Abstract

We present an efficient and faithful hyperentanglement purification protocol (hyper-EPP) for three-photon system in mixed hyperentangled Greenberger-Horne-Zeilinger states with bit-flip errors in both spatial-mode and polarization degrees of freedom (DOFs), resorting to the fidelity-robust quantum gates and hyperentanglement link. Our high-efficiency hyper-EPP comes from two aspects. One is to pump the higher-fidelity hyperentanglement from different three-photon systems into the same three-photon system with fidelity-robust swap gates, the other is to reproduce some hyperentangled three-photon systems from hyperentangled two-photon subsystems based on hyperentanglement link. Moreover, as the infidelity originating from imperfect single-photon scattering can be heralded as a failure by triggering a detector, our hyper-EPP operates faithfully with the present quantum circuits. Furthermore, our hyper-EPP can be directly extended to purify multiple photon systems entangled in one DOF or hyperentangled in multiple DOFs.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2019 (2)

R. Y. Qi, Z. Sun, Z. Lin, P. H. Niu, W. T. Hao, L. Y. Song, Q. Huang, J. C. Gao, L. G. Yin, and G. L. Long, “Implementation and security analysis of practical quantum secure direct communication,” Light: Sci. Appl. 8(1), 22 (2019).
[Crossref]

F. F. Du and Z. R. Shi, “Robust hybrid hyper-controlled-not gates assisted by an input-output process of low-q cavities,” Opt. Express 27(13), 17493–17506 (2019).
[Crossref]

2017 (3)

W. Zhang, D. S. Ding, Y. B. Sheng, L. Zhou, B. S. Shi, and G. C. Guo, “Quantum secure direct communication with quantum memory,” Phys. Rev. Lett. 118(22), 220501 (2017).
[Crossref]

N. Kalb, A. A. Reiserer, P. C. Humphreys, J. J. Bakermans, S. J. Kamerling, N. H. Nickerson, S. C. Benjamin, D. J. Twitchen, M. Markham, and R. Hanson, “Entanglement distillation between solid-state quantum network nodes,” Science 356(6341), 928–932 (2017).
[Crossref]

L. Zhou and Y. B. Sheng, “Polarization entanglement purification for concatenated greenberger-horne-zeilinger state,” Ann. Phys. 385, 10–35 (2017).
[Crossref]

2016 (3)

L. Zhou and Y. B. Sheng, “Purification of logic-qubit entanglement,” Sci. Rep. 6(1), 28813 (2016).
[Crossref]

F. F. Du, T. Li, and G. L. Long, “Refined hyperentanglement purification of two-photon systems for high-capacity quantum communication with cavity-assisted interaction,” Ann. Phys. 375, 105–118 (2016).
[Crossref]

T. Li and G. L. Long, “Hyperparallel optical quantum computation assisted by atomic ensembles embedded in double-sided optical cavities,” Phys. Rev. A 94(2), 022343 (2016).
[Crossref]

2015 (3)

B. C. Ren, G. Y. Wang, and F. G. Deng, “Universal hyperparallel hybrid photonic quantum gates with dipole-induced transparency in the weak-coupling regime,” Phys. Rev. A 91(3), 032328 (2015).
[Crossref]

T. J. Wang, L. L. Liu, R. Zhang, C. Cao, and C. Wang, “One-step hyperentanglement purification and hyperdistillation with linear optics,” Opt. Express 23(7), 9284–9294 (2015).
[Crossref]

Y. Zhang, C. Zeng, H. Zhang, D. P. Li, G. Gao, Q. Z. Huang, Y. Wang, J. Z. Yu, and J. S. Xia, “Single-mode emission from ge quantum dots in photonic crystal nanobeam cavity,” IEEE Photonics Technol. Lett. 27(9), 1026–1029 (2015).
[Crossref]

2014 (1)

I. Shomroni, S. Rosenblum, Y. Lovsky, O. Bechler, G. Guendelman, and B. Dayan, “All-optical routing of single photons by a one-atom switch controlled by a single photon,” Science 345(6199), 903–906 (2014).
[Crossref]

2013 (6)

