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.

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References

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  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. A77, 022321 (2008).
    [CrossRef]
  3. M. M. Wilde, H. Krovi, and T. A. Brun, “Coherent communication with continuous quantum variables,” Phys. Rev. A75,060303(R) (2007).
    [CrossRef]
  4. I. Devetak, “Triangle of dualities between quantum communication protocols,” Phys. Rev. Lett.97, 140503 (2006).
    [CrossRef] [PubMed]
  5. T. A. Brun, I. Devetak, and M-H Hsieh, “Correcting quantum errors with entanglement,” Science314, 436–439 (2006).
    [CrossRef] [PubMed]
  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.47, 777 (1935).
    [CrossRef]
  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.84, 3482 (2000).
    [CrossRef] [PubMed]
  10. R. Filip, P. Marek, and U. L. Andersen, “Measurement-induced continuous-variable quantum interactions,” Phys. Rev. A71, 042308 (2005).
    [CrossRef]

2008 (1)

M. M. Wilde, T. A. Brun, J. P. Dowling, and H. Lee, “Coherent communication with linear optics,” Phys. Rev. A77, 022321 (2008).
[CrossRef]

2007 (1)

M. M. Wilde, H. Krovi, and T. A. Brun, “Coherent communication with continuous quantum variables,” Phys. Rev. A75,060303(R) (2007).
[CrossRef]

2006 (2)

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]

2005 (1)

R. Filip, P. Marek, and U. L. Andersen, “Measurement-induced continuous-variable quantum interactions,” Phys. Rev. A71, 042308 (2005).
[CrossRef]

2004 (1)

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

2000 (1)

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

1989 (1)

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

1935 (1)

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

Andersen, U. L.

R. Filip, P. Marek, and U. L. Andersen, “Measurement-induced continuous-variable quantum interactions,” Phys. Rev. A71, 042308 (2005).
[CrossRef]

Braunstein, S. L.

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

Brun, T. A.

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]

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).

Devetak, I.

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).

Dowling, J. P.

M. M. Wilde, T. A. Brun, J. P. Dowling, and H. Lee, “Coherent communication with linear optics,” Phys. Rev. A77, 022321 (2008).
[CrossRef]

Einstein, A.

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

Filip, R.

R. Filip, P. Marek, and U. L. Andersen, “Measurement-induced continuous-variable quantum interactions,” Phys. Rev. A71, 042308 (2005).
[CrossRef]

Greenberger, D. M.

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

Harrow, A.

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

Horne, M. A.

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

Hsieh, M-H

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).

Krovi, H.

M. M. Wilde, H. Krovi, and T. A. Brun, “Coherent communication with continuous quantum variables,” Phys. Rev. A75,060303(R) (2007).
[CrossRef]

Lee, H.

M. M. Wilde, T. A. Brun, J. P. Dowling, and H. Lee, “Coherent communication with linear optics,” Phys. Rev. A77, 022321 (2008).
[CrossRef]

Marek, P.

R. Filip, P. Marek, and U. L. Andersen, “Measurement-induced continuous-variable quantum interactions,” Phys. Rev. A71, 042308 (2005).
[CrossRef]

Podolsky, B.

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

Rosen, N.

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

van Loock, P.

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

Wilde, M. M.

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]

Zeilinger, A.

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

Bell’s Theorem, Quantum Theory, and Conceptions of the Universe (1)

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

Phys. Rev. (1)

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

Phys. Rev. A (3)

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]

R. Filip, P. Marek, and U. L. Andersen, “Measurement-induced continuous-variable quantum interactions,” Phys. Rev. A71, 042308 (2005).
[CrossRef]

Phys. Rev. Lett. (3)

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

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

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

Science (1)

T. A. Brun, I. Devetak, and M-H Hsieh, “Correcting quantum errors with entanglement,” Science314, 436–439 (2006).
[CrossRef] [PubMed]

Other (1)

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

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Figures (4)

Fig. 1
Fig. 1

The three-party position-quadrature (PQ) protocol requires a four-mode GHZ entangled state, and we label the four modes as A1, A2, B, C. Alice possesses mode A1 and A2, Bob possesses mode B and Claire possesses mode C. Alice has an input mode A to be transmitted. The elements in this figure: AM is an amplitude modulator which displaces the position quadrature of an optical mode [2]. PM is a phase modulator which kicks the momentum quadrature of an optical mode [2]. BS is a beam splitter.

Fig. 2
Fig. 2

On the left side, we present all the scenarios of graph which represents the prepared entanglement resources when n is equal to 3. The number of scenarios is three. On the right side, the graph indicates the best prepared entanglement resources for n-party conat channel.

Fig. 3
Fig. 3

This figure outlines our scheme. The thick red line represents a quantum channel between two parties. The local operations and modes are enclosed by dashed lines with the name. The blue thin rectangle represents phase shifter at an angle of π.

