搜索

x

留言板

尊敬的读者、作者、审稿人, 关于本刊的投稿、审稿、编辑和出版的任何问题, 您可以本页添加留言。我们将尽快给您答复。谢谢您的支持!

姓名
邮箱
手机号码
标题
留言内容
验证码

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

引用本文:
Citation:

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
PDF
HTML
导出引用
  • 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)
    [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]

    van der Lit J, Boneschanscher M P, Vanmaekelbergh D, Ijäs M, Uppstu A, Ervasti M, Harju A, Liljeroth P, Swart I 2013 Nat. Commun. 4 2023Google Scholar

    [35]

    Schmidt M J, Golor M, Lang T C, Wessel S 2013 Phys. Rev. B 87 245431Google Scholar

    [36]

    Golor M, Koop C, Lang T C, Wessel S, Schmidt M J 2013 Phys. Rev. Lett. 111 085504Google Scholar

    [37]

    Gräfe M, Szameit A 2015 2 D Materials 2 034005Google Scholar

    [38]

    Burkard G, Bulaev D V, Trauzettel B, Loss D 2007 Nat. Phys. 3 192Google Scholar

    [39]

    Recher P, Trauzettel B 2010 Nanotechnology 21 302001Google Scholar

    [40]

    Chen C, Chang Y 2015 Phys. Rev. B 92 245406Google Scholar

    [41]

    Cimatti I, Bondì L, Serrano G, Malavolti L, Cortigiani B, Velez-Fort E, Betto D, Ouerghi A, Brookes N B, Loth S, Mannini M, Totti F, Sessoli R 2019 Nanoscale Horizons 4 1202Google Scholar

    [42]

    Guo G, Lin Z, Tu T, Cao G, Li X, Guo G 2009 New J. Phys. 11 123005Google Scholar

    [43]

    Dragoman D, Dragoman M 2015 Nanotechnology 26 485201Google Scholar

    [44]

    Dragoman M, Dinescu A, Dragoman D 2018 IEEE Trans. Nanotechnol. 17 362Google Scholar

    [45]

    Wootters W K 1998 Phys. Rev. Lett. 80 2245Google Scholar

    [46]

    Werner R F 1989 Phys. Rev. A 40 4277Google Scholar

    [47]

    Aolita L, de Melo F, Davidovich L 2015 Rep. Prog. Phys. 78 042001Google Scholar

    [48]

    Bowen G, Bose S 2001 Phys. Rev. Lett. 87 267901Google Scholar

    [49]

    Jozsa R 1994 J. Mod. Opt. 41 2315Google Scholar

  • 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]

    van der Lit J, Boneschanscher M P, Vanmaekelbergh D, Ijäs M, Uppstu A, Ervasti M, Harju A, Liljeroth P, Swart I 2013 Nat. Commun. 4 2023Google Scholar

    [35]

    Schmidt M J, Golor M, Lang T C, Wessel S 2013 Phys. Rev. B 87 245431Google Scholar

    [36]

    Golor M, Koop C, Lang T C, Wessel S, Schmidt M J 2013 Phys. Rev. Lett. 111 085504Google Scholar

    [37]

    Gräfe M, Szameit A 2015 2 D Materials 2 034005Google Scholar

    [38]

    Burkard G, Bulaev D V, Trauzettel B, Loss D 2007 Nat. Phys. 3 192Google Scholar

    [39]

    Recher P, Trauzettel B 2010 Nanotechnology 21 302001Google Scholar

    [40]

    Chen C, Chang Y 2015 Phys. Rev. B 92 245406Google Scholar

    [41]

    Cimatti I, Bondì L, Serrano G, Malavolti L, Cortigiani B, Velez-Fort E, Betto D, Ouerghi A, Brookes N B, Loth S, Mannini M, Totti F, Sessoli R 2019 Nanoscale Horizons 4 1202Google Scholar

    [42]

    Guo G, Lin Z, Tu T, Cao G, Li X, Guo G 2009 New J. Phys. 11 123005Google Scholar

    [43]

    Dragoman D, Dragoman M 2015 Nanotechnology 26 485201Google Scholar

    [44]

    Dragoman M, Dinescu A, Dragoman D 2018 IEEE Trans. Nanotechnol. 17 362Google Scholar

    [45]

    Wootters W K 1998 Phys. Rev. Lett. 80 2245Google Scholar

    [46]

    Werner R F 1989 Phys. Rev. A 40 4277Google Scholar

    [47]

    Aolita L, de Melo F, Davidovich L 2015 Rep. Prog. Phys. 78 042001Google Scholar

    [48]

    Bowen G, Bose S 2001 Phys. Rev. Lett. 87 267901Google Scholar

    [49]

