Entanglement purification and faithful qubit transmission 第四届全国冷原子物理和量子信息青年学者学术讨论会 Entanglement purification and faithful qubit transmission Fu-Guo Deng (邓 富 国) Department of Physics, Beijing Normal University, Beijing 100875 2010. 8. 5, Dalian 量子通信简介（清华大学邓富国）

(Chongqing Unversity) collaborators Yu-Bo Sheng, 盛宇波 (Tsinghua Unversity) Postdoc. Hong-Yu Zhou 周宏余 (Beijing Normal Unversity) Xi-Han Li, 李熙涵 (Chongqing Unversity) 重庆大学

Outline 1. Motivation 2. Entanglement purification 3. Faithful qubit transmission 4. Summary

1. Motivation

R. Ursin et al，Entanglement-based quantum communication over 144km， University of Vienna，Nature Physics 3：481-484 (2007)

Briegel et al., Phys. Rev. Lett. 81, 5932 (1998) Long-distance quantum communication: Quantum repeater Entanglement: a) Generation b) Distribution c) Purification d) Swapping e) Storage Briegel et al., Phys. Rev. Lett. 81, 5932 (1998)

Quantum repeater Single-photon quantum repeater L.M. Duan et al., Nature 414, 413 (2001) two-photon quantum repeater B. Zhao et al., Phys. Rev. Lett. 98, 240502 (2007) Z. B. Chen et al., Phys. Rev. A 76, 022329 (2007)

由于纠缠只能通过局域产生，然后通过信道发送粒子的方式来共享。这样，纠缠粒子和环境的相互作用就不可避免。本来处于最大纠缠态的粒子对经信道传输后就会发生退相干，这种退相干通常有两种结果: 纠缠纯化 由纯的最大纠缠态变成混合态 Entanglement purification 纠缠浓缩 由最大纠缠态变成部分纠缠态 Entanglement concentration

two-photon entanglement purification and concentration C. H. Bennett, et al., Phys. Rev. Lett. 76, 722 (1996). D. Deutsch et al., Phys. Rev. Lett. 77, 2818 (1996). J. W. Pan et al., Nature (London) 410, 1067 (2001). C. Simon, J. W. Pan, Phys. Rev. Lett. 89, 257901 (2002). Y. B. Sheng, F. G. Deng, H. Y. Zhou, Phys. Rev. A 77, 042308 (2008). Y. B. Sheng, F. G. Deng, Phys. Rev. A 81, 032307 (2010). (a’) C. H. Bennett, et al., Phys. Rev. A 53, 2046 (1996). (b’) T. Yamamoto, et al., Phys. Rev. A 64, 012304 (2001). (c’) Z. Zhao, J. W. Pan, M. S. Zhan, Phys. Rev. A 64, 014301 (2001). (d’) Y. B. Sheng, F. G. Deng, H. Y. Zhou, Phys. Rev. A 77, 062325 (2008).

After entanglement generation: Single-photon entanglement purification and concentration After entanglement generation: Partial entanglement state (a) N. Sangouard, C. Simon, T. Coudreau, N. Gisin, Purification of single-photon entanglement with linear optics, Phys. Rev. A 78, 50301(R) (2008). (b) Y. B. Sheng, F. G. Deng, H. Y. Zhou, Single-photon entanglement concentration for long-distance quantum communication, Quantum information & computation 10, 272 (2010).

(1) EPP based on CNOT gates 2. Entanglement purification protocol (EPP) (1) EPP based on CNOT gates Charles H. Bennett. et al., Phys. Rev. Lett. 76, 722 (1996)

改进的CNOT门纯化方案 D. Deutsch et al., Phys. Rev. Lett. 77, 2818 (1996)

(2) EPP based on linear optical elements J. W. Pan et al., Nature 410, 1067 (2001)

(3) EPP based on spatial entanglement 单对光子： 两对光子： C. Simon, Jian-Wei Pan, Phys. Rev. Lett. 89, 257901 (2002)

利用空间纠缠纯化极化纠缠在产生单对纠缠光子的时候，可以完全纯化比特错误，而且可以完全纠正。但是对于相位错误，依然要通过后续的手段，也就是常规的纠缠纯化方案来实现。 如果产生两对纠缠光子，则与理想源一样，采取选择四个空间模式都有光子的事件，可以纯化比特翻转错误。

(4) EPP based on cross-Kerr nonlinearity a. The principle of cross-Kerr nonlinearity b. The principle of PDC entanglement source Yu-Bo Sheng, Fu-Guo Deng, Hong-Yu Zhou, Phys. Rev. A 77,  042308 (2008)

