G. Elias, “Entanglement and interaction in a topological quantum walk,” SciPost Phys. 5, 19 (2018).
[Crossref]

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).
[Crossref]

B. Wang, T. Chen, and X. Zhang, “Experimental observation of topologically protected bound states with vanishing Chern numbers in a two-dimensional quantum walk,” Phys. Rev. Lett. 121, 100501 (2018).
[Crossref]

C. Chen, X. Ding, J. Qin, Y. He, Y.-H. Luo, M.-C. Chen, C. Liu, X.-L. Wang, W.-J. Zhang, H. Li, L.-X. You, Z. Wang, D.-W. Wang, B. C. Sanders, C.-Y. Lu, and J.-W. Pan, “Observation of topologically protected edge states in a photonic two-dimensional quantum walk,” Phys. Rev. Lett. 121, 100502 (2018).
[Crossref]

X.-Y. Xu, Q.-Q. Wang, W.-W. Pan, K. Sun, J.-S. Xu, G. Chen, J.-S. Tang, M. Gong, Y.-J. Han, C.-F. Li, and G.-C. Guo, “Measuring the winding number in a large-scale chiral quantum walk,” Phys. Rev. Lett. 120, 260501 (2018).
[Crossref]

Y.-R. Zhang, Y. Zeng, H. Fan, J. Q. You, and F. Nori, “Characterization of topological states via dual multipartite entanglement,” Phys. Rev. Lett. 120, 250501 (2018).
[Crossref]

K. Choo, C. W. von Keyserlingk, N. Regnault, and T. Neupert, “Measurement of the entanglement spectrum of a symmetry-protected topological state using the IBM quantum computer,” Phys. Rev. Lett. 121, 086808 (2018).
[Crossref]

2017 (4)

L. Pezzè, M. Gabbrielli, L. Lepori, and A. Smerzi, “Multipartite entanglement in topological quantum phases,” Phys. Rev. Lett. 119, 250401 (2017).
[Crossref]

X. Zhan, L. Xiao, Z.-H. Bian, K.-K. Wang, X.-Z. Qiu, B. C. Sanders, W. Yi, and P. Xue, “Detecting topological invariants in nonunitary discrete-time quantum walks,” Phys. Rev. Lett. 119, 130501 (2017).
[Crossref]

F. Cardano, A. D’Errico, A. Dauphin, M. Maffei, B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato, L. Marrucci, M. Lewenstein, and P. Massignan, “Detection of Zak phases and topological invariants in a chiral quantum walk of twisted photons,” Nat. Commun. 8, 15516 (2017).
[Crossref]

W.-W. Zhang, B. C. Sanders, S. Apers, S. K. Goyal, and D. L. Feder, “Detecting topological transitions in two dimensions by Hamiltonian evolution,” Phys. Rev. Lett. 119, 197401 (2017).
[Crossref]

2016 (4)

F. Cardano, M. Maffei, F. Massa, B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato, and L. Marrucci, “Statistical moments of quantum-walk dynamics reveal topological quantum transitions,” Nat. Commun. 7, 11439 (2016).
[Crossref]

Z. Liu and R. N. Bhatt, “Quantum entanglement as a diagnostic of phase transitions in disordered fractional quantum hall liquids,” Phys. Rev. Lett. 117, 206801 (2016).
[Crossref]

N. Goldman, J. C. Budich, and P. Zoller, “Topological quantum matter with ultracold gases in optical lattices,” Nat. Phys. 12, 639–645 (2016).
[Crossref]

C. K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu, “Classification of topological quantum matter with symmetries,” Rev. Mod. Phys. 88, 035005 (2016).
[Crossref]

2015 (4)

R. Islam, R. Ma, P. M. Preiss, M. E. Tai, A. Lukin, M. Rispoli, and M. Greiner, “Measuring entanglement entropy in a quantum many-body system,” Nature 528, 77–83 (2015).
[Crossref]

F. Cardano, F. Massa, H. Qassim, E. Karimi, S. Slussarenko, D. Paparo, C. de Lisio, F. Sciarrino, E. Santamato, R. W. Boyd, and L. Marrucci, “Quantum walks and wavepacket dynamics on a lattice with twisted photons,” Sci. Adv. 1, e1500087 (2015).
[Crossref]

H. Obuse, J. K. Asbóth, Y. Nishimura, and N. Kawakami, “Unveiling hidden topological phases of a one-dimensional Hadamard quantum walk,” Phys. Rev. B 92, 045424 (2015).
[Crossref]

