Abstract

Entanglement distillation is an efficient method to retrieve high-quality entanglement from a large number of noise-damped entanglement states. Many investigations on continuous variable entanglement distillation have focused on low-loss or medium-loss channels. Herein, we primarily study entanglement distillation in extremely lossy (total channel transmittance $\eta\ll 1$) or even near-zero transmittance ($\eta\approx 0$) channels. In particular, we analyze the manner in which entanglement and success probability decrease as $\eta\to 0$. The performances of conventional biside photon subtraction and one-time photon subtraction on two optical modes separated by $L$ km are investigated. The motivation for our study includes the recent advancements in twin-field quantum key distribution, wherein single-photon detection is sufficient to provide a useful private key. In the same vein, we propose an entanglement distillation scheme with an efficiency of $O(\sqrt\eta)$ using one-time photon subtraction and weak coherent state injection. Thus, our proposed scheme provides a method for distilling extremely damped entanglement states for continuous variable quantum information processing.

© 2020 Optical Society of America

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2019 (1)

M. Curty, K. Azuma, and H.-K. Lo, “Simple security proof of twin-field type quantum key distribution protocol,” npj Quantum Inf. 5, 64 (2019).
[Crossref]

2018 (3)

A. Boaron, G. Boso, D. Rusca, C. Vulliez, C. Autebert, M. Caloz, M. Perrenoud, G. Gras, F. Bussières, M.-J. Li, D. Nolan, A. Martin, and H. Zbinden, “Secure quantum key distribution over 421 km of optical fiber,” Phys. Rev. Lett. 121, 190502 (2018).
[Crossref]

M. Lucamarini, Z. L. Yuan, J. F. Dynes, and A. J. Shields, “Overcoming the rate-distance limit of quantum key distribution without quantum repeaters,” Nature 557, 400–403 (2018).
[Crossref]

X. Ma, P. Zeng, and H. Zhou, “Phase-matching quantum key distribution,” Phys. Rev. X 8, 031043 (2018).
[Crossref]

2017 (3)

S.-K. Liao, W.-Q. Cai, W.-Y. Liu, L. Zhang, Y. Li, J.-G. Ren, J. Yin, Q. Shen, Y. Cao, Z.-P. Li, F.-Z. Li, X.-W. Chen, L.-H. Sun, J.-J. Jia, J.-C. Wu, X.-J. Jiang, J.-F. Wang, Y.-M. Huang, Q. Wang, Y.-L. Zhou, L. Deng, T. Xi, L. Ma, T. Hu, Q. Zhang, Y.-A. Chen, N.-L. Liu, X.-B. Wang, Z.-C. Zhu, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, “Satellite-to-ground quantum key distribution,” Nature 549, 43–47 (2017).
[Crossref]

J.-G. Ren, P. Xu, H.-L. Yong, L. Zhang, S.-K. Liao, J. Yin, W.-Y. Liu, W.-Q. Cai, M. Yang, L. Li, K.-X. Yang, X. Han, Y.-Q. Yao, J. Li, H.-Y. Wu, S. Wan, L. Liu, D.-Q. Liu, Y.-W. Kuang, Z.-P. He, P. Shang, C. Guo, R.-H. Zheng, K. Tian, Z.-C. Zhu, N.-L. Liu, C.-Y. Lu, R. Shu, Y.-A. Chen, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, “Ground-to-satellite quantum teleportation,” Nature 549, 70–73 (2017).
[Crossref]

S. Pirandola, R. Laurenza, C. Ottaviani, and L. Banchi, “Fundamental limits of repeaterless quantum communications,” Nat. Commun. 8, 15043 (2017).
[Crossref]

2016 (1)

H.-L. Yin, T.-Y. Chen, Z.-W. Yu, H. Liu, L.-X. You, Y.-H. Zhou, S.-J. Chen, Y. Mao, M.-Q. Huang, W.-J. Zhang, H. Chen, M. J. Li, D. Nolan, F. Zhou, X. Jiang, Z. Wang, Q. Zhang, X.-B. Wang, and J.-W. Pan, “Measurement-device-independent quantum key distribution over a 404 km optical fiber,” Phys. Rev. Lett. 117, 190501 (2016).
[Crossref]

2014 (1)

H.-K. Lo, M. Curty, and K. Tamaki, “Secure quantum key distribution,” Nat. Photonics 8, 595–604 (2014).
[Crossref]

2013 (1)

2012 (2)

