Abstract

Recently, the strong single-mode squeezed vacuum state of light with $-15$ dB of classic noise has been created [Phys. Rev. Lett. 117, 110801 (2016) [CrossRef]  ]. This is an important resource in continuous variable (CV) quantum communication and fault-tolerant quantum computing. Nevertheless, strong squeezing means a non-negligible population in high-photon-number subspace and may pose a strong challenge to photon-based technology, for example, entanglement distillation. Entanglement distillation is an efficient method for retrieving a higher quality of entanglement from a weakly entangled state. Up till now, almost all the schemes for entanglement distillation are restricted to the low-squeezing regime. The distillation of strong-squeezing induced entanglement is an interesting but open topic in the near future. Here, we take the single-mode squeezed entangled state (SMSE) as an example and show that conventional photon subtraction based distillation fails to generate a higher quality of entanglement with experimentally feasible optical beam splitters. To this point, we show that the superposition of photon annihilation and creation could be an effective method. A practical scheme for implementing such a superposition with on-line squeezing is suggested. Our method is verified numerically by the calculation of a 25 dB SMSE state with an ideal photon detector and for distillation of 20.5 dB SMSE state with practical dichotic on-off detectors. The analysis of the distillation of the strong squeezing effect could be extended straightforwardly to other squeezing-based quantum information processing, such as quantum metrology and quantum illumination.

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

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References

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2019 (1)

L. Barsotti, J. Harms, and R. Schnabel, “Squeezed vacuum states of light for gravitational wave detectors,” Rep. Prog. Phys. 82(1), 016905 (2019).
[Crossref]

2018 (1)

E. Molnar, P. Adam, G. Mogyorosi, and M. Mechler, “Quantum state engineering via coherent-state superpositions in traveling optical fields,” Phys. Rev. A 97(2), 023818 (2018).
[Crossref]

2017 (1)

L. Y. Hu, Z. Y. Liao, and M. Suhail Zubairy, “Continuous-variable entanglement via multiphoton catalysis,” Phys. Rev. A 95(1), 012310 (2017).
[Crossref]

2016 (1)

H. Vahlbruch, M. Mehmet, K. Danzmann, and R. Schnabel, “Detection of 15 dB Squeezed States of Light and their Application for the Absolute Calibration of Photoelectric Quantum Efficiency,” Phys. Rev. Lett. 117(11), 110801 (2016).
[Crossref]

2014 (1)

N. C. Menicucci, “Fault-Tolerant Measurement-Based Quantum Computing with Continuous-Variable Cluster States,” Phys. Rev. Lett. 112(12), 120504 (2014).
[Crossref]

2013 (4)

S. L. Zhang, Y. L. Dong, J. H. Shi, X. B. Zou, and G. C. Guo, “Roles of thermal noise and detector efficiency in distillation of continuous variable entanglement state,” J. Opt. Soc. Am. B 30(10), 2704–2709 (2013).
[Crossref]

A. Tipsmark, J. S. Neergaard-Nielsen, and U. L. Andersen, “Displacement-enhanced entanglement distillation of single-mode-squeezed entangled states,” Opt. Express 21(6), 6670–6680 (2013).
[Crossref]

J. Lee and H. Nha, “Entanglement distillation for continuous variables in a thermal environment: Effectiveness of a non-Gaussian operation,” Phys. Rev. A 87(3), 032307 (2013).
[Crossref]

S.-Y. Lee, S.-W. Ji, and C.-W. Lee, “Increasing and decreasing entanglement characteristics for continuous variables by a local photon subtraction,” Phys. Rev. A 87(5), 052321 (2013).
[Crossref]

2012 (3)

H.-J. Kim, S.-Y. Lee, S.-W. Ji, and H. Nha, “Quantum linear amplifier enhanced by photon subtraction and addition,” Phys. Rev. A 85(1), 013839 (2012).
[Crossref]

C.-W. Lee, J. Lee, H. Nha, and H. Jeong, “Generating a Schrödinger-cat-like state via a coherent superposition of photonic operations,” Phys. Rev. A 85(6), 063815 (2012).
[Crossref]

