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退相位环境下Werner态在石墨烯基量子通道中的隐形传输

张乐 袁训锋 谭小东

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退相位环境下Werner态在石墨烯基量子通道中的隐形传输

张乐, 袁训锋, 谭小东

Teleportation of Werner state via graphene-based quantum channels under dephasing environment

Zhang Le, Yuan Xun-Feng, Tan Xiao-Dong
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  • 基于有效低能理论, 研究了退相位环境下Werner态在石墨烯基量子通道中的隐形传输. 结果表明, 输出态纠缠度总是随着输入态纠缠度的增大而增大, 而相应的保真度却正好相反; 对于给定的输入态, 量子通道中的纠缠越大, 输出态的品质就越高. 对于石墨烯基量子通道, 低温和弱库仑排斥势可以减缓其纠缠资源在退相位环境中的衰减, 且温度低于40 K, 电子间库仑排斥势小于6 eV时, 输出态的平均保真度可以达到80%以上. 这就说明石墨烯材料在量子信息领域中具有潜在的应用价值.
    The teleportation of Werner state in the graphene-based quantum channels under the dephasing environment is studied through the effective low-energy theory in this paper. The results show that the output entanglement normally reaches a higher level as the input entanglement increases, while the performance of the corresponding fidelity is opposite. Given the input state, the greater entanglement in the quantum channel can provide the higher-quality output state. For graphene-based quantum channels, the low temperature and weak Coulomb repulsive potential can decelerate the attenuation of entanglement resources in the dephasing environment. Moreover, when the temperature is lower than 40 K and the coulomb repulsive potential between electrons is less than 6 eV, the average fidelity of the output state reaches more than 80%. These results indicate that graphene has potential applications in quantum information.
      通信作者: 谭小东, txd10@163.com
    • 基金项目: 陕西省自然科学基础研究计划项目(批准号: 2021JQ-837)、国家自然科学基金(批准号: 11847042)、商洛学院科学与技术研究项目(批准号: 19SKY025)和商洛市科技局创新团队(批准号: SK2017-46)资助的课题
      Corresponding author: Tan Xiao-Dong, txd10@163.com
    • Funds: Project supported by the Natural Science Basic Research Program of Shaanxi Province, China (Grant No. 2021JQ-837), the National Natural Science Foundation of China (Grant No. 11847042), the Science and Technology Research Program of Shangluo University, China (Grant No. 19SKY025), and the Innovation Team of Science and Technology Bureau in Shangluo, China (Grant No. SK2017-46)
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  • 图 1  退相位环境下Werner态在石墨烯基量子通道中的隐形传输原理图 (a) 构建量子通道的特殊石墨烯纳米带(SGNR)几何结构. 带长(L)和带宽(W)分别用沿着扶手椅形和锯齿形边缘的六方格子的数目表征. 红色和蓝色小球表示一对呈反铁磁耦合的电子自旋, 它们就是构建量子通道的物理比特. (b) Werner态的隐形传输原理图. 黑色小球(1, 2)表示产生Werner态的物理比特. 量子通道的物理比特分别由两个尺寸完全相同的SGNR锯齿端上的纠缠粒子对(3, 5)和(4, 6)承担

    Fig. 1.  Schematic illustration of teleporting the Werner state via the graphene-based quantum channels under the dephasing environment: (a) Lattice geometry of the special graphene nanoribbon (SGNR) used to form quantum channels. The ribbon length (L) and width (W) are characterized by the number of hexagons along the armchair and zigzag direction, respectively. The red and blue particles denote a pair of spins with the antiferromagnetic coupling, which serve as physical qubits to support quantum channels. (b) Schematic illustration of teleporting the Werner state. The black particles (1, 2) are physical qubits used to prepare the Werner state. The physical qubits of quantum channels are supported by two pairs of the entangled spins (3, 5) and (4, 6) in two same SGNRs, respectively.

    图 2  退相位环境下通道态$ {\boldsymbol{\rho}} _{{\text{ch}}}^{{\text{Pha}}} $的纠缠度$C_{{\text{ch}}}^{{\text{Pha}}}$随温度T和出错概率p的变化 (a) U = 2.0 eV; (b) U = 3.5 eV; (c) U = 4.5 eV; (d) U = 6.0 eV

    Fig. 2.  Concurrence $C_{{\text{ch}}}^{{\text{Pha}}}$ for the channel state $ {\boldsymbol{\rho}} _{{\text{ch}}}^{{\text{Pha}}} $ in the dephasing environment as a function of temperature T and probability p: (a) U = 2.0 eV; (b) U = 3.5 eV; (c) U = 4.5 eV; (d) U = 6.0 eV.