    Jozsa R 1994 J. Mod. Opt. 41 2315Google Scholar

  • [1] 文镇南, 易有根, 徐效文, 郭迎. 无噪线性放大的连续变量量子隐形传态. 物理学报, 2022, 71(13): 130307. doi: 10.7498/aps.71.20212341
    [2] 武莹, 李锦芳, 刘金明. 基于部分测量增强量子隐形传态过程的量子Fisher信息. 物理学报, 2018, 67(14): 140304. doi: 10.7498/aps.67.20180330
    [3] 贾芳, 刘寸金, 胡银泉, 范洪义. 量子隐形传态保真度的新公式及应用. 物理学报, 2016, 65(22): 220302. doi: 10.7498/aps.65.220302
    [4] 郑伯昱, 董慧龙, 陈非凡. 基于量子修正的石墨烯纳米带热导率分子动力学表征方法. 物理学报, 2014, 63(7): 076501. doi: 10.7498/aps.63.076501
    [5] 刘世右, 郑凯敏, 贾芳, 胡利云, 谢芳森. 单-双模组合压缩热态的纠缠性质及在量子隐形传态中的应用. 物理学报, 2014, 63(14): 140302. doi: 10.7498/aps.63.140302
    [6] 张沛, 周小清, 李智伟. 基于量子隐形传态的无线通信网络身份认证方案. 物理学报, 2014, 63(13): 130301. doi: 10.7498/aps.63.130301
    [7] 曾永昌, 田文, 张振华. 周期性纳米洞内边缘氧饱和石墨烯纳米带的电子特性. 物理学报, 2013, 62(23): 236102. doi: 10.7498/aps.62.236102
    [8] 杨平, 王晓亮, 李培, 王欢, 张立强, 谢方伟. 氮掺杂和空位对石墨烯纳米带热导率影响的分子动力学模拟. 物理学报, 2012, 61(7): 076501. doi: 10.7498/aps.61.076501
    [9] 乔盼盼, 艾合买提·阿不力孜, 蔡江涛, 路俊哲, 麦麦提依明·吐孙, 日比古·买买提明. 利用热平衡态超导电荷量子比特实现量子隐形传态. 物理学报, 2012, 61(24): 240303. doi: 10.7498/aps.61.240303
    [10] 林琦, 陈余行, 吴建宝, 孔宗敏. N掺杂对zigzag型石墨烯纳米带的能带结构和输运性质的影响. 物理学报, 2011, 60(9): 097103. doi: 10.7498/aps.60.097103
    [11] 顾芳, 张加宏, 杨丽娟, 顾斌. 应变石墨烯纳米带谐振特性的分子动力学研究. 物理学报, 2011, 60(5): 056103. doi: 10.7498/aps.60.056103
    [12] 王志勇, 胡慧芳, 顾林, 王巍, 贾金凤. 含Stone-Wales缺陷zigzag型石墨烯纳米带的电学和光学性能研究. 物理学报, 2011, 60(1): 017102. doi: 10.7498/aps.60.017102
    [13] 潘长宁, 方见树, 彭小芳, 廖湘萍, 方卯发. 耗散系统中实现原子态量子隐形传态的保真度. 物理学报, 2011, 60(9): 090303. doi: 10.7498/aps.60.090303
    [14] 何锐, Bing He. 量子隐形传态的新方案. 物理学报, 2011, 60(6): 060302. doi: 10.7498/aps.60.060302
    [15] 谭长玲, 谭振兵, 马丽, 陈军, 杨帆, 屈凡明, 刘广同, 杨海方, 杨昌黎, 吕力. 石墨烯纳米带量子点中的量子混沌现象. 物理学报, 2009, 58(8): 5726-5729. doi: 10.7498/aps.58.5726
    [16] 唐有良, 刘 翔, 张小伟, 唐筱芳. 用一个纠缠态实现多粒子纠缠态的量子隐形传送. 物理学报, 2008, 57(12): 7447-7451. doi: 10.7498/aps.57.7447
    [17] 夏云杰, 王光辉, 杜少将. 双模最小关联混合态作为量子信道实现量子隐形传态的保真度. 物理学报, 2007, 56(8): 4331-4336. doi: 10.7498/aps.56.4331
    [18] 周小清, 邬云文. 利用三粒子纠缠态建立量子隐形传态网络的探讨. 物理学报, 2007, 56(4): 1881-1887. doi: 10.7498/aps.56.1881
    [19] 张 茜, 李福利, 李宏荣. 基于双模压缩信道的双模高斯态量子隐形传态. 物理学报, 2006, 55(5): 2275-2280. doi: 10.7498/aps.55.2275
    [20] 宋同强. 利用双模压缩真空态实现量子态的远程传输. 物理学报, 2004, 53(10): 3358-3362. doi: 10.7498/aps.53.3358
计量
  • 文章访问数:  615
  • PDF下载量:  20
  • 被引次数: 0
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

退相位环境下Werner态在石墨烯基量子通道中的隐形传输

  • 商洛学院电子信息与电气工程学院, 商洛 726000
  • 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

摘要: 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.

English Abstract

    • 量子隐形传态(quantum teleportation, QT)是以量子纠缠作为资源, 并结合经典通信将一未知量子态从一个地方传送到另外一个地方的全新通信技术[1]. 它不仅是一种重要的量子通信方式, 而且是发展量子计算[1]和量子网络[2]的基础. 自1993年Bennett等[3]首次提出QT以来, 许多新的量子技术也随之发展起来, 如量子中继[4,5]、测量为基础的量子计算[6,7]、量子门的远程传输[8,9]以及无需纠缠资源协助的量子态安全传输等[10]. 在过去的十几年里, QT已经在各种各样的物理系统中实现, 如光子系统[11-15]、囚禁原子系统[16,17]、原子系综[18,19]、固态系统[20-22]以及IBM Q Experience在线量子平台等[23,24]. 2017年, 我国建成了全球首条量子通信“京沪干线”, 结合“墨子号”量子卫星实现了天地一体化的广域量子通信体系, 并完成了星地间的QT实验[25]. 在这些量子系统中, 固态量子系统易与现代微纳加工工艺接轨, 且大规模可扩展性问题能自然得到解决, 是未来发展固态量子计算的重要研究方向. 进而固态量子系统中量子信息的传递与存储也引起了人们极大的研究兴趣[26]. 光子到固态物质的量子态传输已经在光子比特和固态量子存储为基础的实验中实现[20,27]. 固态物质间的量子态传输也已经在超导量子电路中实现[21]. 最近, Llewellyn等[28]还在硅基光子芯片间实现了量子隐形传态和多光子纠缠. 这些对于发展固态量子计算与量子通信具有深远的意义. 然而, 由于外界环境的影响, 量子退相干是所有量子系统所面临的共同难题. 因此, 要发展固态量子系统, 就必须找到能够保持较长相干时间的材料.

