Abstract

As the interaction between the photons and the environment which will make the entangled photon pairs in less entangled states or even in mixed states, the security and the efficiency of quantum communication will decrease. We present an efficient hyperentanglement purification protocol that distills nonlocal high-fidelity hyper-entangled Bell states in both polarization and spatial-mode degrees of freedom from ensembles of two-photon system in mixed states using linear optics. Here, we consider the influence of the photon loss in the channel which generally is ignored in the conventional entanglement purification and hyperentanglement purification (HEP) schemes. Compared with previous HEP schemes, our HEP scheme decreases the requirement for nonlocal resources by employing high-dimensional mode-check measurement, and leads to a higher fidelity, especially in the range where the conventional HEP schemes become invalid but our scheme still can work.

© 2017 Optical Society of America

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References

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2016 (2)

G.-Y. Wang, Q. Liu, and F.-G. Deng, “Hyperentanglement purification for two-photon six-qubit quantum systems,” Phys. Rev. A 94, 032319 (2016).
[Crossref]

K. Li, F.-Z. Kong, M. Yang, Q. Yang, and Z.-L. Cao, “Qubit-loss-free fusion of W states,” Phys. Rev. A 94, 062315 (2016).
[Crossref]

2015 (3)

2014 (2)

T.-J. Wang, C. Cao, and C. Wang, “Linear-optical implementation of hyper-distillation from photon loss,” Phys. Rev. A 89, 052303 (2014).
[Crossref]

I. M. Mirza and J. C. Schotland, “Multiqubit entanglement in bidirectional-chiral-waveguide QED,” Phys. Rev. A 94, 012302 (2014).
[Crossref]

2013 (8)

I. M. Mirza, S. J. van Enk, and H. J. Kimble, “Single-photon time-dependent spectra in coupled cavity arrays,” JOSA B 10, 2640 (2013).
[Crossref]

B. Calkins, P. L. Mennea, A. E. Lita, B. J. Metcalf, W. S. Kolthammer, A. Lamas-Lineras, J. B. Spring, P. C. Humphreys, R. P. Mirin, J. C. Gates, P. G. R. Smith, I. A. Walmsley, T. Gerrits, and S. W. Nam, “High quantum-efficiency photon-number-resolving detector for photonic on-chip information processing,” Opt. Express 21, 22657 (2013).
[Crossref] [PubMed]

E. Meyer-Scott, M. Bula, K. Bartkiewicz, A. Černoch, J. Soubusta, T. Jennewein, and K. Lemr, “Entanglementbased linear-optical qubit amplifier,” Phys. Rev. A 88, 012327 (2013).
[Crossref]

K. Bartkiewicz, A. Černoch, and K. Lemr, “State-dependent linear-optical qubit amplifier,” Phys. Rev. A 88, 062304 (2013).
[Crossref]

B. C. Ren, F. -F. Du, and F. G. Deng, “Hyperentanglement concentration for two-photon four-qubit systems with linear optics,” Phys. Rev. A 88, 012302 (2013).
[Crossref]

E. Meyer-Scott, M. Bula, K. Bartkiewicz, A. Cernoch, J. Soubusta, T. Jennewein, and K. Lemr, “Entanglement-based linear-optical qubit amplifier,” Phys. Rev. A 88, 012327(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, 115201 (2013).
[Crossref]

S. Bugu, C. Yesilyurt, and F. Ozaydin, “Enhancing the w-state quantum-network-fusion process with a single fredkin gate,” Phys. Rev. A 87, 032331 (2013).
[Crossref]

2012 (2)

P. B. Dixon, G. A. Howland, J. Schneeloch, and J. C. Howell, “Quantum mutual information capacity for high-dimensional entangled states,” Phys. Rev. Lett. 108, 143603 (2012).
[Crossref] [PubMed]

B.-C. Ren, H.-R. Wei, M. Hua, T. Li, and F.-G. Deng, ”Complete hyperentangled-Bell-state analysis for photon systems assisted by quantum-dot spins in optical microcavities,” Opt. Express 20, 24664 (2012).
[Crossref] [PubMed]

2010 (2)

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

X. H. Li, “Deterministic polarization-entanglement purification using spatial entanglement,” Phys. Rev. A 82, 044304 (2010).
[Crossref]

2009 (3)

M. M. Wilde and D. B. Uskov, “Linear-optical hyperentanglement-assisted quantum error-correcting code,” Phys. Rev. A 79, 022305 (2009).
[Crossref]