V. D’Auria, O. Morin, C. Fabre, and J. Laurat, “Effect of the heralding detector properties on the conditional generation of single-photon states,” Eur. Phys. J. D 66, 249 (2012).
[Crossref]

S. Wang, W. Chen, J.-F. Guo, Z.-Q. Yin, H.-W. Li, Z. Zhou, G.-C. Guo, and Z.-F. Han, “2 GHz clock quantum key distribution over 260 km of standard telecom fiber,” Opt. Lett. 37, 1008–1010 (2012).
[Crossref]

2011 (2)

S. Zhang and P. van Loock, “Local Gaussian operations can enhance continuous-variable entanglement distillation,” Phys. Rev. A 84, 062309 (2011).
[Crossref]

J. Fiurášek, “Improving entanglement concentration of Gaussian states by local displacements,” Phys. Rev. A 84, 012335 (2011).
[Crossref]

2010 (2)

S. L. Zhang and P. van Loock, “Distillation of mixed-state continuous-variable entanglement by photon subtraction,” Phys. Rev. A 82, 062316 (2010).
[Crossref]

H. Takahashi, J. S. Neergaard-Nielsen, M. Takeuchi, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, “Entanglement distillation from Gaussian input states,” Nat. Photonics 4, 178–181 (2010).
[Crossref]

2009 (1)

A. Ourjoumtsev, F. Ferreyrol, R. Tualle-Brouri, and P. Grangier, “Preparation of non-local superpositions of quasi-classical light states,” Nat. Phys. 5, 189–192 (2009).
[Crossref]

2008 (1)

A. A. Semenov, A. V. Turchin, and H. V. Gomonay, “Detection of quantum light in the presence of noise,” Phys. Rev. A 78, 055803 (2008).
[Crossref]

2007 (1)

A. Ourjoumtsev, A. Dantan, R. Tualle-Brouri, and P. Grangier, “Increasing entanglement between Gaussian states by coherent photon subtraction,” Phys. Rev. Lett. 98, 030502 (2007).
[Crossref]

2006 (2)

A. Kitagawa, M. Takeoka, M. Sasaki, and A. Chefles, “Entanglement evaluation of non-Gaussian states generated by photon subtraction from squeezed states,” Phys. Rev. A 73, 042310 (2006).
[Crossref]

M. Sasaki and S. Suzuki, “Multimode theory of measurement-induced non-Gaussian operation on wideband squeezed light: analytical formula,” Phys. Rev. A 73, 043807 (2006).
[Crossref]

2005 (5)

M. S. Kim, E. Park, P. L. Knight, and H. Jeong, “Nonclassicality of a photon-subtracted Gaussian field,” Phys. Rev. A 71, 043805 (2005).
[Crossref]

X.-B. Wang, “Beating the photon-number-splitting attack in practical quantum cryptography,” Phys. Rev. Lett. 94, 230503 (2005).
[Crossref]

H.-K. Lo, X. Ma, and K. Chen, “Decoy state quantum key distribution,” Phys. Rev. Lett. 94, 230504 (2005).
[Crossref]

M. B. Plenio, “Logarithmic negativity: a full entanglement monotone that is not convex,” Phys. Rev. Lett. 95, 090503 (2005).
[Crossref]

S. L. Braunstein and P. van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513–577 (2005).
[Crossref]

2004 (1)

D. Gottesman, H.-K. Lo, N. Lükenhaus, and J. Preskill, “Simple security proof of twin-field type quantum key distribution protocol,” Quant. Inf. Comput. 5, 325–360 (2004).
[Crossref]

2003 (2)

W.-Y. Hwang, “Quantum key distribution with high loss: toward global secure communication,” Phys. Rev. Lett. 91, 057901 (2003).
[Crossref]

S. Olivares, M. G. A. Paris, and R. Bonifacio, “Teleportation improvement by inconclusive photon subtraction,” Phys. Rev. A 67, 032314 (2003).
[Crossref]

2002 (4)

G. Vidal and R. F. Werner, “Computable measure of entanglement,” Phys. Rev. A 65, 032314 (2002).
[Crossref]

J. Eisert, S. Scheel, and M. B. Plenio, “Distilling Gaussian states with Gaussian operations is impossible,” Phys. Rev. Lett. 89, 137903 (2002).
[Crossref]

J. Fiurášek, “Gaussian transformations and distillation of entangled Gaussian states,” Phys. Rev. Lett. 89, 137904 (2002).
[Crossref]

G. Giedke and J. I. Cirac, “Characterization of Gaussian operations and distillation of Gaussian states,” Phys. Rev. A 66, 032316 (2002).
[Crossref]

2000 (2)

N. Lütkenhaus, “Security against individual attacks for realistic quantum key distribution,” Phys. Rev. A 61, 052304 (2000).
[Crossref]

T. Opatrný, G. Kurizki, and D.-G. Welsch, “Improvement on teleportation of continuous variables by photon subtraction via conditional measurement,” Phys. Rev. A 61, 032302 (2000).
[Crossref]

Andersen, U. L.