QND based cross-Kerr nonlinearity for parity check：

bit-flip error： After parity-check measurement with QND：

Two photon pairs： No errors:

A bit-flip error on one of the two pairs：

Bit-flip errors take place on the two photon pairs： fidelity： p1：the probability for one photon pair from PDC p2：two photon pairs F0：the fidelity of the original state transmitted

Entanglement purification for an ideal source (PDC源的后续纯化)

对应PBS方案的 四个空间模式 出射事件 提高效率！对应PBS方案 光子从两个空间模式出射

(5) Deterministic entanglement purification (DEPP) Difference between conventional EPPs (CEPP) and DEPP： CEPP(渐进式纠缠纯化)： 从一个处于混合纠缠态的系综中，渐进式地得到一部分保真度高的纠缠光子对 DEPP： 一纠缠光子对经噪声信道传输后，可以纯化为一对最大纠缠光子态 Yu-Bo Sheng, Fu-Guo Deng， Phys. Rev. A 81, 032307 (2010)

Hyperentanglement state After transformation in a noisy channel, the whole state: The part of polarization Mixed state The part of frequency and spatial, do not suffer from the noise: Robust

Four admixtures: a phase-flip bit & phase error bit-flip no error probability b c d error type

a) bit-flip error purification No error same phase shift same phase shift 0 output mode a2b2 output mode a1b1

different phase shift and 0 0 and output mode a2b1 output mode a1b2 Bit-flip error different phase shift and 0 0 and output mode a2b1 output mode a1b2

has the same result with With different phase shifts and output modes, the two parties in quantum communication can exactly correct the bit-flip error.

b) phase-flip error purification WDM: leads the photon to different spatial mode according to its frequency After first step, the initial state will become: The spatial entanglement has consumed The part of polarization: Hardamard Phase-flip error can not be purified directly

same phase shift 0 output mode c2d2 output mode c1d1 same phase shift different phase shift and 0 0 and output mode c2d1 output mode c1d2

Finally, the two parties will get a maximally entangled state or Using quantum frequency upconversion, the two parties can erase distinguishability for frequency

3. Faithful qubit transmission

信道噪声 噪声来源：热涨落、介质的不均匀性、双折射等 对抗噪声的方法：反馈控制 VS 建模处理 量子纠错码(Quantum Error Correction Code-QECC) 纠缠纯化(Entanglement purification) 退相干无关子空间(Decoherence Free Subspace-DFS) 量子态避错传输…… 联合噪声(collective noise)假设 噪声随时间是缓变的，即同时或时间间隔很短的几个光子或波包在同一噪声信道中传输时受到的影响相同。 量子通信简介（清华大学邓富国） 37

T. Yamamoto. et al. (Japan)，Phys. Rev. Lett. 95, 040503 (2005) (1) 借助辅助粒子的量子态避错传输 方案特点：1. 基于辅助粒子；2. 成功率低，1/16→1/8 T. Yamamoto. et al. (Japan)，Phys. Rev. Lett. 95, 040503 (2005) 1. 引入频率自由度标识光子； 2. 成功率为 PRL95 最大值的 4 倍，即为 1/2 Xi-Han Li, et al., Opt. Commun. 282, 4025 (2009) 量子通信简介（清华大学邓富国） 38 38

对抗退相位噪声实验 T. Yamamoto. et al ，Experimental ancilla-assisted qubit transmission against correlated noise using quantum parity checking， New J. Phys. 9：191 (2007)

(2) 单量子态自避错传输 Xi-Han Li, Fu-Guo Deng, Hong-Yu Zhou，Appl. Phys. Lett.  91, 144101 (2007)

Success probability ： 1/2

General self-error-rejecting qubit transmission

PS=75%

Fu-Guo Deng, Xi-Han Li, Hong-Yu Zhou，2010

(3) Faithful entanglement distribution Initial state Collective noise channel

Coupled with coherent states： 其作用类似将噪声作用通过不同的空间模式分离 根据不同的相干态相移情况，可以得到不同的频率纠缠态，且可以确定其出射的路径。

Yu-Bo Sheng, Fu-Guo Deng, Phys. Rev. A 81, 042332 (2010) Entanglement transformation： 从c2d2出射 得到确定的纠缠态，没有损耗 Yu-Bo Sheng, Fu-Guo Deng, Phys. Rev. A 81, 042332 (2010)

Faithful entanglement distribution for quantum key distribution Hui-Chong Niu, Fu-Guo Deng, 2010

Summary 1. Entanglement purification based nonlinear optics. 2. Deterministic entanglement purification . 3. Self-error-rejecting qubit transmission. 4. Faithful entanglement distribution.

谢谢大家！