T. Tashima, T. Wakatsuki, S. K. Özdemir, T. Yamamoto, M. Koashi, and N. Imoto, ”Local transformation of two einstein-podolsky-rosen photon pairs into a three-photon w state,” Phys. Rev. Lett. 102, 130502 (2009).
[Crossref] [PubMed]

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

2008 (2)

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, 042308 (2008).
[Crossref]

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

2007 (1)

M. Barbieri, G. Vallone, P. Mataloni, and F. De Martini, “Complete and deterministic discrimination of polarization Bell states assisted by momentum entanglement,” Phys. Rev. A 75, 042317 (2007).
[Crossref]

2006 (1)

C. Schuck, G. Huber, C. Kurtsiefer, and H. Weinfurter, “Complete Deterministic Linear Optics Bell State Analysis,” Phys. Rev. Lett. 96, 190501 (2006).
[Crossref] [PubMed]

2005 (1)

C. Wang, F. G. Deng, Y. S. Li, X. S. Liu, and G. L. Long, “Quantum secure direct communication with high-dimension quantum superdense coding,” Phys. Rev. A 71, 044305 (2005).
[Crossref]

2004 (1)

A. Yabushita and T. Kobayashi, “Spectroscopy by frequency-entangled photon pairs,” Phys. Rev. A 69, 013806 (2004).
[Crossref]

2003 (1)

J. W. Pan, S. Gasparoni, R. Ursin, G. Weihs, and A. Zeilinger, “Experimental entanglement purification of arbitrary unknown states,” Nature 423, 417 (2003).
[Crossref] [PubMed]

2002 (5)

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

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

S. Massar, “Nonlocality, closing the detection loophole, and communication complexity,” Phys. Rev. A 65, 032121 (2002).
[Crossref]

D. Bruss and C. Macchiavello, “Optimal eavesdropping in cryptography with three-dimensional quantum states,” Phys. Rev. Lett. 88, 127901 (2002).
[Crossref] [PubMed]

N. J. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, “Security of quantum key distribution using d-level systems,” Phys. Rev. Lett. 88, 127902 (2002).
[Crossref] [PubMed]

2001 (1)

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

1997 (1)

D. Bouwmeester, J. W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature 390, 575–579 (1997).
[Crossref]

1996 (1)

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, 722–725 (1996).
[Crossref] [PubMed]

1992 (2)

C. H. Bennett, G. Brassard, and N. D. Mermin, “Quantum cryptography without Bell’s theorem,” Phys. Rev. Lett. 68, 557–560 (1992).
[Crossref] [PubMed]

C. H. Bennett and S. J. Wiesner, “Communication via one- and two-particle operators on Einstein-Podolsky- Rosen states,¹œ Phys,” Phys. Rev. Lett. 69, 2881–2884 (1992).
[Crossref] [PubMed]

1991 (1)

A. K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett. 67, 661–664 (1991).
[Crossref] [PubMed]

1987 (1)

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044 (1987).
[Crossref] [PubMed]

Afzelius, M.

Barbieri, M.

M. Barbieri, G. Vallone, P. Mataloni, and F. De Martini, “Complete and deterministic discrimination of polarization Bell states assisted by momentum entanglement,” Phys. Rev. A 75, 042317 (2007).
[Crossref]

Barreiro, J. T.

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

Bartkiewicz, K.

E. Meyer-Scott, M. Bula, K. Bartkiewicz, A. Cernoch, J. Soubusta, T. Jennewein, and K. Lemr, “Entanglement-based linear-optical qubit amplifier,” Phys. Rev. A 88, 012327(2013).
[Crossref]

E. Meyer-Scott, M. Bula, K. Bartkiewicz, A. Černoch, J. Soubusta, T. Jennewein, and K. Lemr, “Entanglementbased linear-optical qubit amplifier,” Phys. Rev. A 88, 012327 (2013).
[Crossref]

K. Bartkiewicz, A. Černoch, and K. Lemr, “State-dependent linear-optical qubit amplifier,” Phys. Rev. A 88, 062304 (2013).
[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, 722–725 (1996).
[Crossref] [PubMed]

C. H. Bennett, G. Brassard, and N. D. Mermin, “Quantum cryptography without Bell’s theorem,” Phys. Rev. Lett. 68, 557–560 (1992).
[Crossref] [PubMed]

C. H. Bennett and S. J. Wiesner, “Communication via one- and two-particle operators on Einstein-Podolsky- Rosen states,¹œ Phys,” Phys. Rev. Lett. 69, 2881–2884 (1992).
[Crossref] [PubMed]

Bourennane, M.

