高维量子逻辑门及高维量子信息处理
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摘要
量子计算与量子信息处理是涵盖了信息理论、计算机理论与量子力学的交叉学科,在信息、物理以及计算机等众多领域有着非常大的影响.量子特性在信息安全、信息容量以及计算速度的提高等方面都具有独特的优势.量子逻辑门是量子计算与量子信息处理中的一个关键模块,因此,如何构建一个合适的逻辑门也是现阶段热门的研究领域.此外,研究量子信息科学,纠缠光子对也是一个不可或缺的元素.目前有很多种产生纠缠光子对的理论和实验方案,例如参量下转换等方案.高维量子系统可以在很大程度上提高量子信道容量和信息存储空间,通过实现高维度量子逻辑门,能够提高量子计算与量子信息处理的速度.然而,直接由两个高维子系统相互作用构建高维逻辑门是很困难的.在这种情况下,即使要实现一个很小的高维逻辑电路,也会耗费大量二维逻辑门.本文主要介绍了利用纠缠光子对的偏振、频率和空间模式自由度实现的二维以及高维单自由度和多自由度的量子逻辑门方案,并探讨了这些方案在量子信息处理和量子计算等方面的应用以及发展趋势.
Quantum information science has extensive applications in various research fields, such as information, physics and computer science. It is well known that the quantum properties of a system exhibit many unique advantages in the security of information transmission and processing, computation, and the improvement of channel capacity. Quantum information includes several research fields, for example, quantum communication, quantum computing and so on. In the past 30 years,the development of quantum communication, quantum computing and quantum information processing has made great progress in both theory and experiments.As a fundamental resource for quantum information processing, entanglement is a key element for many applications,such as quantum key distribution, quantum teleportation, quantum dense coding, quantum secure direct communication and quantum metrology. Because of the weak interaction with the environment and the degrees of freedom, entangled photons are the excellent candidates as the quantum-information carrier. Several remarkable experiments have been performed with the photonic entanglement. However, it is necessary to share a pair of entangled photons between the communicating parties in advance in these quantum information processes. Furthermore, the application of hyperentanglement, the entanglement of photon pairs simultaneously existing in more than one degree of freedom, has been widely studied for the reason that it can improve the channel capacity of quantum communications and implement hyperparallel computing, such as the quantum error-correcting code, quantum repeater and deterministic entanglement purification. Moreover, the quantum logic gates play an essential role in quantum information processing which attracts much attention on the designing of the quantum logic gates. Depending on the coherent dynamics of the cavity quantum electrodynamical system, deterministic quantum gate operations between the quantum systems can be realized.Here in this review, by considering high-dimensional quantum systems, several high-dimensional quantum protocols are presented. They can greatly improve the capacity of quantum channel and noise immunity, which exhibits potential applications in quantum information processing. This review describes the development of quantum computing and quantum information processing using hyperentanglement. And the entangled photons in single degree of freedom and multi-degrees of freedom are used to illustrate the two-dimensional and high-dimensional quantum information processing schemes. Finally, the latest progress and further applications of quantum computing and quantum information processing are introduced.
Quantum information science has extensive applications in various research fields, such as information, physics and computer science. It is well known that the quantum properties of a system exhibit many unique advantages in the security of information transmission and processing, computation, and the improvement of channel capacity. Quantum information includes several research fields, for example, quantum communication, quantum computing and so on. In the past 30 years,the development of quantum communication, quantum computing and quantum information processing has made great progress in both theory and experiments.As a fundamental resource for quantum information processing, entanglement is a key element for many applications,such as quantum key distribution, quantum teleportation, quantum dense coding, quantum secure direct communication and quantum metrology. Because of the weak interaction with the environment and the degrees of freedom, entangled photons are the excellent candidates as the quantum-information carrier. Several remarkable experiments have been performed with the photonic entanglement. However, it is necessary to share a pair of entangled photons between the communicating parties in advance in these quantum information processes. Furthermore, the application of hyperentanglement, the entanglement of photon pairs simultaneously existing in more than one degree of freedom, has been widely studied for the reason that it can improve the channel capacity of quantum communications and implement hyperparallel computing, such as the quantum error-correcting code, quantum repeater and deterministic entanglement purification. Moreover, the quantum logic gates play an essential role in quantum information processing which attracts much attention on the designing of the quantum logic gates. Depending on the coherent dynamics of the cavity quantum electrodynamical system, deterministic quantum gate operations between the quantum systems can be realized.Here in this review, by considering high-dimensional quantum systems, several high-dimensional quantum protocols are presented. They can greatly improve the capacity of quantum channel and noise immunity, which exhibits potential applications in quantum information processing. This review describes the development of quantum computing and quantum information processing using hyperentanglement. And the entangled photons in single degree of freedom and multi-degrees of freedom are used to illustrate the two-dimensional and high-dimensional quantum information processing schemes. Finally, the latest progress and further applications of quantum computing and quantum information processing are introduced.
