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## Continuous-variable entanglement purification with atomic systems

Stojan Rebic, Stefano Mancini, Giovanna Morigi, and David Vitali »View Author Affiliations

^{1,}

^{*}Stefano Mancini,

^{2}Giovanna Morigi,

^{3,}

^{4}and David Vitali

^{2}

^{1}Centre for Quantum Computing Technology, Macquarie University, Sydney, New South Wales 2109, Australia

^{2}Physics Department, University of Camerino, I-62032 Camerino, Italy

^{3}Departament de Física, Universitat Autònoma de Barcelona, E-08193 Bellaterra, Spain

^{4}Theoretische Physik, Universität des Saarlandes, D-66041 Saarbrücken, Germany

^{*}Corresponding author: srebic@ics.mq.edu.au

JOSA B, Vol. 27, Issue 6, pp. A198-A207 (2010)

http://dx.doi.org/10.1364/JOSAB.27.00A198

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### Abstract

We present a proposal for entanglement purification of the continuous-variable quantum state of two propagating optical fields. The scheme is based on each field interacting with a local node-atomic ensemble whose internal collective excitation plays the role of an ancillary continuous-variable resource. Entanglement purification is achieved by a dichotomic measurement, representing the required non-Gaussian element, which consists of detecting the presence or absence of collective excitations in the atomic ensemble. This scheme can be extended to networks, where the nodes are single trapped atoms, and constitutes an important building block for the implementation of a continuous-variable quantum repeater.

© 2010 Optical Society of America

**OCIS Codes**

(270.0270) Quantum optics : Quantum optics

(270.6570) Quantum optics : Squeezed states

(270.5565) Quantum optics : Quantum communications

(270.5585) Quantum optics : Quantum information and processing

**ToC Category:**

Continuous Variables

**History**

Original Manuscript: December 14, 2009

Revised Manuscript: March 16, 2010

Manuscript Accepted: April 16, 2010

Published: May 28, 2010

**Citation**

Stojan Rebic, Stefano Mancini, Giovanna Morigi, and David Vitali, "Continuous-variable entanglement purification with atomic systems," J. Opt. Soc. Am. B **27**, A198-A207 (2010)

http://www.opticsinfobase.org/josab/abstract.cfm?URI=josab-27-6-A198

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### References

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- D. E. Browne, J. Eisert, S. Scheel, and M. B. Plenio, “Driving non-Gaussian to Gaussian states with linear optics,” Phys. Rev. A 67, 062320 (2003). [CrossRef]
- H. Zeng and F. Lin, “Quantum conversion between the cavity fields and the center-of-mass motion of ions in a quantized trap,” Phys. Rev. A 50, R3589–R3592 (1994). [CrossRef] [PubMed]
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- G. Morigi, J. Eschner, S. Mancini, and D. Vitali, “Entangled light pulses from single cold atoms,” Phys. Rev. Lett. 96, 023601 (2006). [CrossRef] [PubMed]
- G. Morigi, J. Eschner, S. Mancini, and D. Vitali, “Coherent generation of EPR-entangled light pulses mediated by a single trapped atom,” Phys. Rev. A 73, 033822 (2006). [CrossRef]
- D. Vitali, P. Cañizares, J. Eschner, and G. Morigi, “Time-separated entangled light pulses from a single-atom emitter,” New J. Phys. 10, 033025 (2008). [CrossRef]
- J. Eschner, G. Morigi, F. Schmidt-Kaler, and R. Blatt, “Laser cooling of trapped ions,” J. Opt. Soc. Am. B 20, 1003–1015 (2003). [CrossRef]
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- D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, “Quantum dynamics of single trapped ions,” Rev. Mod. Phys. 75, 281–324 (2003). [CrossRef]
- A. Imamoglu, “High efficiency photon counting using stored light,” Phys. Rev. Lett. 89, 163602 (2002). [CrossRef] [PubMed]
- D. F. V. James and P. G. Kwiat, “Atomic vapor-based high efficiency optical detectors with photon number resolution,” Phys. Rev. Lett. 89, 183601 (2002). [CrossRef] [PubMed]
- S. Pirandola, S. Mancini, D. Vitali, and P. Tombesi, “Continuous-variable entanglement and quantum-state teleportation between optical and macroscopic vibrational modes through radiation pressure,” Phys. Rev. A 68, 062317 (2003). [CrossRef]
- D. F. Walls and G. J. Milburn, Quantum Optics, 2nd ed. (Springer, 2008). [CrossRef]
- S. Olivares, M. G. A. Paris, and R. Bonifacio, “Teleportation improvement by inconclusive photon subtraction,” Phys. Rev. A 67, 032314 (2003). [CrossRef]
- S. L. Braunstein and H. J. Kimble, “Teleportation of continuous quantum variables,” Phys. Rev. Lett. 80, 869–872 (1998). [CrossRef]
- A. Furusawa, J. Sorensen, S. L. Braunstein, C. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282, 706–709 (1998). [CrossRef] [PubMed]
- S. Pirandola and S. Mancini, “Quantum teleportation with continuous variables: a survey,” Laser Phys. 16, 1418–1438 (2006). [CrossRef]
- K. E. Cahill and R. J. Glauber, “Density operators and quasiprobability distributions,” Phys. Rev. 177, 1882–1902 (1969). [CrossRef]
- G. Adesso and F. Illuminati, “Continuous variable tangle, monogamy inequality, and entanglement sharing in Gaussian states of continuous variable systems,” New J. Phys. 8, 15 (2006). [CrossRef]
- 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]
- 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]