N. J. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, “Security of quantum key distribution using d-level systems,” Phys. Rev. Lett. 88, 127902 (2002).
[Crossref] [PubMed]

Bouwmeester, D.

D. Bouwmeester, J. W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature 390, 575–579 (1997).
[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, 722–725 (1996).
[Crossref] [PubMed]

C. H. Bennett, G. Brassard, and N. D. Mermin, “Quantum cryptography without Bell’s theorem,” Phys. Rev. Lett. 68, 557–560 (1992).
[Crossref] [PubMed]

Brukner, C.

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

Bruss, D.

D. Bruss and C. Macchiavello, “Optimal eavesdropping in cryptography with three-dimensional quantum states,” Phys. Rev. Lett. 88, 127901 (2002).
[Crossref] [PubMed]

Bugu, S.

S. Bugu, C. Yesilyurt, and F. Ozaydin, “Enhancing the w-state quantum-network-fusion process with a single fredkin gate,” Phys. Rev. A 87, 032331 (2013).
[Crossref]

Bula, M.

E. Meyer-Scott, M. Bula, K. Bartkiewicz, A. Černoch, J. Soubusta, T. Jennewein, and K. Lemr, “Entanglementbased linear-optical qubit amplifier,” Phys. Rev. A 88, 012327 (2013).
[Crossref]

E. Meyer-Scott, M. Bula, K. Bartkiewicz, A. Cernoch, J. Soubusta, T. Jennewein, and K. Lemr, “Entanglement-based linear-optical qubit amplifier,” Phys. Rev. A 88, 012327(2013).
[Crossref]

Bussières, F.

Calkins, B.

Cao, C.

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

T.-J. Wang, C. Cao, and C. Wang, “Linear-optical implementation of hyper-distillation from photon loss,” Phys. Rev. A 89, 052303 (2014).
[Crossref]

Cao, Z.-L.

K. Li, F.-Z. Kong, M. Yang, Q. Yang, and Z.-L. Cao, “Qubit-loss-free fusion of W states,” Phys. Rev. A 94, 062315 (2016).
[Crossref]

Ceccarelli, R.

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

Cerf, N. J.

N. J. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, “Security of quantum key distribution using d-level systems,” Phys. Rev. Lett. 88, 127902 (2002).
[Crossref] [PubMed]

Cernoch, A.

K. Bartkiewicz, A. Černoch, and K. Lemr, “State-dependent linear-optical qubit amplifier,” Phys. Rev. A 88, 062304 (2013).
[Crossref]

E. Meyer-Scott, M. Bula, K. Bartkiewicz, A. Černoch, J. Soubusta, T. Jennewein, and K. Lemr, “Entanglementbased linear-optical qubit amplifier,” Phys. Rev. A 88, 012327 (2013).
[Crossref]

E. Meyer-Scott, M. Bula, K. Bartkiewicz, A. Cernoch, J. Soubusta, T. Jennewein, and K. Lemr, “Entanglement-based linear-optical qubit amplifier,” Phys. Rev. A 88, 012327(2013).
[Crossref]

Chuang, I. L.

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2000).

De Martini, F.

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

M. Barbieri, G. Vallone, P. Mataloni, and F. De Martini, “Complete and deterministic discrimination of polarization Bell states assisted by momentum entanglement,” Phys. Rev. A 75, 042317 (2007).
[Crossref]

Deng, F. G.

B. C. Ren, F. -F. Du, and F. G. Deng, “Hyperentanglement concentration for two-photon four-qubit systems with linear optics,” Phys. Rev. A 88, 012302 (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, 115201 (2013).
[Crossref]

Y. B. Sheng and F. G. Deng, “Deterministic entanglement purification and complete nonlocal Bell-state analysis with hyperentanglement,” Phys. Rev. A 81, 032307 (2010).
[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, 042308 (2008).
[Crossref]

C. Wang, F. G. Deng, Y. S. Li, X. S. Liu, and G. L. Long, “Quantum secure direct communication with high-dimension quantum superdense coding,” Phys. Rev. A 71, 044305 (2005).
[Crossref]

Deng, F.-G.

Dixon, P. B.

P. B. Dixon, G. A. Howland, J. Schneeloch, and J. C. Howell, “Quantum mutual information capacity for high-dimensional entangled states,” Phys. Rev. Lett. 108, 143603 (2012).
[Crossref] [PubMed]

Du, F. -F.