引文
1 Zhang L,Silberhorn C,Walmsley I A.Secure quantum key distribution using continuous variables of single photons.Phys Rev Lett,2008,100:110504
2 Shor P W.Algorithms for quantum computation:Discrete logarithm and factoring.In:Goldwasser S,ed.Proceedings of the 35th Annual Symposium on Foundations of Computer Science.Los Alamotos:IEEE Computer Society Press,1994.124-134
3 Dixon P B,Howland G A,Schneeloch J,et al.Quantum mutual information capacity for high-dimensional entangled states.Phys Rev Lett,2012,108:143603
4 Bennett C H,Brassard G.Quantum cryptography:Public key distribution and coin tossing.Theor Comput Sci,2014,560:7-11
5 Hillery M,Bu?ek V,Berthiaume A.Quantum secret sharing.Phys Rev A,1999,59:1829-1834
6 Long G L,Liu X S.Theoretically efficient high-capacity quantum-key-distribution scheme.Phys Rev A,2002,65:032302
7 Deng F G,Long G L,Liu X S.Two-step quantum direct communication protocol using the Einstein-Podolsky-Rosen pair block.Phys Rev A,2003,68:042317
8 Grover L K.Quantum mechanics helps in searching for a needle in a haystack.Phys Rev Lett,1997,79:325-328
9 Gottesman D,Chuang I L.Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations.Nature,1999,402:390-393
10 Bennett C H,Brassard G,Crepeau C,et al.Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels.Phys Rev Lett,1993,70:1895-1899
11 Bouwmeester D,Pan J W,Mattle K,et al.Experimental quantum teleportation.Nature,1997,390:575-579
12 Cirac J I,Zoller P,Kimble H J,et al.Quantum state transfer and entanglement distribution among distant nodes in a quantum network.Phys Rev Lett,1997,78:3221-3224
13 Walln?fer J,Zwerger M,Muschik C,et al.Two-dimensional quantum repeaters.Phys Rev A,2016,94:052307
14 Bennett C H,Brassard G,Popescu S,et al.Purification of noisy entanglement and faithful teleportation via noisy channels.Phys Rev Lett,1996,76:722-725
15 Bennett C H,Bernstein H J,Popescu S,et al.Concentrating partial entanglement by local operations.Phys Rev A,1996,53:2046-2052
16 Pan J W,Simon C,Brukner C,et al.Entanglement purification for quantum communication.Nature,2001,410:1067-1070
17 Poyatos J F,Cirac J I,Zoller P.Complete characterization of a quantum process:The two-bit quantum gate.Phys Rev Lett,1997,78:390-393
18 Browne D E,Plenio M B,Huelga S F.Robust creation of entanglement between ions in spatially separate cavities.Phys Rev Lett,2003,91:067901
19 Serafini A,Mancini S,Bose S.Distributed quantum computation via optical fibers.Phys Rev Lett,2006,96:010503
20 Deng Z J,Zhang X L,Wei H,et al.Implementation of a nonlocal N-qubit conditional phase gate by single-photon interference.Phys Rev A,2007,76:044305
21 Barrett S D,Rohde P P,Stace T M.Scalable quantum computing with atomic ensembles.New J Phys,2010,12:093032
22 Beige A,Lim Y L,Kwek L C.A repeat-until-success quantum computing scheme.New J Phys,2007,9:197
23 Barrett S D,Kok P.Efficient high-fidelity quantum computation using matter qubits and linear optics.Phys Rev A,2005,71:060310
24 Rohde P P,Barrett S D.Strategies for the preparation of large cluster states using non-deterministic gates.New J Phys,2007,9:198
25 Calsamiglia J.Generalized measurements by linear elements.Phys Rev A,2002,65:030301
26 Ursin R,Jennewein T,Aspelmeyer M,et al.Quantum teleportation across the Danube.Nature,2004,430:849
27 van Houwelingen J A W,Brunner N,Beveratos A,et al.Quantum teleportation with a three-bell-state analyzer.Phys Rev Lett,2006,96:130502
28 Duan L M,Wang B,Kimble H J.Robust quantum gates on neutral atoms with cavity-assisted photon scattering.Phys Rev A,2005,72:032333