      石墨烯[29]中自旋-轨道耦合和精细作用非常微弱, 电子自旋能够保持较长的相干时间, 因此是非常理想的固态量子材料. 为了获得更加优异的电磁学性能, 经常把石墨烯裁剪成准一维的结构, 即石墨烯纳米带(graphene nanoribbon, GNR). 根据边缘结构的不同, 常见的GNR有呈金属性但无磁性的扶手椅型GNR和呈半导体性且有边缘磁性的锯齿型GNR[30-32]. 目前实验上已经能够制备出高品质的带有两个锯齿端的扶手椅型GNR[33,34]. 基于有效低能理论[35], Golor等[36]研究了该GNR两个锯齿端间的磁性关联, 结果发现纳米带两端的自旋是相互纠缠的. 进一步, Gräfe和Szameit[37]发现石墨烯的边缘几何形状对边缘态中的量子关联演化有非常显著的影响. 为了探索石墨烯材料在量子信息领域中的应用, 相关学者提出在石墨烯量子点[38,39]、石墨烯纳米带量子点[40]和钒基酞菁/石墨烯/SiC(0001)杂化结构中[41]制备自旋量子比特的方案. 基于自旋量子比特, Guo等[42]在“Z”字形GNR中设计了一种高效率和高保真度的可扩展量子计算方案. 后来, Dragoman等在理论上提出了在室温条件下实现量子逻辑门的方案和改进的Deutsch-Jozsa算法[43], 并在晶片尺寸的石墨烯基结构上进行了实验验证[44]. 以上结果均表明石墨烯材料在量子信息领域有潜在的应用价值.

      考虑到石墨烯诸多优良的物理特性, 且带有两个锯齿端的高品质扶手椅型GNR已经可以在实验上制备, 本文将研究该特殊结构的石墨烯纳米带(special graphene nanoribbon, SGNR)在量子通信方面的应用—在SGNR构建的量子通道中实现Werner态的隐形传输. 考虑到真实的量子通道总是会受消相干环境的影响, 简单起见, 本文只考虑退相位环境对量子通道及输出态的影响. 通过计算输出态的纠缠度和保真度分析输出态的品质, 同时讨论温度和电子间的库仑排斥势对输出态的影响, 进而说明该量子通道的鲁棒性, 推动石墨烯材料在量子信息领域中的应用.

    • 本文考虑的特殊结构的石墨烯纳米带SGNR如图1(a)所示. Golor等[36]的工作已经证明, 对于带宽W = 3且带长$ L \geqslant 8 $的SGNR来讲, 两锯齿端上的自旋粒子间呈反铁磁耦合, 有效相互作用的哈密顿量$ {H_0} $可以用海森伯模型描述[36]:

      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

      Figure 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.

      $ {H}_{0}=J{{\boldsymbol{\sigma}} }_{\text{L}}\cdot {{\boldsymbol{\sigma}} }_{\text{R}}=J({\sigma }_{\text{L}}^{x}{\sigma }_{\text{R}}^{x}+{\sigma }_{\text{L}}^{y}{\sigma }_{\text{R}}^{y}+{\sigma }_{\text{L}}^{z}{\sigma }_{\text{R}}^{z}), $

      其中$ {{\boldsymbol{\sigma}} _{{\text{L}}\left( {\text{R}} \right)}} $表示作用在左(右)锯齿端有效自旋粒子上的泡利算符, 它的3个分量分别由$ \sigma _{{\text{L(R)}}}^x $, $ \sigma _{{\text{L(R)}}}^y $, $ \sigma _{{\text{L(R)}}}^z $表示, $ J = t^{*2}/U^*$表示这两个自旋粒子间的反铁磁耦合系数. 这里$ \mathop t\nolimits^* \approx 1.29\mathop {\text{e}}\nolimits^{{{ - L} / {1.86}}} \;{\text{eV}} $表示电子从一个锯齿端跳跃到另外一个锯齿端所需的能量, $ \mathop U\nolimits^* \approx 0.1 U $表示电子之间有效库仑排斥势. 当系统处于绝对温度为T的热平衡态时, 密度算符${\rho _T}$

      $ {\rho _T} = {{\exp ( - \beta {H_0})} / {\text{Z}}}, $

      其中, $ \beta =1/({k}_{\text{B}}T) $, ${k_{\text{B}}}$为玻尔兹曼常数, $Z = $ ${{\rm{Tr}}} [\exp ( - \beta {H_0})]$为该系统的配分函数. 在标准基组$\left\{ {\left| {00} \right\rangle, \;\left| {01} \right\rangle, \;\left| {10} \right\rangle, \;\left| {11} \right\rangle } \right\}$下, (2)式可以表示为