B. C. Ren, F. -F. Du, and F. G. Deng, “Hyperentanglement concentration for two-photon four-qubit systems with linear optics,” Phys. Rev. A 88, 012302 (2013).
[Crossref]

Eibl, M.

D. Bouwmeester, J. W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature 390, 575–579 (1997).
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C. Wang, F. G. Deng, Y. S. Li, X. S. Liu, and G. L. Long, “Quantum secure direct communication with high-dimension quantum superdense coding,” Phys. Rev. A 71, 044305 (2005).
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G.-Y. Wang, Q. Liu, and F.-G. Deng, “Hyperentanglement purification for two-photon six-qubit quantum systems,” Phys. Rev. A 94, 032319 (2016).
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T.-J. Wang, L.-L. Liu, R. Zhang, C. Cao, and C. Wang, “One-step hyperentanglement purification and hyperdistillation with linear optics,” Optics Express 23,9284 (2015).
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J. W. Pan, S. Gasparoni, R. Ursin, G. Weihs, and A. Zeilinger, “Experimental entanglement purification of arbitrary unknown states,” Nature 423, 417 (2003).
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C. Schuck, G. Huber, C. Kurtsiefer, and H. Weinfurter, “Complete Deterministic Linear Optics Bell State Analysis,” Phys. Rev. Lett. 96, 190501 (2006).
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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, 722–725 (1996).
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K. Li, F.-Z. Kong, M. Yang, Q. Yang, and Z.-L. Cao, “Qubit-loss-free fusion of W states,” Phys. Rev. A 94, 062315 (2016).
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Yang, Q.

K. Li, F.-Z. Kong, M. Yang, Q. Yang, and Z.-L. Cao, “Qubit-loss-free fusion of W states,” Phys. Rev. A 94, 062315 (2016).
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S. Bugu, C. Yesilyurt, and F. Ozaydin, “Enhancing the w-state quantum-network-fusion process with a single fredkin gate,” Phys. Rev. A 87, 032331 (2013).
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Zeilinger, A.

J. W. Pan, S. Gasparoni, R. Ursin, G. Weihs, and A. Zeilinger, “Experimental entanglement purification of arbitrary unknown states,” Nature 423, 417 (2003).
[Crossref] [PubMed]

D. Bouwmeester, J. W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature 390, 575–579 (1997).
[Crossref]

Zellinger, A.

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

Zhang, R.

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

Zhou, H. Y.

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, 042308 (2008).
[Crossref]

JOSA B (1)

I. M. Mirza, S. J. van Enk, and H. J. Kimble, “Single-photon time-dependent spectra in coupled cavity arrays,” JOSA B 10, 2640 (2013).
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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, 115201 (2013).
[Crossref]

Nature (3)

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

D. Bouwmeester, J. W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature 390, 575–579 (1997).
[Crossref]

J. W. Pan, S. Gasparoni, R. Ursin, G. Weihs, and A. Zeilinger, “Experimental entanglement purification of arbitrary unknown states,” Nature 423, 417 (2003).
[Crossref] [PubMed]

Nature Phys. (1)

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

Opt. Express (3)

Optica (1)

Optics Express (1)

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

Phys. Rev A (1)

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

Phys. Rev. A (18)

C. Wang, F. G. Deng, Y. S. Li, X. S. Liu, and G. L. Long, “Quantum secure direct communication with high-dimension quantum superdense coding,” Phys. Rev. A 71, 044305 (2005).
[Crossref]

S. Massar, “Nonlocality, closing the detection loophole, and communication complexity,” Phys. Rev. A 65, 032121 (2002).
[Crossref]

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

M. Barbieri, G. Vallone, P. Mataloni, and F. De Martini, “Complete and deterministic discrimination of polarization Bell states assisted by momentum entanglement,” Phys. Rev. A 75, 042317 (2007).
[Crossref]

A. Yabushita and T. Kobayashi, “Spectroscopy by frequency-entangled photon pairs,” Phys. Rev. A 69, 013806 (2004).
[Crossref]

M. M. Wilde and D. B. Uskov, “Linear-optical hyperentanglement-assisted quantum error-correcting code,” Phys. Rev. A 79, 022305 (2009).
[Crossref]

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

X. H. Li, “Deterministic polarization-entanglement purification using spatial entanglement,” Phys. Rev. A 82, 044304 (2010).
[Crossref]