Autebert, C.

A. Boaron, G. Boso, D. Rusca, C. Vulliez, C. Autebert, M. Caloz, M. Perrenoud, G. Gras, F. Bussières, M.-J. Li, D. Nolan, A. Martin, and H. Zbinden, “Secure quantum key distribution over 421 km of optical fiber,” Phys. Rev. Lett. 121, 190502 (2018).
[Crossref]

Azuma, K.

M. Curty, K. Azuma, and H.-K. Lo, “Simple security proof of twin-field type quantum key distribution protocol,” npj Quantum Inf. 5, 64 (2019).
[Crossref]

Banchi, L.

S. Pirandola, R. Laurenza, C. Ottaviani, and L. Banchi, “Fundamental limits of repeaterless quantum communications,” Nat. Commun. 8, 15043 (2017).
[Crossref]

Boaron, A.

A. Boaron, G. Boso, D. Rusca, C. Vulliez, C. Autebert, M. Caloz, M. Perrenoud, G. Gras, F. Bussières, M.-J. Li, D. Nolan, A. Martin, and H. Zbinden, “Secure quantum key distribution over 421 km of optical fiber,” Phys. Rev. Lett. 121, 190502 (2018).
[Crossref]

Bonifacio, R.

S. Olivares, M. G. A. Paris, and R. Bonifacio, “Teleportation improvement by inconclusive photon subtraction,” Phys. Rev. A 67, 032314 (2003).
[Crossref]

Boso, G.

A. Boaron, G. Boso, D. Rusca, C. Vulliez, C. Autebert, M. Caloz, M. Perrenoud, G. Gras, F. Bussières, M.-J. Li, D. Nolan, A. Martin, and H. Zbinden, “Secure quantum key distribution over 421 km of optical fiber,” Phys. Rev. Lett. 121, 190502 (2018).
[Crossref]

Braunstein, S. L.

S. L. Braunstein and P. van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513–577 (2005).
[Crossref]

Bussières, F.

A. Boaron, G. Boso, D. Rusca, C. Vulliez, C. Autebert, M. Caloz, M. Perrenoud, G. Gras, F. Bussières, M.-J. Li, D. Nolan, A. Martin, and H. Zbinden, “Secure quantum key distribution over 421 km of optical fiber,” Phys. Rev. Lett. 121, 190502 (2018).
[Crossref]

Cai, W.-Q.

S.-K. Liao, W.-Q. Cai, W.-Y. Liu, L. Zhang, Y. Li, J.-G. Ren, J. Yin, Q. Shen, Y. Cao, Z.-P. Li, F.-Z. Li, X.-W. Chen, L.-H. Sun, J.-J. Jia, J.-C. Wu, X.-J. Jiang, J.-F. Wang, Y.-M. Huang, Q. Wang, Y.-L. Zhou, L. Deng, T. Xi, L. Ma, T. Hu, Q. Zhang, Y.-A. Chen, N.-L. Liu, X.-B. Wang, Z.-C. Zhu, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, “Satellite-to-ground quantum key distribution,” Nature 549, 43–47 (2017).
[Crossref]

J.-G. Ren, P. Xu, H.-L. Yong, L. Zhang, S.-K. Liao, J. Yin, W.-Y. Liu, W.-Q. Cai, M. Yang, L. Li, K.-X. Yang, X. Han, Y.-Q. Yao, J. Li, H.-Y. Wu, S. Wan, L. Liu, D.-Q. Liu, Y.-W. Kuang, Z.-P. He, P. Shang, C. Guo, R.-H. Zheng, K. Tian, Z.-C. Zhu, N.-L. Liu, C.-Y. Lu, R. Shu, Y.-A. Chen, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, “Ground-to-satellite quantum teleportation,” Nature 549, 70–73 (2017).
[Crossref]

Caloz, M.