      $ {{\boldsymbol{\rho}} _T} = \frac{1}{4}\left[ {\begin{array}{*{20}{c}} {1 + \left\langle {\sigma _{\text{L}}^z\sigma _{\text{R}}^z} \right\rangle }&0&0&0 \\ 0&{1 - \left\langle {\sigma _{\text{L}}^z\sigma _{\text{R}}^z} \right\rangle }&{\left\langle {\sigma _{\text{L}}^x\sigma _{\text{R}}^x} \right\rangle + \left\langle {\sigma _{\text{L}}^y\sigma _{\text{R}}^y} \right\rangle }&0 \\ 0&{\left\langle {\sigma _{\text{L}}^x\sigma _{\text{R}}^x} \right\rangle + \left\langle {\sigma _{\text{L}}^y\sigma _{\text{R}}^y} \right\rangle }&{1 - \left\langle {\sigma _{\text{L}}^z\sigma _{\text{R}}^z} \right\rangle }&0 \\ 0&0&0&{1 + \left\langle {\sigma _{\text{L}}^z\sigma _{\text{R}}^z} \right\rangle } \end{array}} \right], $

      其中$ \left\langle {\sigma _{\text{L}}^i\sigma _{\text{R}}^i} \right\rangle = {{\rm{Tr}}} \left( {\sigma _{\text{L}}^i\sigma _{\text{R}}^i{{\boldsymbol{\rho}} _T}} \right) $称为自旋关联函数, $ i = x, y, z $. 由于哈密顿量$ {H_0} $中的耦合系数J各向同性, 于是计算得到

      $\begin{split} \left\langle {\sigma _{\text{L}}^x\sigma _{\text{R}}^x} \right\rangle = \;&\left\langle {\sigma _{\text{L}}^y\sigma _{\text{R}}^y} \right\rangle = \left\langle {\sigma _{\text{L}}^z\sigma _{\text{R}}^z} \right\rangle = {{\rm{Tr}}} \left( {\sigma _{\text{L}}^z\sigma _{\text{R}}^z{{\boldsymbol{\rho}} _T}} \right) \\ =\;& c = ({{1 - {\eta ^4}}})/({{3 + {\eta ^4}}}), \end{split}$

      这里η是与自旋关联函数相关的参数, 表达式为

      $ \eta = \exp \left( {\frac{J}{{{k_{\text{B}}}T}}} \right) = \exp \left( {\frac{{7.7244 \times {{10}^5}{{\text{e}}^{ - \tfrac{L}{{0.93}}}}}}{{TU}}} \right), $

      其中U表示电子间的库仑排斥势. 于是, (3)式用自旋关联函数c可简化为

      $ {{\boldsymbol{\rho}} _T} = \frac{1}{4}\left[ {\begin{array}{*{20}{c}} {1 + c}&0&0&0 \\ 0&{1 - c}&{2c}&0 \\ 0&{2c}&{1 - c}&0 \\ 0&0&0&{1 + c} \end{array}} \right] . $

      利用共生纠缠度Concurrence可以对任意一个两体量子态$ {\boldsymbol{\rho}} $中的量子纠缠进行度量, 其定义为[45]

      $ C({\boldsymbol{\rho}} ) = \max \left\{ {0,{\lambda _1} - {\lambda _2} - {\lambda _3} - {\lambda _4}} \right\}, $

      其中${\lambda _k}$(k = 1, 2, 3, 4)是算符$R = {\boldsymbol{\rho}} ( {\sigma _y} \otimes {\sigma _y} )\times $$ {{\boldsymbol{\rho}} ^*} ({{\sigma _y} \otimes {\sigma _y}})$本征值的方根, 且满足${\lambda _1} \geqslant {\lambda _2} \geqslant {\lambda _3} \geqslant {\lambda _4}$, ${{\boldsymbol{\rho}} ^*}$${\boldsymbol{\rho}} $的复共轭. 通过计算, 得到热平衡态$ {{{\rho}} _T} $的纠缠度为

      $ {C_T} = C({{\boldsymbol{\rho}} _T}) = \max \left\{ {0,\;{{ - \left( {1 + 3c} \right)} / 2}} \right\}. $

      由于SGNR两锯齿端上的自旋粒子间呈反铁磁耦合, 所以自旋关联函数$ c = \left\langle {\sigma _{\text{L}}^z\sigma _{\text{R}}^z} \right\rangle $满足$- 1 \leqslant $$ c \leqslant 0$. 根据(8)式可得, 当$ - 1 \leqslant c \lt {{ - 1} / 3}$时, 一定有${C_T} \ne 0$. 因此, 对于给定的排斥势U, 只要外界温度不高于临界值

      $ {T_{\text{c}}} = {{2.6375 \times {{10}^6}{{\text{e}}^{ - \tfrac{L}{{0.93}}}}}}/{U}, $

      那么SGNR两锯齿端上的自旋粒子间就一定存在量子纠缠${C_T}$. 例如, 对于宽度W = 3长度L = 10的SGNR, 当温度$T = 5\;{\text{K}}$, $U = 3\;{\text{eV}}$时, 将(4)式和(5)式代入(8)式计算可得两自旋粒子间的纠缠度为${C_T} \approx {\text{0}}{\text{.9293}}$. 因此可以利用该纠缠粒子对作为量子通道去实现量子态的隐形传输. 在后续的讨论中固定选择宽度W = 3长度L = 10的两条SGNR来构建量子通道. 以Werner态作为输入态来考察该量子通道的性能. Werner态的一般形式为[46]${{\boldsymbol{\rho}} _{{\text{in}}}} \;=\; (1 - b)I +b\left| {{\varPsi ^ - }} \right\rangle \left\langle {{\varPsi ^ - }} \right|$, 其中$\left| {{\varPsi ^ - }} \right\rangle \;=\; ( \left| {01} \right\rangle - $$ \left| {10} \right\rangle ) / {\sqrt 2 }$, $b \in \left[ {0, \;1} \right]$. 在标准基组$\{ \left| {00} \right\rangle, \;\left| {01} \right\rangle, \;\left| {10} \right\rangle, $$ \;\left| {11} \right\rangle \}$下, Werner态可以表示为