J. M. Zeuner, M. C. Rechtsman, Y. Plotnik, Y. Lumer, S. Nolte, M. S. Rudner, M. Segev, and A. Szameit, “Observation of a topological transition in the bulk of a non-Hermitian system,” Phys. Rev. Lett. 115, 040402 (2015).
[Crossref]

2014 (2)

G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat, T. Uehlinger, D. Greif, and T. Esslinger, “Experimental realization of the topological Haldane model with ultracold fermions,” Nature 515, 237–240 (2014).
[Crossref]

T. P. Oliveira and P. D. Sacramento, “Entanglement modes and topological phase transitions in superconductors,” Phys. Rev. B 89, 094512 (2014).
[Crossref]

2013 (6)

A. M. Childs, D. Gosset, and Z. Webb, “Universal computation by multiparticle quantum walk,” Science 339, 791–794 (2013).
[Crossref]

R. Vieira, E. P. M. Amorim, and G. Rigolin, “Dynamically disordered quantum walk as a maximal entanglement generator,” Phys. Rev. Lett. 111, 180503 (2013).
[Crossref]

A. Crespi, R. Osellame, R. Ramponi, V. Giovannetti, R. Fazio, L. Sansoni, F. De Nicola, F. Sciarrino, and P. Mataloni, “Anderson localization of entangled photons in an integrated quantum walk,” Nat. Photonics 7, 322–328 (2013).
[Crossref]

X.-J. Liu, K.-T. Law, T.-K. Ng, and P. A. Lee, “Detecting topological phases in cold atoms,” Phys. Rev. Lett. 111, 120402 (2013).
[Crossref]

J. K. Asbóth and H. Obuse, “Bulk-boundary correspondence for chiral symmetric quantum walks,” Phys. Rev. B 88, 121406 (2013).
[Crossref]

S. Moulieras, M. Lewenstein, and G. Puentes, “Entanglement engineering and topological protection by discrete-time quantum walks,” J. Phys. B 46, 104005 (2013).
[Crossref]

2012 (3)

O. Gamel and D. F. V. James, “Measures of quantum state purity and classical degree of polarization,” Phys. Rev. A 86, 033830 (2012).
[Crossref]

T. Kitagawa, “Topological phenomena in quantum walks: elementary introduction to the physics of topological phases,” Quantum Inf. Process. 11, 1107–1148 (2012).
[Crossref]

T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, and A. G. White, “Observation of topologically protected bound states in photonic quantum walks,” Nat. Commun. 3, 882 (2012).
[Crossref]

2011 (4)

J. Zhang, T.-C. Wei, and R. Laflamme, “Experimental quantum simulation of entanglement in many-body systems,” Phys. Rev. Lett. 107, 010501 (2011).
[Crossref]

S. D. Berry and J. B. Wang, “Two-particle quantum walks: entanglement and graph isomorphism testing,” Phys. Rev. A 83, 042317 (2011).
[Crossref]

S. Diehl, E. Rico, M. A. Baranov, and P. Zoller, “Topology by dissipation in atomic quantum wires,” Nat. Phys. 7, 971–977 (2011).
[Crossref]

A. Schreiber, K. N. Cassemiro, V. Potocek, A. Gabris, I. Jex, and C. Silberhorn, “Decoherence and disorder in quantum walks: from ballistic spread to localization,” Phys. Rev. Lett. 106, 180403 (2011).
[Crossref]

2010 (3)

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. Q. Zhou, Y. Lahini, N. Ismail, K. Worhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. O’Brien, “Quantum walks of correlated photons,” Science 329, 1500–1503 (2010).
[Crossref]

T. Kitagawa, M. S. Rudner, E. Berg, and E. Demler, “Exploring topological phases with quantum walks,” Phys. Rev. A 82, 033429 (2010).
[Crossref]

L. A. Wray, S. Y. Xu, Y. Q. Xia, Y. S. Hor, D. Qian, A. V. Fedorov, H. Lin, A. Bansil, R. J. Cava, and M. Z. Hasan, “Observation of topological order in a superconducting doped topological insulator,” Nat. Phys. 6, 855–859 (2010).
[Crossref]

2009 (2)

Y. Xia, D. Qian, D. Hsieh, L. Wray, A. Pal, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, “Observation of a large-gap topological-insulator class with a single Dirac cone on the surface,” Nat. Phys. 5, 398–402 (2009).
[Crossref]

M. S. Rudner and L. S. Levitov, “Topological transition in a non-Hermitian quantum walk,” Phys. Rev. Lett. 102, 065703 (2009).
[Crossref]

