## The multiparty coherent channel and its implementation with linear optics |

Optics Express, Vol. 21, Issue 17, pp. 19790-19798 (2013)

http://dx.doi.org/10.1364/OE.21.019790

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### Abstract

The continuous-variable coherent (conat) channel is a useful resource for coherent communication, supporting coherent teleportation and coherent superdense coding. We extend the conat channel to multiparty conditions by proposing definitions on multiparty position-quadrature and momentum-quadrature conat channel. We additionally provide two methods to implement this channel using linear optics. One method is the multiparty version of coherent communication assisted by entanglement and classical communication (CCAECC). The other is multiparty coherent superdense coding.

© 2013 OSA

## 1. Introduction

1. A. Harrow, “Coherent communication of classical messages,” Phys. Rev. Lett. **92**, 097902 (2004) [CrossRef] [PubMed] .

1. A. Harrow, “Coherent communication of classical messages,” Phys. Rev. Lett. **92**, 097902 (2004) [CrossRef] [PubMed] .

*x*〉

*→ |*

^{A}*x*〉

*({|*

^{B}*x*〉}

_{x}_{∈ {0,1}}is a basis for ℂ

^{2}. A is sender Alice, and B is receiver Bob). A cbit channel is: |

*x*〉

*→ |*

^{A}*x*〉

*|*

^{B}*x*〉

*(E is inaccessible environment).*

^{E}*x*〉

*→ |*

^{A}*x*〉

*|*

^{A}*x*〉

*. For example, if Alice possesses an arbitrary qubit: |*

^{B}*ψ*〉

*=*

^{A}*α*|0〉

*+*

^{A}*β*|1〉

*and transmits it through a cobit channel, the channel generates: |*

^{A}*ϕ*〉

*=*

^{AB}*α*|0〉

*|0〉*

^{A}*+*

^{B}*β*|1〉

*|1〉*

^{A}*. The process maintains the coherent superposition property of Alice’s original state, this is the reason for the channel’s name[1*

^{B}1. A. Harrow, “Coherent communication of classical messages,” Phys. Rev. Lett. **92**, 097902 (2004) [CrossRef] [PubMed] .

2. M. M. Wilde, T. A. Brun, J. P. Dowling, and H. Lee, “Coherent communication with linear optics,” Phys. Rev. A **77**, 022321 (2008) [CrossRef] .

3. M. M. Wilde, H. Krovi, and T. A. Brun, “Coherent communication with continuous quantum variables,” Phys. Rev. A **75**,060303(R) (2007) [CrossRef] .

*x*〉

*→ |*

^{A}*x*〉

*|*

^{A}*x*〉

*and |*

^{B}*p*〉

*→ |*

^{A}*p*〉

*|*

^{A}*p*〉

*respectively, where |*

^{B}*x*〉 and |

*p*〉 represents position and momentum eigenstate. Nonideal (finitely squeezed) conat channels are also discussed in the Heisenberg picture.

**92**, 097902 (2004) [CrossRef] [PubMed] .

4. I. Devetak, “Triangle of dualities between quantum communication protocols,” Phys. Rev. Lett. **97**, 140503 (2006) [CrossRef] [PubMed] .

**92**, 097902 (2004) [CrossRef] [PubMed] .

5. T. A. Brun, I. Devetak, and M-H Hsieh, “Correcting quantum errors with entanglement,” Science **314**, 436–439 (2006) [CrossRef] [PubMed] .

7. A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. **47**, 777 (1935) [CrossRef] .

## 2. Definitions of multiparty conat channel

*n*receivers (Alice, Bob, Claire ⋯ and Nick). Notice that the sender is also among the receivers. We denote sender Alice as

*A*and the received massages through the channel as

*A′*,

*B′*,

*C′*,⋯ ,

*N′*. The multiparty PQ conat channel Δ̃

*copies the position quadrature of the sender to all the receivers with noise. The resulting multimode state is similar to multiparty Greenberger-Horne-Zeilinger (GHZ) entangled states [8]. The difference is that total momentum of the output modes is close to the original momentum*

_{X}*p̂*, encoding messages into all the receivers involved, while the total momentum of GHZ entangled state is zero. Multiparty MQ conat channel is similarly defined.

_{A}*.*

_{X}*p̂*

_{ΔX}. The parameters

*ε*

_{1},

*ε*

_{2}⋯ and

*ε*in (3) describe the performance of channel in the meaning of noise.

_{n}*.*

_{P}## 3. Implementations of multiparty conat channel using linear optics

2. M. M. Wilde, T. A. Brun, J. P. Dowling, and H. Lee, “Coherent communication with linear optics,” Phys. Rev. A **77**, 022321 (2008) [CrossRef] .