      $ {{\boldsymbol{\rho}} _{{\text{in}}}} = \frac{1}{4}\left[ {\begin{array}{*{20}{c}} {1 - b}&0&0&0 \\ 0&{1 + b}&{ - 2b}&0 \\ 0&{ - 2b}&{1 + b}&0 \\ 0&0&0&{1 - b} \end{array}} \right] . $

      Werner态的传输原理如图1(b)所示. 假设Alice要将一个未知的Werner态发送给Bob, 首先Alice对A1A2处的量子比特(1, 3)和(2, 4)进行联合贝尔基测量, 结果将等概率地得到表1中16个态中的一个. 标准的贝尔基为 ${E_0} = | {{\varPsi ^ - }} \rangle \langle {{\varPsi ^ - }} |$, ${E_{^1}} = | {{\varPhi ^ - }} \rangle \langle {{\varPhi ^ - }} |$, ${E_{^2}} = |{{\varPhi ^ + }} \rangle \langle {{\varPhi ^ + }} | $, ${E_{^3}} = |\varPsi ^ + \rangle \langle \varPsi ^ + |$, 其中, $| \varPsi ^ \pm \rangle = ( | 01\rangle \pm | 10\rangle )/{\sqrt 2 }$, $|\varPhi ^ \pm \rangle = (| 00 \rangle \pm | 11\rangle ) /{\sqrt 2 }$. 然后, Alice将测量结果通过经典通道(如打电话、发邮件等方式)告诉Bob. 最后, Bob根据Alice的测量结果分别对B1B2处的量子比特(5, 6)进行相应的幺正操作(见表1), 如$I = $$ \left| 0 \right\rangle \left\langle 0 \right| + \left| 1 \right\rangle \left\langle 1 \right|$, ${\sigma _x} = |0 \rangle \langle 1 | + |1 \rangle \langle 0 |$, ${\sigma _y} = {\text{i}} |1 \rangle \langle 0 | - {\text{i}} | 0 \rangle \langle 1 |$, $ {\sigma _z} = \left| 0 \right\rangle \left\langle 0 \right| - $$ \left| 1 \right\rangle \left\langle 1 \right| $, 就能复原Werner态. 从整体效果来看, Werner态会在量子比特1和2上消失, 最后在量子比特5和6上出现. 以上就是实现两体量子态隐形传输的基本方案, 而在实际操作过程中由于环境的影响经常导致量子退相干现象的发生, 这会对量子通道带来极大的影响. 常见的退相干环境下的量子通道有振幅阻尼通道, 退相位通道和退极化通道[47]. 本文主要考察退相位环境对SGNR量子通道的影响.

      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}}$, $\sigma _z^{{B_2}}$

      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.

    • 假设SGNR锯齿端上的两自旋粒子各自分别与退相位环境作用. 在这样的一个集体退相位环境下, 量子态${{{\rho}} _T}$将演化为

      $ {\boldsymbol{\rho}} _{{\text{ch}}}^{{\text{Pha}}} = \varepsilon ({{\boldsymbol{\rho}} _T}) = \sum\limits_{ij} {{{\boldsymbol{K}}_i} \otimes {{\boldsymbol{K}}_j}{{\boldsymbol{\rho}} _T}{\boldsymbol{K}}_i^\dagger \otimes {\boldsymbol{K}}_j^\dagger }, $

      其中${{\boldsymbol{K}}_1} = \sqrt {1 - {p / 2}} \left[ {\begin{array}{*{20}{c}} 1&0 \\ 0&1 \end{array}} \right]$, ${{\boldsymbol{K}}_2} = \sqrt {{p / 2}} \left[ {\begin{array}{*{20}{c}} 1&0 \\ 0&{ - 1} \end{array}} \right]$称为Kraus算符[1,47], 满足$\displaystyle\sum\nolimits_i {{\boldsymbol{K}}_i^\dagger {{\boldsymbol{K}}_i}} = {\boldsymbol{I}}$, I为单位算符. 这里p表示量子比特发生错误的概率, 它是关于时间的函数, 其表达式依赖于具体的物理系统, 理论上很难给出. 不失一般性, 在下面的讨论中以p作为基本参数来描述量子通道的动力学演化. 在标准基组$\left\{ {\left| {00} \right\rangle, \;\left| {01} \right\rangle, \;\left| {10} \right\rangle, \;\left| {11} \right\rangle } \right\}$下, (11)式可以进一步表示为

      $ {\boldsymbol{\rho}} _{{\text{ch}}}^{{\text{Pha}}} = \frac{1}{4}\left[ {\begin{array}{*{20}{c}} {1 + c}&0&0&0 \\ 0&{1 - c}&{2{{(1 - p)}^2}c}&0 \\ 0&{2{{(1 - p)}^2}c}&{1 - c}&0 \\ 0&0&0&{1 + c} \end{array}} \right] . $

      在退相位量子通道下, 最终的输出态为[48]

      $\begin{split} {\boldsymbol{\rho}} _{{\text{out}}}^{{\text{Pha}}} =\;& \sum\limits_{i,j = 0}^3 {{{\rm{Tr}}} \left( {{E^i}{\boldsymbol{\rho}} _{{\text{ch}}}^{{\text{Pha}}}} \right)} {{\rm{Tr}}} \left( {{E^j}{\boldsymbol{\rho}} _{{\text{ch}}}^{{\text{Pha}}}} \right) \\ &\times \left( {{\sigma ^i} \otimes {\sigma ^j}} \right){{\boldsymbol{\rho}} _{{\text{in}}}}\left( {{\sigma ^i} \otimes {\sigma ^j}} \right), \end{split}$

