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.

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References

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  1. A. Ekert, "From quantum-codemaking to 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 quantummechanics," 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 J. F. Poyatos, J. I. Cirac and P. Zoller, "Quantum Computations with Trapped Ions at non-zero Temperature," (unpublished).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

Other

A. Ekert, "From quantum-codemaking to 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

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]

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

William K. Wootters, "A Wigner-function formulation of finite-state quantummechanics," Ann. Phys. 176, 1-21, (1987)

[CrossRef]

J. F. Poyatos, J. I. Cirac and P. Zoller, "Quantum Computations with Trapped Ions at non-zero Temperature," (unpublished).

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

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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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