**Adesso, G.**

- G. Adesso and F. Illuminati, “Continuous variable tangle, monogamy inequality, and entanglement sharing in Gaussian states of continuous variable systems,” New J. Phys. 8, 15 (2006). [CrossRef]

**Afzelius, M.**

- See, for instance, Ch. Simon, M. Afzelius, J. Appel, A. Boyer de la Giroday, S. J. Dewhurst, N. Gisin, C. Y. Hu, F. Jelezko, S. Kroll, J. H. Muller, J. Nunn, E. Polzik, J. Rarity, H. de Riedmatten, W. Rosenfeld, A. J. Shields, N. Skold, R. M. Stevenson, R. Thew, I. Walmsley, M. Weber, H. Weinfurter, J. Wrachtrup, and R. J. Young, “Quantum memories: a review based on the European Integrated Project ‘Qubit Applications (QAP)’,” Eur. Phys. J. D 58, 1–22 (2010). [CrossRef]

**Appel, J.**

- See, for instance, Ch. Simon, M. Afzelius, J. Appel, A. Boyer de la Giroday, S. J. Dewhurst, N. Gisin, C. Y. Hu, F. Jelezko, S. Kroll, J. H. Muller, J. Nunn, E. Polzik, J. Rarity, H. de Riedmatten, W. Rosenfeld, A. J. Shields, N. Skold, R. M. Stevenson, R. Thew, I. Walmsley, M. Weber, H. Weinfurter, J. Wrachtrup, and R. J. Young, “Quantum memories: a review based on the European Integrated Project ‘Qubit Applications (QAP)’,” Eur. Phys. J. D 58, 1–22 (2010). [CrossRef]

**Behroozi, C. H.**

- C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, “Observation of coherent optical information storage in an atomic medium using halted light,” Nature 409, 490–493 (2001). [CrossRef] [PubMed]

**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]

**Blatt, R.**

- J. Eschner, G. Morigi, F. Schmidt-Kaler, and R. Blatt, “Laser cooling of trapped ions,” J. Opt. Soc. Am. B 20, 1003–1015 (2003). [CrossRef]
- D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, “Quantum dynamics of single trapped ions,” Rev. Mod. Phys. 75, 281–324 (2003). [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]

**Boyer de la Giroday, A.**

**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]

**Braunstein, S. L.**

- S. L. Braunstein and P. van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513–577 (2005). [CrossRef]
- S. L. Braunstein and H. J. Kimble, “Teleportation of continuous quantum variables,” Phys. Rev. Lett. 80, 869–872 (1998). [CrossRef]
- A. Furusawa, J. Sorensen, S. L. Braunstein, C. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282, 706–709 (1998). [CrossRef] [PubMed]

**Browne, D. E.**

- D. E. Browne, J. Eisert, S. Scheel, and M. B. Plenio, “Driving non-Gaussian to Gaussian states with linear optics,” Phys. Rev. A 67, 062320 (2003). [CrossRef]

