Abstract

We illustrate an entanglement distillation protocol (EDP) for a mixed photon-ensemble which composed of four kinds of entangled states and vacuum states. Exploiting the linear optics and local entanglement resource (four-qubit entangled GHZ state), we design the nondemolition parity-checking and qubit amplifying (PCQA) setup for photonic polarization degree of freedom which are the key device of our scheme. With the PCQA setup, a high-fidelity entangled photon-pair system can be achieved against the transmission losses and the decoherence in noisy channels. And in the available purification range for our EDP, the fidelity of this ensemble can be improved to the maximal value through iterated operations. Compared to the conventional entanglement purification schemes, our scheme largely reduces the initialization requirement of the distilled mixed quantum system, and overcomes the difficulties posed by inherent channel losses during photon transmission. All these advantages make this scheme more useful in the practical applications of long-distance quantum communication.

© 2015 Optical Society of America

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    [Crossref]
  27. C. Wang, Y. Zhang, and R. Zhang, “Entanglement purification based on hybrid entangled state using quantum-dot and microcavity coupled system,” Opt. Exp. 19, 25685–25695 (2011).
    [Crossref]
  28. Y. B. Sheng, L. Zhou, and G. L. Long, “Hybrid entanglement purification for quantum repeaters,” Phys. Rev. A 88, 022302 (2013).
    [Crossref]
  29. Y. B. Sheng and L. Zhou, “Deterministic entanglement distillation for secure double-server blind quantum computation,” Sci. Rep. 5, 7815 (2015).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
  32. S. L. Zhang, S. Yang, X. B. Zou, B. S. Shi, and G. C. Guo, “Protecting single-photon entangled state from photon loss with noiseless linear amplification,” Phys. Rev. A 86, 034302 (2012).
    [Crossref]
  33. L. Zhou and Y. B. Sheng, “Recyclable amplification protocol for the single-photon entangled state,” Laser Phys. Lett. 12, 045203 (2015).
    [Crossref]
  34. Y. Ou-Yang, Z. F. Feng, L. Zhou, and Y. B. Sheng, “Protecting single-photon entanglement with imperfect single-photon source,” Quant. Inf. Process. 14, 635–651 (2015).
    [Crossref]
  35. 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]
  36. K. Vollbrecht, C. A. Muschik, and J. I. Cirac, “Entanglement distillation by dissipation and continuous quantum repeaters,” Phys. Rev. Lett. 107, 120502 (2011).
    [Crossref] [PubMed]
  37. M. Asano, M. Bechu, M. Tame, Ş. K. Özdemir, R. Ikuta, D. Ö. Güney, T. Yamamoto, L. Yang, M. Wegener, and N. Imoto, “Distillation of photon entanglement using a plasmonic metamaterial,” arXiv1507.07948 (2015).
  38. M. A. Farooqui, J. Breeland, M. I. Aslam, M. Sadatgol, Ş. K. Özdemir, M. Tame, L. Yang, and D. Ö. Güney, “Quantum entanglement distillation with metamaterials,” Opt. Exp. 23, 17941–17954 (2015).
    [Crossref]
  39. T. Yamamoto, R. Nagase, J. Shimamura, S. K. Özdemir, M. Koashi, and N Imot, “Experimental ancilla-assisted qubit transmission against correlated noise using quantum parity checking,” New Journal of Physics 9, 191 (2007).
    [Crossref]
  40. T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Probabilistic quantum logic operations using polarizing beam splitters,” Phys. Rev. A 64, 062311 (2001).
    [Crossref]
  41. Ş. K. Özdemir, K. Bartkiewicz, Y. Liu, and A. Miranowicz, “Teleportation of qubit states through dissipative channels: Conditions for surpassing the no-cloning limit,” Phys. Rev. A 76, 042325 (2007).
    [Crossref]
  42. Ş. K. Özdemir, A. Miranowicz, M. Koashi, and N. Imoto, “Pulse-mode quantum projection synthesis: Effects of mode mismatch on optical state truncation and preparation,” Phys. Rev. A 66, 053809 (2002).