B. C. Ren, F. -F. Du, and F. G. Deng, “Hyperentanglement concentration for two-photon four-qubit systems with linear optics,” Phys. Rev. A 88, 012302 (2013).
[Crossref]

E. Meyer-Scott, M. Bula, K. Bartkiewicz, A. Cernoch, J. Soubusta, T. Jennewein, and K. Lemr, “Entanglement-based linear-optical qubit amplifier,” Phys. Rev. A 88, 012327(2013).
[Crossref]

T.-J. Wang, C. Cao, and C. Wang, “Linear-optical implementation of hyper-distillation from photon loss,” Phys. Rev. A 89, 052303 (2014).
[Crossref]

E. Meyer-Scott, M. Bula, K. Bartkiewicz, A. Černoch, J. Soubusta, T. Jennewein, and K. Lemr, “Entanglementbased linear-optical qubit amplifier,” Phys. Rev. A 88, 012327 (2013).
[Crossref]

K. Bartkiewicz, A. Černoch, and K. Lemr, “State-dependent linear-optical qubit amplifier,” Phys. Rev. A 88, 062304 (2013).
[Crossref]

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Figures (6)

Fig. 1
Fig. 1 The schematic diagram of the 4-dimensional mode-checking operation. S-part is used to perform non-demolition parity-check in spatial-mode DOF [32], and P-part is used to perform parity-check on polarization qubits [21, 33].
Fig. 2
Fig. 2 The 4-dimensional flip operation device which can perform a |0〉 → |3〉 →|2〉 →|1〉 →|0〉 operation a photon. The half-wave plate (HWP) which perform the flip operation in the polarization DOF (|H〉 ↔ |V〉).
Fig. 3
Fig. 3 The fidelity Four (a) for our scheme and the fidelity Fcon (b) for the conventional HEP operation in the case α β = 1. In order to clearly illustrate the advantage of our scheme, we also numerically simulated the value of the formulas (c) F o u r F 0 and (d) F c o n F 0 .
Fig. 4
Fig. 4 The fidelity Four (a) for our scheme and the fidelity Fcon (b) for the conventional HEP operation in the case α β = 3.16. In order to clearly illustrate the advantage of our scheme, we also numerically simulated the value of the formulas FourFcon in the cases α β = 1 (c) and α β = 3.16 (d).
Fig. 5
Fig. 5 The distillation ranges for the conventional HEP schemes (blue dashed lines area) and our scheme (purple area where contains blue dashed part) in the case of α β = 1 (a) and α β = 3.16 (b).
Fig. 6
Fig. 6 The fidelities of our HEP schemes for the iteration times n = 1 (a), 2 (b), and 3 (c), respectively, in the cases of α β = 1.

Equations (22)