D. O’Shea, C. Junge, J. Volz, and A. Rauschenbeutel, “Fiber-optical switch controlled by a single atom,” Phys. Rev. Lett. 111(19), 193601 (2013).
[Crossref]

H. Bernien, B. Hensen, W. Pfaff, G. Koolstra, M. S. Blok, L. Robledo, T. Taminiau, M. Markham, D. J. Twitchen, and L. Childress, “Heralded entanglement between solid-state qubits separated by three metres,” Nature 497(7447), 86–90 (2013).
[Crossref]

Y. B. Sheng, L. Zhou, and G. L. Long, “Hybrid entanglement purification for quantum repeaters,” Phys. Rev. A 88(2), 022302 (2013).
[Crossref]

H. R. Wei and F. G. Deng, “Compact quantum gates on electron-spin qubits assisted by diamond nitrogen-vacancy centers inside cavities,” Phys. Rev. A 88(4), 042323 (2013).
[Crossref]

B. C. Ren and F. G. Deng, “Hyperentanglement purification and concentration assisted by diamond nv centers inside photonic crystal cavities,” Laser Phys. Lett. 10(11), 115201 (2013).
[Crossref]

T. Li, B. C. Ren, H. R. Wei, M. Hua, and F. G. Deng, “High-efficiency multipartite entanglement purification of electron-spin states with charge detection,” Quantum Inf. Process. 12(2), 855–876 (2013).
[Crossref]

2012 (1)

2011 (2)

J. R. Maze, A. Gali, E. Togan, Y. Chu, A. Trifonov, E. Kaxiras, and M. D. Lukin, “Properties of nitrogen-vacancy centers in diamond: the group theoretic approach,” New J. Phys. 13(2), 025025 (2011).
[Crossref]

F. G. Deng, “Efficient multipartite entanglement purification with the entanglement link from a subspace,” Phys. Rev. A 84(5), 052312 (2011).
[Crossref]

2010 (3)

E. Togan, Y. Chu, A. S. Trifonov, L. Jiang, J. Maze, L. Childress, M. V. G. Dutt, A. S. Sørensen, P. R. Hemmer, and A. S. Zibrov, “Quantum entanglement between an optical photon and a solid-state spin qubit,” Nature 466(7307), 730–734 (2010).
[Crossref]

Y. B. Sheng and F. G. Deng, “One-step deterministic polarization-entanglement purification using spatial entanglement,” Phys. Rev. A 82(4), 044305 (2010).
[Crossref]

Y. B. Sheng and F. G. Deng, “Deterministic entanglement purification and complete nonlocal bell-state analysis with hyperentanglement,” Phys. Rev. A 81(3), 032307 (2010).
[Crossref]

2009 (5)

C. Y. Hu, W. J. Munro, J. L. O’Brien, and J. G. Rarity, “Proposed entanglement beam splitter using a quantum-dot spin in a double-sided optical microcavity,” Phys. Rev. B 80(20), 205326 (2009).
[Crossref]

Y. B. Sheng, F. G. Deng, B. K. Zhao, T. J. Wang, and H. Y. Zhou, “Multipartite entanglement purification with quantum nondemolition detectors,” Eur. Phys. J. D 55(1), 235–242 (2009).
[Crossref]

R. Ceccarelli, G. Vallone, F. De Martini, P. Mataloni, and A. Cabello, “Experimental entanglement and nonlocality of a two-photon six-qubit cluster state,” Phys. Rev. Lett. 103(16), 160401 (2009).
[Crossref]

G. Vallone, R. Ceccarelli, F. De Martini, and P. Mataloni, “Hyperentanglement of two photons in three degrees of freedom,” Phys. Rev. A 79(3), 030301 (2009).
[Crossref]

G. Fuchs, V. Dobrovitski, D. Toyli, F. Heremans, and D. Awschalom, “Gigahertz dynamics of a strongly driven single quantum spin,” Science 326(5959), 1520–1522 (2009).
[Crossref]

2008 (2)

J. T. Barreiro, T. C. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nat. Phys. 4(4), 282–286 (2008).
[Crossref]

Y. B. Sheng, F. G. Deng, and H. Y. Zhou, “Efficient polarization-entanglement purification based on parametric down-conversion sources with cross-kerr nonlinearity,” Phys. Rev. A 77(4), 042308 (2008).
[Crossref]

