Abstract

We show how the dynamics of open quantum systems can be fully characterized by using quantum tomography methods. We apply these methods to the case of an ion trap quantum computer, which does not operate under ideal conditions due to coupling to several environments. We study the performance of a fundamental two–bit quantum gate as a function of various parameters related to the interaction of the ions with external laser fields.

© 1998 Optical Society of America

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References

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  1. A. Ekert, “From quantum-codemakingto quantum code-breaking,” Report No. quant-ph/9703035, (Los Alamos Nat. Laboratory, Albuquerky, NM, 1997), and references therein. http://xxx.lanl.gov/abs/quant-ph/9703035
  2. D. P. Divincenzo, “Topics in Quantum Computers,” Report No. cond-mat/9612126 (Los Alamos Nat. Laboratory, Albuquerky, NM, 1996), and references therein. http://xxx.lanl.gov/abs/cond-mat/9612126
  3. J. F. Poyatos, J. I. Cirac, and P. Zoller, “Complete Characterization of a Quantum Process: The two-bit Quantum Gate,” Phys. Rev. Lett. 78 , 390, (1997). http://info.uibk.ac.at/c/c7/c705/qo/pub/pub96.html
    [Crossref]
  4. J. I. Cirac and P. Zoller, “Quantum Computations with Cold Trapped Ions,” Phys. Rev. Lett. 74, 4091, (1995).
    [Crossref] [PubMed]
  5. I. Chuang and M. A. Nielsen, “Prescription for experimental determination of the dynamics of a quantum black box,” Report No. quant-ph/9610001 (Los Alamos Nat. Laboratory, Albuquerky, NM, 1997).http://xxx.lanl.gov/abs/quant-ph/9610001
  6. William K. Wootters, “A Wigner-function formulation of finite-state quantum mechanics,” Ann. Phys. 176, 1-21, (1987)
    [Crossref]
  7. J. F. Poyatos, J. I. Cirac, and P. Zoller, “Quantum Computations with Trapped Ions at non-zero Temperature,” (unpublished).
  8. C. H Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. Wooters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. 76, 722, (1996)
    [Crossref] [PubMed]
  9. A. Peres, “Separability Criterion for Density Matrices,” Phys. Rev. Lett. 77, 1413 (1996). http://xxx.lanl.gov/abs/quant-ph/9604005
    [Crossref] [PubMed]
  10. M. Horodecki, P. Horodecki, and R. Horodecki, “Separability of Mixed States: Necessary and Sufficient Conditions,” Report No. quant-ph/9605038 (Los Alamos Nat. Laboratory,Albuquerky, NM, 1996). http://xxx.lanl.gov/abs/quant-ph/9605038

1997 (2)

J. F. Poyatos, J. I. Cirac, and P. Zoller, “Complete Characterization of a Quantum Process: The two-bit Quantum Gate,” Phys. Rev. Lett. 78 , 390, (1997). http://info.uibk.ac.at/c/c7/c705/qo/pub/pub96.html
[Crossref]

I. Chuang and M. A. Nielsen, “Prescription for experimental determination of the dynamics of a quantum black box,” Report No. quant-ph/9610001 (Los Alamos Nat. Laboratory, Albuquerky, NM, 1997).http://xxx.lanl.gov/abs/quant-ph/9610001

1996 (3)

C. H Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. Wooters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. 76, 722, (1996)
[Crossref] [PubMed]

A. Peres, “Separability Criterion for Density Matrices,” Phys. Rev. Lett. 77, 1413 (1996). http://xxx.lanl.gov/abs/quant-ph/9604005
[Crossref] [PubMed]

M. Horodecki, P. Horodecki, and R. Horodecki, “Separability of Mixed States: Necessary and Sufficient Conditions,” Report No. quant-ph/9605038 (Los Alamos Nat. Laboratory,Albuquerky, NM, 1996). http://xxx.lanl.gov/abs/quant-ph/9605038

1995 (1)

J. I. Cirac and P. Zoller, “Quantum Computations with Cold Trapped Ions,” Phys. Rev. Lett. 74, 4091, (1995).
[Crossref] [PubMed]

1987 (1)

William K. Wootters, “A Wigner-function formulation of finite-state quantum mechanics,” Ann. Phys. 176, 1-21, (1987)
[Crossref]

Bennett, C. H

C. H Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. Wooters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. 76, 722, (1996)
[Crossref] [PubMed]

Brassard, G.

C. H Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. Wooters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. 76, 722, (1996)
[Crossref] [PubMed]

Chuang, I.