      其中, ${\sigma ^0} = I$, ${\sigma ^1} = {\sigma _x}$, ${\sigma ^2} = {\sigma _y}$, ${\sigma ^3} = {\sigma _z}$, ${E^0} = $$ \left| {{\varPsi ^ - }} \right\rangle \left\langle {{\varPsi ^ - }} \right|$, ${E^1} = \left| {{\varPhi ^ - }} \right\rangle \left\langle {{\varPhi ^ - }} \right|$, ${E^2} = \left| {{\varPhi ^ + }} \right\rangle \left\langle {{\varPhi ^ + }} \right|$, ${E^3} = $$ \left| {{\varPsi ^ + }} \right\rangle \left\langle {{\varPsi ^ + }} \right|$, $ \left| {{\varPsi ^ \pm }} \right\rangle = {{\left( {\left| {01} \right\rangle \pm \left| {10} \right\rangle } \right)} / {\sqrt 2 }} $, $\left| {{\varPhi ^ \pm }} \right\rangle = ( \left| {00} \right\rangle \pm $$ \left| {11} \right\rangle ) / {\sqrt 2 }$. 在标准基组$\left\{ {\left| {00} \right\rangle, \;\left| {01} \right\rangle, \;\left| {10} \right\rangle, \;\left| {11} \right\rangle } \right\}$下, (13)式可以进一步表示为

      $ {\boldsymbol{\rho}} _{{\text{out}}}^{{\text{Pha}}} = \frac{1}{4}\left[ {\begin{array}{*{20}{c}} {1 - b{c^2}}&0&0&0 \\ 0&{1 + b{c^2}}&{ - 2b{c^2}{{(1 - p)}^4}}&0 \\ 0&{ - 2b{c^2}{{(1 - p)}^4}}&{1 + b{c^2}}&0 \\ 0&0&0&{1 - b{c^2}} \end{array}} \right] . $

      根据(7)式, 通道态$ {\boldsymbol{\rho}} _{{\text{ch}}}^{{\text{Pha}}} $和输出态$ {\boldsymbol{\rho}} _{{\text{out}}}^{{\text{Pha}}} $的纠缠度分别为

      $ C({\boldsymbol{\rho}} _{{\text{ch}}}^{{\text{Pha}}}) = \max \left\{ {\begin{array}{*{20}{c}} {0,}&{{{ - \left[ {2{{(1 - p)}^2}c + c + 1} \right]} / 2}} \end{array}} \right\}, $

      $ C({\boldsymbol{\rho}} _{{\text{out}}}^{{\text{Pha}}}) = \left\{ {\begin{aligned} &{\max \left\{ {\begin{aligned} {0,}&{{{b{c^2}\left[ {2{{(1 - p)}^4} + 1} \right]} / 2} - {1 / 2}} \end{aligned}} \right\},}&{b > 0}, \\ & {0,}&{b \leqslant 0}. \end{aligned}} \right. $

      为了检验输出态的品质, 需要计算保真度, 其定义为[49]

      $ F({{\boldsymbol{\rho}} _{{\text{in}}}},{{\boldsymbol{\rho}} _{{\text{out}}}}) = {\left( {{{\rm{Tr}}} \sqrt {\sqrt {{{\boldsymbol{\rho}} _{{\text{in}}}}} {{\boldsymbol{\rho}} _{{\text{out}}}}\sqrt {{{\boldsymbol{\rho}} _{{{\rm{in}}} }}} } }\; \right)^2}, $

      其中${{\boldsymbol{\rho}} _{{\text{in}}}}$$ {{\boldsymbol{\rho}} _{{\text{out}}}} $分别表示输入态和输出态的密度矩阵. 将(10)式和(14)式代入(17)式, 最终计算得到

      $ \begin{split} \;& F({{\boldsymbol{\rho}} _{{\text{in}}}},\;{\boldsymbol{\rho}} _{{\text{out}}}^{{\text{Pha}}}) =\frac{{3 - (1 - 3b)b{c^2} - b}}{8} + \frac{{{b^2}{c^2}{{(1 - p)}^4}}}{2} + \frac{1}{8}\sqrt {(1 - b)(1 + 3b)\left[ {{{(1 + b{c^2})}^2} - 4{b^2}{c^4}{{(1 - p)}^8}} \right]} \\ & \qquad + \frac{{1 - b}}{4}\sqrt {(1 - b{c^2})\left[ {1 + b{c^2} - 2b{c^2}{{(1 - p)}^4}} \right]} + \frac{1}{4}\sqrt {(1 - b)(1 + 3b)(1 - b{c^2})\left[ {1 + b{c^2} + 2b{c^2}{{(1 - p)}^4}} \right]} . \end{split}$