2005 (1)

J. T. Barreiro, N. K. Langford, N. A. Peters, and P. G. Kwiat, “Generation of hyperentangled photon pairs,” Phys. Rev. Lett. 95(26), 260501 (2005).
[Crossref]

2002 (3)

G. L. Long and X. S. Liu, “Theoretically efficient high-capacity quantum-key-distribution scheme,” Phys. Rev. A 65(3), 032302 (2002).
[Crossref]

X. S. Liu, G. L. Long, D. M. Tong, and F. Li, “General scheme for superdense coding between multiparties,” Phys. Rev. A 65(2), 022304 (2002).
[Crossref]

C. Simon and J. W. Pan, “Polarization entanglement purification using spatial entanglement,” Phys. Rev. Lett. 89(25), 257901 (2002).
[Crossref]

2001 (2)

J. W. Pan, C. Simon, Č. Brukner, and A. Zeilinger, “Entanglement purification for quantum communication,” Nature 410(6832), 1067–1070 (2001).
[Crossref]

F. Jelezko, C. Tietz, A. Gruber, I. Popa, A. Nizovtsev, S. Kilin, and J. Wrachtrup, “Spectroscopy of single n-v centers in diamond,” Single Mol. 2(4), 255–260 (2001).
[Crossref]

1999 (1)

M. Hillery, V. Bužek, and A. Berthiaume, “Quantum secret sharing,” Phys. Rev. A 59(3), 1829–1834 (1999).
[Crossref]

1998 (2)

H. J. Briegel, W. Dür, J. I. Cirac, and P. Zoller, “Quantum repeaters: the role of imperfect local operations in quantum communication,” Phys. Rev. Lett. 81(26), 5932–5935 (1998).
[Crossref]

M. Murao, M. B. Plenio, S. Popescu, V. Vedral, and P. L. Knight, “Multiparticle entanglement purification protocols,” Phys. Rev. A 57(6), R4075–R4078 (1998).
[Crossref]

1997 (1)

1996 (2)

C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. 76(5), 722–725 (1996).
[Crossref]

D. Deutsch, A. Ekert, R. Jozsa, C. Macchiavello, S. Popescu, and A. Sanpera, “Quantum privacy amplification and the security of quantum cryptography over noisy channels,” Phys. Rev. Lett. 77(13), 2818–2821 (1996).
[Crossref]

1994 (1)

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73(1), 58–61 (1994).
[Crossref]

1992 (2)

C. H. Bennett, G. Brassard, and N. D. Mermin, “Quantum cryptography without bell’s theorem,” Phys. Rev. Lett. 68(5), 557–559 (1992).
[Crossref]

C. H. Bennett and S. J. Wiesner, “Communication via one-and two-particle operators on einstein-podolsky-rosen states,” Phys. Rev. Lett. 69(20), 2881–2884 (1992).
[Crossref]

1991 (1)

A. K. Ekert, “Quantum cryptography based on bell’s theorem,” Phys. Rev. Lett. 67(6), 661–663 (1991).
[Crossref]

Awschalom, D.

G. Fuchs, V. Dobrovitski, D. Toyli, F. Heremans, and D. Awschalom, “Gigahertz dynamics of a strongly driven single quantum spin,” Science 326(5959), 1520–1522 (2009).
[Crossref]

Bakermans, J. J.

N. Kalb, A. A. Reiserer, P. C. Humphreys, J. J. Bakermans, S. J. Kamerling, N. H. Nickerson, S. C. Benjamin, D. J. Twitchen, M. Markham, and R. Hanson, “Entanglement distillation between solid-state quantum network nodes,” Science 356(6341), 928–932 (2017).
[Crossref]

Barreiro, J. T.

J. T. Barreiro, T. C. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nat. Phys. 4(4), 282–286 (2008).
[Crossref]

J. T. Barreiro, N. K. Langford, N. A. Peters, and P. G. Kwiat, “Generation of hyperentangled photon pairs,” Phys. Rev. Lett. 95(26), 260501 (2005).
[Crossref]

Bechler, O.