O. Černotík and J. Fiurá šek, “Displacement-enhanced continuous-variable entanglement concentration,” Phys. Rev. A 86(5), 052339 (2012).
[Crossref]

2011 (4)

S.-Y. Lee, S.-W. Ji, H.-J. Kim, and H. Nha, “Enhancing quantum entanglement for continuous variables by a coherent superposition of photon subtraction and addition,” Phys. Rev. A 84(1), 012302 (2011).
[Crossref]

S. L. Zhang and P. van Loock, “Local Gaussian operations can enhance continuous-variable entanglement distillation,” Phys. Rev. A 84(6), 062309 (2011).
[Crossref]

R. Ukai, N. Iwata, Y. Shimokawa, S. C. Armstrong, A. Politi, J.-i. Yoshikawa, P. van Loock, and A. Furusawa, “Demonstration of Unconditional One-Way Quantum Computations for Continuous Variables,” Phys. Rev. Lett. 106(24), 240504 (2011).
[Crossref]

J. Fiurášek, “Improving entanglement concentration of Gaussian states by local displacements,” Phys. Rev. A 84(1), 012335 (2011).
[Crossref]

2010 (4)

S.-Y. Lee and H. Nha, “Quantum state engineering by a coherent superposition of photon subtraction and addition,” Phys. Rev. A 82(5), 053812 (2010).
[Crossref]

L. Lydersen, C. Wiechers, C. Wittmann, D. Elser, J. Skaar, and V. Makarov, “Hacking commercial quantum cryptography systems by tailored bright illumination,” Nat. Photonics 4(10), 686–689 (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(3), 178–181 (2010).
[Crossref]

B. Hage, A. Samblowski, J. DiGuglielmo, J. Fiurášek, and R. Schnabel, “Iterative Entanglement Distillation: Approaching the Elimination of Decoherence,” Phys. Rev. Lett. 105(23), 230502 (2010).
[Crossref]

2009 (1)

A. P. Lund and T. C. Ralph, “Continuous-variable entanglement distillation over a general lossy channel,” Phys. Rev. A 80(3), 032309 (2009).
[Crossref]

2008 (2)

B. Hage, A. Samblowski, J. Diguglielmo, A. Franzen, J. Fiurášek, and R. Schnabel, “Preparation of distilled and purified continuous- variable entangled states,” Nat. Phys. 4(12), 915–918 (2008).
[Crossref]

A. A. Semenov, A. V. Turchin, and H. V. Gomonay, “Detection of quantum light in the presence of noise,” Phys. Rev. A 78(5), 055803 (2008).
[Crossref]

2007 (2)

V. Parigi, A. Zavatta, M. S. Kim, and M. Bellini, “Probing Quantum Commutation Rules by Addition and Subtraction of Single Photons to/from a Light Field,” Science 317(5846), 1890–1893 (2007).
[Crossref]

A. Zavatta, V. Parigi, and M. Bellini, “Experimental nonclassicality of single-photon-added thermal light states,” Phys. Rev. A 75(5), 052106 (2007).
[Crossref]

2006 (3)

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(4), 042310 (2006).
[Crossref]

K. Sanaka, K. J. Resch, and A. Zeilinger, “Filtering Out Photonic Fock States,” Phys. Rev. Lett. 96(8), 083601 (2006).
[Crossref]

R. Reichle, D. Leibfried, E. Knill, J. Britton, and R. Blakestad, “Experimental purification of two-atom entanglement,” Nature 443(7113), 838–841 (2006).
[Crossref]

2005 (5)

R. Filip, P. Marek, and U. L. Andersen, “Measurement-induced continuous-variable quantum interactions,” Phys. Rev. A 71(4), 042308 (2005).
[Crossref]

M. B. Plenio, “Logarithmic Negativity: A Full Entanglement Monotone That is not Convex,” Phys. Rev. Lett. 95(9), 090503 (2005).
[Crossref]