29 Hu C Y,Young A,O’Brien J L,et al.Giant optical Faraday rotation induced by a single-electron spin in a quantum dot:Applications to entangling remote spins via a single photon.Phys Rev B,2008,78:085307
30 Zhou X F,Zhang Y S,Guo G C.Nonlocal gate of quantum network via cavity quantum electrodynamics.Phys Rev A,2005,71:064302
31 Cao C,Chen X,Duan Y W,et al.Concentrating partially entangled W-class states on nonlocal atoms using low-Q optical cavity and linear optical elements.Sci China-Phys Mech Astron,2016,59:100315
32 Cerf N J,Bourennane M,Karlsson A,et al.Security of quantum key distribution using d-level systems.Phys Rev Lett,2002,88:127902
33 Wang C,Deng F G,Li Y S,et al.Quantum secure direct communication with high-dimension quantum superdense coding.Phys Rev A,2005,71:044305
34 Barreiro J T,Langford N K,Peters N A,et al.Generation of hyperentangled photon pairs.Phys Rev Lett,2005,95:260501
35 Wilde M M,Uskov D B.Linear-optical hyperentanglement-assisted quantum error-correcting code.Phys Rev A,2009,79:022305
36 Sheng Y B,Deng F G.Deterministic entanglement purification and complete nonlocal Bell-state analysis with hyperentanglement.Phys Rev A,2010,81:032307
37 Vallone G,Ceccarelli R,de Martini F,et al.Hyperentanglement of two photons in three degrees of freedom.Phys Rev A,2009,79:030301
38 Kim J S,Das A,Sanders B C.Entanglement monogamy of multipartite higher-dimensional quantum systems using convex-roof extended negativity.Phys Rev A,2009,79:012329
39 Richart D,Fischer Y,Weinfurter H.Experimental implementation of higher dimensional time-energy entanglement.Appl Phys B,2012,106:543-550
40 Neves L,Lima G,Aguirre Gómez J G,et al.Generation of entangled states of qudits using twin photons.Phys Rev Lett,2005,94:100501
41 O’Sullivan-Hale M N,Ali Khan I,Boyd R W,et al.Pixel entanglement:Experimental realization of optically entangled d=3 and d=6 qudits.Phys Rev Lett,2005,94:220501
42 Vaziri A,Weihs G,Zeilinger A.Experimental two-photon,three-dimensional entanglement for quantum communication.Phys Rev Lett,2002,89:240401
43 Peruzzo A,Lobino M,Matthews J C F,et al.Quantum walks of correlated photons.Science,2010,329:1500-1503
44 Edgar M P,Tasca D S,Izdebski F,et al.Imaging high-dimensional spatial entanglement with a camera.Nat Commun,2012,3:984
45 Yokoyama S,Ukai R,Armstrong S C,et al.Ultra-large-scale continuous-variable cluster states multiplexed in the time domain.Nat Photon,2013,7:982-986
46 D'Ambrosio V,Bisesto F,Sciarrino F,et al.Device-independent certification of high-dimensional quantum systems.Phys Rev Lett,2014,112:140503
47 Krenn M,Huber M,Fickler R,et al.Generation and confirmation of a(100×100)-dimensional entangled quantum system.Proc Natl Acad Sci USA,2014,111:6243-6247
48 Wang X L,Luo Y H,Huang H L,et al.18-Qubit entanglement with six photons’three degrees of freedom.Phys Rev Lett,2018,120:260502
49 Zahedinejad E,Ghosh J,Sanders B C.High-fidelity single-shot Toffoli gate via quantum control.Phys Rev Lett,2015,114:200502
50 Giesz V,Somaschi N,Hornecker G,et al.Coherent manipulation of a solid-state artificial atom with few photons.Nat Commun,2016,7:11986
51 Anderson B E,Sosa-Martinez H,Riofrío C A,et al.Accurate and robust unitary transformations of a high-dimensional quantum system.Phys Rev Lett,2015,114:240401
52 Bonato C,Haupt F,Oemrawsingh S S R,et al.CNOT and Bell-state analysis in the weak-coupling cavity QED regime.Phys Rev Lett,2010,104:160503