I. Shomroni, S. Rosenblum, Y. Lovsky, O. Bechler, G. Guendelman, and B. Dayan, “All-optical routing of single photons by a one-atom switch controlled by a single photon,” Science 345(6199), 903–906 (2014).
[Crossref]

Benjamin, S. C.

N. Kalb, A. A. Reiserer, P. C. Humphreys, J. J. Bakermans, S. J. Kamerling, N. H. Nickerson, S. C. Benjamin, D. J. Twitchen, M. Markham, and R. Hanson, “Entanglement distillation between solid-state quantum network nodes,” Science 356(6341), 928–932 (2017).
[Crossref]

Bennett, C. H.

C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. 76(5), 722–725 (1996).
[Crossref]

C. H. Bennett, G. Brassard, and N. D. Mermin, “Quantum cryptography without bell’s theorem,” Phys. Rev. Lett. 68(5), 557–559 (1992).
[Crossref]

C. H. Bennett and S. J. Wiesner, “Communication via one-and two-particle operators on einstein-podolsky-rosen states,” Phys. Rev. Lett. 69(20), 2881–2884 (1992).
[Crossref]

Bernien, H.

H. Bernien, B. Hensen, W. Pfaff, G. Koolstra, M. S. Blok, L. Robledo, T. Taminiau, M. Markham, D. J. Twitchen, and L. Childress, “Heralded entanglement between solid-state qubits separated by three metres,” Nature 497(7447), 86–90 (2013).
[Crossref]

Bernstein, H. J.

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73(1), 58–61 (1994).
[Crossref]

Bertani, P.

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73(1), 58–61 (1994).
[Crossref]

Berthiaume, A.

M. Hillery, V. Bužek, and A. Berthiaume, “Quantum secret sharing,” Phys. Rev. A 59(3), 1829–1834 (1999).
[Crossref]

Blok, M. S.

H. Bernien, B. Hensen, W. Pfaff, G. Koolstra, M. S. Blok, L. Robledo, T. Taminiau, M. Markham, D. J. Twitchen, and L. Childress, “Heralded entanglement between solid-state qubits separated by three metres,” Nature 497(7447), 86–90 (2013).
[Crossref]

Brassard, G.

C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. 76(5), 722–725 (1996).
[Crossref]

C. H. Bennett, G. Brassard, and N. D. Mermin, “Quantum cryptography without bell’s theorem,” Phys. Rev. Lett. 68(5), 557–559 (1992).
[Crossref]

Briegel, H. J.

H. J. Briegel, W. Dür, J. I. Cirac, and P. Zoller, “Quantum repeaters: the role of imperfect local operations in quantum communication,” Phys. Rev. Lett. 81(26), 5932–5935 (1998).
[Crossref]

Brukner, C.

J. W. Pan, C. Simon, Č. Brukner, and A. Zeilinger, “Entanglement purification for quantum communication,” Nature 410(6832), 1067–1070 (2001).
[Crossref]

Bužek, V.

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L. Zhou and Y. B. Sheng, “Polarization entanglement purification for concatenated greenberger-horne-zeilinger state,” Ann. Phys. 385, 10–35 (2017).
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Appl. Opt. (1)

Eur. Phys. J. D (1)

Y. B. Sheng, F. G. Deng, B. K. Zhao, T. J. Wang, and H. Y. Zhou, “Multipartite entanglement purification with quantum nondemolition detectors,” Eur. Phys. J. D 55(1), 235–242 (2009).
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IEEE Photonics Technol. Lett. (1)

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Laser Phys. Lett. (1)

B. C. Ren and F. G. Deng, “Hyperentanglement purification and concentration assisted by diamond nv centers inside photonic crystal cavities,” Laser Phys. Lett. 10(11), 115201 (2013).
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Light: Sci. Appl. (1)

R. Y. Qi, Z. Sun, Z. Lin, P. H. Niu, W. T. Hao, L. Y. Song, Q. Huang, J. C. Gao, L. G. Yin, and G. L. Long, “Implementation and security analysis of practical quantum secure direct communication,” Light: Sci. Appl. 8(1), 22 (2019).
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Nat. Phys. (1)

J. T. Barreiro, T. C. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nat. Phys. 4(4), 282–286 (2008).
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Nature (3)