    [Crossref]
  43. Y. Liu, Ş. K. Özdemir, A. Miranowicz, and N. Imoto, “Kraus representation of a damped harmonic oscillator and its application,” Phys. Rev. A 70, 042308 (2004).
    [Crossref]
  44. We give a discussion of the reason why we perform the Hadamard operation in step 2 in detail in Section 4.
  45. M. Horodecki and P. Horodecki, “Reduction criterion of separability and limits for a class of distillation protocols,” Phys. Rev. A 59, 4206–4216 (1999).
    [Crossref]
  46. R. Horodeckia, M. Horodeckib, and P. Horodeckic, “Teleportation, Bell’s inequalities and inseparability,” Phys. Lett. A 222, 21–25 (1996).
    [Crossref]
  47. S. Bandyopadhyay, “Origin of noisy states whose teleportation fidelity can be enhanced through dissipation,” Phys. Rev. A 65, 022302 (2002).
    [Crossref]
  48. If after the nosiy channle, the transmitted state is pure state, our scheme also can be used for distilling the entanglement. For example, when p = 0, the initial state of the photons ABCD is a pure state |ψ−〉AB⊗|ψ−〉CD=12(|hhvv〉+|vvhh〉−|hvvh〉−|vhhv〉)ACBD. According to the step 1 of our EDP, as only odd-parity states are reserved, Alice and Bob share a new ensemble ρp which state is |ψ+⟩CD, and p′2 = 1, p′1 = 0. Then, Alice and Bob perform the Hadamard operation on the photons of the new ensemble, the state of the new ensemble shared by Alice and Bob is transformed to |ϕ−⟩CD. Finally, after the step 3 in which Alice and Bob only pick up the odd-parity states of the new ensemble, according to Eq. (14) and (15), p1″=1, and the final state shared by Alice and Bob is |ϕ+⟩.
  49. J. S. Bell, Speakable and Unspeakable in Quantum Mechanics (Cambridge University, 1987).
  50. T. J. Wang, C. Cao, and C. Wang, “Linear-optical implementation of hyperdistillation from photon loss,” Phys. Rev. A 89, 052303 (2014).
    [Crossref]
  51. A. Cabello and F. Sciarrino, “Loophole-free Bell test based on local precertification of photons presence,” Phys. Rev. X. 2, 021010 (2012).
  52. P. M. Pearle, “Hidden-variable example based upon data rejection,” Phys. Rev. D 2, 1418–1425 (1970).
    [Crossref]
  53. E. Meyer-Scott, M. Bula, K. Bartkiewicz, A. Černoch, J. Soubusta, T. Jennewein, and K. Lemr, “Entanglement-based linear-optical qubit amplifier,” Phys. Rev. A 88, 012327 (2013).
    [Crossref]
  54. K. Bartkiewicz, A. Černoch, and K. Lemr, “State-dependent linear-optical qubit amplifier,” Phys. Rev. A 88, 062304 (2013).
    [Crossref]
  55. If and only if p1=p2=12, the intial state of our scheme is in a Werner state, that is, ρ=14(|ϕp+〉〈ϕp+|+|ϕp−〉〈ϕp−|+|ψp+〉〈ψp+|+|ψp−〉〈ψp−|) which fidelity can’t be improved by both the Bennett et al’s EP scheme and our EDP. Actually, when p = 1/2, the state is maximally mixed state, a separable state from which no entanglement can be distilled.
  56. X.-C. Yao, T.-X. Wang, P. Xu, H. Lu, G.-S. Pan, X.-H. Bao, C.-Z. Peng, C.-Y. Lu, Y.-A. Chen, and J.-W. Pan, “Observation of eight-photon entanglement,” Nature Photonics 6, 225–228 (2012).
    [Crossref]

2015 (4)

Y. B. Sheng and L. Zhou, “Deterministic entanglement distillation for secure double-server blind quantum computation,” Sci. Rep. 5, 7815 (2015).
[Crossref] [PubMed]

L. Zhou and Y. B. Sheng, “Recyclable amplification protocol for the single-photon entangled state,” Laser Phys. Lett. 12, 045203 (2015).
[Crossref]

Y. Ou-Yang, Z. F. Feng, L. Zhou, and Y. B. Sheng, “Protecting single-photon entanglement with imperfect single-photon source,” Quant. Inf. Process. 14, 635–651 (2015).
[Crossref]

M. A. Farooqui, J. Breeland, M. I. Aslam, M. Sadatgol, Ş. K. Özdemir, M. Tame, L. Yang, and D. Ö. Güney, “Quantum entanglement distillation with metamaterials,” Opt. Exp. 23, 17941–17954 (2015).
[Crossref]

