Abstract

Entanglement generation in discrete-time quantum walks is deemed to be another key property beyond the transport behaviors. The latter has been widely used in investigating the localization or topology in quantum walks. However, there are few experiments involving the former for addressing the challenges in full reconstruction of the final wave function. Here, we report an experiment demonstrating enhancement of the entanglement in quantum walks using dynamic disorder. Through reconstructing the local spinor state for each site, von Neumann entropy can be obtained and used to quantify the coin-position entanglement. We find that the enhanced entanglement in the dynamically disordered quantum walks is independent of the initial state, which is different from the entanglement generation in Hadamard quantum walks. Our results are inspirational for achieving quantum computing based on quantum walks.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (1)

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]

2017 (2)

S. Barkhofen, T. Nitsche, F. Elster, L. Lorz, A. Gábris, I. Jex, and C. Silberhorn, “Measuring topological invariants in disordered discrete-time quantum walks,” Phys. Rev. A 96, 033846 (2017).
[Crossref]

M. Zeng and E. H. Yong, “Discrete-time quantum walk with phase disorder: localization and entanglement entropy,” Sci. Rep. 7, 12024 (2017).
[Crossref]

2016 (4)

U. Mishra, D. Rakshit, R. Prabhu, A. S. De, and U. Sen, “Constructive interference between disordered couplings enhances multiparty entanglement in quantum Heisenberg spin glass models,” New J. Phys. 18, 083044 (2016).
[Crossref]

J. Boutari, A. Feizpour, S. Barz, C. Di Franco, M. S. Kim, W. S. Kolthammer, and I. A. Walmsley, “Large scale quantum walks by means of optical fiber cavities,” J. Opt. 18, 094007 (2016).
[Crossref]

M. C. Rechtsman, Y. Lumer, Y. Plotnik, A. Perez-Leija, A. Szameit, and M. Segev, “Topological protection of photonic path entanglement,” Optica 3, 925–930 (2016).
[Crossref]

J. G. Titchener, A. S. Solntsev, and A. A. Sukhorukov, “Two-photon tomography using on-chip quantum walks,” Opt. Lett. 41, 4079–4082 (2016).
[Crossref]

2015 (1)

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]

2014 (1)

R. Vieira, E. P. M. Amorim, and G. Rigolin, “Entangling power of disordered quantum walks,” Phys. Rev. A 89, 042307 (2014).
[Crossref]

2013 (5)

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

C. M. Chandrashekar, “Two-component Dirac-like Hamiltonian for generating quantum walk on one-, two- and three-dimensional lattices,” Sci. Rep. 3, 2829 (2013).
[Crossref]

O. Morin, J. D. Bancal, M. Ho, P. Sekatski, V. D’Auria, N. Gisin, J. Laurat, and N. Sangouard, “Witnessing trustworthy single-photon entanglement with local homodyne measurements,” Phys. Rev. Lett. 110, 130401 (2013).
[Crossref]

T. Eichelkraut, R. Heilmann, S. Weimann, S. Stützer, F. Dreisow, D. N. Christodoulides, S. Nolte, and A. Szameit, “Mobility transition from ballistic to diffusive transport in non-Hermitian lattices,” Nat. Commun. 4, 2533 (2013).
[Crossref]

2012 (5)

L. Sansoni, F. Sciarrino, G. Vallone, P. Mataloni, A. Crespi, R. Ramponi, and R. Osellame, “Two-particle Bosonic-Fermionic quantum walk via integrated photonics,” Phys. Rev. Lett. 108, 010502 (2012).
[Crossref]

A. Schreiber, A. Gábris, P. P. Rohde, K. Laiho, M. Štefaňák, V. Potoček, C. Hamilton, I. Jex, and C. Silberhorn, “A 2D quantum walk simulation of two-particle dynamics,” Science 336, 55–58 (2012).
[Crossref]

S. E. Venegas-Andraca, “Quantum walks: a comprehensive review,” Quantum Inf. Process. 11, 1015–1106 (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]

J. P. Crutchfield, “Between order and chaos,” Nat. Phys. 8, 17–24 (2012).
[Crossref]

2011 (4)