A. Boaron, G. Boso, D. Rusca, C. Vulliez, C. Autebert, M. Caloz, M. Perrenoud, G. Gras, F. Bussières, M.-J. Li, D. Nolan, A. Martin, and H. Zbinden, “Secure quantum key distribution over 421 km of optical fiber,” Phys. Rev. Lett. 121, 190502 (2018).
[Crossref]

Cao, Y.

S.-K. Liao, W.-Q. Cai, W.-Y. Liu, L. Zhang, Y. Li, J.-G. Ren, J. Yin, Q. Shen, Y. Cao, Z.-P. Li, F.-Z. Li, X.-W. Chen, L.-H. Sun, J.-J. Jia, J.-C. Wu, X.-J. Jiang, J.-F. Wang, Y.-M. Huang, Q. Wang, Y.-L. Zhou, L. Deng, T. Xi, L. Ma, T. Hu, Q. Zhang, Y.-A. Chen, N.-L. Liu, X.-B. Wang, Z.-C. Zhu, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, “Satellite-to-ground quantum key distribution,” Nature 549, 43–47 (2017).
[Crossref]

Chefles, A.

A. Kitagawa, M. Takeoka, M. Sasaki, and A. Chefles, “Entanglement evaluation of non-Gaussian states generated by photon subtraction from squeezed states,” Phys. Rev. A 73, 042310 (2006).
[Crossref]

Chen, H.

H.-L. Yin, T.-Y. Chen, Z.-W. Yu, H. Liu, L.-X. You, Y.-H. Zhou, S.-J. Chen, Y. Mao, M.-Q. Huang, W.-J. Zhang, H. Chen, M. J. Li, D. Nolan, F. Zhou, X. Jiang, Z. Wang, Q. Zhang, X.-B. Wang, and J.-W. Pan, “Measurement-device-independent quantum key distribution over a 404 km optical fiber,” Phys. Rev. Lett. 117, 190501 (2016).
[Crossref]

Chen, K.

H.-K. Lo, X. Ma, and K. Chen, “Decoy state quantum key distribution,” Phys. Rev. Lett. 94, 230504 (2005).
[Crossref]

Chen, S.-J.

H.-L. Yin, T.-Y. Chen, Z.-W. Yu, H. Liu, L.-X. You, Y.-H. Zhou, S.-J. Chen, Y. Mao, M.-Q. Huang, W.-J. Zhang, H. Chen, M. J. Li, D. Nolan, F. Zhou, X. Jiang, Z. Wang, Q. Zhang, X.-B. Wang, and J.-W. Pan, “Measurement-device-independent quantum key distribution over a 404 km optical fiber,” Phys. Rev. Lett. 117, 190501 (2016).
[Crossref]

Chen, T.-Y.

H.-L. Yin, T.-Y. Chen, Z.-W. Yu, H. Liu, L.-X. You, Y.-H. Zhou, S.-J. Chen, Y. Mao, M.-Q. Huang, W.-J. Zhang, H. Chen, M. J. Li, D. Nolan, F. Zhou, X. Jiang, Z. Wang, Q. Zhang, X.-B. Wang, and J.-W. Pan, “Measurement-device-independent quantum key distribution over a 404 km optical fiber,” Phys. Rev. Lett. 117, 190501 (2016).
[Crossref]

Chen, W.

Chen, X.-W.

S.-K. Liao, W.-Q. Cai, W.-Y. Liu, L. Zhang, Y. Li, J.-G. Ren, J. Yin, Q. Shen, Y. Cao, Z.-P. Li, F.-Z. Li, X.-W. Chen, L.-H. Sun, J.-J. Jia, J.-C. Wu, X.-J. Jiang, J.-F. Wang, Y.-M. Huang, Q. Wang, Y.-L. Zhou, L. Deng, T. Xi, L. Ma, T. Hu, Q. Zhang, Y.-A. Chen, N.-L. Liu, X.-B. Wang, Z.-C. Zhu, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, “Satellite-to-ground quantum key distribution,” Nature 549, 43–47 (2017).
[Crossref]

Chen, Y.-A.