2014 (2)

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

T. Li, G. J. Yang, and F. G. Deng, “Entanglement distillation for quantum communication network with atomicensemble memories,” Opt. Exp. 22, 23897–23911 (2014).
[Crossref]

2013 (5)

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

T. J. Wang and G. L. Long, “Entanglement concentration for arbitrary unknown less-entangled three-photon W states with linear optics,” J. Opt. Soc. Am. B 30, 1069–1076 (2013).
[Crossref]

E. Meyer-Scott, M. Bula, K. Bartkiewicz, A. Černoch, J. Soubusta, T. Jennewein, and K. Lemr, “Entanglement-based 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]

2012 (4)

X.-C. Yao, T.-X. Wang, P. Xu, H. Lu, G.-S. Pan, X.-H. Bao, C.-Z. Peng, C.-Y. Lu, Y.-A. Chen, and J.-W. Pan, “Observation of eight-photon entanglement,” Nature Photonics 6, 225–228 (2012).
[Crossref]

A. Cabello and F. Sciarrino, “Loophole-free Bell test based on local precertification of photons presence,” Phys. Rev. X. 2, 021010 (2012).

S. L. Zhang, S. Yang, X. B. Zou, B. S. Shi, and G. C. Guo, “Protecting single-photon entangled state from photon loss with noiseless linear amplification,” Phys. Rev. A 86, 034302 (2012).
[Crossref]

Y. B. Sheng, L. Zhou, S. M. Zhao, and B. Y. Zheng, “Efficient single-photon-assisted entanglement concentration for partially entangled photon pairs,” Phys. Rev. A 85, 012307 (2012).
[Crossref]

2011 (2)

K. Vollbrecht, C. A. Muschik, and J. I. Cirac, “Entanglement distillation by dissipation and continuous quantum repeaters,” Phys. Rev. Lett. 107, 120502 (2011).
[Crossref] [PubMed]

C. Wang, Y. Zhang, and R. Zhang, “Entanglement purification based on hybrid entangled state using quantum-dot and microcavity coupled system,” Opt. Exp. 19, 25685–25695 (2011).
[Crossref]

2010 (1)

G.Y. Xiang, T.C. Ralph, A.P. Lund, N. Walk, and G.J. Pryde, “Heralded noiseless linear amplification and distillation of entanglement,” Nature Photonics 4, 316–319 (2010).
[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]

T. Yamamoto, K. Hayashi, Ş. K. Őzdemir, M. Koashi, and N. Imoto, “Robust photonic entanglement distribution by state-independent encoding onto decoherence-free subspace,” Nature Photonics 2, 488–491 (2008).
[Crossref]

2007 (5)

R. Prevedel, M. S. Tame, A. Stefanov, M. Paternostro, M. S. Kim, and A. Zeilinger, “Experimental demonstration of decoherence-free one-way information transfer,” Phys. Rev. Lett. 99, 250503 (2007).
[Crossref]

D. W. Leung, M. A. Nielsen, I. L. Chuang, and Y. Yamamoto, “Approximate quantum error correction can lead to better codes,” Phys. Rev. A 56, 2567–2573(2007).
[Crossref]

Y. W. Cheong, S. W. Lee, J. Lee, and H.-W. Lee, “Entanglement purification for high-dimensional multipartite systems,” Phys. Rev. A. 76, 042314 (2007).
[Crossref]

T. Yamamoto, R. Nagase, J. Shimamura, S. K. Özdemir, M. Koashi, and N Imot, “Experimental ancilla-assisted qubit transmission against correlated noise using quantum parity checking,” New Journal of Physics 9, 191 (2007).
[Crossref]

Ş. K. Özdemir, K. Bartkiewicz, Y. Liu, and A. Miranowicz, “Teleportation of qubit states through dissipative channels: Conditions for surpassing the no-cloning limit,” Phys. Rev. A 76, 042325 (2007).
[Crossref]

2005 (3)

A. Miyake and H. J. Briegel, “Distillation of multipartite entanglement by complementary stabilizer measurements,” Phys. Rev. Lett. 95, 220501 (2005).
[Crossref] [PubMed]