I. Chuang and M. A. Nielsen, “Prescription for experimental determination of the dynamics of a quantum black box,” Report No. quant-ph/9610001 (Los Alamos Nat. Laboratory, Albuquerky, NM, 1997).http://xxx.lanl.gov/abs/quant-ph/9610001

Cirac, J. I.

J. F. Poyatos, J. I. Cirac, and P. Zoller, “Complete Characterization of a Quantum Process: The two-bit Quantum Gate,” Phys. Rev. Lett. 78 , 390, (1997). http://info.uibk.ac.at/c/c7/c705/qo/pub/pub96.html
[Crossref]

J. I. Cirac and P. Zoller, “Quantum Computations with Cold Trapped Ions,” Phys. Rev. Lett. 74, 4091, (1995).
[Crossref] [PubMed]

J. F. Poyatos, J. I. Cirac, and P. Zoller, “Quantum Computations with Trapped Ions at non-zero Temperature,” (unpublished).

Divincenzo, D. P.

D. P. Divincenzo, “Topics in Quantum Computers,” Report No. cond-mat/9612126 (Los Alamos Nat. Laboratory, Albuquerky, NM, 1996), and references therein. http://xxx.lanl.gov/abs/cond-mat/9612126

Ekert, A.

A. Ekert, “From quantum-codemakingto quantum code-breaking,” Report No. quant-ph/9703035, (Los Alamos Nat. Laboratory, Albuquerky, NM, 1997), and references therein. http://xxx.lanl.gov/abs/quant-ph/9703035

Horodecki, M.

M. Horodecki, P. Horodecki, and R. Horodecki, “Separability of Mixed States: Necessary and Sufficient Conditions,” Report No. quant-ph/9605038 (Los Alamos Nat. Laboratory,Albuquerky, NM, 1996). http://xxx.lanl.gov/abs/quant-ph/9605038

Horodecki, P.

M. Horodecki, P. Horodecki, and R. Horodecki, “Separability of Mixed States: Necessary and Sufficient Conditions,” Report No. quant-ph/9605038 (Los Alamos Nat. Laboratory,Albuquerky, NM, 1996). http://xxx.lanl.gov/abs/quant-ph/9605038

Horodecki, R.

M. Horodecki, P. Horodecki, and R. Horodecki, “Separability of Mixed States: Necessary and Sufficient Conditions,” Report No. quant-ph/9605038 (Los Alamos Nat. Laboratory,Albuquerky, NM, 1996). http://xxx.lanl.gov/abs/quant-ph/9605038

Nielsen, M. A.

I. Chuang and M. A. Nielsen, “Prescription for experimental determination of the dynamics of a quantum black box,” Report No. quant-ph/9610001 (Los Alamos Nat. Laboratory, Albuquerky, NM, 1997).http://xxx.lanl.gov/abs/quant-ph/9610001

Peres, A.

A. Peres, “Separability Criterion for Density Matrices,” Phys. Rev. Lett. 77, 1413 (1996). http://xxx.lanl.gov/abs/quant-ph/9604005
[Crossref] [PubMed]

Popescu, S.

C. H Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. Wooters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. 76, 722, (1996)
[Crossref] [PubMed]

Poyatos, J. F.

J. F. Poyatos, J. I. Cirac, and P. Zoller, “Complete Characterization of a Quantum Process: The two-bit Quantum Gate,” Phys. Rev. Lett. 78 , 390, (1997). http://info.uibk.ac.at/c/c7/c705/qo/pub/pub96.html
[Crossref]

J. F. Poyatos, J. I. Cirac, and P. Zoller, “Quantum Computations with Trapped Ions at non-zero Temperature,” (unpublished).

Schumacher, B.

C. H Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. Wooters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. 76, 722, (1996)
[Crossref] [PubMed]

Smolin, J. A.

C. H Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. Wooters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. 76, 722, (1996)
[Crossref] [PubMed]

Wooters, W.

C. H Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. Wooters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. 76, 722, (1996)
[Crossref] [PubMed]

Wootters, William K.

William K. Wootters, “A Wigner-function formulation of finite-state quantum mechanics,” Ann. Phys. 176, 1-21, (1987)
[Crossref]

Zoller, P.

J. F. Poyatos, J. I. Cirac, and P. Zoller, “Complete Characterization of a Quantum Process: The two-bit Quantum Gate,” Phys. Rev. Lett. 78 , 390, (1997). http://info.uibk.ac.at/c/c7/c705/qo/pub/pub96.html
[Crossref]

J. I. Cirac and P. Zoller, “Quantum Computations with Cold Trapped Ions,” Phys. Rev. Lett. 74, 4091, (1995).
[Crossref] [PubMed]

J. F. Poyatos, J. I. Cirac, and P. Zoller, “Quantum Computations with Trapped Ions at non-zero Temperature,” (unpublished).