    图 3  退相位环境下通道态$ {\boldsymbol{\rho}} _{{\text{ch}}}^{{\text{Pha}}} $的纠缠度$C_{{\text{ch}}}^{{\text{Pha}}}$随库仑排斥势U和出错概率p的变化 (a) T = 0 K, (b) T = 5 K, (c) T = 10 K, (d) T = 15 K

    Fig. 3.  Concurrence $C_{{\text{ch}}}^{{\text{Pha}}}$ for the channel state $ {\boldsymbol{\rho}} _{{\text{ch}}}^{{\text{Pha}}} $ in the dephasing environment as a function of Coulomb repulsion U and probability p: (a) T = 0 K; (b) T = 5 K; (c) T = 10 K; (d) T = 15 K.

    图 4  退相位环境下输出态$ {{\rho}} _{out}^{Pha} $的纠缠度$C_{out}^{Pha}$随参数b和出错概率p的变化 (a) T = 5 K, U = 3.5 eV; (b) T = 5 K, U = 6.0 eV; (c) T = 10 K, U = 3.5 eV; (d) T = 10 K, U = 6.0 eV

    Fig. 4.  Concurrence $C_{{\text{out}}}^{{\text{Pha}}}$ for the output state $ {{\rho}} _{{\text{out}}}^{{\text{Pha}}} $ under the dephasing channel as a function of parameter b and probability p: (a) T = 5 K and U = 3.5 eV; (b) T = 5 K and U = 6.0 eV; (c) T = 10 K and U = 3.5 eV; (d) T = 10 K and U = 6.0 eV.

    图 5  退相位环境下输出态$ {\boldsymbol{\rho}} _{{\text{out}}}^{{\text{Pha}}} $的保真度$F({{\boldsymbol{\rho}} _{{\text{in}}}}, \;{\boldsymbol{\rho}} _{{\text{out}}}^{{\text{Pha}}})$随参数b和出错概率p的变化 (a) T = 5 K, U = 3.5 eV; (b) T = 5 K, U = 6.0 eV; (c) T = 10 K, U = 3.5 eV; (d) T = 10 K, U = 6.0 eV.

    Fig. 5.  Fidelity $F({{\boldsymbol{\rho}} _{{\text{in}}}}, \;{\boldsymbol{\rho}} _{{\text{out}}}^{{\text{Pha}}})$ for the output state $ {\boldsymbol{\rho}} _{{\text{out}}}^{{\text{Pha}}} $ under the dephasing channel as a function of parameter b and probability p: (a) T = 5 K and U = 3.5 eV; (b) T = 5 K and U = 6.0 eV; (c) T = 10 K and U = 3.5 eV; (d) T = 10 K and U = 6.0 eV.

    图 6  退相位环境下输出态$ {\boldsymbol{\rho}} _{{\text{out}}}^{{\text{Pha}}} $的平均保真度${F_A}({{\boldsymbol{\rho}} _{{{\rm{in}}} }}, \;{\boldsymbol{\rho}} _{{\text{out}}}^{{\text{Pha}}})$随出错概率p和温度T的变化 (a) U = 2.0 eV; (b) U = 3.5 eV; (c) U = 4.5 eV; (d) U = 6.0 eV.

    Fig. 6.  Average fidelity ${F_A}({{\boldsymbol{\rho}} _{{\text{in}}}}, \;{\boldsymbol{\rho}} _{{\text{out}}}^{{\text{Pha}}})$ of the output state under the dephasing channel as a function of probability p and temperature T: (a) U = 2.0 eV; (b) U = 3.5 eV; (c) U = 4.5 eV; (d) U = 6.0 eV.

    图 7  退相位环境下输出态$ {\boldsymbol{\rho}} _{{\text{out}}}^{{\text{Pha}}} $的平均保真度${F_A}({{\boldsymbol{\rho}} _{{\text{in}}}}, \;{\boldsymbol{\rho}} _{{\text{out}}}^{{\text{Pha}}})$随出错概率p和库仑排斥势U的变化 (a) T = 0 K; (b) T = 5 K; (c) T = 10 K; (d) T = 15 K

    Fig. 7.  Average fidelity ${F_A}({{\boldsymbol{\rho}} _{{\text{in}}}}, \;{\boldsymbol{\rho}} _{{\text{out}}}^{{\text{Pha}}})$ of the output state $ {\boldsymbol{\rho}} _{{\text{out}}}^{{\text{Pha}}} $ under the dephasing channel as a function of probability p and Coulomb repulsion U: (a) T = 0 K; (b) T = 5 K; (c) T = 10 K; (d) T = 15 K, respectively.

    表 1  Alice执行联合贝尔基测量所得的16种可能结果与对应每种测量结果Bob为复原Werner态所执行的幺正操作

    Table 1.  Sixteen possible results of joint Bell-state measurements performed by Alice and the unitary operations performed by Bob according to each measurement result for restoring the Werner state.