T. Yamamoto, J. Shimamura, Ş. K. Őzdemir, M. Koashi, and N. Imoto, “Faithful qubit distribution assisted by one additional qubit against collective noise,” Phys. Rev. Lett. 95, 040503 (2005).
[Crossref] [PubMed]

F. L. Yan and T. Gao, “Quantum secret sharing between multiparty and multiparty without entanglement,” Phys. Rev. A 72, 012304 (2005).
[Crossref]

2004 (2)

L. Xiao, G. L. Long, F. G. Deng, and J. W. Pan, “Efficient multiparty quantum-secret-sharing schemes,” Phys. Rev. A 69, 052307 (2004).
[Crossref]

Y. Liu, Ş. K. Özdemir, A. Miranowicz, and N. Imoto, “Kraus representation of a damped harmonic oscillator and its application,” Phys. Rev. A 70, 042308 (2004).
[Crossref]

2003 (2)

T. Yamamoto, M. Koashi, S. K. Ozdemir, and N. Imoto, “Experimental extraction of an entangled photon pair from two identically decohered pairs,” Nature 421343–346 (2003).
[Crossref] [PubMed]

J. W. Pan, S. Gasparonl, R. Ursin, G. Weihs, and A. Zellinger, “Experimental entanglement purification of arbitrary unknown states,” Nature 423, 417–422 (2003).
[Crossref] [PubMed]

2002 (4)

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

C. Simon and J. W. Pan, “Polarization entanglement purification using spatial entanglement,” Phys. Rev. A 89, 257901 (2002).

S. Bandyopadhyay, “Origin of noisy states whose teleportation fidelity can be enhanced through dissipation,” Phys. Rev. A 65, 022302 (2002).
[Crossref]

Ş. K. Özdemir, A. Miranowicz, M. Koashi, and N. Imoto, “Pulse-mode quantum projection synthesis: Effects of mode mismatch on optical state truncation and preparation,” Phys. Rev. A 66, 053809 (2002).
[Crossref]

2001 (2)

T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Probabilistic quantum logic operations using polarizing beam splitters,” Phys. Rev. A 64, 062311 (2001).
[Crossref]

H. K. Lo and S. Popescu, “Concentrating entanglement by local actions:?Beyond mean values,” Phys. Rev. A 63, 022301 (2001).
[Crossref]

1999 (3)

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

M. Horodecki and P. Horodecki, “Reduction criterion of separability and limits for a class of distillation protocols,” Phys. Rev. A 59, 4206–4216 (1999).
[Crossref]

M. Horodecki and P. Horodecki, “Reduction criterion of separability and limits for a class of distillation protocols,” Phys. Rev. A 59, 4206–4216 (1999).
[Crossref]

1998 (1)

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, 5932–5935 (1998).
[Crossref]

1996 (5)

C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, “Concentrating partial entanglement by local operations,” Phys. Rev. A 53, 2046–2052 (1996).
[Crossref] [PubMed]

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]

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, 2818–2821 (1996).
[Crossref] [PubMed]

C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, “Mixed-state entanglement and quantum error correction,” Phys. Rev. A. 54, 3824–3851 (1996).
[Crossref] [PubMed]

R. Horodeckia, M. Horodeckib, and P. Horodeckic, “Teleportation, Bell’s inequalities and inseparability,” Phys. Lett. A 222, 21–25 (1996).
[Crossref]

1993 (1)

C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1993).
[Crossref] [PubMed]

1970 (1)

P. M. Pearle, “Hidden-variable example based upon data rejection,” Phys. Rev. D 2, 1418–1425 (1970).
[Crossref]

Asano, M.

M. Asano, M. Bechu, M. Tame, Ş. K. Özdemir, R. Ikuta, D. Ö. Güney, T. Yamamoto, L. Yang, M. Wegener, and N. Imoto, “Distillation of photon entanglement using a plasmonic metamaterial,” arXiv1507.07948 (2015).

Aslam, M. I.

M. A. Farooqui, J. Breeland, M. I. Aslam, M. Sadatgol, Ş. K. Özdemir, M. Tame, L. Yang, and D. Ö. Güney, “Quantum entanglement distillation with metamaterials,” Opt. Exp. 23, 17941–17954 (2015).
[Crossref]

Bandyopadhyay, S.

S. Bandyopadhyay, “Origin of noisy states whose teleportation fidelity can be enhanced through dissipation,” Phys. Rev. A 65, 022302 (2002).
[Crossref]

Bao, X.-H.