2008 (3)

K. S. Choi, H. Deng, J. Laurat, and H. Kimble, “Mapping photonic entanglement into and out of a quantum memory,” Nature 452, 67–71 (2008).
[Crossref]

L. Amico, R. Fazio, A. Osterloh, and V. Vedral, “Entanglement in many-body systems,” Rev. Mod. Phys. 80, 517–576 (2008).
[Crossref]

C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. DasSarma, “Non-abelian anyons and topological quantum computation,” Rev. Mod. Phys. 80, 1083–1159 (2008).
[Crossref]

2006 (4)

D. N. Sheng, Z. Y. Weng, L. Sheng, and F. D. M. Haldane, “Quantum spin-Hall effect and topologically invariant Chern numbers,” Phys. Rev. Lett. 97, 036808 (2006).
[Crossref]

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Supplementary Material (1)

NameDescription
» Supplement 1       We provide the detailed physical descriptions of the connection between the topological robustness and the resulting robustness of entanglement, and show our work can be extended to some other Floquet systems.

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Figures (4)

Fig. 1.
Fig. 1. (a) Phase diagram of the split-step quantum walks and (b) asymptotic entanglement as a function of control parameters $ {\theta _1} $ and $ {\theta _2} $ for the given initial state $ | {{ \downarrow _y}} \rangle \otimes | {x = 0} \rangle $. The red solid lines and green dashed lines in the phase diagram mean the phase-transition lines where a gap closed at quasi-energy $ E = 0 $ and $ E = \pi $, respectively. The distinct robustness of entanglement against $ {\theta _2} $ is separated by the orange dashed lines in (b).
Fig. 2.
Fig. 2. Scheme of the experimental setup. (a) The heralded single photons are prepared via a beam-like SPDC in BBO2. (b) The initial coin state is prepared by means of a polarization-dependent beam splitter (PBS), half-wave plate (HWP), and quarter-wave plate (QWP). (c) QW is realized by a sequence of HWPs (for realizing coin operations) and calcite crystals (for realizing shift operations). (d) The reduced density matrix of the coin can be reconstructed via standard tomography, realized by a set of QWP-HWP-PBS. IF, interference filter; SMF, single-mode fiber; SPAD, single-photon avalanche diode.
Fig. 3.
Fig. 3. Entanglement measured (blue and orange symbols) against the number of steps for the initial coin states (a) $ \sqrt {0.5} (|H\rangle + i|V\rangle ) $ and (b) $ \sqrt {0.7} |H\rangle + \sqrt {0.3} i|V\rangle $ located at the original site ($ x = 0 $). The blue and orange solid lines show the theoretical predictions for each step with $ \{ {\theta _1},{\theta _2}\} = \{ \pi /2,\pi /5\} $ and $ \{ \pi /2,4\pi /5\} $, and the dashed horizontal lines show the asymptotic entanglement in the long-time limit. For the two scenarios, we add a small perturbation $ \delta {\theta _2} $ to $ {\theta _2} $. The colored dashed and dashed-dotted lines represent the theoretical predictions, and colored diamonds and triangles show the experimental results for $ \delta {\theta _2} = \pm \pi /15 $ and $ \pm \pi /30 $, respectively. The entire offsets of entanglement $ \delta {S_E} $ in trivial topological phase, as indicated by the shaded regions, are obviously higher than the offsets in non-trivial topological phase, which are zero in the long-time limit for the two different initial states. Error bars are smaller than the size of the symbols.
Fig. 4.
Fig. 4. Trajectories in the phase diagram are presented in (a). The entanglement measured (red symbols) as a function of the parameter $ {\theta _2} $ when (b) $ {\theta _1} = \pi /5 $ (c) $ {\theta _1} = \pi /2 $, and (d) $ {\theta _1} = 4\pi /5 $ after a walk of 13 steps. The blue dashed lines represent theoretical predictions after a 13-step walk, vertical black dashed lines at $ {\theta _2} = \pi /2 $ mark the phase boundaries, and the orange solid lines show the asymptotic entanglement in the long-time limit. Error bars are smaller than the size of the symbols.

Equations (2)

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S E [ ρ ( t ) ] = 1 2 P [ ρ C ( t ) ] log 2 [ 1 + P [ ρ C ( t ) ] 1 P [ ρ C ( t ) ] ] 1 2 log 2 [ 1 P [ ρ C ( t ) ] 2 4 ] .
P [ ρ ¯ C ] = π π d k 2 π | n y ( k ) | 2 ,