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[ α 1 α 2 | γ 1 | H D 3 + ξ 1 | V D 2 ( γ 2 | H + ξ 2 | V ) c 1 + β 1 β 2 ( γ 1 | H + ξ 1 | V ) a 2 ( γ 2 | H D 4 + ξ 2 | V ) D 1 ] 1 2 ( | V D 3 | V D 4 | H D 1 | V a 1 o + | V D 3 | V D 4 | V c 2 o | H D 2 + | V D 3 | H c 2 o | H D 1 | H D 2 + | H a 1 o | V D 4 | H D 1 | H D 2 ) .
1 2 [ α 1 α 2 ( γ 1 γ 2 | H H a 1 o c 1 | H D 1 | H D 2 | H D 3 | V D 4 + ξ 1 ξ 2 | V V a 1 o c 1 | H D 1 | V D 2 | V D 3 | V D 4 ) + β 1 β 2 ( γ 1 γ 2 | H H a 2 c 2 o | H D 1 | H D 2 | V D 3 | H D 4 + ξ 1 ξ 2 | V V a 2 c 2 o | V D 1 | H D 2 | V D 3 | V D 4 ) + α 1 α 2 ( γ 1 ξ 2 | H V a 1 o c 1 | H D 1 | H D 2 | H D 3 | V D 4 + ξ 1 γ 2 | V H a 1 o c 1 | H D 1 | V D 2 | V D 3 | V D 4 ) + β 1 β 2 ( γ 1 ξ 2 | H V a 2 c 2 o | V D 1 | H D 2 | V D 3 | V D 4 + ξ 1 γ 2 | V H a 2 c 2 o | H D 1 | H D 2 | V D 3 | H D 4 ) ] .
1 2 [ α 1 α 2 ( γ 1 γ 2 | H a 1 | + H | H D 1 | H D 2 | H D 3 | H D 4 + ξ 1 ξ 2 | V a 1 | + V | H D 1 | V D 2 | V D 3 | V D 4 ) + β 1 β 2 ( γ 1 γ 2 | H a 2 | H | H D 1 | H D 2 | V D 3 | H D 4 + ξ 1 ξ 2 | V a 2 | V | V D 1 | H D 2 | V D 3 | V D 4 ) ] ,
1 8 [ | H D 5 I P I S + | V D 5 σ P I S + | H D 6 σ S I P + | V D 6 σ P σ S ] [ ( | H H H H | V V V V ) D 1 D 2 D 3 D 4 I P I S + ( | V V V V | H H H V ) D 1 D 2 D 3 D 4 ( σ S | H a H | + I | V a V | ) + ( | H V H H | V H V V ) D 1 D 2 D 3 D 4 ( I | H a H | σ S | V a V | ) + ( | H H V H | V V H V ) D 1 D 2 D 3 D 4 ( σ S | H a H | I | V a V | ) + ( | V H H H | H V V V ) D 1 D 2 D 3 D 4 ( I | H a H | + σ S | V a V | ) + ( | V V H H | H H V V ) D 1 D 2 D 3 D 4 σ P I S + ( | H V V H | V H H V ) D 1 D 2 D 3 D 4 σ S I P + ( | V H V H | H V H V ) D 1 D 2 D 3 D 4 σ P σ S ] | v 0 a ,
| 0 a = | a 1 , H , | 1 a = | a 1 , V , | 2 a = | a 2 , H , | 3 a = | a 2 , V , | 0 b = | b 1 , H , | 1 b = | b 1 , V , | 2 b = | b 2 , H , | 3 b = | b 2 , V .
ρ a b S = 1 T 1 2 | a 1 a a 1 | + 1 T 2 2 | a 2 a a 2 | + T 1 + T 2 2 [ P 1 | φ + a b φ + | + ( 1 P 1 ) | φ a b φ | ] ,
ρ a b S P 1 | φ a b φ | + ( 1 P 1 ) | φ a b φ | ,
ρ a b P P 2 | ϕ + a P b P ϕ + | + ( 1 P 2 ) | ψ + a P b P ψ + | .
| 0 = | k 1 , H , | 1 = | k 1 , V , | 2 = | k 2 , H , | 3 = | k 2 , V .
ρ A B = [ P 1 P 2 | χ 0 χ 0 | + P 1 ( 1 P 2 ) | χ 1 χ 1 | + P 1 ( 1 P 2 ) | χ 2 χ 2 | + ( 1 P 1 ) ( 1 P 2 ) | χ 3 χ 3 | ] A B ,