S. Olivares and M. G. A. Paris, “Photon subtracted states and enhancement of nonlocality in the presence of noise,” J. Opt. B: Quantum Semiclassical Opt. 7(10), S392–S397 (2005).
[Crossref]

S. Olivares and M. G. A. Paris, “Squeezed Fock state by inconclusive photon subtraction,” J. Opt. B: Quantum Semiclassical Opt. 7(12), S616–S621 (2005).
[Crossref]

A. Zavatta, S. Viciani, and M. Bellini, “Single-photon excitation of a coherent state: Catching the elementary step of stimulated light emission,” Phys. Rev. A 72(2), 023820 (2005).
[Crossref]

2004 (1)

A. Zavatta, S. Viciani, and M. Bellin, “Quantum-to-Classical Transition with Single-Photon-Added Coherent States of Light,” Science 306(5696), 660–662 (2004).
[Crossref]

2003 (1)

T. Yamamoto, M. Koashi, Ş. K. Özdemir, and N. Imoto, “Experimental extraction of an entangled photon pair from two identically decohered pairs,” Nature 421(6921), 343–346 (2003).
[Crossref]

2002 (5)

A. I. Lvovsky and J. Mlynek, “Quantum-Optical Catalysis: Generating Nonclassical States of Light by Means of Linear Optics,” Phys. Rev. Lett. 88(25), 250401 (2002).
[Crossref]

J. Eisert, S. Scheel, and M. B. Plenio, “Distilling Gaussian States with Gaussian Operations is Impossible,” Phys. Rev. Lett. 89(13), 137903 (2002).
[Crossref]

J. Fiurášek, “Gaussian Transformations and Distillation of Entangled Gaussian States,” Phys. Rev. Lett. 89(13), 137904 (2002).
[Crossref]

G. Giedke and J. I. Cirac, “Characterization of Gaussian operations and distillation of Gaussian states,” Phys. Rev. A 66(3), 032316 (2002).
[Crossref]

G. Vidal and R. F. Werner, “omputable measure of entanglement,” Phys. Rev. A 65(3), 032314 (2002).
[Crossref]

2001 (2)

P. G. Kwiat, S. Barraza-Lopez, A. Stefanov, and N. Gisin, “Experimental entanglement distillation and ’hidden’ non-locality,” Nature 409(6823), 1014–1017 (2001).
[Crossref]

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

2000 (2)

L. M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Entanglement Purification of Gaussian Continuous Variable Quantum States,” Phys. Rev. Lett. 84(17), 4002–4005 (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(3), 032302 (2000).
[Crossref]

1998 (1)

S. L. Braunstein and H. J. Kimble, “Teleportation of Continuous Quantum Variables,” Phys. Rev. Lett. 80(4), 869–872 (1998).
[Crossref]

1991 (1)

G. S. Agarwal and K. Tara, “Nonclassical properties of states generated by the excitations on a coherent state,” Phys. Rev. A 43(1), 492–497 (1991).
[Crossref]

Adam, P.

E. Molnar, P. Adam, G. Mogyorosi, and M. Mechler, “Quantum state engineering via coherent-state superpositions in traveling optical fields,” Phys. Rev. A 97(2), 023818 (2018).
[Crossref]

Agarwal, G. S.

G. S. Agarwal and K. Tara, “Nonclassical properties of states generated by the excitations on a coherent state,” Phys. Rev. A 43(1), 492–497 (1991).
[Crossref]

Andersen, U. L.

A. Tipsmark, J. S. Neergaard-Nielsen, and U. L. Andersen, “Displacement-enhanced entanglement distillation of single-mode-squeezed entangled states,” Opt. Express 21(6), 6670–6680 (2013).
[Crossref]

R. Filip, P. Marek, and U. L. Andersen, “Measurement-induced continuous-variable quantum interactions,” Phys. Rev. A 71(4), 042308 (2005).
[Crossref]

Armstrong, S. C.

R. Ukai, N. Iwata, Y. Shimokawa, S. C. Armstrong, A. Politi, J.-i. Yoshikawa, P. van Loock, and A. Furusawa, “Demonstration of Unconditional One-Way Quantum Computations for Continuous Variables,” Phys. Rev. Lett. 106(24), 240504 (2011).
[Crossref]

Barraza-Lopez, S.