      下面通过数值计算来具体讨论退相位环境对SGNR量子通道的影响. 图2给出的是退相位环境下通道态$ {\boldsymbol{\rho}} _{{\text{ch}}}^{{\text{Pha}}} $的纠缠度$C_{{\text{ch}}}^{{\text{Pha}}}$随温度T和出错概率p的变化. 从可以看出, 对于给定的库仑排斥势U, 在低温且p比较小(即演化时间比较短)的条件下, 通道态$ {\boldsymbol{\rho}} _{{\text{ch}}}^{{\text{Pha}}} $的纠缠度$C_{{\text{ch}}}^{{\text{Pha}}}$几乎趋近于最大值1; 当温度T升高时, $C_{{\text{ch}}}^{{\text{Pha}}}$快速衰减, 最终当温度达到某一确定值${T_{\text{c}}}$时衰减为零; 随着p的增大, $C_{{\text{ch}}}^{{\text{Pha}}}$不断衰减, 最终在p = 1处衰减为零; 通过比较图2(a)图2(d)发现, 对于给定的p, 纠缠消失的临界温度${T_{\text{c}}}$会随着U的增大而减小; 当p = 0时, ${T_{\text{c}}}$的变化由(9)式决定. 为了进一步考察U对量子通道的影响, 计算了$C_{{\text{ch}}}^{{\text{Pha}}}$Up的变化, 结果如图3所示. 当T = 0 K时, 如图3(a)所示, $C_{{\text{ch}}}^{{\text{Pha}}}$随着p逐渐衰减, 但其完全不受U的影响. 通过计算绝对零度下的自旋关联函数, 得到$c = - 1$, 于是由(15)式可得$C_{{\text{ch}}}^{{\text{Pha}}} = {(1 - p)^2}$, 这就解释了图3(a)的结果. 当T = 5 K时, 从图3(b)可以看出, $C_{{\text{ch}}}^{{\text{Pha}}}$随着U的增大而减弱. 当温度进一步升高时, 通过与图3(c)图3(d)的比较发现: 温度越高, $C_{{\text{ch}}}^{{\text{Pha}}}$U的衰减就越剧烈. 这是因为当TU增大时, 由(5)式可知η在不断减小, 于是由(4)式可知自旋关联函数c在不断增大. 根据(15)式, 对于给定的p, $C_{{\text{ch}}}^{{\text{Pha}}}$是关于c的单调递减函数, 因此高温和强库仑排斥势对$C_{{\text{ch}}}^{{\text{Pha}}}$有着非常强的抑制作用. 这也就意味着, 在实际量子通信过程中, 要尽量降低温度或者减弱库仑排斥势, 这样才能使量子通道处在比较理想的状态.

      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

      Figure 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

      Figure 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.

      在退相位环境下Werner态经量子和经典通道的传输, 最终得到的输出态$ {{\rho}} _{{\text{out}}}^{{\text{Pha}}} $的纠缠度$C_{{\text{out}}}^{{\text{Pha}}}$随参数b和出错概率p的变化如图4所示. 对于Werner态, 当$0 \leqslant b \leqslant {1 / 3}$时, 结合(7)式与(10)式可得$C({{\boldsymbol{\rho}} _{{\text{in}}}}) = 0$, 而当${1 / 3} \lt b \leqslant 1$时, $C({{\boldsymbol{\rho}} _{{\text{in}}}}) = {{\left( {3 b - 1} \right)} / 2}$. 显然, 输入态的纠缠度是关于b的单调递增函数. 从图4(a)可以看出, 当T = 5 K, U = 3.5 eV时, $C_{{\text{out}}}^{{\text{Pha}}}$随着b的增大而增大, 随着p的增大而不断衰减; 只有当输入态的纠缠比较鲁棒且演化时间比较短的情况下才有较为显著的纠缠输出. 当T = 5 K, U = 6.0 eV时, 如图4(b)所示, $C_{{\text{out}}}^{{\text{Pha}}}$被进一步削弱. 当T = 10 K, U = 3.5 eV时, 只有在b = 1且p = 0附近才有微弱的纠缠输出, 见图4(c). 当T = 10 K, U = 6.0 eV时, 如图4(d)所示, 输出纠缠$C_{{\text{out}}}^{{\text{Pha}}} = 0$, 即在此条件下无法完成纠缠态的传输, 原因是此时量子通道中已经没了纠缠资源.

      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

      Figure 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4对应, 图5给出的是$ {\boldsymbol{\rho}} _{{\text{out}}}^{{\text{Pha}}} $的保真度$F({{\boldsymbol{\rho}} _{{\text{in}}}}, \;{\boldsymbol{\rho}} _{{\text{out}}}^{{\text{Pha}}})$bp的变化. 从图5可以看到, 对于给定的b值, $F({{\boldsymbol{\rho}} _{{\text{in}}}}, \;{\boldsymbol{\rho}} _{{\text{out}}}^{{\text{Pha}}})$会随着p的增大而缓慢衰减; 只有对于较大的b值, $F({{\boldsymbol{\rho}} _{{\text{in}}}}, \;{\boldsymbol{\rho}} _{{\text{out}}}^{{\text{Pha}}})$p才有显著的衰减. 当$0 \leqslant b \leqslant {1 / 3}$时, $C({{\boldsymbol{\rho}} _{{\text{in}}}}) = 0$, 这时输出态的保真度不会低于90%. 从图5还可以看到, $F({{\boldsymbol{\rho}} _{{\text{in}}}}, \;{\boldsymbol{\rho}} _{{\text{out}}}^{{\text{Pha}}})$总是随着b的增大而减小. 前文已经指出输入态${{\boldsymbol{\rho}} _{{\text{in}}}}$的纠缠度$ C({{\boldsymbol{\rho}} _{{\text{in}}}}) $是关于b的增函数. 因此, 较大的b值对应纠缠度较大的Werner态. 于是可以得到以下结论: $F({{\boldsymbol{\rho}} _{{\text{in}}}}, \;{\boldsymbol{\rho}} _{{\text{out}}}^{{\text{Pha}}})$会随着输入态纠缠度$C({{\boldsymbol{\rho}} _{{\text{in}}}})$的增大而减小, 且$C({{\boldsymbol{\rho}} _{{\text{in}}}})$越大$F({{\boldsymbol{\rho}} _{{\text{in}}}}, \;{\boldsymbol{\rho}} _{{\text{out}}}^{{\text{Pha}}})$p的衰减也就越显著. 这是因为受环境的影响, SGNR量子通道处于混合态, 并非最大纠缠纯态. 而量子通道处于非最大纠缠态时, 输入态的纠缠度越大, 传输其需要利用量子通道中的纠缠资源就越多. 所以, 对于给定的SGNR量子通道, 当输入态纠缠度不断增大甚至超过量子通道所能提供的最大纠缠资源时, 一定会导致输出态的品质下降. 通过比较还可以发现: 相同的温度下, 库仑排斥势越大, $F({{\boldsymbol{\rho}} _{{\text{in}}}}, \;{\boldsymbol{\rho}} _{{\text{out}}}^{{\text{Pha}}})$越小; 相同的库仑排斥势下, 温度越高$F({{\boldsymbol{\rho}} _{{\text{in}}}}, \;{\boldsymbol{\rho}} _{{\text{out}}}^{{\text{Pha}}})$越小. 因此, 在实际应用中要提高输出态的保真度就需要想办法降低高温或减弱库仑排斥势.