S.-K. Liao, W.-Q. Cai, W.-Y. Liu, L. Zhang, Y. Li, J.-G. Ren, J. Yin, Q. Shen, Y. Cao, Z.-P. Li, F.-Z. Li, X.-W. Chen, L.-H. Sun, J.-J. Jia, J.-C. Wu, X.-J. Jiang, J.-F. Wang, Y.-M. Huang, Q. Wang, Y.-L. Zhou, L. Deng, T. Xi, L. Ma, T. Hu, Q. Zhang, Y.-A. Chen, N.-L. Liu, X.-B. Wang, Z.-C. Zhu, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, “Satellite-to-ground quantum key distribution,” Nature 549, 43–47 (2017).
[Crossref]

J.-G. Ren, P. Xu, H.-L. Yong, L. Zhang, S.-K. Liao, J. Yin, W.-Y. Liu, W.-Q. Cai, M. Yang, L. Li, K.-X. Yang, X. Han, Y.-Q. Yao, J. Li, H.-Y. Wu, S. Wan, L. Liu, D.-Q. Liu, Y.-W. Kuang, Z.-P. He, P. Shang, C. Guo, R.-H. Zheng, K. Tian, Z.-C. Zhu, N.-L. Liu, C.-Y. Lu, R. Shu, Y.-A. Chen, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, “Ground-to-satellite quantum teleportation,” Nature 549, 70–73 (2017).
[Crossref]

Cirac, J. I.

G. Giedke and J. I. Cirac, “Characterization of Gaussian operations and distillation of Gaussian states,” Phys. Rev. A 66, 032316 (2002).
[Crossref]

Curty, M.

M. Curty, K. Azuma, and H.-K. Lo, “Simple security proof of twin-field type quantum key distribution protocol,” npj Quantum Inf. 5, 64 (2019).
[Crossref]

H.-K. Lo, M. Curty, and K. Tamaki, “Secure quantum key distribution,” Nat. Photonics 8, 595–604 (2014).
[Crossref]

D’Auria, V.

V. D’Auria, O. Morin, C. Fabre, and J. Laurat, “Effect of the heralding detector properties on the conditional generation of single-photon states,” Eur. Phys. J. D 66, 249 (2012).
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Dantan, A.

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S.-K. Liao, W.-Q. Cai, W.-Y. Liu, L. Zhang, Y. Li, J.-G. Ren, J. Yin, Q. Shen, Y. Cao, Z.-P. Li, F.-Z. Li, X.-W. Chen, L.-H. Sun, J.-J. Jia, J.-C. Wu, X.-J. Jiang, J.-F. Wang, Y.-M. Huang, Q. Wang, Y.-L. Zhou, L. Deng, T. Xi, L. Ma, T. Hu, Q. Zhang, Y.-A. Chen, N.-L. Liu, X.-B. Wang, Z.-C. Zhu, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, “Satellite-to-ground quantum key distribution,” Nature 549, 43–47 (2017).
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Figures (8)