P. G. Kwiat, S. Barraza-Lopez, A. Stefanov, and N. Gisin, “Experimental entanglement distillation and ’hidden’ non-locality,” Nature 409(6823), 1014–1017 (2001).
[Crossref]

Barsotti, L.

L. Barsotti, J. Harms, and R. Schnabel, “Squeezed vacuum states of light for gravitational wave detectors,” Rep. Prog. Phys. 82(1), 016905 (2019).
[Crossref]

Bellin, M.

A. Zavatta, S. Viciani, and M. Bellin, “Quantum-to-Classical Transition with Single-Photon-Added Coherent States of Light,” Science 306(5696), 660–662 (2004).
[Crossref]

Bellini, M.

V. Parigi, A. Zavatta, M. S. Kim, and M. Bellini, “Probing Quantum Commutation Rules by Addition and Subtraction of Single Photons to/from a Light Field,” Science 317(5846), 1890–1893 (2007).
[Crossref]

A. Zavatta, V. Parigi, and M. Bellini, “Experimental nonclassicality of single-photon-added thermal light states,” Phys. Rev. A 75(5), 052106 (2007).
[Crossref]

A. Zavatta, S. Viciani, and M. Bellini, “Single-photon excitation of a coherent state: Catching the elementary step of stimulated light emission,” Phys. Rev. A 72(2), 023820 (2005).
[Crossref]

Blakestad, R.

R. Reichle, D. Leibfried, E. Knill, J. Britton, and R. Blakestad, “Experimental purification of two-atom entanglement,” Nature 443(7113), 838–841 (2006).
[Crossref]

Braunstein, S. L.

S. L. Braunstein and H. J. Kimble, “Teleportation of Continuous Quantum Variables,” Phys. Rev. Lett. 80(4), 869–872 (1998).
[Crossref]

Britton, J.

R. Reichle, D. Leibfried, E. Knill, J. Britton, and R. Blakestad, “Experimental purification of two-atom entanglement,” Nature 443(7113), 838–841 (2006).
[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]

Cernotík, O.

O. Černotík and J. Fiurá šek, “Displacement-enhanced continuous-variable entanglement concentration,” Phys. Rev. A 86(5), 052339 (2012).
[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(4), 042310 (2006).
[Crossref]

Cirac, J. I.

G. Giedke and J. I. Cirac, “Characterization of Gaussian operations and distillation of Gaussian states,” Phys. Rev. A 66(3), 032316 (2002).
[Crossref]

L. M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Entanglement Purification of Gaussian Continuous Variable Quantum States,” Phys. Rev. Lett. 84(17), 4002–4005 (2000).
[Crossref]

Danzmann, K.

H. Vahlbruch, M. Mehmet, K. Danzmann, and R. Schnabel, “Detection of 15 dB Squeezed States of Light and their Application for the Absolute Calibration of Photoelectric Quantum Efficiency,” Phys. Rev. Lett. 117(11), 110801 (2016).
[Crossref]

DiGuglielmo, J.

B. Hage, A. Samblowski, J. DiGuglielmo, J. Fiurášek, and R. Schnabel, “Iterative Entanglement Distillation: Approaching the Elimination of Decoherence,” Phys. Rev. Lett. 105(23), 230502 (2010).
[Crossref]

B. Hage, A. Samblowski, J. Diguglielmo, A. Franzen, J. Fiurášek, and R. Schnabel, “Preparation of distilled and purified continuous- variable entangled states,” Nat. Phys. 4(12), 915–918 (2008).
[Crossref]

Dong, Y. L.

Duan, L. M.