X.-C. Yao, T.-X. Wang, P. Xu, H. Lu, G.-S. Pan, X.-H. Bao, C.-Z. Peng, C.-Y. Lu, Y.-A. Chen, and J.-W. Pan, “Observation of eight-photon entanglement,” Nature Photonics 6, 225–228 (2012).
[Crossref]

Bartkiewicz, K.

E. Meyer-Scott, M. Bula, K. Bartkiewicz, A. Černoch, J. Soubusta, T. Jennewein, and K. Lemr, “Entanglement-based 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]

Ş. K. Özdemir, K. Bartkiewicz, Y. Liu, and A. Miranowicz, “Teleportation of qubit states through dissipative channels: Conditions for surpassing the no-cloning limit,” Phys. Rev. A 76, 042325 (2007).
[Crossref]

Bechu, M.

M. Asano, M. Bechu, M. Tame, Ş. K. Özdemir, R. Ikuta, D. Ö. Güney, T. Yamamoto, L. Yang, M. Wegener, and N. Imoto, “Distillation of photon entanglement using a plasmonic metamaterial,” arXiv1507.07948 (2015).

Bell, J. S.

J. S. Bell, Speakable and Unspeakable in Quantum Mechanics (Cambridge University, 1987).

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, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, “Mixed-state entanglement and quantum error correction,” Phys. Rev. A. 54, 3824–3851 (1996).
[Crossref] [PubMed]

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If and only if p1=p2=12, the intial state of our scheme is in a Werner state, that is, ρ=14(|ϕp+〉〈ϕp+|+|ϕp−〉〈ϕp−|+|ψp+〉〈ψp+|+|ψp−〉〈ψp−|) which fidelity can’t be improved by both the Bennett et al’s EP scheme and our EDP. Actually, when p = 1/2, the state is maximally mixed state, a separable state from which no entanglement can be distilled.

We give a discussion of the reason why we perform the Hadamard operation in step 2 in detail in Section 4.

If after the nosiy channle, the transmitted state is pure state, our scheme also can be used for distilling the entanglement. For example, when p = 0, the initial state of the photons ABCD is a pure state |ψ−〉AB⊗|ψ−〉CD=12(|hhvv〉+|vvhh〉−|hvvh〉−|vhhv〉)ACBD. According to the step 1 of our EDP, as only odd-parity states are reserved, Alice and Bob share a new ensemble ρp which state is |ψ+⟩CD, and p′2 = 1, p′1 = 0. Then, Alice and Bob perform the Hadamard operation on the photons of the new ensemble, the state of the new ensemble shared by Alice and Bob is transformed to |ϕ−⟩CD. Finally, after the step 3 in which Alice and Bob only pick up the odd-parity states of the new ensemble, according to Eq. (14) and (15), p1″=1, and the final state shared by Alice and Bob is |ϕ+⟩.

J. S. Bell, Speakable and Unspeakable in Quantum Mechanics (Cambridge University, 1987).

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Figures (4)

Fig. 1
Fig. 1 The non-demolition parity-checking and qubit amplifying (PCQA) setup. The polarizing beam splitter (PBS) can transmit horizontally polarized photon (|H⟩) and reflect vertically polarized photon (|V⟩). The Hadamard operation could be performed using H plate which is a half-wave plate with the angle of 22.5° to the horizontal direction, that is | H 1 2 ( | H + | V ), | V 1 2 ( | H + | V ). The F mode plate is a half-wave plate with the axis at 45° with respect to the horizontal direction rotates the photon polarization as |H⟩⇌|V⟩. The eight detectors D1D1D2D2D3D3D4D4 belong to four different groups D 1 D 2 D 3 D 4, i.e., {Dj,Dj} ∈ D j ( j = 1 , 2 , 3 o r 4 ).
Fig. 2
Fig. 2 The fidelity of the two-photon entangled Bell state in our polarized EDP. Fp,n alters with the iteration number n of EDP processes and the initial the coefficient of the mixed less-entangled Bell state p = p1 = p2. Here p1(p2) is the initial fidelity against the bit-(phase-) flip error.
Fig. 3
Fig. 3 The fidelity fb,m (a), fp,m (b), f b , m (c) and f p , m (d) are altered with the iteration number of entanglement concentration processes m and the coefficients p1 and p2.
Fig. 4
Fig. 4 The fidelities of the two-photon entangled Bell states in the conventional EP schemes Fs,m and in our polarized EDP Fp,n in the case of p1 = p2 = p

Tables (1)

Tables Icon

Table 1 Relation between the final states of D1D2D3D4D1D2D3D4 and the corresponding single qubit operations on the photon at the output port b′.