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E. Togan, Y. Chu, A. S. Trifonov, L. Jiang, J. Maze, L. Childress, M. V. G. Dutt, A. S. Sørensen, P. R. Hemmer, and A. S. Zibrov, “Quantum entanglement between an optical photon and a solid-state spin qubit,” Nature 466(7307), 730–734 (2010).
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New J. Phys. (1)

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Opt. Express (3)

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T. Li and G. L. Long, “Hyperparallel optical quantum computation assisted by atomic ensembles embedded in double-sided optical cavities,” Phys. Rev. A 94(2), 022343 (2016).
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Y. B. Sheng, L. Zhou, and G. L. Long, “Hybrid entanglement purification for quantum repeaters,” Phys. Rev. A 88(2), 022302 (2013).
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Y. B. Sheng, F. G. Deng, and H. Y. Zhou, “Efficient polarization-entanglement purification based on parametric down-conversion sources with cross-kerr nonlinearity,” Phys. Rev. A 77(4), 042308 (2008).
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Phys. Rev. B (1)

C. Y. Hu, W. J. Munro, J. L. O’Brien, and J. G. Rarity, “Proposed entanglement beam splitter using a quantum-dot spin in a double-sided optical microcavity,” Phys. Rev. B 80(20), 205326 (2009).
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Phys. Rev. Lett. (12)

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C. Simon and J. W. Pan, “Polarization entanglement purification using spatial entanglement,” Phys. Rev. Lett. 89(25), 257901 (2002).
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W. Zhang, D. S. Ding, Y. B. Sheng, L. Zhou, B. S. Shi, and G. C. Guo, “Quantum secure direct communication with quantum memory,” Phys. Rev. Lett. 118(22), 220501 (2017).
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C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. 76(5), 722–725 (1996).
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[Crossref]

H. J. Briegel, W. Dür, J. I. Cirac, and P. Zoller, “Quantum repeaters: the role of imperfect local operations in quantum communication,” Phys. Rev. Lett. 81(26), 5932–5935 (1998).
[Crossref]

Quantum Inf. Process. (1)

T. Li, B. C. Ren, H. R. Wei, M. Hua, and F. G. Deng, “High-efficiency multipartite entanglement purification of electron-spin states with charge detection,” Quantum Inf. Process. 12(2), 855–876 (2013).
[Crossref]

Sci. Rep. (1)

L. Zhou and Y. B. Sheng, “Purification of logic-qubit entanglement,” Sci. Rep. 6(1), 28813 (2016).
[Crossref]

Science (3)

N. Kalb, A. A. Reiserer, P. C. Humphreys, J. J. Bakermans, S. J. Kamerling, N. H. Nickerson, S. C. Benjamin, D. J. Twitchen, M. Markham, and R. Hanson, “Entanglement distillation between solid-state quantum network nodes,” Science 356(6341), 928–932 (2017).
[Crossref]

I. Shomroni, S. Rosenblum, Y. Lovsky, O. Bechler, G. Guendelman, and B. Dayan, “All-optical routing of single photons by a one-atom switch controlled by a single photon,” Science 345(6199), 903–906 (2014).
[Crossref]

G. Fuchs, V. Dobrovitski, D. Toyli, F. Heremans, and D. Awschalom, “Gigahertz dynamics of a strongly driven single quantum spin,” Science 326(5959), 1520–1522 (2009).
[Crossref]

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F. Jelezko, C. Tietz, A. Gruber, I. Popa, A. Nizovtsev, S. Kilin, and J. Wrachtrup, “Spectroscopy of single n-v centers in diamond,” Single Mol. 2(4), 255–260 (2001).
[Crossref]

Other (1)

M. O. Scully and M. S. Zubairy, “Quantum optics,” (1999).

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Figures (7)