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L. Barsotti, J. Harms, and R. Schnabel, “Squeezed vacuum states of light for gravitational wave detectors,” Rep. Prog. Phys. 82(1), 016905 (2019).
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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(3), 178–181 (2010).
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R. Ukai, N. Iwata, Y. Shimokawa, S. C. Armstrong, A. Politi, J.-i. Yoshikawa, P. van Loock, and A. Furusawa, “Demonstration of Unconditional One-Way Quantum Computations for Continuous Variables,” Phys. Rev. Lett. 106(24), 240504 (2011).
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S.-Y. Lee, S.-W. Ji, H.-J. Kim, and H. Nha, “Enhancing quantum entanglement for continuous variables by a coherent superposition of photon subtraction and addition,” Phys. Rev. A 84(1), 012302 (2011).
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T. Yamamoto, M. Koashi, Ş. K. Özdemir, and N. Imoto, “Experimental extraction of an entangled photon pair from two identically decohered pairs,” Nature 421(6921), 343–346 (2003).
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S.-Y. Lee, S.-W. Ji, and C.-W. Lee, “Increasing and decreasing entanglement characteristics for continuous variables by a local photon subtraction,” Phys. Rev. A 87(5), 052321 (2013).
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J. Lee and H. Nha, “Entanglement distillation for continuous variables in a thermal environment: Effectiveness of a non-Gaussian operation,” Phys. Rev. A 87(3), 032307 (2013).
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S.-Y. Lee, S.-W. Ji, and C.-W. Lee, “Increasing and decreasing entanglement characteristics for continuous variables by a local photon subtraction,” Phys. Rev. A 87(5), 052321 (2013).
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H.-J. Kim, S.-Y. Lee, S.-W. Ji, and H. Nha, “Quantum linear amplifier enhanced by photon subtraction and addition,” Phys. Rev. A 85(1), 013839 (2012).
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L. Lydersen, C. Wiechers, C. Wittmann, D. Elser, J. Skaar, and V. Makarov, “Hacking commercial quantum cryptography systems by tailored bright illumination,” Nat. Photonics 4(10), 686–689 (2010).
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H. Vahlbruch, M. Mehmet, K. Danzmann, and R. Schnabel, “Detection of 15 dB Squeezed States of Light and their Application for the Absolute Calibration of Photoelectric Quantum Efficiency,” Phys. Rev. Lett. 117(11), 110801 (2016).
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J. Lee and H. Nha, “Entanglement distillation for continuous variables in a thermal environment: Effectiveness of a non-Gaussian operation,” Phys. Rev. A 87(3), 032307 (2013).
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C.-W. Lee, J. Lee, H. Nha, and H. Jeong, “Generating a Schrödinger-cat-like state via a coherent superposition of photonic operations,” Phys. Rev. A 85(6), 063815 (2012).
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S.-Y. Lee, S.-W. Ji, H.-J. Kim, and H. Nha, “Enhancing quantum entanglement for continuous variables by a coherent superposition of photon subtraction and addition,” Phys. Rev. A 84(1), 012302 (2011).
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T. Opatrný, G. Kurizki, and D. G. Welsch, “Improvement on teleportation of continuous variables by photon subtraction via conditional measurement,” Phys. Rev. A 61(3), 032302 (2000).
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T. Yamamoto, M. Koashi, Ş. K. Özdemir, and N. Imoto, “Experimental extraction of an entangled photon pair from two identically decohered pairs,” Nature 421(6921), 343–346 (2003).
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V. Parigi, A. Zavatta, M. S. Kim, and M. Bellini, “Probing Quantum Commutation Rules by Addition and Subtraction of Single Photons to/from a Light Field,” Science 317(5846), 1890–1893 (2007).
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R. Ukai, N. Iwata, Y. Shimokawa, S. C. Armstrong, A. Politi, J.-i. Yoshikawa, P. van Loock, and A. Furusawa, “Demonstration of Unconditional One-Way Quantum Computations for Continuous Variables,” Phys. Rev. Lett. 106(24), 240504 (2011).
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A. P. Lund and T. C. Ralph, “Continuous-variable entanglement distillation over a general lossy channel,” Phys. Rev. A 80(3), 032309 (2009).
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R. Reichle, D. Leibfried, E. Knill, J. Britton, and R. Blakestad, “Experimental purification of two-atom entanglement,” Nature 443(7113), 838–841 (2006).
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B. Hage, A. Samblowski, J. DiGuglielmo, J. Fiurášek, and R. Schnabel, “Iterative Entanglement Distillation: Approaching the Elimination of Decoherence,” Phys. Rev. Lett. 105(23), 230502 (2010).
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B. Hage, A. Samblowski, J. Diguglielmo, A. Franzen, J. Fiurášek, and R. Schnabel, “Preparation of distilled and purified continuous- variable entangled states,” Nat. Phys. 4(12), 915–918 (2008).
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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(4), 042310 (2006).
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J. Eisert, S. Scheel, and M. B. Plenio, “Distilling Gaussian States with Gaussian Operations is Impossible,” Phys. Rev. Lett. 89(13), 137903 (2002).
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L. Barsotti, J. Harms, and R. Schnabel, “Squeezed vacuum states of light for gravitational wave detectors,” Rep. Prog. Phys. 82(1), 016905 (2019).
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H. Vahlbruch, M. Mehmet, K. Danzmann, and R. Schnabel, “Detection of 15 dB Squeezed States of Light and their Application for the Absolute Calibration of Photoelectric Quantum Efficiency,” Phys. Rev. Lett. 117(11), 110801 (2016).
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B. Hage, A. Samblowski, J. DiGuglielmo, J. Fiurášek, and R. Schnabel, “Iterative Entanglement Distillation: Approaching the Elimination of Decoherence,” Phys. Rev. Lett. 105(23), 230502 (2010).
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B. Hage, A. Samblowski, J. Diguglielmo, A. Franzen, J. Fiurášek, and R. Schnabel, “Preparation of distilled and purified continuous- variable entangled states,” Nat. Phys. 4(12), 915–918 (2008).
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Shimokawa, Y.