**Cahill, K. E.**

- K. E. Cahill and R. J. Glauber, “Density operators and quasiprobability distributions,” Phys. Rev. 177, 1882–1902 (1969). [CrossRef]

**Cañizares, P.**

- D. Vitali, P. Cañizares, J. Eschner, and G. Morigi, “Time-separated entangled light pulses from a single-atom emitter,” New J. Phys. 10, 033025 (2008). [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]

**Choi, K. S.**

- K. S. Choi, H. Deng, J. Laurat, and H. J. Kimble, “Mapping photonic entanglement into and out of a quantum memory,” Nature 452, 67–71 (2008). [CrossRef] [PubMed]

**Cirac, J. I.**

- L.-M. Duan, J. I. Cirac, and P. Zoller, “Three-dimensional theory for interaction between atomic ensembles and free-space light,” Phys. Rev. A 66, 023818 (2002). [CrossRef]
- L.-M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long- distance quantum communication with atomic ensembles and linear optics,” Nature 414, 413–418 (2001). [CrossRef] [PubMed]

**de Riedmatten, H.**

**Deng, H.**

- K. S. Choi, H. Deng, J. Laurat, and H. J. Kimble, “Mapping photonic entanglement into and out of a quantum memory,” Nature 452, 67–71 (2008). [CrossRef] [PubMed]

**Dewhurst, S. J.**

**Duan, L. -M.**

- L.-M. Duan, J. I. Cirac, and P. Zoller, “Three-dimensional theory for interaction between atomic ensembles and free-space light,” Phys. Rev. A 66, 023818 (2002). [CrossRef]
- L.-M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long- distance quantum communication with atomic ensembles and linear optics,” Nature 414, 413–418 (2001). [CrossRef] [PubMed]

**Dutton, Z.**

- C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, “Observation of coherent optical information storage in an atomic medium using halted light,” Nature 409, 490–493 (2001). [CrossRef] [PubMed]

**Eisert, J.**

- D. E. Browne, J. Eisert, S. Scheel, and M. B. Plenio, “Driving non-Gaussian to Gaussian states with linear optics,” Phys. Rev. A 67, 062320 (2003). [CrossRef]
- J. Eisert, S. Scheel, and M. B. Plenio, “Distilling Gaussian states with Gaussian operations is impossible,” Phys. Rev. Lett. 89, 137903 (2002). [CrossRef] [PubMed]

**Eschner, J.**

- D. Vitali, P. Cañizares, J. Eschner, and G. Morigi, “Time-separated entangled light pulses from a single-atom emitter,” New J. Phys. 10, 033025 (2008). [CrossRef]
- G. Morigi, J. Eschner, S. Mancini, and D. Vitali, “Coherent generation of EPR-entangled light pulses mediated by a single trapped atom,” Phys. Rev. A 73, 033822 (2006). [CrossRef]
- G. Morigi, J. Eschner, S. Mancini, and D. Vitali, “Entangled light pulses from single cold atoms,” Phys. Rev. Lett. 96, 023601 (2006). [CrossRef] [PubMed]
- J. Eschner, G. Morigi, F. Schmidt-Kaler, and R. Blatt, “Laser cooling of trapped ions,” J. Opt. Soc. Am. B 20, 1003–1015 (2003). [CrossRef]

**Fleischhauer, A.**

- D. F. Phillips, A. Fleischhauer, A. Mair, R. L. Walsworth, and M. D. Lukin, “Storage of light in atomic vapor,” Phys. Rev. Lett. 86, 783–786 (2001). [CrossRef] [PubMed]

**Fleischhauer, M.**

- M. D. Lukin, S. F. Yelin, and M. Fleischhauer, “Entanglement of atomic ensembles by trapping correlated photon states,” Phys. Rev. Lett. 84, 4232–4235 (2000). [CrossRef] [PubMed]
- M. Fleischhauer and M. D. Lukin, “Dark-state polaritons in electromagnetically induced transparency,” Phys. Rev. Lett. 84, 5094–5097 (2000). [CrossRef] [PubMed]

**Fuchs, C.**

- A. Furusawa, J. Sorensen, S. L. Braunstein, C. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282, 706–709 (1998). [CrossRef] [PubMed]

**Furusawa, A.**

- 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]

**Giovannetti, V.**

- V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. 96, 010401 (2006). [CrossRef] [PubMed]