| χ 0 = 1 2 [ γ | ϕ + + ξ | ψ + ] A S B S | ϕ + A P B P = 1 2 [ γ | φ 0 + ξ | φ 1 ] A B , | χ 1 = 1 2 [ γ | ϕ + + ξ | ψ + ] A S B S | ψ + A P B P = 1 2 [ γ | φ 2 + ξ | φ 3 ] A B , | χ 2 = 1 2 [ ξ | ϕ + + γ | ψ + ] A S B S | ϕ + A P B P = 1 2 [ ξ | φ 0 + γ | φ 1 ] A B , | χ 3 = 1 2 [ ξ | ϕ + + γ | ψ + ] A S B S | ψ + A P B P = 1 2 [ ξ | φ 2 + γ | φ 3 ] A B .
ρ C D = [ P 1 P 2 | χ 0 χ 0 | + P 1 ( 1 P 2 ) | χ 1 χ 1 | + P 1 ( 1 P 2 ) | χ 2 χ 2 | + ( 1 P 1 ) ( 1 P 2 ) | χ 3 χ 3 | ] C D ,
ρ 0 = ρ A B ρ C D = ( P 1 2 P 2 2 | v 0 v 0 | + P 1 2 ( 1 P 2 ) 2 | v 1 v 1 | _ + P 2 2 ( 1 P 1 ) 2 | v 2 v 2 | + ( 1 P 2 ) 2 ( 1 P 1 ) 2 | v 3 v 3 | _ + ( 1 P 1 ) 2 P 2 ( 1 P 2 ) ( | v 4 v 4 | + | v 5 v 5 | ) + P 1 2 P 2 ( 1 P 2 ) ( | v 6 v 6 | + | v 7 v 7 | ) + P 1 ( 1 P 1 ) P 2 2 ( | v 8 v 8 | + | v 9 v 9 | ) _ + P 1 P 2 ( 1 P 1 ) ( 1 P 2 ) ( | v 10 v 10 | + | v 11 v 11 | ) + | v 12 v 12 | + | v 13 v 13 | + P 1 ( 1 P 1 ) ( 1 P 2 ) 2 ( | v 14 v 14 | + | v 15 v 15 | _ ) A B C D ,
| v 0 A B C D = | χ 0 A B | χ 0 C D = 1 2 [ γ 2 | φ 0 | φ 0 + ξ 2 | φ 1 | φ 1 + γ ξ ( | φ 0 | φ 1 + | φ 1 | φ 0 ) ] , | v 1 A B C D = | χ 1 A B | χ 1 C D = 1 2 [ γ 2 | φ 2 | φ 2 + ξ 2 | φ 3 | φ 3 + γ ξ ( | φ 2 | φ 3 + | φ 3 | φ 2 ) ] , | v 2 A B C D = | χ 2 A B | χ 2 C D = 1 2 [ γ 2 | φ 1 | φ 1 + ξ 2 | φ 0 | φ 0 + γ ξ ( | φ 0 | φ 1 + | φ 1 | φ 0 ) ] , | v 3 A B C D = | χ 3 A B | χ 3 C D = 1 2 [ γ 2 | φ 3 | φ 3 + ξ 2 | φ 2 | φ 2 + γ ξ ( | φ 2 | φ 3 + | φ 3 | φ 2 ) ] , | v 4 A B C D = | χ 2 A B | χ 3 C D = 1 2 [ γ 2 | φ 1 | φ 3 + ξ 2 | φ 0 | φ 2 + γ ξ ( | φ 1 | φ 2 + | φ 3 | φ 0 ) ] , | v 5 A B C D = | χ 3 A B | χ 2 C D = 1 2 [ γ 2 | φ 3 | φ 1 + ξ 2 | φ 2 | φ 0 + γ ξ ( | φ 3 | φ 0 + | φ 2 | φ 1 ) ] , | v 6 A B C D = | χ 0 A B | χ 1 C D = 1 2 [ γ 2 | φ 0 | φ 2 + ξ 2 | φ 1 | φ 3 + γ ξ ( | φ 0 | φ 3 + | φ 1 | φ 2 ) ] , | v 7 A B C D = | χ 1 A B | χ 0 C D = 1 2 [ γ 2 | φ 2 | φ 0 + ξ 2 | φ 3 | φ 1 + γ ξ ( | φ 2 | φ 1 + | φ 3 | φ 0 ) ] , | v 8 A B C D = | χ 0 A B | χ 2 C D = 1 2 [ γ 2 | φ 0 | φ 1 + ξ 2 | φ 1 | φ 0 + γ ξ ( | φ 0 | φ 0 + | φ 1 | φ 1 ) ] , | v 9 A B C D = | χ 2 A B | χ 0 C D = 1 2 [ γ 2 | φ 1 | φ 0 + ξ 2 | φ 0 | φ 1 + γ ξ ( | φ 0 | φ 0 + | φ 1 | φ 1 ) ] , | v 10 A B C D = | χ 0 A B | χ 3 C D = 1 2 [ γ 2 | φ 0 | φ 3 + ξ 2 | φ 1 | φ 2 + γ ξ ( | φ 0 | φ 2 + | φ 1 | φ 3 ) ] , | v 11 A B C D = | χ 3 A B | χ 0 C D = 1 2 [ γ 2 | φ 3 | φ 0 + ξ 2 | φ 2 | φ 1 + γ ξ ( | φ 2 | φ 0 + | φ 3 | φ 1 ) ] , | v 12 A B C D = | χ 1 A B | χ 2 C D = 1 2 [ γ 2 | φ 2 | φ 1 + ξ 2 | φ 3 | φ 0 + γ ξ ( | φ 2 | φ 0 + | φ 3 | φ 1 ) ] , | v 13 A B C D = | χ 2 A B | χ 1 C D = 1 2 [ γ 2 | φ 1 | φ 2 + ξ 2 | φ 0 | φ 3 + γ ξ ( | φ 0 | φ 3 + | φ 1 | φ 0 ) ] , | v 14 A B C D = | χ 1 A B | χ 3 C D = 1 2 [ γ 2 | φ 2 | φ 3 + ξ 2 | φ 3 | φ 2 + γ ξ ( | φ 2 | φ 2 + | φ 3 | φ 3 ) ] , | v 15 A B C D = | χ 3 A B | χ 1 C D = 1 2 [ γ 2 | φ 3 | φ 2 + ξ 2 | φ 2 | φ 3 + γ ξ ( | φ 2 | φ 2 + | φ 3 | φ 3 ) ] .