Fig. 1.
Fig. 1. (a) Schemes of biside photon subtraction on a biside decayed two-mode squeezed state (TMSS). Successful distribution is established when both single-photon detectors register a click: $({D_E},{D_F})=(1,1)$. (b) Scheme for one-time photon subtraction on biside decayed TMSS. Successful distribution is realized when the detectors register $({D_E},{D_F})=(0,1)$ or (1,0). Beam splitters BS1, BS2, BS5, and BS6 have a transmittance of $\sqrt\eta$; they are used to simulate channel loss on both half-way decays. BS3 and BS4 have a transmittance $T$ for photon subtraction. BS7 has a transmittance 0.5. For comparison, ${D_E}$ an ${D_F}$ are assumed for the ideal single-photon detector. The trash boxes denote the discarding operation.
Fig. 2.
Fig. 2. (a) ${e_1},{e_2}$ and (b) ${p_1},-{p_2},{p_3}$ (in logarithmic scale) as a function of $\lambda$. Other parameter: $T=0.9$.
Fig. 3.
Fig. 3. Entanglement (a) and success probability (b) in logarithmic scale as a function of transmission distance $L$. (c) Success probability as a function of distilled entanglement in biside PS scheme [Fig. 1(a)] and one-time PS scheme [Fig. 1(b)]. Photon in each optical mode is truncated within the $\{|0\rangle ,|1\rangle ,\cdots ,|D-1\rangle\}$ subspace for numerical convergence and computing feasibility. Other parameters: $T=0.9,D=6,\lambda=0.1$. (d) Entanglement as a function of dimension of photon number subsapce $D$. Other parameters: $T=0.9,L=10.52\;{\rm km}$, $\lambda=0.10$.
Fig. 4.
Fig. 4. (a) Entanglement and (b) success probability in logarithmic scale as a function of transmission distance $L$. (c) Success probability as a function of distilled entanglement in biside PS scheme [Fig. 1(a)] and one-time PS scheme [Fig. 1(b)]. Photon in each optical mode is truncated within the $\{|0\rangle ,|1\rangle ,\cdots ,|D-1\rangle\}$ subspace for numerical convergence and computing feasibility. Other parameters: $T=0.9, D=18,\lambda=0.71$. (d) Entanglement as a function of dimension of photon number subsapce $D$. Other parameters: $T=0.9, L=10.52\;{\rm km}$, $\lambda=0.71$.
Fig. 5.
Fig. 5. (a) Entanglement and (b) success probability (in logarithmic scale) of distilled state with on–off photon detectors as a function of transmission distance $L$. (c) Success probability as a function of distilled entanglement in biside PS scheme [Fig. 1(a)] and one-time PS scheme [Fig. 1(b)]. Photon in each optical mode is truncated within the $\{|0\rangle ,|1\rangle ,\cdots ,|D-1\rangle\}$ subspace for numerical convergence and computing feasibility. Other parameters: $T=0.9,D=6,\lambda=0.10$. (d) Entanglement as a function of dimension of photon number subsapce $D$. Other parameters: $T=0.9,L=10.52\;{\rm km}$, $\lambda=0.10$.
Fig. 6.
Fig. 6. (a) Entanglement and (b) success probability (in logarithmic scale) of distilled state with biside PS scheme [Fig. 1(a)] and one-time PS scheme [Fig. 1(b)] for $\alpha=0,0.10,0.20,$ and 0.32. Photon in each optical mode is truncated within the $\{|0\rangle ,|1\rangle ,\cdots ,|7\rangle\}$ subspace. Other parameters: $T=0.9,D=8,\lambda=0.5$.
Fig. 7.
Fig. 7. (a) Entanglement and (b) success probability (in logarithmic scale) of distilled state with biside PS scheme and one-time PS scheme for $\alpha :0\sim 0.32$. (c) Success probability as a function of distilled entanglement in Figs. 1(a) and 1(b). Photon in each optical mode is truncated within the $\{|0\rangle ,|1\rangle ,\cdots ,|D-1\rangle\}$ subspace for numerical convergence and computing feasibility. Other parameters: $T=0.9,D=8,\lambda=0.3$. (d) Entanglement as a function of increasing $D$, for $\alpha=0.32,T=0.9,\lambda=0.3,L=14.29\;{\rm km}$.
Fig. 8.
Fig. 8. (a) Entanglement before and after one-time PS distillation of ${\rho_{{\rm mix}}}$ for increasing values of $\lambda$. Other parameters: $T=0.90,\epsilon=0.20,L=5\;{\rm km}$, ${N_B}=0.1$. Photon in each optical mode is truncated at $D=8$. (b) Entanglement as a function of the size of photon number subspace. Other parameters: $T=0.90,\epsilon=0.20,L=5\;{\rm km}$, ${N_B}=0.1.\lambda=\tanh (0.4)=0.3799$.

Equations (74)