Equations (19)

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| φ A B | G H Z 4 1 2 [ α ( | D 1 + | D 1 ) ( | D 4 + | D 4 ) + β ( | D 3 + | D 3 ) ( | D 2 + | D 2 ) + γ ( | D 1 + | D 1 ) ( | D 2 + | D 2 ) + ξ ( | D 3 + | D 3 ) ( | D 4 + | D 4 ) ] A B [ ( | D 3 | D 3 ) ( | D 2 | D 2 ) | H V a b | V H a b + ( | D 4 | D 4 ) ( | D 1 | D 1 ) ] .
| o d d A B | G H Z 4 α ( | D 1 + | D 1 ) ( | D 4 + | D 4 ) ( | D 3 | D 3 ) ( | D 2 | D 2 ) | H V a b + β ( | D 3 + | D 3 ) ( | D 2 + | D 2 ) ( | D 4 | D 4 ) ( | D 1 | D 1 ) | V H a b = 1 4 [ ( | D 1 D 2 D 3 D 4 + | D 1 D 2 D 3 D 4 + | D 1 D 2 D 3 D 4 + | D 1 D 2 D 3 D 4 | D 1 D 2 D 3 D 4 | D 1 D 2 D 3 D 4 | D 1 D 2 D 3 D 4 | D 1 D 2 D 3 D 4 ) ( a | H V + β | V H ) a b + ( | D 1 D 2 D 3 D 4 + | D 1 D 2 D 3 D 4 | D 1 D 2 D 3 D 4 | D 1 D 2 D 3 D 4 + | D 1 D 2 D 3 D 4 + | D 1 D 2 D 3 D 4 | D 1 D 2 D 3 D 4 | D 1 D 2 D 3 D 4 ) ( a | H V β | V H ) a b .
| H A | G H Z 4 ( | D 1 + | D 1 ) [ ( | D 3 | D 3 ) ( | D 2 | D 2 ) | H V A B + | V H A B ( | D 4 | D 4 ) ( | D 1 | D 1 ) ] ,
| V A | G H Z 4 ( | D 3 + | D 3 ) [ ( | D 3 | D 3 ) ( | D 2 | D 2 ) | H V A B + | V H A B ( | D 4 | D 4 ) ( | D 1 | D 1 ) ] .
ρ a b 0 = ( 1 T ) | χ a χ | | v a c b v a c | + T ρ a b p ,
ρ a b p = p 1 p 2 | ϕ + a b ϕ + | + p 1 ( 1 p 2 ) | ϕ a b ϕ | + p 2 ( 1 p 1 ) | ψ + a b ψ + | + ( 1 p 1 ) ( 1 p 2 ) | ψ a b ψ | ,
| ϕ a b = 1 2 ( | H H | V V ) a b , | ψ ± a b = 1 2 ( | H V ± | V H ) a b .
ρ A B 0 = ( 1 T ) | χ a χ | | v a c b v a c | + T ρ A B p , ρ C D 0 = ( 1 T ) | χ a χ | | v a c b v a c | + T ρ C D p
ρ A B p ρ C D p = { p 1 [ p 2 | ϕ + A B ϕ + | + ( 1 p 2 ) | ϕ A B ϕ | ] + ( 1 p 1 ) [ p 2 | ψ + A B ψ + | + ( 1 p 2 ) | ψ A B ψ | ] } { p 1 [ p 2 | ϕ + C D ϕ + | + ( 1 p 2 ) | ϕ C D ϕ | ] + ( 1 p 1 ) [ p 2 | ψ + C D ψ + | + ( 1 p 2 ) | ψ C D ψ | ] } .
ρ o d d = p 1 2 { [ p 2 2 + ( 1 p 2 ) 2 ] | φ 1 A B C D φ 1 | + 2 p 2 ( 1 p 2 ) | φ 2 A B C D φ 2 | } + ( 1 p 1 ) 2 { [ p 2 2 + ( 1 p 2 ) 2 ] | φ 3 A B C D φ 3 | + 2 p 2 ( 1 p 2 ) | φ 4 A B C D φ 4 | } ,