    Alice对A1处的量子比特(1, 3)
    A2处的量子比特(2, 4)执行
    联合贝尔基测量后所得结果
    为了复原Werner态Bob对B1B2处的量子比特(5,6)
    所执行的相应幺正操作
    $E_0^{{A_1}} \otimes E_0^{{A_2}}$${I^{{B_1}}}$, ${I^{{B_2}}}$
    $E_0^{{A_1}} \otimes E_1^{{A_2}}$${I^{{B_1}}}$, $\sigma _x^{{B_2}}$
    $E_0^{{A_1}} \otimes E_2^{{A_2}}$${I^{{B_1}}}$, $\sigma _y^{{B_2}}$
    $E_0^{{A_1}} \otimes E_3^{{A_2}}$${I^{{B_1}}}$, $\sigma _z^{{B_2}}$
    $E_1^{{A_1}} \otimes E_0^{{A_2}}$$\sigma _x^{{B_1}}$, ${I^{{B_2}}}$
    $E_1^{{A_1}} \otimes E_1^{{A_2}}$$\sigma _x^{{B_1}}$, $\sigma _x^{{B_2}}$
    $E_1^{{A_1}} \otimes E_2^{{A_2}}$$\sigma _x^{{B_1}}$, $\sigma _y^{{B_2}}$
    $E_1^{{A_1}} \otimes E_3^{{A_2}}$$\sigma _x^{{B_1}}$, $\sigma _z^{{B_2}}$
    $E_2^{{A_1}} \otimes E_0^{{A_2}}$$\sigma _y^{{B_1}}$, ${I^{{B_2}}}$
    $E_2^{{A_1}} \otimes E_1^{{A_2}}$$\sigma _y^{{B_1}}$, $\sigma _x^{{B_2}}$
    $E_2^{{A_1}} \otimes E_2^{{A_2}}$$\sigma _y^{{B_1}}$, $\sigma _y^{{B_2}}$
    $E_2^{{A_1}} \otimes E_3^{{A_2}}$$\sigma _y^{{B_1}}$, $\sigma _z^{{B_2}}$
    $E_3^{{A_1}} \otimes E_0^{{A_2}}$$\sigma _z^{{B_1}}$, ${I^{{B_2}}}$
    $E_3^{{A_1}} \otimes E_1^{{A_2}}$$\sigma _z^{{B_1}}$, $\sigma _x^{{B_2}}$
    $E_3^{{A_1}} \otimes E_2^{{A_2}}$$\sigma _z^{{B_1}}$, $\sigma _y^{{B_2}}$
    $E_3^{{A_1}} \otimes E_3^{{A_2}}$$\sigma _z^{{B_1}}$, $\sigma _z^{{B_2}}$
    下载: 导出CSV
  • [1]

    Nielsen M A, Chuang I L 2002 Quantum Computation and Quantum Information (Cambridge: Cambridge University Press)

    [2]

    Davis S I, Peña C, Xie S, Lauk N, Narváez L, Valivarthi R, Allmaras J P, Beyer A D, Gim Y, Hussein M, Iskander G, Kim H L, Korzh B, Mueller A, Rominsky M, Shaw M, Tang D, Wollman E E, Simon C, Spentzouris P, Oblak D, Sinclair N, Spiropulu M 2020 PRX Quantum 1 020317Google Scholar

    [3]

    Bennett C H, Brassard G, Crépeau C, Jozsa R, Peres A, Wootters W K 1993 Phys. Rev. Lett. 70 1895Google Scholar

    [4]

    Briegel H J, Dür W, Cirac J I, Zoller P 1998 Phys. Rev. Lett. 81 5932Google Scholar

    [5]

    Sangouard N, Simon C, de Riedmatten H, Gisin N 2011 Rev. Mod. Phys. 83 33Google Scholar

    [6]

    Gottesman D, Chuang I L 1999 Nature 402 390Google Scholar

    [7]

    Raussendorf R, Briegel H J 2001 Phys. Rev. Lett. 86 5188Google Scholar

    [8]

    Chou K S, Blumoff J Z, Wang C S, Reinhold P C, Axline C J, Gao Y Y, Frunzio L, Devoret M H, Jiang L, Schoelkopf R J 2018 Nature 561 368Google Scholar

    [9]

    Wan Y, Kienzler D, Erickson S D, Mayer K H, Leibfried D 2019 Science 364 875Google Scholar

    [10]

    王明宇, 王馨德, 阮东, 龙桂鲁 2021 物理学报 70 190301Google Scholar

    Wang M Y, Wang X D, Ruan D, Long G L 2021 Acta Phys. Sin. 70 190301Google Scholar

    [11]

    Bouwmeester D, Pan J W, Mattle K, Eibl M, Zeilinger A 1997 Nature 390 575Google Scholar