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| ψ T M S S = n = 0 1 λ 2 λ n | n A | n B , λ = tanh ( r ) .
P ( 1 , 1 ) = 1 λ 2 1 λ 2 T ~ 2 [ λ η ( 1 T ) 1 λ 2 T ~ 2 ] 2 ( 1 + ( λ T ~ ) 2 ) ,
T ~ = 1 η ( 1 T ) ,
E N ( ρ ) = log 2 ρ Γ A = log 2 [ 1 + 2 N ( ρ Γ A ) ] ,
E N = 3 log 2 [ 1 + λ T ~ 1 λ ( η T + η 1 ) ] + log 2 [ ( λ λ η ) 2 + ( 1 λ η T ) 2 ] log 2 [ 1 + ( λ T ~ ) 2 ] .
P ( 1 , 1 ) = p 1 η + p 2 η 3 2 + p 3 η 2 + O ( η 5 2 ) ,
p 1 = ( 1 T ) 2 λ 2 ( 1 + λ 2 ) ( 1 + λ 2 ) 2 ,
p 2 = 4 ( 1 T ) 3 λ 4 ( 2 + λ 2 ) ( 1 λ 2 ) 3 ,
p 3 = 2 ( 1 T ) 4 λ 4 ( 2 + 17 λ 2 + 5 λ 4 ) ( 1 λ 2 ) 4 ,
E N = e 1 η + e 2 η + O ( η 3 2 ) ,
e 1 = 4 T λ ( 1 λ + λ 2 ) ( 1 + λ + λ 2 + λ 3 ) ln 2 ,
e 2 = 2 T λ 2 ( T ( 1 + λ ) ( 1 + λ ) 3 + 2 ( 2 + λ 2 2 λ 3 + λ 4 ) ) ( 1 + λ ) 2 ( 1 + λ 2 ) 2 ln 2 .
U kl ( t ) = exp [ arctan ( 1 t t ) ( a k a l a k a l ) ] .
| ψ ABCD = U AC ( η ) U BD ( η ) ( | ψ T M S S | 00 CD ) .
| ψ ABCDEF = U AE ( T ) U BF ( T ) ( | ψ ABCD | 00 EF ) .
ρ ABEF = T r CDGH [ U ( | ψ ABCDEF ψ ABCDEF | | 00 GH 00 | ) U ] , U = U EG ( η ) U FH ( η ) .
P m 1 , m 2 ρ AB = T r EF [ U EF ( 1 2 ) ρ ABEF U EF ( 1 2 ) ( I AB | m 1 , m 2 EF m 1 , m 2 | ) ] ,
P m 1 , m 2 = T r [ U EF ( 1 2 ) ρ ABEF U EF ( 1 2 ) ( I AB | m 1 , m 2 EF m 1 , m 2 | ) ] ,
P ( 0 , 1 ) ( λ 1 ) λ 2 ( 1 T ) η ,
P ( 0 , 1 ) ( λ O ( 1 ) ) ( 1 T ) η ,
P ( a ) ( o n , o n ) = λ 2 ( 1 T ~ ) 2 ( 1 + λ 2 T ~ ) ( 1 λ 2 T ~ ) ( 1 λ 2 T ~ 2 ) = q 1 η + q 2 η 3 2 + q 3 η 2 + O ( η 5 2 ) ,
E N ( a ) ( o n , o n ) = log 2 [ ( 1 λ 2 ) η R ( 1 λ η T ) 2 λ 2 ( 1 η ) R ~ ] + log 2 [ R ~ ( 1 λ ) ( 1 λ ( 2 η T 1 ) ) 1 η ( 1 λ η T ) 2 λ 2 ( 1 η ) 2 ] + log 2 [ ( 1 λ 2 T ~ ) ( 1 λ 2 T ~ 2 ) ( 1 T ~ ) 2 ( 1 + λ 2 T ~ ) ] = f 1 η + f 2 η + O ( η 3 2 ) ,
q 1 = ( 1 + T ) 2 λ 2 ( 1 + λ 2 ) ( 1 + λ 2 ) 2 ,
q 2 = 2 ( 1 T ) 3 λ 4 ( 2 + λ 2 ) ( 1 λ 2 ) 3 ,
q 3 = ( 1 T ) 4 λ 4 ( 1 + 10 λ 2 + 3 λ 4 ) ( 1 λ 2 ) 4 ,
f 1 = 4 T λ ( 1 λ + λ 2 ) ( 1 + λ + λ 2 + λ 3 ) ln 2 ,
f 2 = 2 T λ 2 ( 2 + λ 2 2 λ 3 + λ 4 + T ( 1 2 λ + λ 2 + 2 λ 4 ) ) ( 1 + λ + λ 2 + λ 3 ) 2 ln 2 .
W ( x ) = R 2 N d 2 N ξ ( 2 π ) 2 N exp [ i x ξ T ] χ ( ξ ) ,
χ ( ξ , V , R ¯ ) = exp [ 1 2 ξ V ξ T + i R ¯ ξ T ] ,
S t ( k l ) = ( t 1 t t 1 t 1 t t 1 t t ) .
V AB = c 2 I 4 + s 2 σ x σ z ,
V ABCDEFGH = S ( V AB + 1 2 I 12 ) S T ,
S = S 1 / 2 EF ( S η EG S η FH ) ( S T AE S T BF ) ( S η AC S η BD ) .
V ABEF = V ABCDEFGH ( 1 , 2 , 3 , 4 , 9 , 10 , 11 , 12 ) .