| φ 1 A C B D = 1 2 ( | H V H V + | V H V H ) A C B D , | φ 2 A C B D = 1 2 ( | H V H V | V H V H ) A C B D , | φ 3 A C B D = 1 2 ( | H V V H + | V H H V ) A C B D , | φ 4 A C B D = 1 2 ( | H V V H | V H H V ) A C B D .
ρ 1 p = p 1 p 2 | ϕ p + C D ϕ p + | + p 1 ( 1 p 2 ) | ϕ p C D ϕ p | + p 2 ( 1 p 1 ) | ψ p + C D ψ p + | + ( 1 p 1 ) ( 1 p 2 ) | ψ p C D ψ p | ,
p 1 = p 1 2 p 1 2 + ( 1 p 1 ) 2 , p 2 = p 2 2 + ( 1 p 2 ) 2 .
ρ 1 , C D = p 2 [ p 1 | ϕ p + C D ϕ p + | + ( 1 p 1 ) | ϕ p C D ϕ p | ] + ( 1 p 2 ) [ p 1 | ψ p + C D ψ p + | + ( 1 p 1 ) | ψ p C D ψ p | ] .
ρ 1 , C D = p 2 [ p 1 | ϕ p + C D ϕ p + | + ( 1 p 1 ) | ϕ p C D ϕ p | ] + ( 1 p 2 ) [ p 1 | ψ p + C D ψ p + | + ( 1 p 1 ) | ψ p C D ψ p | ] , ρ 1 , C D = p 2 [ p 1 | ϕ p + C D ϕ p + | + ( 1 p 1 ) | ϕ p C D ϕ p | ] + ( 1 p 2 ) [ p 1 | ψ p + C D ψ p + | + ( 1 p 1 ) | ψ p C D ψ p | ] .
ρ o d d , C C D D = p 2 2 { [ p 1 2 + ( 1 p 1 ) 2 ] | φ 1 C C D D φ 1 | + 2 p 1 ( 1 p 1 ) | φ 2 C C D D φ 2 | } + ( 1 p 2 ) 2 { [ p 1 2 + ( 1 p 1 ) 2 ] | φ 3 C C D D φ 3 | + 2 p 1 ( 1 p 1 ) | φ 4 C C D D φ 4 | } .
ρ 2 p = p 1 [ p 2 | ϕ p + C D ϕ p + | + ( 1 p 2 ) | ϕ p C D ϕ p | ] + ( 1 p 1 ) [ p 2 | ψ p + C D ψ p + | + ( 1 p 2 ) | ψ p C D ψ p | ] ,
p 1 = p 2 2 p 2 2 + ( 1 p 2 ) 2 = [ p 2 2 + ( 1 p 2 ) 2 ] 2 [ p 2 2 + ( 1 p 2 ) 2 ] 2 + [ 1 p 2 2 ( 1 p 2 ) 2 ] 2 , p 2 = p 1 2 + ( 1 p 1 ) 2 = [ p 1 2 p 1 2 + ( 1 p 1 ) 2 ] 2 + [ 1 p 1 2 p 1 2 + ( 1 p 1 ) 2 ] 2 .
| φ A B | ψ + E 1 E 2 | ψ + E 3 E 4 = 1 2 ( α | H V + β | V H + γ | H H + ξ | V V ) A B ( | H V H V + | V H V H + | H V V H + | V H H V ) E 1 E 2 E 3 E 4 1 2 [ α ( | D 1 + | D 1 ) ( | D 4 + | D 4 ) + β ( | D 3 + | D 3 ) ( | D 2 + | D 2 ) + γ ( | D 1 + | D 1 ) ( | D 2 + | D 2 ) + ξ ( | D 3 + | D 3 ) ( | D 4 + | D 4 ) ] [ ( | D 3 | D 3 ) ( | D 2 | D 2 ) | H V a b + ( | D 4 | D 4 ) ( | D 1 | D 1 ) | V H a b + ( | D 4 | D 4 ) ( | D 2 | D 2 ) | H V a a + ( | D 3 | D 3 ) ( | D 1 | D 1 ) | V H b b ] .

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