53 Wei H R,Deng F G.Universal quantum gates for hybrid systems assisted by quantum dots inside double-sided optical microcavities.Phys Rev A,2013,87:022305
54 Ren B C,Wei H R,Deng F G.Deterministic photonic spatial-polarization hyper-controlled-not gate assisted by a quantum dot inside a one-side optical microcavity.Laser Phys Lett,2013,10:095202
55 Ren B C,Wang G Y,Deng F G.Universal hyperparallel hybrid photonic quantum gates with dipole-induced transparency in the weak-coupling regime.Phys Rev A,2015,91:032328
56 Xia B Y,Cao C,Han Y H,et al.Universal photonic three-qubit quantum gates with two degrees of freedom assisted by charged quantum dots inside single-sided optical microcavities.Laser Phys,2018,28:095201
57 Wang T J,Wang C.Universal hybrid three-qubit quantum gates assisted by a nitrogen-vacancy center coupled with a whispering-gallery-mode microresonator.Phys Rev A,2014,90:052310
58 Wang T J,Zhang Y,Wang C.Universal hybrid hyper-controlled quantum gates assisted by quantum dots in opticaldouble-sided microcavities.Laser Phys Lett,2014,11:025203
59 Wang T J,Wang C.Parallel quantum computing teleportation for spin qubits in quantum dot and microcavity coupled system.IEEE J Sel Top Quantum Electron,2015,21:91-97
60 He L Y,Wang T J,Wang C.Construction of high-dimensional universal quantum logic gates using aΛsystem coupled with a whispering-gallerymode microresonator.Opt Express,2016,24:15429
61 Pan J W,Gasparoni S,Ursin R,et al.Experimental entanglement purification of arbitrary unknown states.Nature,2003,423:417-422
62 Wang T J,Cao C,Wang C.Linear-optical implementation of hyperdistillation from photon loss.Phys Rev A,2014,89:052303
63 Wang T J,Wang C.High-efficient entanglement distillation from photon loss and decoherence.Opt Express,2015,23:31550
64 Wang T J,Mi S C,Wang C.Hyperentanglement purification using imperfect spatial entanglement.Opt Express,2017,25:2969
2 Shor P W.Algorithms for quantum computation:Discrete logarithm and factoring.In:Goldwasser S,ed.Proceedings of the 35th Annual Symposium on Foundations of Computer Science.Los Alamotos:IEEE Computer Society Press,1994.124-134
3 Dixon P B,Howland G A,Schneeloch J,et al.Quantum mutual information capacity for high-dimensional entangled states.Phys Rev Lett,2012,108:143603
4 Bennett C H,Brassard G.Quantum cryptography:Public key distribution and coin tossing.Theor Comput Sci,2014,560:7-11
5 Hillery M,Bu?ek V,Berthiaume A.Quantum secret sharing.Phys Rev A,1999,59:1829-1834
6 Long G L,Liu X S.Theoretically efficient high-capacity quantum-key-distribution scheme.Phys Rev A,2002,65:032302
7 Deng F G,Long G L,Liu X S.Two-step quantum direct communication protocol using the Einstein-Podolsky-Rosen pair block.Phys Rev A,2003,68:042317
8 Grover L K.Quantum mechanics helps in searching for a needle in a haystack.Phys Rev Lett,1997,79:325-328
9 Gottesman D,Chuang I L.Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations.Nature,1999,402:390-393
10 Bennett C H,Brassard G,Crepeau C,et al.Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels.Phys Rev Lett,1993,70:1895-1899
11 Bouwmeester D,Pan J W,Mattle K,et al.Experimental quantum teleportation.Nature,1997,390:575-579
12 Cirac J I,Zoller P,Kimble H J,et al.Quantum state transfer and entanglement distribution among distant nodes in a quantum network.Phys Rev Lett,1997,78:3221-3224
13 Walln?fer J,Zwerger M,Muschik C,et al.Two-dimensional quantum repeaters.Phys Rev A,2016,94:052307
14 Bennett C H,Brassard G,Popescu S,et al.Purification of noisy entanglement and faithful teleportation via noisy channels.Phys Rev Lett,1996,76:722-725
