We study a two-dimensional quantum walk with only one walker alternatively walking along the horizontal and vertical directions driven by a single two-side coin. We find the analytical expressions of the first two moments of the walker’s position distribution in the long-time limit, which indicates that the variance of the position distribution grows quadratically with walking steps, showing a ballistic spreading typically for quantum walks. Besides, we analyze the correlation by calculating the quantum mutual information and the measurement-induced disturbance respectively as the outcome of the walk in one dimension is correlated to the other with the coin as a bridge. It is shown that the quantum correlation between walker spaces increases gradually with the walking steps.
Received: 30 June 2020
Revised: 04 August 2020
Accepted manuscript online: 13 August 2020
Fund: the National Natural Science Foundation of China (Grant Nos. 11674056 and U1930402) and the startup fund from Beijing Computational Science Research Center. KKW acknowledges support from the Project funded by China Postdoctoral Science Foundation (Grant No. 2019M660016).
Quan Lin(林泉), Hao Qin(秦豪) Kun-Kun Wang(王坤坤), Lei Xiao(肖磊), and Peng Xue(薛鹏) A two-dimensional quantum walk driven by a single two-side coin 2020 Chin. Phys. B 29 110303
Fig. 1.
Schematic for the two-dimensional QW. At the beginning of the walk, the single walker, starts from the position (x0 = 0,y0 = 0) with a two-side quantum coin in his hand. Then the walker flips the coin and takes a step along the x axis to the left or right if the coin turns out to be head or tail. Hence, the walker is possible to appear at positions (−1,0) and (1,0), i.e., the orange points. After this, the walker flips the coin again and the outcome of the coin flipping decides the walker’s movement along the y axis, up or down. Therefore, we may find the walker at positions (−1,−1), (−1,1), (1,−1), and (1,1), i.e., the yellow points in the figure with certain probabilities. We consider a step along the x axis and a step along the y axis respectively of the walker as a single course of our walk. During the second course, the walker possibly gets to the positions of the green points after the first coin flipping and finally arrives at the positions of the pink points and also the initial red point, which are 9 possible positions in total, after the second coin flipping. We omit the intermediate positions the walker may reach from the third course of the walk in the figure. It can be seen that after the n-th course the walk shows a spreading of position distribution including (n + 1)2 possible points on the two-dimensional lattice.
Fig. 2.
(a) The position distribution of the two-dimensional CRW after the 10th step. (b) The variance of the position distributions of the two-dimensional CRW for the fist 10 steps.
Fig. 3.
(a) The position distribution of the two-dimensional QW after the 10th step. (b) The variance of the position distributions of the two-dimensional QW for the fist 10 steps.
Fig. 4.
(a) The quantum mutual information of two-dimensional QW (squares) and of the maximal entangled state (dots). (b) The ln–ln plot of the quantum mutual information of two-dimensional QW (squares) and of the maximal entangled state (dots).
Fig. 5.
(a) The measurement-induced disturbance of two-dimensional QW (squares) and of the maximal entangled state (dots). (b) The ln–ln plot of the measurement-induced disturbance of two-dimensional QW (squares) and of the maximal entangled state (dots).
[1]
Dür W, Raussendorf R, Kendon V M, Briegel H J 2002 Phys. Rev. A66 052319 DOI: 10.1103/PhysRevA.66.052319
Xiao L, Zhan X, Bian Z H, Wang K K, Zhang X, Wang X P, Li J, Mochizuki K, Kim D, Kawakami N, Yi W, Obuse H, Sanders B C, Xue P 2017 Nat. Phys.13 1117 DOI: 10.1038/nphys4204
[16]
Xiao L, Deng T S, Wang K K, Zhu G Y, Wang Z, Yi W, Xue P 2020 Nat. Phys.16 761 DOI: 10.1038/s41567-020-0836-6
Cardano F, Massa F, Qassim H, Karimi E, Slussarenko S, Paparo D, de Lisio C, Sciarrino F, Santamato E, Boyd R W, Marrucci L 2015 Sci. Adv.1 e1500087 DOI: 10.1126/sciadv.1500087
[21]
Defienne H, Barbieri M, Walmsley I A, Smith B J, Gigan S 2016 Sci. Adv.1 e1501054 DOI: 10.1126/sciadv.1501054
[22]
Wang K K, Qiu X, Xiao L, Zhan X, Bian Z H, Yi W, Xue P 2019 Phys. Rev. Lett.122 020501 DOI: 10.1103/PhysRevLett.122.020501
Ryan C A, Laforest M, Boileau J C, Laflamme R 2005 Phys. Rev. A72 062317 DOI: 10.1103/PhysRevA.72.062317
[25]
Broome M A, Fedrizzi A, Lanyon B P, Kassal I, Aspuru-Guzik A, White A G 2010 Phys. Rev. Lett.104 153602 DOI: 10.1103/PhysRevLett.104.153602
[26]
Schreiber A, Cassemiro K N, Potocek V, Gabris A, Mosley P J, Andersson E, Jex I, Silberhorn C 2010 Phys. Rev. Lett.104 050502 DOI: 10.1103/PhysRevLett.104.050502
[27]
Kitagawa T, Broome M A, Fedrizzi A, Rudner M S, Berg E, Kassal I, Aspuru-Guzik A, Demler E, White A G 2012 Nat. Commun.3 882 DOI: 10.1038/ncomms1872
Zahringer F, Kirchmair G, Gerritsma R, Solano E, Blatt R, Roos C F 2010 Phys. Rev. Lett.104 100503 DOI: 10.1103/PhysRevLett.104.100503
[43]
Peruzzo A, Lobino M, Matthews J C F, Matsuda N, Politi A, Poulios K, Zhou X, Lahini Y, Ismail N, Worhoff K, Bromberg Y, Silberberg Y, Thompson M G, OBrien J L 2010 Science329 1500 DOI: 10.1126/science.1193515
[44]
Schreiber A, Gabris A, Rohde P P, Laiho K, Stefanak M, Potocek V, Hamilton C, Jex I, Silberhorn C 2012 Science336 55 DOI: 10.1126/science.1218448
[45]
Crespi A, Osellame R, Ramponi R, Giovannetti V, Fazio R, Sansoni L, Nicola F D, Sciarrino F, Mataloni P 2013 Nat. Photon.7 322 DOI: 10.1038/nphoton.2013.26
[46]
Mohseni M, Rebentrost P, Lloyd S, Aspuru-Guzik A 2008 J. Chem. Phys.129 174106 DOI: 10.1063/1.3002335
DErrico A, Cardano F, Maffei M, Dauphin A, Barboza R, Esposito C, Piccirillo B, Lewenstein M, Massignan P, Marrucci L 2020 Optica7 108 DOI: 10.1364/OPTICA.365028
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