Fig. 1.
Fig. 1. (a) Schematic diagrams of an NV-center-cavity system, and the optical transitions of the NV center with the circularly polarized photons. $R^{\uparrow } (R^{\downarrow })$ and $L^{\uparrow } (L^{\downarrow })$ represent the right- and left-circularly polarized photons propagating along (against) the quantization axis $z$, respectively. (b) Schematic diagram of a modified NV union. SW is an optical switch, which makes the photons entering into and going out of the circuit unit in sequence. M is a mirror and D is a single-photon detector. BS is a $50:50$ beam splitter, which performs the Hadamard operation, that is, $|i_{1}\rangle \rightarrow (|j_{1}\rangle +|j_{2}\rangle )/\sqrt {2}$, or $|i_{2}\rangle \rightarrow (|j_{1}\rangle -|j_{2}\rangle )/\sqrt {2}$, in the spatial-mode DOF of one photon. H represents a quarter-wave plate, which performs the Hadamard operation, that is, $|R\rangle \rightarrow (|R\rangle +|L\rangle )/\sqrt {2}$, or $|L\rangle \rightarrow (|R\rangle -|L\rangle )/\sqrt {2}$, in the polarization DOF of one photon.
Fig. 2.
Fig. 2. Schematic diagram of the fidelity-robust spatial-polarization parity-check gate (S-P-PCG) for a two-photon system. CPBS is a circularly polarizing beam splitter, which reflects the left-circular-polarization photon $\vert L\rangle$ and transmits the right-circular-polarization photon $\vert R\rangle$, respectively. X is a half-wave plate, which performs a bit-flip operation on the polarization DOF of the photon. T is a partially transmitting mirror.
Fig. 3.
Fig. 3. (a) Schematic diagram of the fidelity-robust spatial-spatial-swap (S-S swap ) gate. (b) Schematic diagram of the fidelity-robust polarization-polarization-swap (P-P-swap) gate. The red quantum circuit represent that the photons will enter again for the second round.
Fig. 4.
Fig. 4. (a) Schematic diagram of the first step of our hyper-EPP with S-P-PCGs. (b) Schematic diagram of a single-photon measurement device (SPMD).
Fig. 5.
Fig. 5. (a) Schematic diagram of the second step of our hyper-EPP with P-P-swap gates. (b) Schematic diagram of the second step of our hyper-EPP with hyperentanglement link (HL).
Fig. 6.
Fig. 6. (a) The efficiency $\eta _{1}$ and (b) The efficiency $\eta _{2}$ versus the initial spatial-mode fidelity $g_{0}$ and polarization fidelity $f_{0}$, respectively.
Fig. 7.
Fig. 7. (a) The efficiency $\eta _{p}$ of our fidelity-robust S-PCG (or P-PCG) vs the Purcell factor $F_{ P}$ and the cavity decay rate $\lambda$ with the condition $\omega =\omega _{c}=\omega _{X^{-}}$. (b) The efficiency $\eta _{s}$ of our fidelity-robust S-S-swap (or P-P-swap) gate vs the same condition.

Tables (2)

Tables Icon

Table 1. The states of the two-photon systems obtained from cross-combinations and their probabilities (suppose $x=A_{1}B_{1}C_{1}$ and $y=A_{2}B_{2}C_{2}$ for simplification).

Tables Icon

Table 2. The states of the three-photon hyperentangled systems obtained from two two-photon systems and their probabilities with HL (suppose $x=A_{1}B_{1}$ , $y=A_{2}C_{1})$.

Equations (16)