R. Ukai, N. Iwata, Y. Shimokawa, S. C. Armstrong, A. Politi, J.-i. Yoshikawa, P. van Loock, and A. Furusawa, “Demonstration of Unconditional One-Way Quantum Computations for Continuous Variables,” Phys. Rev. Lett. 106(24), 240504 (2011).
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J. W. Pan, C. Simon, Č. Brukner, and A. Zeilinger, “Entanglement purification for quantum communication,” Nature 410(6832), 1067–1070 (2001).
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L. Lydersen, C. Wiechers, C. Wittmann, D. Elser, J. Skaar, and V. Makarov, “Hacking commercial quantum cryptography systems by tailored bright illumination,” Nat. Photonics 4(10), 686–689 (2010).
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P. G. Kwiat, S. Barraza-Lopez, A. Stefanov, and N. Gisin, “Experimental entanglement distillation and ’hidden’ non-locality,” Nature 409(6823), 1014–1017 (2001).
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L. Y. Hu, Z. Y. Liao, and M. Suhail Zubairy, “Continuous-variable entanglement via multiphoton catalysis,” Phys. Rev. A 95(1), 012310 (2017).
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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(3), 178–181 (2010).
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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(3), 178–181 (2010).
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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(3), 178–181 (2010).
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R. Ukai, N. Iwata, Y. Shimokawa, S. C. Armstrong, A. Politi, J.-i. Yoshikawa, P. van Loock, and A. Furusawa, “Demonstration of Unconditional One-Way Quantum Computations for Continuous Variables,” Phys. Rev. Lett. 106(24), 240504 (2011).
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H. Vahlbruch, M. Mehmet, K. Danzmann, and R. Schnabel, “Detection of 15 dB Squeezed States of Light and their Application for the Absolute Calibration of Photoelectric Quantum Efficiency,” Phys. Rev. Lett. 117(11), 110801 (2016).
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R. Ukai, N. Iwata, Y. Shimokawa, S. C. Armstrong, A. Politi, J.-i. Yoshikawa, P. van Loock, and A. Furusawa, “Demonstration of Unconditional One-Way Quantum Computations for Continuous Variables,” Phys. Rev. Lett. 106(24), 240504 (2011).
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T. Opatrný, G. Kurizki, and D. G. Welsch, “Improvement on teleportation of continuous variables by photon subtraction via conditional measurement,” Phys. Rev. A 61(3), 032302 (2000).
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Wittmann, C.