15 Bennett C H,Bernstein H J,Popescu S,et al.Concentrating partial entanglement by local operations.Phys Rev A,1996,53:2046-2052
16 Pan J W,Simon C,Brukner C,et al.Entanglement purification for quantum communication.Nature,2001,410:1067-1070
17 Poyatos J F,Cirac J I,Zoller P.Complete characterization of a quantum process:The two-bit quantum gate.Phys Rev Lett,1997,78:390-393
18 Browne D E,Plenio M B,Huelga S F.Robust creation of entanglement between ions in spatially separate cavities.Phys Rev Lett,2003,91:067901
19 Serafini A,Mancini S,Bose S.Distributed quantum computation via optical fibers.Phys Rev Lett,2006,96:010503
20 Deng Z J,Zhang X L,Wei H,et al.Implementation of a nonlocal N-qubit conditional phase gate by single-photon interference.Phys Rev A,2007,76:044305
21 Barrett S D,Rohde P P,Stace T M.Scalable quantum computing with atomic ensembles.New J Phys,2010,12:093032
22 Beige A,Lim Y L,Kwek L C.A repeat-until-success quantum computing scheme.New J Phys,2007,9:197
23 Barrett S D,Kok P.Efficient high-fidelity quantum computation using matter qubits and linear optics.Phys Rev A,2005,71:060310
24 Rohde P P,Barrett S D.Strategies for the preparation of large cluster states using non-deterministic gates.New J Phys,2007,9:198
25 Calsamiglia J.Generalized measurements by linear elements.Phys Rev A,2002,65:030301
26 Ursin R,Jennewein T,Aspelmeyer M,et al.Quantum teleportation across the Danube.Nature,2004,430:849
27 van Houwelingen J A W,Brunner N,Beveratos A,et al.Quantum teleportation with a three-bell-state analyzer.Phys Rev Lett,2006,96:130502
28 Duan L M,Wang B,Kimble H J.Robust quantum gates on neutral atoms with cavity-assisted photon scattering.Phys Rev A,2005,72:032333
29 Hu C Y,Young A,O’Brien J L,et al.Giant optical Faraday rotation induced by a single-electron spin in a quantum dot:Applications to entangling remote spins via a single photon.Phys Rev B,2008,78:085307
30 Zhou X F,Zhang Y S,Guo G C.Nonlocal gate of quantum network via cavity quantum electrodynamics.Phys Rev A,2005,71:064302
31 Cao C,Chen X,Duan Y W,et al.Concentrating partially entangled W-class states on nonlocal atoms using low-Q optical cavity and linear optical elements.Sci China-Phys Mech Astron,2016,59:100315
32 Cerf N J,Bourennane M,Karlsson A,et al.Security of quantum key distribution using d-level systems.Phys Rev Lett,2002,88:127902
33 Wang C,Deng F G,Li Y S,et al.Quantum secure direct communication with high-dimension quantum superdense coding.Phys Rev A,2005,71:044305
34 Barreiro J T,Langford N K,Peters N A,et al.Generation of hyperentangled photon pairs.Phys Rev Lett,2005,95:260501
35 Wilde M M,Uskov D B.Linear-optical hyperentanglement-assisted quantum error-correcting code.Phys Rev A,2009,79:022305
36 Sheng Y B,Deng F G.Deterministic entanglement purification and complete nonlocal Bell-state analysis with hyperentanglement.Phys Rev A,2010,81:032307
37 Vallone G,Ceccarelli R,de Martini F,et al.Hyperentanglement of two photons in three degrees of freedom.Phys Rev A,2009,79:030301
38 Kim J S,Das A,Sanders B C.Entanglement monogamy of multipartite higher-dimensional quantum systems using convex-roof extended negativity.Phys Rev A,2009,79:012329
39 Richart D,Fischer Y,Weinfurter H.Experimental implementation of higher dimensional time-energy entanglement.Appl Phys B,2012,106:543-550
40 Neves L,Lima G,Aguirre Gómez J G,et al.Generation of entangled states of qudits using twin photons.Phys Rev Lett,2005,94:100501