| φ 0 A B | φ 0 C D = 1 4 ( | 100 + | 11 + | 22 + | 33 ) A B ( | 00 + | 11 + | 22 + | 33 ) C D = 1 4 ( | 0000 + | 1111 + | 2222 + | 3333 + | 0101 + | 1212 + | 2323 + | 3030 _ + | 0202 + | 1313 + | 2020 + | 3131 + | 0303 + | 1010 + | 2121 + | 3232 ) A C B D
| φ 1 A B | φ 1 C D = 1 4 ( | 02 + | 13 + | 20 + | 31 ) A B ( | 02 + | 13 + | 20 + | 31 ) C D = 1 4 ( | 0022 + | 1133 + | 2200 + | 3311 + | 0123 + | 2301 + | 1230 + | 3012 _ + | 0220 + | 1331 + | 2002 + | 3113 + | 0321 + | 2103 + | 1032 + | 3210 ) A C B D .
| φ 2 A B | φ 2 C D = 1 4 ( | 01 + | 10 + | 23 + | 32 ) A B ( | 01 + | 10 + | 23 + | 32 ) C D = 1 4 ( | 0011 + | 1100 + | 2233 + | 3322 + | 0110 + | 2332 + | 1203 + | 3021 + | 0213 + | 1302 + | 2031 + | 3120 + | 0312 + | 1001 + | 2130 + | 3223 ) A C B D .
| φ 3 A B | φ 3 C D = 1 4 ( | 03 + | 21 + | 12 + | 30 ) A B ( | 03 + | 21 + | 12 + | 30 ) C D = 1 4 ( | 0033 + | 1122 + | 2211 + | 3300 + | 0132 + | 1212 + | 2310 + | 3003 + | 0231 + | 2013 + | 1320 + | 3102 + | 0330 + | 1023 + | 2112 + | 3201 ) A C B D ,
| φ 2 A B | φ 3 C D = 1 4 ( | 01 + | 10 + | 23 + | 32 ) A B ( | 03 + | 21 + | 12 + | 30 ) C D = 1 4 ( | 0013 + | 1102 + | 2231 + | 3302 + | 0112 + | 2230 + | 1201 + | 3023 _ + | 0211 + | 1300 + | 2033 + | 3122 + | 0310 + | 1003 + | 2132 + | 3221 ) A B C D ,
| φ 3 A B | φ 2 C D = 1 4 ( | 03 + | 21 + | 12 + | 30 ) A B ( | 01 + | 10 + | 23 + | 32 ) C D = 1 4 ( | 0031 + | 1120 + | 2213 + | 3320 + | 3001 + | 0130 + | 1223 + | 2312 _ + | 2011 + | 3100 + | 0233 + | 1322 + | 1021 + | 3203 + | 2110 + | 0332 ) A B C D ,
ρ 0 = ( P 1 2 P 2 2 | v 0 v 0 | + ( P 1 2 + ( 1 P 1 ) 2 ) ( 1 P 2 ) 2 | v 1 v 1 | + P 2 2 ( 1 P 1 ) 2 | v 2 v 2 | + 2 P 1 ( 1 P 1 ) P 2 2 ( | v 3 v 3 | + P 1 ( 1 P 1 ) ( 1 P 2 ) 2 ( | v 4 v 4 | + | v 5 v 5 | ) a b ,
| v 0 a b = 1 2 ( γ 2 | φ 0 _ + ξ 2 | φ 1 ) a b , | v 1 a b = 1 2 γ ξ ( | φ 2 + | φ 3 ) a b , | v 2 a b = 1 2 ( ξ 2 | φ 0 _ + γ 2 | φ 1 ) a b , | v 3 a b = 1 2 γ ξ ( | φ 0 _ + | φ 1 ) a b , | v 4 a b = 1 2 ( γ 2 | φ 2 + ξ 2 | φ 3 ) a b , | v 5 a b = 1 2 ( γ 2 | φ 3 + ξ 2 | φ 2 ) a b .

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