**Gisin, N.**

**Glauber, R. J.**

- K. E. Cahill and R. J. Glauber, “Density operators and quasiprobability distributions,” Phys. Rev. 177, 1882–1902 (1969). [CrossRef]

**Hau, L. V.**

**Hayasaka, K.**

- 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]

**Holstein, T.**

- T. Holstein and H. Primakoff, “Field dependence of the intrinsic domain magnetization of a ferromagnet,” Phys. Rev. 58, 1098–1113 (1940). [CrossRef]

**Horodecki, K.**

- R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Rev. Mod. Phys. 81, 865–942 (2009). [CrossRef]

**Horodecki, M.**

- R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Rev. Mod. Phys. 81, 865–942 (2009). [CrossRef]

**Horodecki, P.**

**Horodecki, R.**

**Hu, C. Y.**

**Illuminati, F.**

- G. Adesso and F. Illuminati, “Continuous variable tangle, monogamy inequality, and entanglement sharing in Gaussian states of continuous variable systems,” New J. Phys. 8, 15 (2006). [CrossRef]

**Imamoglu, A.**

- A. Imamoglu, “High efficiency photon counting using stored light,” Phys. Rev. Lett. 89, 163602 (2002). [CrossRef] [PubMed]

**James, D. F. V.**

- D. F. V. James and P. G. Kwiat, “Atomic vapor-based high efficiency optical detectors with photon number resolution,” Phys. Rev. Lett. 89, 183601 (2002). [CrossRef] [PubMed]

**Jelezko, F.**

**Julsgaard, B.**

- B. Julsgaard, A. Kozhekin, and E. S. Polzik, “Experimental long-lived entanglement of two macroscopic objects,” Nature 413, 400–403 (2001). [CrossRef] [PubMed]

**Kimble, H. J.**

- A. S. Parkins and H. J. Kimble, “Quantum state transfer between motion and light,” J. Opt. B: Quantum Semiclassical Opt. 1, 496–504 (1999). [CrossRef]
- S. L. Braunstein and H. J. Kimble, “Teleportation of continuous quantum variables,” Phys. Rev. Lett. 80, 869–872 (1998). [CrossRef]

**Kitagawa, 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]

**Kozhekin, A.**

- B. Julsgaard, A. Kozhekin, and E. S. Polzik, “Experimental long-lived entanglement of two macroscopic objects,” Nature 413, 400–403 (2001). [CrossRef] [PubMed]

**Kroll, S.**

**Kwiat, P. G.**

- D. F. V. James and P. G. Kwiat, “Atomic vapor-based high efficiency optical detectors with photon number resolution,” Phys. Rev. Lett. 89, 183601 (2002). [CrossRef] [PubMed]

**Laurat, J.**

**Leibfried, D.**

- D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, “Quantum dynamics of single trapped ions,” Rev. Mod. Phys. 75, 281–324 (2003). [CrossRef]

**Lin, F.**

- H. Zeng and F. Lin, “Quantum conversion between the cavity fields and the center-of-mass motion of ions in a quantized trap,” Phys. Rev. A 50, R3589–R3592 (1994). [CrossRef] [PubMed]

**Liu, C.**

**Lloyd, S.**

- V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. 96, 010401 (2006). [CrossRef] [PubMed]

**Lukin, M. D.**

- D. F. Phillips, A. Fleischhauer, A. Mair, R. L. Walsworth, and M. D. Lukin, “Storage of light in atomic vapor,” Phys. Rev. Lett. 86, 783–786 (2001). [CrossRef] [PubMed]
- M. D. Lukin, S. F. Yelin, and M. Fleischhauer, “Entanglement of atomic ensembles by trapping correlated photon states,” Phys. Rev. Lett. 84, 4232–4235 (2000). [CrossRef] [PubMed]
- M. Fleischhauer and M. D. Lukin, “Dark-state polaritons in electromagnetically induced transparency,” Phys. Rev. Lett. 84, 5094–5097 (2000). [CrossRef] [PubMed]