Fig. 4
Fig. 4

This figure outlines our scheme. The thick red line represents a quantum channel between two parties. The local operations and modes are enclosed by dashed lines with the name. The blue thin rectangle represents phase shifter at an angle of π.

Equations (21)

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[ x ^ A p ^ A ] T [ x ^ A , p ^ A x ^ B p ^ B x ^ N p ^ N ] T
{ x ^ A = x ^ A x ^ B = x ^ A + x ^ Δ X 1 x ^ C = x ^ A + x ^ Δ X 2 x ^ N = x ^ A + x ^ Δ X n 1 p ^ A = p ^ A + p ^ Δ X
{ x ^ Δ X 1 = x ^ Δ X 2 = = x ^ Δ X n 1 = 0 p ^ A + p ^ B + p ^ C + + p ^ N = p ^ A p ^ Δ X + p ^ B + p ^ C + p ^ N = 0 ( x ^ Δ X 1 ) 2 ε 1 ( x ^ Δ X 2 ) 2 ε 2 ( x ^ Δ X n 1 ) 2 ε n 1 ( p ^ Δ X + p ^ B + p ^ C + + p ^ N ) 2 ε n
[ x ^ A , p ^ A ] = [ x ^ B , p ^ B ] = = [ x ^ N , p ^ N ] = i
[ x ^ A p ^ A ] T [ x ^ A p ^ A x ^ B p ^ B x ^ N p ^ N ] T
{ p ^ A = p ^ A p ^ B = p ^ A + p ^ Δ P 1 p ^ C = p ^ A + p ^ Δ P 2 p ^ N = p ^ A + p ^ Δ P n 1 x ^ A = x ^ A + x ^ Δ P
{ p ^ Δ P 1 = p ^ Δ P 2 = = p ^ Δ P n 1 = 0 x ^ A + x ^ B + x ^ C + + x ^ N = x ^ A x ^ Δ P + x ^ B + x ^ C + + x ^ N = 0 ( p ^ Δ P 1 ) 2 ε 1 ( p ^ Δ P 2 ) 2 ε 2 ( p ^ Δ P n 1 ) 2 ε n 1 ( x ^ Δ P + x ^ B + x ^ C + + x ^ N ) 2 ε n
[ x ^ A , p ^ A ] = [ x ^ B , p ^ B ] = = [ x ^ N , p ^ N ] = i
{ x ^ A 1 = 1 4 e + r 1 x ^ 1 ( 0 ) + 3 4 e r 2 x ^ 2 ( 0 ) p ^ A 1 = 1 4 e r 1 p ^ 1 ( 0 ) + 3 4 e + r 2 p ^ 2 ( 0 ) x ^ A 2 = 1 4 e + r 1 x ^ 1 ( 0 ) 1 12 e r 2 x ^ 2 ( 0 ) + 2 3 e r 3 x ^ 3 ( 0 ) p ^ A 2 = 1 4 e r 1 p ^ 1 ( 0 ) 1 12 e + r 2 p ^ 2 ( 0 ) + 2 3 e + r 3 p ^ 3 ( 0 ) x ^ B = 1 4 e + r 1 x ^ 1 ( 0 ) 1 12 e r 2 x ^ 2 ( 0 ) 1 6 e r 3 x ^ 3 ( 0 ) + 1 2 e r 4 x ^ 4 ( 0 ) p ^ B = 1 4 e r 1 p ^ 1 ( 0 ) 1 12 e + r 2 p ^ 2 ( 0 ) 1 6 e + r 3 p ^ 3 ( 0 ) + 1 2 e + r 4 p ^ 4 ( 0 ) x ^ C = 1 4 e + r 1 x ^ 1 ( 0 ) 1 12 e r 2 x ^ 2 ( 0 ) 1 6 e r 3 x ^ 3 ( 0 ) 1 2 e r 4 x ^ 4 ( 0 ) p ^ C = 1 4 e r 1 p ^ 1 ( 0 ) 1 12 e + r 2 p ^ 2 ( 0 ) 1 6 e + r 3 p ^ 3 ( 0 ) 1 2 e + r 4 p ^ 4 ( 0 )
{ x ^ + = ( x ^ A + x ^ A 1 ) / 2 , p ^ + = ( p ^ A + p ^ A 1 ) / 2 x ^ = ( x ^ A x ^ A 1 ) / 2 , p ^ = ( p ^ A p ^ A 1 ) / 2