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d a ^ d t = [ i ( ω c ω ) + μ + κ 2 ] a ^ g σ ^ μ ( a ^ i n + a ^ i n ) , d σ ^ d t = [ i ( ω k ω ) + γ 2 ] σ ^ g σ ^ z a ^ , a ^ o u t = a ^ i n + μ a ^ , a ^ o u t = a ^ i n + μ a ^ .
t q ( ω ) = μ [ i ( ω k ω ) + γ 2 ] [ i ( ω k ω ) + γ 2 ] [ i ( ω c ω ) + μ + κ 2 ] + q g 2 , r q ( ω ) = 1 + t q ( ω ) .
| R ( L ) , 1 r ( ω ) | L ( R ) , 1 + t ( ω ) | R ( L ) , 1 , | R ( L ) , 1 r 0 ( ω ) | L ( R ) , 1 + t 0 ( ω ) | R ( L ) , 1 , | R ( L ) , + 1 r 0 ( ω ) | L ( R ) , + 1 + t 0 ( ω ) | R ( L ) , + 1 , | R ( L ) , + 1 r ( ω ) | L ( R ) , + 1 + t ( ω ) | R ( L ) , + 1 .
| R , i 1 | φ + P 1 | R , i 1 | φ + + P 2 | L , i 2 | φ , | R , i 1 | φ P 1 | R , i 1 | φ + P 2 | L , i 2 | φ + ,
| ϕ 1 ± S | ϕ 1 ± P | φ 1 + | φ 2 + S-P-PCG ( P 2 2 | ϕ 1 ± S | φ 1 + ) ( P 2 2 | ϕ 1 ± P | φ 2 + ) , | ϕ 2 ± S | ϕ 1 ± P | φ 1 + | φ 2 + S-P-PCG ( P 2 2 | ϕ 1 ± S | φ 1 ) ( P 2 2 | ϕ 1 ± P | φ 2 + ) , | ϕ 1 ± S | ϕ 2 ± P | φ 1 + | φ 2 + S-P-PCG ( P 2 2 | ϕ 1 ± S | φ 1 + ) ( P 2 2 | ϕ 1 ± P | φ 2 ) , | ϕ 2 ± S | ϕ 2 ± P | φ 1 + | φ 2 + S-P-PCG ( P 2 2 | ϕ 1 ± S | φ 1 ) ( P 2 2 | ϕ 1 ± P | φ 2 ) .
| ϕ A = | ϕ A S | ϕ A P = ( sin α 1 | a 1 + cos α 1 | a 2 ) ( sin α 2 | R + cos α 2 | L ) , | ϕ B = | ϕ B S | ϕ B P = ( sin β 1 | b 1 + cos β 1 | b 2 ) ( sin β 2 | R + cos β 2 | L ) .
| Φ = P 2 4 2 [ ( sin β 1 | a 1 + cos β 1 | a 2 ) ( sin α 1 | b 1 + cos α 1 | b 2 ) | + 1 + ( sin β 1 | a 1 cos β 1 | a 2 ) ( sin α 1 | b 1 cos α 1 | b 2 ) | 1 ] | ϕ A P | ϕ B P .
| Φ s w a p = P 2 4 ( sin β 1 | a 1 + cos β 1 | a 2 ) ( sin α 1 | b 1 + cos α 1 | b 2 ) | ϕ A P | ϕ B P .
| Φ s w a p = P 2 4 ( sin β 2 | R + cos β 2 | L ) ( sin α 2 | R + cos α 2 | L ) | ϕ A S | ϕ B S .
| ψ 0 ± A B C S = 1 2 ( | a 1 b 1 c 1 ± | a 2 b 2 c 2 ) A B C , | ψ 1 ± A B C S = 1 2 ( | a 2 b 1 c 1 ± | a 1 b 2 c 2 ) A B C , | ψ 2 ± A B C S = 1 2 ( | a 1 b 2 c 1 ± | a 2 b 1 c 2 ) A B C , | ψ 3 ± A B C S = 1 2 ( | a 1 b 1 c 2 ± | a 2 b 2 c 1 ) A B C , | ψ 0 ± A B C P = 1 2 ( | R R R ± | L L L ) A B C , | ψ 1 ± A B C P = 1 2 ( | L R R ± | R L L ) A B C , | ψ 2 ± A B C P = 1 2 ( | R L R ± | L R L ) A B C , | ψ 3 ± A B C P = 1 2 ( | R R L ± | L L R ) A B C .