R. Prabhu, S. Pradhan, A. Sen(De), and U. Sen, “Disorder overtakes order in information concentration over quantum networks,” Phys. Rev. A 84, 042334 (2011).
[Crossref]

A. Schreiber, K. N. Cassemiro, V. Potoček, 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]

C. Di Franco, M. Mc Gettrick, and T. Busch, “Mimicking the probability distribution of a two-dimensional Grover walk with a single-qubit coin,” Phys. Rev. Lett. 106, 080502 (2011).
[Crossref]

A. Regensburger, C. Bersch, B. Hinrichs, G. Onishchukov, A. Schreiber, C. Silberhorn, and U. Peschel, “Photon propagation in a discrete fiber network: an interplay of coherence and losses,” Phys. Rev. Lett. 107, 233902 (2011).
[Crossref]

2010 (4)

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]

A. Schreiber, K. N. Cassemiro, V. Potocek, A. Gabris, P. J. Mosley, E. Andersson, I. Jex, and C. Silberhorn, “Photons walking the line: a quantum walk with adjustable coin operations,” Phys. Rev. Lett. 104, 050502 (2010).
[Crossref]

A. Niederberger, M. M. Rams, J. Dziarmaga, F. M. Cucchietti, J. Wehr, and M. Lewenstein, “Disorder-induced order in quantum XY chains,” Phys. Rev. A 82, 013630 (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]

2009 (2)

A. M. Childs, “Universal computation by quantum walk,” Phys. Rev. Lett. 102, 180501 (2009).
[Crossref]

M. Karski, L. Förster, J.-M. Choi, A. Steffen, W. Alt, D. Meschede, and A. Widera, “Quantum walk in position space with single optically trapped atoms,” Science 325, 174–177 (2009).
[Crossref]

2008 (2)

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

H. B. Perets, Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, and Y. Silberberg, “Realization of quantum walks with negligible decoherence in waveguide lattices,” Phys. Rev. Lett. 100, 170506 (2008).
[Crossref]

2007 (1)

D. Binosi, G. De Chiara, S. Montangero, and A. Recati, “Increasing entanglement through engineered disorder in the random Ising chain,” Phys. Rev. B 76, 140405 (2007).
[Crossref]

2005 (1)

I. Carneiro, M. Loo, X. Xu, M. Girerd, V. Kendon, and P. L. Knight, “Entanglement in coined quantum walks on regular graphs,” New J. Phys. 7, 156 (2005).
[Crossref]

2004 (2)

P. Ribeiro, P. Milman, and R. Mosseri, “Aperiodic quantum random walks,” Phys. Rev. Lett. 93, 190503 (2004).
[Crossref]

P. Blanchard and M.-O. Hongler, “Quantum random walks and piecewise deterministic evolutions,” Phys. Rev. Lett. 92, 120601 (2004).
[Crossref]

2002 (1)

T. D. Mackay, S. D. Bartlett, L. T. Stephenson, and B. C. Sanders, “Quantum walks in higher dimensions,” J. Phys. A 35, 2745–2753 (2002).
[Crossref]

1993 (1)

Y. Aharonov, L. Davidovich, and N. Zagury, “Quantum random walks,” Phys. Rev. A 48, 1687–1690 (1993).
[Crossref]

1991 (1)

A. K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett. 67, 661–663 (1991).
[Crossref]

1976 (1)

A. Lempel and J. Ziv, “On the complexity of finite sequences,” IEEE Trans. Inf. Theory 22, 75–81 (1976).
[Crossref]

1958 (1)

P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109, 1492–1505 (1958).
[Crossref]

1905 (1)

K. Pearson, “The problem of the random walk,” Nature 72, 294 (1905).
[Crossref]

Aharonov, Y.

Y. Aharonov, L. Davidovich, and N. Zagury, “Quantum random walks,” Phys. Rev. A 48, 1687–1690 (1993).
[Crossref]

Alt, W.

M. Karski, L. Förster, J.-M. Choi, A. Steffen, W. Alt, D. Meschede, and A. Widera, “Quantum walk in position space with single optically trapped atoms,” Science 325, 174–177 (2009).
[Crossref]

Amorim, E. P. M.