**Method 1:**

*n*+ 1)-party GHZ entangled state and classical communication channel, as shown in Fig. 1. For simplicity, we implement a three-party PQ conat channel as a demonstration. We will generalize to

*n*-party PQ conat channel later (multiparty MQ conat channel is similar).

9. P. van Loock and S. L. Braunstein, “Multipartite entanglement for continuous variables: a quantum teleportation network,” Phys. Rev. Lett. **84**, 3482 (2000) [CrossRef] [PubMed] .

*r*

_{1},

*r*

_{2},

*r*

_{3},

*r*

_{4}, and generates entangled states

*A*

_{1},

*A*

_{2},

*B*and

*C*. The equations with coefficients calculated are given:

*r*.

*Step 1.*Alice mixes mode

*A*and

*A*

_{1}locally on a balanced (50%) beam splitter (BS), generating modes (+) and (−).

*Step 2.*We express

*x̂*

_{A2},

*p̂*

_{A2},

*x̂*and

_{B}*x̂*in terms of

_{C}*x̂*

_{−}and

*p̂*

_{+}.

*x̂*

_{−}and

*p̂*

_{+}using homodyne detection, and operator

*x̂*

_{−},

*p̂*

_{+}collapse to value

*x*

_{−},

*p*

_{+}. Then she sends the value

*x*

_{−}to Bob and Claire over a classical communication channel. Suppose the photodetectors have efficiency

*η*.

*Step 3.*Alice displaces the position quadrature of her mode

*A*

_{2}by

*B*and

*C*by

*A′*,

*B′*and

*C′*:

*e*

^{−2}

*+ 2(1 −*

^{r}*η*)/

*η*]-approximate PQ coherent channel [2

2. M. M. Wilde, T. A. Brun, J. P. Dowling, and H. Lee, “Coherent communication with linear optics,” Phys. Rev. A **77**, 022321 (2008) [CrossRef] .

*n*is equal to 3. As

*n*increases, we can easily calculate that:

*ε*

_{1},

*ε*

_{2}⋯

*ε*

_{n}_{−1}remain unchanged (=2

*e*

^{−2}

*) and*

^{r}*ε*amounts to (

_{n}*n*+ 1)

*e*

^{−2}

*+ 2(1 −*

^{r}*η*)/

*η*. When it comes to the general

*n*-party conditions, we can implement an [(

*n*+ 1)

*e*

^{−2}

*+ 2(1 −*

^{r}*η*)/

*η*] approximate coherent channel similarly by using the (

*n*+ 1)-party GHZ entanglement.

*x̂*and

*p̂*. The required resource is a GHZ-like entangled state with total position

*x̂*

_{A1}+

*x̂*

_{A2}+

*x̂*+ ⋯ +

_{B}*x̂*→ 0 and certain momenta equal. So we omit these discussions here.

_{N}**Method 2:**

3. M. M. Wilde, H. Krovi, and T. A. Brun, “Coherent communication with continuous quantum variables,” Phys. Rev. A **75**,060303(R) (2007) [CrossRef] .

*n*receivers obtained the two modes respectively. In addition, (

*n*− 1) prepared EPR pairs among the receivers are required.

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

*χ*is coupling strength. The transformation of an ideal continuous-variable QND interaction with unit gain on two input optical modes, denoted as

*Q̂*operation, is expressed as:

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

*Q̂*operation:

^{p}*n*−1) EPR pairs for receivers. We illustrate these requirements in Fig. 2, on the condition that

*n*equals to 3. We use a graph to illustrate the entanglement relations among the parties involved. A node in the graph represents an individual party in the channel, and the edge between two nodes indicates EPR entanglement relation between two parties. We will introduce concepts from graph theory: when two nodes are terminals of a edge, they are called ’adjacent’; we define two nodes as ‘connected’ when a path exists between them; a connected graph is a graph in which any two of the nodes are connected. This method requires that the graph of entanglement resources be a connected graph. The channel has

*n*parties involved. The (

*n*− 1) EPR pairs prepared among the

*n*parties ensure that the representing graph is connected, and any two nodes of the graph has only one path.

*n*is equal to 3, the channel involves one sender and three receivers, two EPR pairs is required.

*Q̂*

_{2,3}, and then

*Q̂*

_{3,4}, then

*Q̂*

_{4,5}, finally

*Q̂*

_{5,6}. then we can get the resulting modes in Heisenberg picture. Modes 1’, 3’, 5’ implement a three-party MQ conat channel by satisfying the constraints in definition 2; and modes 2’, 4’, 6’ satisfy definition 1 and work as a three-party PQ conat channel.