    [12]

    Yin J, Ren J, Lu H, Cao Y, Pan J 2012 Nature 488 185Google Scholar

    [13]

    Ma X S, Herbst T, Scheidl T, Wang D, Kropatschek S, Naylor W, Wittmann B, Mech A, Kofler J, Anisimova E, Makarov V, Jennewein T, Ursin R, Zeilinger A 2012 Nature 489 269Google Scholar

    [14]

    Metcalf B J, Spring J B, Humphreys P C, Thomas-Peter N, Barbieri M, Kolthammer W S, Jin X, Langford N K, Kundys D, Gates J C, Smith B J, Smith P G R, Walmsley I A 2014 Nat. Photonics 8 770Google Scholar

    [15]

    Im D, Lee C, Kim Y, Nha H, Kim M S, Lee S, Kim Y 2021 npj Quantum Inf. 7 86Google Scholar

    [16]

    Barrett M D, Chiaverini J, Schaetz T, Britton J, Itano W M, Jost J D 2004 Nature 429 737Google Scholar

    [17]

    Nölleke C, Neuzner A, Reiserer A, Hahn C, Rempe G, Ritter S 2013 Phys. Rev. Lett. 110 140403Google Scholar

    [18]

    Bao X H, Xu X F, Li C M, Yuan Z S, Lu C Y, Pan J W 2012 Proc. Natl. Acad. Sci. U. S. A. 109 20347Google Scholar

    [19]

    Krauter H, Salart D, Muschik C A, Petersen J M, Shen H, Fernholz T, Polzik E S 2013 Nat. Phys. 9 400Google Scholar

    [20]

    Gao W B, Fallahi P, Togan E, Delteil A, Chin Y S, Miguel-Sanchez J, Imamoğlu A 2013 Nat. Commun. 4 2744Google Scholar

    [21]

    Steffen L, Salathe Y, Oppliger M, Kurpiers P, Baur M, Lang C, Eichler C, Puebla-Hellmann G, Fedorov A, Wallraff A 2013 Nature 500 319Google Scholar

    [22]

    Pfaff W, Hensen B J, Bernien H, van Dam S B, Blok M S, Taminiau T H, Tiggelman M J, Schouten R N, Markham M, Twitchen D J, Hanson R 2014 Science 345 532Google Scholar

    [23]

    Huang N, Huang W, Li C 2020 Sci. Rep. 10 3093Google Scholar

    [24]

    Yu Y, Zhao N, Pei C X, Li W 2021 Commun. Theor. Phys. 73 085103Google Scholar

    [25]

    Ren J G, Xu P, Yong H L, Zhang L, Liao S K, Yin J, Liu W Y, Cai W Q, Yang M, Li L 2017 Nature 549 70Google Scholar

    [26]

    Pirandola S, Eisert J, Weedbrook C, Furusawa A, Braunstein S L 2015 Nat. Photonics 9 641Google Scholar

    [27]

    Bussières F, Clausen C, Tiranov A, Korzh B, Verma V B, Nam S W, Marsili F, Ferrier A, Goldner P, Herrmann H, Silberhorn C, Sohler W, Afzelius M, Gisin N 2014 Nat. Photonics 8 775Google Scholar

    [28]

    Llewellyn D, Ding Y, Faruque I I, Paesani S, Bacco D, Santagati R, Qian Y, Li Y, Xiao Y, Huber M, Malik M, Sinclair G F, Zhou X, Rottwitt K, O Brien J L, Rarity J G, Gong Q, Oxenlowe L K, Wang J, Thompson M G 2020 Nat. Phys. 16 148Google Scholar

    [29]

    Novoselov K S, Geim A K, Morozov S V, Jiang D, Zhang Y, Dubonos S V, Grigorieva I V, Firsov A A 2004 Science 306 666Google Scholar

    [30]

    Castro Neto A H, Guinea F, Peres N M R, Novoselov K S, Geim A K 2009 Rev. Mod. Phys. 81 109Google Scholar

    [31]

    Wakabayashi K, Sasaki K, Nakanishi T, Enoki T 2010 Sci. Technol. Adv. Mater. 11 054504Google Scholar

    [32]

    Yazyev O V 2010 Rep. Prog. Phys. 73 056501Google Scholar

    [33]

    Talirz L, Söde H, Cai J, Ruffieux P, Blankenburg S, Jafaar R, Berger R, Feng X, Müllen K, Passerone D, Fasel R, Pignedoli C A 2013 J. Am. Chem. Soc. 135 2060Google Scholar

    [34]

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出版历程
  • 收稿日期:  2021-10-10
  • 修回日期:  2021-12-10
  • 上网日期:  2022-01-26
  • 刊出日期:  2022-04-05

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