R. Vieira, E. P. M. Amorim, and G. Rigolin, “Entangling power of disordered quantum walks,” Phys. Rev. A 89, 042307 (2014).
[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]

Anderson, P. W.

P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109, 1492–1505 (1958).
[Crossref]

Andersson, E.

A. Schreiber, K. N. Cassemiro, V. Potocek, A. Gabris, P. J. Mosley, E. Andersson, I. Jex, and C. Silberhorn, “Photons walking the line: a quantum walk with adjustable coin operations,” Phys. Rev. Lett. 104, 050502 (2010).
[Crossref]

Aspuru-Guzik, A.

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]

Bancal, J. D.

O. Morin, J. D. Bancal, M. Ho, P. Sekatski, V. D’Auria, N. Gisin, J. Laurat, and N. Sangouard, “Witnessing trustworthy single-photon entanglement with local homodyne measurements,” Phys. Rev. Lett. 110, 130401 (2013).
[Crossref]

Barkhofen, S.

S. Barkhofen, T. Nitsche, F. Elster, L. Lorz, A. Gábris, I. Jex, and C. Silberhorn, “Measuring topological invariants in disordered discrete-time quantum walks,” Phys. Rev. A 96, 033846 (2017).
[Crossref]

Bartlett, S. D.

T. D. Mackay, S. D. Bartlett, L. T. Stephenson, and B. C. Sanders, “Quantum walks in higher dimensions,” J. Phys. A 35, 2745–2753 (2002).
[Crossref]

Barz, S.

J. Boutari, A. Feizpour, S. Barz, C. Di Franco, M. S. Kim, W. S. Kolthammer, and I. A. Walmsley, “Large scale quantum walks by means of optical fiber cavities,” J. Opt. 18, 094007 (2016).
[Crossref]

Berg, E.

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]

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

Bersch, C.

A. Regensburger, C. Bersch, B. Hinrichs, G. Onishchukov, A. Schreiber, C. Silberhorn, and U. Peschel, “Photon propagation in a discrete fiber network: an interplay of coherence and losses,” Phys. Rev. Lett. 107, 233902 (2011).
[Crossref]

Binosi, D.

D. Binosi, G. De Chiara, S. Montangero, and A. Recati, “Increasing entanglement through engineered disorder in the random Ising chain,” Phys. Rev. B 76, 140405 (2007).
[Crossref]

Blanchard, P.

P. Blanchard and M.-O. Hongler, “Quantum random walks and piecewise deterministic evolutions,” Phys. Rev. Lett. 92, 120601 (2004).
[Crossref]

Boutari, J.

J. Boutari, A. Feizpour, S. Barz, C. Di Franco, M. S. Kim, W. S. Kolthammer, and I. A. Walmsley, “Large scale quantum walks by means of optical fiber cavities,” J. Opt. 18, 094007 (2016).
[Crossref]

Boyd, R. W.

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]

Bromberg, Y.

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]

Broome, M. A.

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]

Busch, T.

C. Di Franco, M. Mc Gettrick, and T. Busch, “Mimicking the probability distribution of a two-dimensional Grover walk with a single-qubit coin,” Phys. Rev. Lett. 106, 080502 (2011).
[Crossref]

Cardano, F.

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]

Carneiro, I.

I. Carneiro, M. Loo, X. Xu, M. Girerd, V. Kendon, and P. L. Knight, “Entanglement in coined quantum walks on regular graphs,” New J. Phys. 7, 156 (2005).
[Crossref]

Cassemiro, K. N.

A. Schreiber, K. N. Cassemiro, V. Potoček, 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]

A. Schreiber, K. N. Cassemiro, V. Potocek, A. Gabris, P. J. Mosley, E. Andersson, I. Jex, and C. Silberhorn, “Photons walking the line: a quantum walk with adjustable coin operations,” Phys. Rev. Lett. 104, 050502 (2010).
[Crossref]

Chandrashekar, C. M.

C. M. Chandrashekar, “Two-component Dirac-like Hamiltonian for generating quantum walk on one-, two- and three-dimensional lattices,” Sci. Rep. 3, 2829 (2013).
[Crossref]

Chen, G.