L. Lydersen, C. Wiechers, C. Wittmann, D. Elser, J. Skaar, and V. Makarov, “Hacking commercial quantum cryptography systems by tailored bright illumination,” Nat. Photonics 4(10), 686–689 (2010).
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T. Yamamoto, M. Koashi, Ş. K. Özdemir, and N. Imoto, “Experimental extraction of an entangled photon pair from two identically decohered pairs,” Nature 421(6921), 343–346 (2003).
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Rep. Prog. Phys. (1)

L. Barsotti, J. Harms, and R. Schnabel, “Squeezed vacuum states of light for gravitational wave detectors,” Rep. Prog. Phys. 82(1), 016905 (2019).
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A. Zavatta, S. Viciani, and M. Bellin, “Quantum-to-Classical Transition with Single-Photon-Added Coherent States of Light,” Science 306(5696), 660–662 (2004).
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[Crossref]

Other (4)

Current computation servers with $1$1TB$=10^{12}$=1012 Byte computer meomory can save at most $594^4$5944 double-precision float-point numbers (8 Byte for each double-precision number), meaning maximal truncation happens at $\overline {D}=594$D¯=594.

T. C. Ralph and A. P. Lund, “Nondeterministic Noiseless Linear Amplification of Quantum Systems,” in Quantum Communication Measurement and Computing, Proceedings of 9th International Conference, A. Lvovsky, ed. 155–160, (AIP, 2009).

J. Eisert, “Entanglement in quantum information theory,” Ph.D. thesis, University of Potsdam (2001).

$20.5$20.5 dB is the threshold for fault-tolerant measurement-based continuous variable quantum computing [3].

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Figures (14)

Fig. 1.
Fig. 1. Schematic diagram of entanglement distillation with single-side (a) and biside (b) photon annihilation operation. $|\xi \rangle$ and $|0\rangle$ represents single-mode-squeezed state and vacuum state. $BS$ denotes the optical beam splitter with transmittance $T=0.50$.
Fig. 2.
Fig. 2. (a)Average photon number of $|\xi \rangle$ as a function of squeezing $S_{\mathrm {dB}}$. (b) Entanglement (measured with logarithmic negativity) and (c) variance (measured with $\log _{10}\textrm {Var}(P_1+P_2)$ )before and after single-side photon annihilation (lines with circles ) and biside photon annihilation (lines with asterisks). Photon number is truncated at $\overline {D}=1999$.
Fig. 3.
Fig. 3. Practical single-side and biside distillation with optical beam splitters with transmittance $T$ and photon detectors. Successful photon subtractions are heralded when all photon detectors in each scheme register a single photon state. The input of $C$ and $D$ are pure vacuum states.
Fig. 4.
Fig. 4. Distillation SMSE state generated with 15 dB single mode $|\xi \rangle$. Photon number is truncated at $\overline {D}=199$.
Fig. 5.
Fig. 5. Distillation SMSE state generated with 25 dB single mode $|\xi \rangle$. Photon number is truncated at $\overline {D}=1199$.
Fig. 6.
Fig. 6. Squeezing operation assisted single-side distillation of SMSE state with ideal (a) and practical (b) photon subtraction. Success in practical single-side distillation (b) is heralded when photon detector registers the single photon state.
Fig. 7.
Fig. 7. Logarithmic negativity and probability of success for distilling $|\psi \rangle _{\mathrm {{SMS}}}$ with $S_{\mathrm {dB}}=15$ dB (a)(b) and $S_{\mathrm {dB}}=25$ dB (c)(d). Photon number is truncated at $\overline {D}=199$ for (a)(b) and at $\overline {D}=1999$ for (c)(d).
Fig. 8.
Fig. 8. Logarithmic negativity and probability of success for distilling $|\psi \rangle _{\mathrm {{SMS}}}$ versus beam splitter transmittance $T$. Other parameters: (a)(b) $S_{\mathrm {dB}}=15$ dB, $\overline {D}=199$ and (c)(d) $S_{\mathrm {dB}}=25$ dB, $\overline {D}=1999$.
Fig. 9.
Fig. 9. Lower bound $|r_L^\prime | (r_L^\prime \, <\, 0)$ as a function of squeezing $S_{\mathrm {dB}}$. The letters “U,N” represents the regime in which local squeezing is unnecessary and necessary for successful distillation and the letter “F” denote the regime where even local squeezing-enhanced distillation fails to improve the entanglement. Other parameters: (a)$T=0.80$,(b)$T=0.85$,(c)$T=0.90$,(d)$T=0.95$.
Fig. 10.
Fig. 10. Mean photon number that enters the single photon detector in Fig. 6(b). Other parameters: $T=0.90, r^\prime <0$.
Fig. 11.
Fig. 11. Photon detection with non-unit on-off detector. A beam splitter (transmittance $\eta _D$ ) and ancillary mode D (initialized in vacuum) is used to simulate the quantum efficiency $\eta _D$. The trash box denotes the discarding of the ancillar optical mode. We refer to a successful distillation event when the on-off detector registers ‘on’ result.
Fig. 12.
Fig. 12. Performance of practical single-side distillation of SMSE with $S_{\mathrm {dB}}=15$ dB. Entanglement and its contour plot is given for different combination of $(\eta _{_D},r^\prime )$. All optical modes are truncated at $\overline {D}=139$. Other parameter: $T=0.90$.
Fig. 13.
Fig. 13. Performance of practical single-side distillation of SMSE with $S_{\mathrm {dB}}=20.5$ dB. Entanglement and its contour plot is given for different combination of $(\eta _{_D},r^\prime )$. All optical modes are truncated at $\overline {D}=189$. Other paramter: $T=0.95$.
Fig. 14.
Fig. 14. Numerical convergence for calculation of entanglement distill with on-off photon detectors within photon number space of increasing size $\overline {D}$. Other paramter: $T=0.95,\eta _D=0.9990, r^\prime = -1.8824$.