      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.

      Figure 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.

      为了更加客观地反映SGNR量子通道的传输质量, 计算了在此通道下传输一个任意Werner态的平均保真度

      $ {F_A}({{\boldsymbol{\rho}} _{{\text{in}}}},{\boldsymbol{\rho}} _{{\text{out}}}^{{\text{Pha}}}) = \int_0^1 {F({{\boldsymbol{\rho}} _{{\text{in}}}},{\boldsymbol{\rho}} _{{\text{out}}}^{{\text{Pha}}})} {\text{d}}b, $

      其中$F({{\boldsymbol{\rho}} _{{\text{in}}}}, \;{\boldsymbol{\rho}} _{{\text{out}}}^{{\text{Pha}}})$由(18)式给出. 在电子间的库仑排斥势U给定的条件下, 首先计算了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}} _{{\rm{out}}}^{{\rm{Pha}}})$随出错概率p和温度T的变化, 结果如图6所示. 理论上已经证明: 对于任意一个态的经典传输, 其最大的经典保真度为${F_{{\text{class}}}} = {2 / 3}$. 所以, 对于量子隐形传态, 平均保真度大于2/3是最基本的要求. 从图6可以看出, 虽然输出态的平均保真度${F_A}({{\boldsymbol{\rho}} _{{\text{in}}}}, \;{\boldsymbol{\rho}} _{{\text{out}}}^{{\text{Pha}}})$随着T的增大而逐渐减小, 但是在$0 \leqslant T \leqslant 40\;{\text{K}}$这一温度范围内, 很明显${F_A}({{\boldsymbol{\rho}} _{{\text{in}}}}, \;{\boldsymbol{\rho}} _{{\text{out}}}^{{\text{Pha}}}) > 80\% $. 特别是当Tp的取值很小时, ${F_A}({{\boldsymbol{\rho}} _{{\text{in}}}}, \;{\boldsymbol{\rho}} _{{\text{out}}}^{{\text{Pha}}})$几乎接近于最大值1. 总的来看, ${F_A}({{\boldsymbol{\rho}} _{{\text{in}}}}, \;{\boldsymbol{\rho}} _{{\text{out}}}^{{\text{Pha}}})$p的衰减并不是十分明显. 这也就表明, 在退相位环境下SGNR量子通道有着非常好的鲁棒性. 接下来通过数值计算再来考察${F_A}({{\boldsymbol{\rho}} _{{\text{in}}}}, \;{\boldsymbol{\rho}} _{{\text{out}}}^{{\text{Pha}}})$p和库仑排斥势U的变化, 所得结果如图7所示. 当T = 0时, 如图7(a)所示, ${F_A}({{\boldsymbol{\rho}} _{{\text{in}}}}, \;{\boldsymbol{\rho}} _{{\text{out}}}^{{\text{Pha}}})$不受U的影响, 只是随着p的增大略微有些减小. 当$T \ne 0$时, 比较图7(b)图7(c)可以看出, ${F_A}({{\boldsymbol{\rho}} _{{\text{in}}}}, \;{\boldsymbol{\rho}} _{{\text{out}}}^{{\text{Pha}}})$会随着U的增大而略有减小, 且温度越高, ${F_A}({{\boldsymbol{\rho}} _{{\text{in}}}}, \;{\boldsymbol{\rho}} _{{\text{out}}}^{{\text{Pha}}})$取较大值的分布将不断向pU的低值区域收缩.

      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.

      Figure 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

      Figure 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.

    • 本文研究了退相位环境下Werner态在SGNR量子通道中的隐形传输. 结果表明, 输出态的纠缠度和保真度对量子通道中的纠缠资源以及输入态的纠缠度有很强的依赖性. 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; 对于给定的量子通道, 输出态的纠缠度总是随着输入态纠缠度的增大而增大, 而相应的保真度却总是随着输入态纠缠度的增大而减小. 由于受退相位环境的影响, 量子通道中的纠缠资源会随着时间演化而不断损耗, 高温和强库仑排斥势会加剧这种损耗, 从而导致输出态的品质下降. 但是当温度小于40 K, 且电子间的库仑排斥势小于6 eV时, 输出态的平均保真度总可以达到80%以上, 且平均保真度在退相位环境中的衰减并不明显. 因此, SGNR量子通道在退相位环境下有非常好的鲁棒性. 特别是对于低温和微弱的排斥势, 输出态的平均保真度几乎接近最大值1. 以上结果充分说明石墨烯材料在量子信息领域中具有潜在的应用价值.

参考文献 (49)

目录

    /

    返回文章
    返回