P o f f , o n W ( x A B ) = ( 2 π ) 2 R 4 W ( V A B E F , 0 4 × 1 ) ( x A B E F ) W ( o f f , o n ) ( x E F ) d x E F .
W = d 0 I d 0 I d 00 W { V 0 I , 0 4 × 1 } ( x A B ) d 00 d 0 I d 00 W { V 00 , 0 4 × 1 } ( x A B ) ,
d 00 = 1 1 ( 1 T ) η ( 2 + ( 1 T ) η ) sinh ( r ) 2 ,
d 0 I = 1 1 ( 1 T ) η ( 2 + ( 1 T ) η ) sinh ( r ) 2 ,
V 0 I = ( α 1 β 1 δ 1 γ 1 β 1 α 1 γ 1 δ 1 ) ,
V 00 = ( α 2 β 2 α 2 β 2 β 2 α 2 β 2 α 2 ) ,
β 1 = e r T η sinh ( r ) ( 2 cosh ( r ) + ( T 1 ) η sinh ( r ) ) 2 ( cosh ( r ) + ( 1 + ( T 1 ) η ) sinh ( r ) ) ,
γ 1 = T η sinh ( r ) ( 2 cosh ( r ) ( T 1 ) η sinh ( r ) ) 2 + ( 1 + e 2 r ) ( T 1 ) η ,
α 2 = 1 2 ( 1 + 2 T η ( η T η 1 ) sinh ( r ) 2 ( 1 + ( T 1 ) η ) 2 sinh ( r ) 2 cosh ( r ) 2 ) ,
β 2 = T η sinh ( 2 r ) 2 + 2 ( 1 T ) η ( 2 + ( T 1 ) η ) sinh ( r ) 2 ,
α 1 = e r ( x + y + z ) 4 ( cosh ( r ) + ( 1 + ( T 1 ) η ) sinh ( r ) ) ,
δ 1 = e r ( x + y z ) 4 ( cosh ( r ) + ( η T η 1 ) sinh ( r ) ) ,
x = ( 1 T η ) ( 2 + ( T 1 ) η ) ,
y = η ( 2 T ( T 1 ) η + ( T 1 ) T η ) cosh ( 2 r ) ,
z = ( T 1 ) η sinh ( 2 r ) .
P o f f , o n = d 0 I d 00
= 1 ( 1 T ) η ( 2 + ( 1 T ) η ) sinh ( r ) 2 1 1 ( 1 T ) η ( 2 + ( 1 T ) η ) sinh ( r ) 2
= s 1 η + s 2 η 2 + O ( η 2 ) ,
s 1 = ( 1 T ) sinh ( r ) 2 ,
s 2 = 1 4 ( T 1 ) 2 ( 3 + 5 cosh ( 2 r ) ) sinh ( r ) 2 .
n E = n F = ( 1 T ) λ 2 η 1 λ 2 .
W = d 0 I d 0 I d 00 α W { V 0 I , R 0 I } ( x A B ) d 00 α d 0 I d 00 α W { V 00 , R 00 } ( x A B ) ,
d 00 α = d 00 e 2 T α 2 η [ cosh ( r ) + sinh ( r ) ] [ cosh ( r ) + ( η T η 1 ) sinh ( r ) ] ,
R 00 = ( ζ , 0 , ζ , 0 ) ,
R 0 I = ( 2 2 T α , 0 , 2 2 T α , 0 ) ,
ζ = 2 2 T α [ 1 + T η sinh ( r ) cosh ( r ) + ( η T η 1 ) sinh ( r ) ] .
P o f f , o n α = d 0 I d 00 α
= u 1 η + u 2 η + u 3 η 3 / 2 + O ( η 2 ) ,
u 1 = 2 T α 2 ,
u 2 = ( 1 T ) sinh ( r ) 2 2 T 2 α 4 ,
u 3 = T α 2 3 ( 4 T 2 α 4 3 ( 1 T ) ( 3 cosh [ 2 r ] + sinh [ 2 r ] 3 ) ) .
ρ m i x = ϵ | ψ TMSS ψ | + ( 1 ϵ ) ρ t h ρ t h ,
W ρ d i s t = ϵ W + ( 1 ϵ ) W ,
W = d 0 I d 0 I d 00 W { V 0 I , 0 4 × 1 } ( x A B ) d 00 d 0 I d 00 W { V 00 , 0 4 × 1 } ( x A B ) , d 0 I = 1 1 + 1 2 N B ( 1 T ) ,
d 00 = 1 ( 1 + 1 2 N B ( 1 T ) ) 2 ,
V 0 I = ( α 1 β 1 α 1 β 1 β 1 α 1 β 1 α 1 ) ,
V 00 = d i a g ( α 2 , α 2 , α 2 , α 2 ) ,
α 1 = 4 + 2 N B 2 ( 1 + T ) T + N B ( 2 + ( 2 4 2 ) T ) 8 + 4 N B ( 1 + T ) ,
β 1 = N B 2 ( 1 + T ) T 2 2 ( 2 + N B ( 1 + T ) ) ,
α 2 = 1 + N B ( 1 2 T 2 + 2 T ) 2 N B ( 1 + T ) .

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