Tables (1)

Tables Icon

Table 1. Computer memory (in GB) and Computing time (in second) in storage and evaluating eigenvalues of distilled state.

Equations (9)

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| ψ S M S E = U B S ( 1 2 ) ( S ( ξ ) | 0 | 0 ) = 1 cosh r U B S k = 0 ( 2 k ) ! k ! ( tanh ( r ) 2 ) k | 2 k | 0 = 1 cosh r k = 0 tanh k ( r ) 4 k m = 0 2 k ( 2 k k ) ( 2 k m ) | 2 k m | m ,
S d B = 20 r log 10 e , n = sinh ( r ) 2 .
| ψ 1 = 1 P 1 ( a ^ A I ) | ψ S M S E , | ψ 2 = 1 P 2 ( a ^ A a ^ B ) | ψ S M S E ,
U ( T ) | ψ | 0 = n = 1 , m = 0 c n m k = 0 n ( n k ) T n 2 ( 1 T T ) k 2 | n k , m A B | k C .
| ψ 3 = 1 P 3 ( μ a ^ A + ν a ^ A ) | ψ S M S E ,
| ψ 4 = a ^ A S ( r ) I | ψ S M S E = S ( r ) S ( r ) a ^ A S ( r ) I | ψ S M S E = [ S ( r ) ( cosh ( r ) a A + sinh ( r ) a A ) I ] | ψ S M S E = [ S ( r ) I ] [ f ( a A , a A ) I ] | ψ S M S E ,
n = 1 T 8 e 2 ( r + r ) ( e 2 r + 4 r + e 4 ( r + r ) 4 e 2 ( r + r ) + e 2 r + 1 ) .
E N ( ρ i ) = log 2 | | ρ i Γ A | | = log 2 [ 1 + 2 N ( ρ i Γ A ) ] ,       ρ i = | ψ i ψ i | , i = 1 , 2 ,
| ψ = p Λ p ( n n | U | p ) ( m p | V | m ) | n | m = p Λ p | f p | g p ,

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