{ x ^ A 2 = x ^ A x ^ A 1 + x ^ A 2 2 x ^ p ^ A 2 = p ^ A + ( p ^ A 1 + p ^ A 2 + p ^ B + p ^ C ) p ^ B p ^ C 2 p ^ + x ^ B = x ^ A x ^ A 1 + x ^ B 2 x ^ x ^ C = x ^ A x ^ A 1 + x ^ C 2 x ^
{ x ^ A = x ^ A ( x ^ A 1 x ^ A 2 ) 2 ( 1 η ) / η x ^ 1 ( 0 ) p ^ A = p ^ A + ( p ^ A 1 + p ^ A 2 + p ^ B + p ^ C ) p ^ B p ^ C + 2 ( 1 η ) / η p ^ 2 ( 0 ) x ^ B = x ^ A ( x ^ A 1 x ^ B ) 2 ( 1 η ) / η x ^ 1 ( 0 ) p ^ B = p ^ B x ^ C = x ^ A ( x ^ A 1 x ^ C ) 2 ( 1 η ) / η x ^ 1 ( 0 ) p ^ C = p ^ C
{ x ^ B x ^ A ) = x ^ B x ^ A 2 = 0 ( x ^ B x ^ A ) 2 = ( x ^ B x ^ A 2 ) 2 = 2 e 2 r x ^ C = x ^ A = x ^ C x ^ A 2 = 0 ( x ^ C x ^ A ) 2 = ( x ^ C x ^ A 2 ) 2 = 2 e 2 r p ^ Δ X + p ^ B + p ^ C = p ^ A 1 + p ^ A 2 + p ^ B + p ^ C + 2 ( 1 η ) / η p ^ 2 ( 0 ) = 0 ( p ^ Δ X + p ^ B + p ^ C ) 2 = 4 e 2 r + 2 ( 1 η ) / η
{ x ^ 1 = ( e + r x ^ 1 ( 0 ) + e r x ^ 2 ( 0 ) ) / 2 , p ^ 1 = ( e r p ^ 1 ( 0 ) + e + r p ^ 2 ( 0 ) ) / 2 x ^ 2 = ( e + r x ^ 1 ( 0 ) e r x ^ 2 ( 0 ) ) / 2 , p ^ 2 = ( e r p ^ 1 ( 0 ) e + r p ^ 2 ( 0 ) ) / 2
H ^ = h ¯ χ x ^ 1 x ^ 2
Q ^ 1 , 2 { x ^ 1 = x ^ 1 , p ^ 1 = p ^ 1 p ^ 2 x ^ 2 = x ^ 1 + x ^ 2 , p ^ 2 = p ^ 2
Q ^ 1 , 2 p { x ^ 1 = x ^ 1 x ^ 2 , p ^ 1 = p ^ 1 x ^ 2 = x ^ 2 , p ^ 2 = p ^ 1 + p ^ 2
{ x ^ 1 = x ^ 1 ( x ^ 2 + x ^ 3 ) , p ^ 1 = p ^ 1 x ^ 2 = x ^ 2 , p ^ 2 = p ^ 2 p ^ 3 x ^ 3 = x ^ 2 + x ^ 3 ( x ^ 2 + x ^ 3 x ^ 4 + x ^ 5 ) p ^ 3 = p ^ 1 + p ^ 3 + p ^ 4 x ^ 4 = x ^ 2 + x ^ 3 x ^ 4 , p ^ 4 = p ^ 4 p ^ 5 x ^ 5 = x ^ 2 + x ^ 3 x ^ 4 + x ^ 5 , p ^ 5 = p ^ 1 + p ^ 3 + p ^ 4 + p ^ 5 + p ^ 6 x ^ 6 = x ^ 2 + x ^ 3 x ^ 4 + x ^ 5 x ^ 6 , p ^ 6 = p ^ 6
{ x ^ 1 = x ^ 1 ( x ^ 2 + x ^ 3 ) ( x ^ 2 + x ^ 5 ) , p ^ 1 = p ^ 1 x ^ 2 = x ^ 2 , p ^ 2 = p ^ 2 p ^ 3 p ^ 5 x ^ 3 = x ^ 2 + x ^ 3 , p ^ 3 = p ^ 1 + p ^ 3 + p ^ 4 x ^ 4 = x ^ 2 + x ^ 3 x ^ 4 , p ^ 4 = p ^ 4 x ^ 5 = x ^ 2 + x ^ 5 , p ^ 5 = p ^ 1 + p ^ 5 + p ^ 6 x ^ 6 = x ^ 2 + x ^ 5 x ^ 6 , p ^ 6 = p ^ 6
{ P Q conat channel : ε 1 = 2 e 2 r , ε 2 = 4 e 2 r , ε 3 = 4 e 2 r M Q conat channel : ε 1 = 2 e 2 r , ε 2 = 4 e 2 r , ε 3 = 0
{ P Q conat channel : ε 1 = 2 e 2 r , ε 2 = 2 e 2 r , ε 3 = 4 e 2 r M Q conat channel : ε 1 = 2 e 2 r , ε 2 = 2 e 2 r , ε 3 = 0

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