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Supplementary Material (1)

NameDescription
» Supplement 1       Detailed description of the experiment setup and the Lempel-Ziv complexity of sequences.

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Figures (5)

Fig. 1.
Fig. 1. Illustrations for (a) an ordered QW, where only one kind of coin is employed (red rectangle), and for (b) a dynamically disordered QW, where two kinds of coin are employed (red and green rectangles). The coin-position entanglement as a function of the initial state parameters { θ , ϕ } for (c) an ordered Hadamard QW and (d) the dynamically disordered QW with the coin tossing S C 0 (defined in the text). The dependency of the coin-position entanglement on the initial state is significantly higher for ordered QW compared to the dynamically disordered one.
Fig. 2.
Fig. 2. Diagram of the experimental setup (additional details are given in Supplement 1). The system contains four parts: (1) Second-harmonic generation in BBO1 to obtain the ultraviolet pulse; (2) generation of heralded single photons by adopting the beam-like type SPDC in BBO2; (3) time-multiplexing quantum walks realized by birefringent crystals sketched by the inset at the bottom right corner; (4) ultra-fast detection of the arrival time of single photons with an upconversion single-photon detector. BBO, β BaB 2 O 4 ; L, lens; DM, dichroic mirror; PBS, polarization-dependent beam splitter; HWP, half-wave plate; QWP, quarter-wave plate; R, reflector; FC, fiber collimator; PMT, photomultiplier tubes; SPAD, single-photon avalanche diode.
Fig. 3.
Fig. 3. (a) Coin-position entanglement measured (dots) for different initial states in an ordered Hadamard QW. The lines (solid red, dashed blue, and dotted yellow) are the theoretical predictions of the von Neumann entropies. The entanglement generated in an ordered QW shows a significant dependency on the initial state. Note that the entanglement involves periodical oscillation around an asymptotic value (guided by the dashed black horizontal line) in the limit of infinite steps with an amplitude decay. We have not considered the case { 51 ° , 270 ° } , which is the same as { 51 ° , 90 ° } . (b) Coin-position entanglement measured (dots) for different initial states in the dynamically disordered scenario. The lines (solid red, solid blue, solid yellow, and solid green) are the theoretical predictions. The von Neumann entropies are about 0.98 at 20 steps for all initial state clusters, which significantly differ from the ordered case, where they converge to different asymptotic values for different initial states. The periodical oscillation of entanglement with time degenerates. Only statistical errors are considered with total counts of 24,000 in 400 s.
Fig. 4.
Fig. 4. Measured trend (a) of second moment up to 20 steps (dots) for different initial states in ordered and dynamically disordered QWs. Ballistic transport behavior is observed in an ordered QW (solid red line, fitted as 0.29 t 2 ). In a dynamically disordered QW, a sub-ballistic transport behavior is observed (solid blue line, fitted as 0.6 t 1.54 ), and the dotted red line gives the theoretical predictions. The deep yellow line represents the typical diffuse transport behavior in classical random walks starting at the origin. The error bars are smaller than the point size with only consideration of statistical errors. (b) and (c) give the probability distributions after an 18-step QW for an initial state with θ = 51 ° and ϕ = 0 ° . It is seen clearly that the spreading velocity in (b) an ordered QW is significantly faster than (c) a dynamically disordered one.
Fig. 5.
Fig. 5. Coin-position entanglement measured (opaque bars) for different sequences (forms are given in Supplement 1) in a dynamically disordered QW with the theoretical predictions shown by transparent bars. The inset shows the rate of sequences generating different S E { 0 , 1 } at 20 steps with the initial state { 51 ° , 0 ° } , whose entanglement is lowest (shown with the solid black line) in an ordered Hadamard QW. Note that almost 73% of sequences can generate S E > 0.9 . In each entanglement interval (different color), we randomly choose two sequences. The theoretical average entanglement S E = 0.924 (guided by the black-dashed horizontal line) was calculated by averaging all sequences. Only statistical errors are considered.

Equations (2)

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H = 1 2 ( 1 1 1 1 ) , F = 1 2 ( 1 i i 1 ) .
S E ( ρ ( t ) ) = Tr [ ρ C ( t ) log 2 ρ C ( t ) ] ,

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