ρ A B C = [ g 0 | ψ 0 + S ψ 0 + | + g 1 | ψ 1 + S ψ 1 + | + g 2 | ψ 2 + S ψ 2 + | + g 3 | ψ 3 + S ψ 3 + | ] A B C [ f 0 | ψ 0 + P ψ 0 + | + f 1 | ψ 1 + P ψ 1 + | + f 2 | ψ 2 + P ψ 2 + | + f 3 | ψ 3 + P ψ 3 + | ] A B C ,
| Φ 0 S = 1 2 ( | a 1 b 1 c 1 a ¯ 1 b ¯ 1 c ¯ 1 + | a 2 b 2 c 2 a ¯ 2 b ¯ 2 c ¯ 2 ) A 1 B 1 C 1 A 2 B 2 C 2 , | Φ 1 S = 1 2 ( | a 2 b 1 c 1 a ¯ 2 b ¯ 1 c ¯ 1 + | a 1 b 2 c 2 a ¯ 1 b ¯ 2 c ¯ 2 ) A 1 B 1 C 1 A 2 B 2 C 2 , | Φ 2 S = 1 2 ( | a 1 b 2 c 1 a ¯ 1 b ¯ 2 c ¯ 1 + | a 2 b 1 c 2 a ¯ 2 b ¯ 1 c ¯ 2 ) A 1 B 1 C 1 A 2 B 2 C 2 , | Φ 3 S = 1 2 ( | a 1 b 1 c 2 a ¯ 1 b ¯ 1 c ¯ 2 + | a 2 b 2 c 1 a ¯ 2 b ¯ 2 c ¯ 1 ) A 1 B 1 C 1 A 2 B 2 C 2 , | Φ 0 P = 1 2 ( | R R R R R R + | L L L L L L ) A 1 B 1 C 1 A 2 B 2 C 2 , | Φ 1 P = 1 2 ( | L R R L R R + | R L L R L L ) A 1 B 1 C 1 A 2 B 2 C 2 , | Φ 2 P = 1 2 ( | R L R R L R + | L R L L R L ) A 1 B 1 C 1 A 2 B 2 C 2 , | Φ 3 P = 1 2 ( | R R L R R L + | L L R L L R ) A 1 B 1 C 1 A 2 B 2 C 2 ,
ρ A 1 B 1 = [ g 0 g 3 | ϕ 1 + S ϕ 1 + | + g 1 g 2 | ϕ 2 + S ϕ 2 + | ] [ f 0 f 3 | ϕ 1 + P ϕ 1 + | + f 1 f 2 | ϕ 2 + P ϕ 2 + | ] , ρ B 1 C 1 = [ g 0 g 1 | ϕ 1 + S ϕ 1 + | + g 2 g 3 | ϕ 2 + S ϕ 2 + | ] [ f 0 f 1 | ϕ 1 + P ϕ 1 + | + f 2 f 3 | ϕ 2 + P ϕ 2 + | ] , ρ A 1 C 1 = [ g 0 g 2 | ϕ 1 + S ϕ 1 + | + g 1 g 3 | ϕ 2 + S ϕ 2 + | ] [ f 0 f 2 | ϕ 1 + P ϕ 1 + | + f 1 f 3 | ϕ 2 + P ϕ 2 + | ] .
ρ A 1 B 1 ( A 2 C 1 ) = [ g 0 b | ϕ 1 + S ϕ 1 + | + g 1 b | ϕ 2 + S ϕ 2 + | ] [ f 0 b | ϕ 1 + P ϕ 1 + | + f 1 b | ϕ 2 + P ϕ 2 + | ] ,
ρ A 1 B 1 C 1 = [ g 0 t | ψ 0 + S ψ 0 + | + g 1 t | ψ 1 + S ψ 1 + | + g 2 t | ψ 2 + S ψ 2 + | + g 3 t | ψ 3 + S ψ 3 + | ] [ f 0 t | ψ 0 + P ψ 0 + | + f 1 t | ψ 1 + P ψ 1 + | + f 2 t | ψ 2 + P ψ 2 + | + f 3 t | ψ 3 + P ψ 3 + | ] ,
η 1 = l = q = 0 3 g l g q m = n = 0 3 f m f n = ( 1 2 g 0 + 4 g 0 2 ) ( 1 2 f 0 + 4 f 0 2 ) 9 , η S = m i n { 2 l = q = 0 3 g l g q m n = 0 3 f m f n ,   2 l q = 0 3 g l g q m = n = 0 3 f m f n } , η H = 1 4 l q = 0 3 g l g q m n = 0 3 f m f n , η 2 = η 1 + η S + η H = ( 1 + g 0 2 g 0 2 ) ( 1 + f 0 2 f 0 2 ) 9 + m i n { 1 2 g 0 + 4 g 0 2 3 , 1 2 f 0 + 4 f 0 2 3 } ,

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