Ann. Phys. (1)

William K. Wootters, “A Wigner-function formulation of finite-state quantum mechanics,” Ann. Phys. 176, 1-21, (1987)
[Crossref]

Phys. Rev. Lett. (4)

C. H Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. Wooters, “Purification of noisy entanglement and faithful teleportation via noisy channels,” Phys. Rev. Lett. 76, 722, (1996)
[Crossref] [PubMed]

A. Peres, “Separability Criterion for Density Matrices,” Phys. Rev. Lett. 77, 1413 (1996). http://xxx.lanl.gov/abs/quant-ph/9604005
[Crossref] [PubMed]

J. F. Poyatos, J. I. Cirac, and P. Zoller, “Complete Characterization of a Quantum Process: The two-bit Quantum Gate,” Phys. Rev. Lett. 78 , 390, (1997). http://info.uibk.ac.at/c/c7/c705/qo/pub/pub96.html
[Crossref]

J. I. Cirac and P. Zoller, “Quantum Computations with Cold Trapped Ions,” Phys. Rev. Lett. 74, 4091, (1995).
[Crossref] [PubMed]

Other (5)

I. Chuang and M. A. Nielsen, “Prescription for experimental determination of the dynamics of a quantum black box,” Report No. quant-ph/9610001 (Los Alamos Nat. Laboratory, Albuquerky, NM, 1997).http://xxx.lanl.gov/abs/quant-ph/9610001

A. Ekert, “From quantum-codemakingto quantum code-breaking,” Report No. quant-ph/9703035, (Los Alamos Nat. Laboratory, Albuquerky, NM, 1997), and references therein. http://xxx.lanl.gov/abs/quant-ph/9703035

D. P. Divincenzo, “Topics in Quantum Computers,” Report No. cond-mat/9612126 (Los Alamos Nat. Laboratory, Albuquerky, NM, 1996), and references therein. http://xxx.lanl.gov/abs/cond-mat/9612126

M. Horodecki, P. Horodecki, and R. Horodecki, “Separability of Mixed States: Necessary and Sufficient Conditions,” Report No. quant-ph/9605038 (Los Alamos Nat. Laboratory,Albuquerky, NM, 1996). http://xxx.lanl.gov/abs/quant-ph/9605038

J. F. Poyatos, J. I. Cirac, and P. Zoller, “Quantum Computations with Trapped Ions at non-zero Temperature,” (unpublished).

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Figures (3)

Fig. 1.
Fig. 1.

System environment scheme in a realistic process.

Fig. 2.
Fig. 2.

Fidelity, Purity Quantum degree of a Gate and Entanglement Capability as functions of the dissipation rate. Here κ cm/υ=κ r/υ=κ/υ, η=l and Ω/υ=.l and ∆ = -υ.

Fig. 3.
Fig. 3.

Fidelity as a function of the detuning. Here we have chosen Ω/υ=.l, κ/υ=0.

Equations (15)

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ρ ̂ in ε ρ ̂ out = ε [ ρ ̂ in ] ,
ρ ̂ out = i , i ' = 0 N c i [ c i ' ] * R ̂ i ' i ,
F = Ψ in U ̂ ρ ̂ out U ̂ Ψ in ¯ ,
𝑝 = Tr { ( ρ ̂ out ) 2 } ¯ ,
ψ 1 = 0 , ψ 3 = 1 2 ( 0 + 1 ) ,
ψ 2 = 1 , ψ 4 = 1 2 ( 0 + i 1 ) .
ρ ̂ out = q = 0 15 λ q A ̂ q ,
1 1 2 2 ( 1 ) 1 2 1 1 2 2 , ( 1,2 = 0,1 ) ,
ρ ˙ = i [ H , ρ ] + ρ
H = Δ 1 e 11 e Δ 2 e ' 22 e ' + ν a cm a cm + 3 ν a r a r
+ Ω 1 ( t ) 2 [ e 11 g e i η cm ( a cm + a cm ) e i η r ( a r + a r ) + H . c . ]
+ Ω 2 ( t ) 2 [ e ' 22 g e i η cm ( a cm + a cm ) e i η r ( a r + a r ) + H . c . ] ,
ρ = κ cm ( 2 a cm ρ a cm a cm a cm ρ ρ a cm a cm )
+ κ r ( 2 a r ρ a r a r a r ρ ρ a r a r ) .
𝑄 = max ρ ˜ out , | Ψ me Ψ me ρ ˜ out Ψ me ,

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