**Maccone, L.**

**Mair, A.**

**Mancini, S.**

- G. Morigi, J. Eschner, S. Mancini, and D. Vitali, “Entangled light pulses from single cold atoms,” Phys. Rev. Lett. 96, 023601 (2006). [CrossRef] [PubMed]
- G. Morigi, J. Eschner, S. Mancini, and D. Vitali, “Coherent generation of EPR-entangled light pulses mediated by a single trapped atom,” Phys. Rev. A 73, 033822 (2006). [CrossRef]
- S. Pirandola and S. Mancini, “Quantum teleportation with continuous variables: a survey,” Laser Phys. 16, 1418–1438 (2006). [CrossRef]
- S. Pirandola, S. Mancini, D. Vitali, and P. Tombesi, “Continuous-variable entanglement and quantum-state teleportation between optical and macroscopic vibrational modes through radiation pressure,” Phys. Rev. A 68, 062317 (2003). [CrossRef]

**Milburn, G. J.**

- D. F. Walls and G. J. Milburn, Quantum Optics, 2nd ed. (Springer, 2008). [CrossRef]

**Monroe, C.**

**Morigi, G.**

**Muller, J. H.**

**Neergaard-Nielsen, J. S.**

**Nunn, J.**

**Olivares, S.**

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

**Paris, M. G. A.**

**Parkins, A. S.**

- A. S. Parkins and H. J. Kimble, “Quantum state transfer between motion and light,” J. Opt. B: Quantum Semiclassical Opt. 1, 496–504 (1999). [CrossRef]

**Phillips, D. F.**

**Pirandola, S.**

- S. Pirandola and S. Mancini, “Quantum teleportation with continuous variables: a survey,” Laser Phys. 16, 1418–1438 (2006). [CrossRef]
- S. Pirandola, S. Mancini, D. Vitali, and P. Tombesi, “Continuous-variable entanglement and quantum-state teleportation between optical and macroscopic vibrational modes through radiation pressure,” Phys. Rev. A 68, 062317 (2003). [CrossRef]

**Plenio, M. B.**

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

**Polzik, E.**

**Polzik, E. S.**

**Popescu, S.**

**Primakoff, H.**

- T. Holstein and H. Primakoff, “Field dependence of the intrinsic domain magnetization of a ferromagnet,” Phys. Rev. 58, 1098–1113 (1940). [CrossRef]

**Rarity, J.**

**Rosenfeld, W.**

**Sasaki, M.**

**Scheel, S.**

**Schmidt-Kaler, F.**

**Schumacher, B.**

**Shields, A. J.**

**Simon, Ch.**

**Skold, N.**

**Smolin, J. A.**

**Sorensen, J.**

**Stevenson, R. M.**

**Takahashi, H.**

**Takeoka, M.**

**Takeuchi, M.**

**Thew, R.**

**Tombesi, P.**

**van Loock, P.**

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

**Vitali, D.**

**Walls, D. F.**

- D. F. Walls and G. J. Milburn, Quantum Optics, 2nd ed. (Springer, 2008). [CrossRef]

**Walmsley, I.**

**Walsworth, R. L.**

**Weber, M.**

**Weinfurter, H.**

**Wineland, D.**

**Wootters, W. K.**

**Wrachtrup, J.**

**Yelin, S. F.**

**Young, R. J.**

**Zeng, H.**

- H. Zeng and F. Lin, “Quantum conversion between the cavity fields and the center-of-mass motion of ions in a quantized trap,” Phys. Rev. A 50, R3589–R3592 (1994). [CrossRef] [PubMed]

**Zoller, P.**

### Eur. Phys. J. D

### J. Opt. B: Quantum Semiclassical Opt.

### J. Opt. Soc. Am. B

### Laser Phys.

### Nat. Photonics

### Nature

### New J. Phys.

### Phys. Rev.

### Phys. Rev. A

### Phys. Rev. Lett.

- A. Imamoglu, “High efficiency photon counting using stored light,” Phys. Rev. Lett. 89, 163602 (2002). [CrossRef] [PubMed]

### Rev. Mod. Phys.

### Science

### Other

- D. F. Walls and G. J. Milburn, Quantum Optics, 2nd ed. (Springer, 2008). [CrossRef]

**2010, Simon, Eur. Phys. J. D**

**2010, Takahashi, Nat. Photonics**

**2009, Horodecki, Rev. Mod. Phys.**

**2008, Vitali, New J. Phys.**

**2008, Choi, Nature**

**2006, Giovannetti, Phys. Rev. Lett.**

**2006, Morigi, Phys. Rev. Lett.**

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