*Scenario 2.*In this scenario, Alice possesses four modes at the beginning: mode 1, 2, 3, 5, Bob possesses mode 4 and Claire possesses mode 6. Mode 1 and 2 is to be transmitted while modes 3, 4, 5, 6 are the auxiliary modes. Modes 3, 4 are EPR pair, as well as modes 5, 6. We give Fig. 4 to describe this protocol.

*Q̂*

_{2,3}; then

*Q̂*

_{2,5}; finally

*Q̂*

_{5,6}and

*Q̂*

_{3,4}. The resulting modes are given as follows:

*In Scenario 1*:

*In Scenario 2*:

*ε*

_{2}= 4

*e*

^{−2}

*. The longer the path between one party and Alice, the larger the accumulation of the noise this party gets. In Scenario 2, the length of the path between Alice and any party is one, there is no accumulation of the noise, so Scenario 2 is better. For n-party conat channel, the best prepared entanglement resources is the rightmost graph in Fig. 2.*

^{r}## 4. Conclusion

**92**, 097902 (2004) [CrossRef] [PubMed] .

## Acknowledgments

## References and links

1. | A. Harrow, “Coherent communication of classical messages,” Phys. Rev. Lett. |

2. | M. M. Wilde, T. A. Brun, J. P. Dowling, and H. Lee, “Coherent communication with linear optics,” Phys. Rev. A |

3. | M. M. Wilde, H. Krovi, and T. A. Brun, “Coherent communication with continuous quantum variables,” Phys. Rev. A |

4. | I. Devetak, “Triangle of dualities between quantum communication protocols,” Phys. Rev. Lett. |

5. | T. A. Brun, I. Devetak, and M-H Hsieh, “Correcting quantum errors with entanglement,” Science |

6. | T. A. Brun, I. Devetak, and M-H Hsieh, “Catalytic quantum error correction,” arXiv:quant-ph/0608027 (2006). |

7. | A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. |

8. | D. M. Greenberger, M. A. Horne, and A. Zeilinger, in Bell’s Theorem, Quantum Theory, and Conceptions of the Universe (1989). |

9. | P. van Loock and S. L. Braunstein, “Multipartite entanglement for continuous variables: a quantum teleportation network,” Phys. Rev. Lett. |

10. | R. Filip, P. Marek, and U. L. Andersen, “Measurement-induced continuous-variable quantum interactions,” Phys. Rev. A |

**OCIS Codes**

(060.5565) Fiber optics and optical communications : Quantum communications

(270.5565) Quantum optics : Quantum communications

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: May 16, 2013

Revised Manuscript: June 22, 2013

Manuscript Accepted: June 24, 2013

Published: August 15, 2013

**Citation**

Guangqiang He, Taizhi Liu, and Xin Tao, "The multiparty coherent channel and its implementation with linear optics," Opt. Express **21**, 19790-19798 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-17-19790

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### References

- A. Harrow, “Coherent communication of classical messages,” Phys. Rev. Lett.92, 097902 (2004). [CrossRef] [PubMed]
- M. M. Wilde, T. A. Brun, J. P. Dowling, and H. Lee, “Coherent communication with linear optics,” Phys. Rev. A77, 022321 (2008). [CrossRef]
- M. M. Wilde, H. Krovi, and T. A. Brun, “Coherent communication with continuous quantum variables,” Phys. Rev. A75,060303(R) (2007). [CrossRef]
- I. Devetak, “Triangle of dualities between quantum communication protocols,” Phys. Rev. Lett.97, 140503 (2006). [CrossRef] [PubMed]
- T. A. Brun, I. Devetak, and M-H Hsieh, “Correcting quantum errors with entanglement,” Science314, 436–439 (2006). [CrossRef] [PubMed]
- T. A. Brun, I. Devetak, and M-H Hsieh, “Catalytic quantum error correction,” arXiv:quant-ph/0608027 (2006).
- A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev.47, 777 (1935). [CrossRef]
- D. M. Greenberger, M. A. Horne, and A. Zeilinger, in Bell’s Theorem, Quantum Theory, and Conceptions of the Universe (1989).
- P. van Loock and S. L. Braunstein, “Multipartite entanglement for continuous variables: a quantum teleportation network,” Phys. Rev. Lett.84, 3482 (2000). [CrossRef] [PubMed]
- R. Filip, P. Marek, and U. L. Andersen, “Measurement-induced continuous-variable quantum interactions,” Phys. Rev. A71, 042308 (2005). [CrossRef]

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