41 O’Sullivan-Hale M N,Ali Khan I,Boyd R W,et al.Pixel entanglement:Experimental realization of optically entangled d=3 and d=6 qudits.Phys Rev Lett,2005,94:220501
42 Vaziri A,Weihs G,Zeilinger A.Experimental two-photon,three-dimensional entanglement for quantum communication.Phys Rev Lett,2002,89:240401
43 Peruzzo A,Lobino M,Matthews J C F,et al.Quantum walks of correlated photons.Science,2010,329:1500-1503
44 Edgar M P,Tasca D S,Izdebski F,et al.Imaging high-dimensional spatial entanglement with a camera.Nat Commun,2012,3:984
45 Yokoyama S,Ukai R,Armstrong S C,et al.Ultra-large-scale continuous-variable cluster states multiplexed in the time domain.Nat Photon,2013,7:982-986
46 D'Ambrosio V,Bisesto F,Sciarrino F,et al.Device-independent certification of high-dimensional quantum systems.Phys Rev Lett,2014,112:140503
47 Krenn M,Huber M,Fickler R,et al.Generation and confirmation of a(100×100)-dimensional entangled quantum system.Proc Natl Acad Sci USA,2014,111:6243-6247
48 Wang X L,Luo Y H,Huang H L,et al.18-Qubit entanglement with six photons’three degrees of freedom.Phys Rev Lett,2018,120:260502
49 Zahedinejad E,Ghosh J,Sanders B C.High-fidelity single-shot Toffoli gate via quantum control.Phys Rev Lett,2015,114:200502
50 Giesz V,Somaschi N,Hornecker G,et al.Coherent manipulation of a solid-state artificial atom with few photons.Nat Commun,2016,7:11986
51 Anderson B E,Sosa-Martinez H,Riofrío C A,et al.Accurate and robust unitary transformations of a high-dimensional quantum system.Phys Rev Lett,2015,114:240401
52 Bonato C,Haupt F,Oemrawsingh S S R,et al.CNOT and Bell-state analysis in the weak-coupling cavity QED regime.Phys Rev Lett,2010,104:160503
53 Wei H R,Deng F G.Universal quantum gates for hybrid systems assisted by quantum dots inside double-sided optical microcavities.Phys Rev A,2013,87:022305
54 Ren B C,Wei H R,Deng F G.Deterministic photonic spatial-polarization hyper-controlled-not gate assisted by a quantum dot inside a one-side optical microcavity.Laser Phys Lett,2013,10:095202
55 Ren B C,Wang G Y,Deng F G.Universal hyperparallel hybrid photonic quantum gates with dipole-induced transparency in the weak-coupling regime.Phys Rev A,2015,91:032328
56 Xia B Y,Cao C,Han Y H,et al.Universal photonic three-qubit quantum gates with two degrees of freedom assisted by charged quantum dots inside single-sided optical microcavities.Laser Phys,2018,28:095201
57 Wang T J,Wang C.Universal hybrid three-qubit quantum gates assisted by a nitrogen-vacancy center coupled with a whispering-gallery-mode microresonator.Phys Rev A,2014,90:052310
58 Wang T J,Zhang Y,Wang C.Universal hybrid hyper-controlled quantum gates assisted by quantum dots in opticaldouble-sided microcavities.Laser Phys Lett,2014,11:025203
59 Wang T J,Wang C.Parallel quantum computing teleportation for spin qubits in quantum dot and microcavity coupled system.IEEE J Sel Top Quantum Electron,2015,21:91-97
60 He L Y,Wang T J,Wang C.Construction of high-dimensional universal quantum logic gates using aΛsystem coupled with a whispering-gallerymode microresonator.Opt Express,2016,24:15429
61 Pan J W,Gasparoni S,Ursin R,et al.Experimental entanglement purification of arbitrary unknown states.Nature,2003,423:417-422
62 Wang T J,Cao C,Wang C.Linear-optical implementation of hyperdistillation from photon loss.Phys Rev A,2014,89:052303
63 Wang T J,Wang C.High-efficient entanglement distillation from photon loss and decoherence.Opt Express,2015,23:31550
64 Wang T J,Mi S C,Wang C.Hyperentanglement purification using imperfect spatial entanglement.Opt Express,2017,25:2969