Abstract

We study the stochastic decoherence of qubits using the Bloch equations and the Bloch sphere description of a two-level atom. We show that it is possible to describe a general decoherence process of a qubit by a stochastic map that is dependent on 12 independent parameters. Such a stochastic map is constructed with the help of the damping basis associated with a Master equation that describes the decoherence process of a qubit.

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References

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  1. L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover, New York, 1987).
  2. G. S. Agarwal, Quantum Statistical Theories of Spontaneous Emission and Their Relation to Other Approaches (Springer, Berlin, Heidelberg, 1974), Vol. 70.
  3. C. W. Gardiner, Handbook of Stochastic Processes (Springer, Berlin, Heidelberg, 1984).
  4. M. B. Plenio and P. L. Knight, "Realistic lower bounds for the factorisation time of large numbers on a quantum computer," Phys. Rev. A 53, 2986-2990 (1996).
    [CrossRef] [PubMed]
  5. C. H. Benett and P. W. Shore, "Quantum Information Theory," IEEE Trans. Info. Theory 44, 2724-2748 (1998).
    [CrossRef]
  6. M. B. Ruskai, S. Szarek and E. Werner, "A Characterisation of Completely-Positive Trace Pre-serving Maps on M 2," preprint quantum-ph/0005004, http://xxx.lanl.gov/
  7. K. W�dkiewicz and J. H. Eberly, "Random telegraph theory of effective Bloch equations with applications to free induction decay," Phys. Rev. A 32, 992-1001 (1985).
    [CrossRef] [PubMed]
  8. K. Kraus, States, Effects and Operations: Fundamental Notions of Quantum Theory (Springer-Verlag, Berlin Heidelberg, 1983).
    [CrossRef]
  9. C. W. Gardiner, Quantum Noise (Springer, Berlin, Heidelberg,1991).
  10. H. J. Briegel and B. -G. Englert, "Quantum optical master equations: The use of damping bases," Phys. Rev. A 47, 3311-3328 (1993).
    [CrossRef] [PubMed]
  11. C. King and M. B. Ruskai, "Minimal Entropy of States Emerging from Noisy Channels," preprint quantum-phy/9911079, http://xxx.lanl.gov/

Other (11)

L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover, New York, 1987).

G. S. Agarwal, Quantum Statistical Theories of Spontaneous Emission and Their Relation to Other Approaches (Springer, Berlin, Heidelberg, 1974), Vol. 70.

C. W. Gardiner, Handbook of Stochastic Processes (Springer, Berlin, Heidelberg, 1984).

M. B. Plenio and P. L. Knight, "Realistic lower bounds for the factorisation time of large numbers on a quantum computer," Phys. Rev. A 53, 2986-2990 (1996).
[CrossRef] [PubMed]

C. H. Benett and P. W. Shore, "Quantum Information Theory," IEEE Trans. Info. Theory 44, 2724-2748 (1998).
[CrossRef]

M. B. Ruskai, S. Szarek and E. Werner, "A Characterisation of Completely-Positive Trace Pre-serving Maps on M 2," preprint quantum-ph/0005004, http://xxx.lanl.gov/

K. W�dkiewicz and J. H. Eberly, "Random telegraph theory of effective Bloch equations with applications to free induction decay," Phys. Rev. A 32, 992-1001 (1985).
[CrossRef] [PubMed]

K. Kraus, States, Effects and Operations: Fundamental Notions of Quantum Theory (Springer-Verlag, Berlin Heidelberg, 1983).
[CrossRef]

C. W. Gardiner, Quantum Noise (Springer, Berlin, Heidelberg,1991).

H. J. Briegel and B. -G. Englert, "Quantum optical master equations: The use of damping bases," Phys. Rev. A 47, 3311-3328 (1993).
[CrossRef] [PubMed]

C. King and M. B. Ruskai, "Minimal Entropy of States Emerging from Noisy Channels," preprint quantum-phy/9911079, http://xxx.lanl.gov/

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Equations (42)

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u ˙ = 1 T 2 u Δ v
v ˙ = 1 T 2 v + Δ u + χ w
w ˙ = 1 T 1 ( w w eq ) χ v .
Φ : b b
B = b · σ = [ w u i v u + i v w ] .
det B = ( u 2 + v 2 + w 2 ) .
Φ : B B .
det B det B .
Φ : ρ Φ ( ρ ) .
Tr { Φ ( ρ ) } = Tr { ρ } = 1
Φ A : b b = M b + b 0
N A = [ Λ 1 0 0 0 Λ 2 0 0 0 Λ 3 ]
Λ 1 = Λ 2 = exp ( A t 2 ) , Λ 3 = exp ( A t ) .
b 0 = ( 0 , 0 , Λ 3 1 ) .
Λ 1 = Λ 2 = exp ( Γ t ) , Λ 3 = 1 .
Φ : b b = M b + b 0
ρ = 1 2 ( I + b · σ ) = 1 2 ( I + B ) .
Φ : ρ Φ ( ρ ) = 1 2 ( I + B )
Φ ( I ) = I + b 0 σ .
Φ ( ρ ) = e i h R S L e i h T ρ e i h T S R e i h R
Φ ( ρ ) = A ρ A
A A = I .
d ρ d t = 1 i [ H , ρ ] + ρ .
ρ = i [ F i F i ρ + ρ F i F i 2 F i ρ F i ]
R = R L = L .
Tr { R L } = δ , .
ρ ( t ) = Λ r ( 0 ) L = Λ l ( 0 ) R
r ( 0 ) = Tr { R ρ ( 0 ) } and l ( 0 ) = Tr { L ρ ( 0 ) } .
S L ρ S R = Λ Tr { R ρ } L = Λ R Tr { ρ L } .
ρ = 1 4 T 1 ( 1 w eq ) [ σ σ ρ + ρ σ σ 2 σ ρ σ ]
1 4 T 1 ( 1 + w eq ) [ σ σ ρ + ρ σ σ 2 σ ρ σ ]
( 1 2 T 2 1 4 T 1 ) [ ρ σ 3 ρ σ 3 ] .
R 0 = 1 2 ( I + w eq σ 3 ) , R 1 = 1 2 ( σ + σ ) , R 2 = 1 2 ( σ σ ) , R 3 = 1 2 σ 3
L 0 = 1 2 I , L 1 = 1 2 ( σ + σ ) , L 2 = 1 2 ( σ σ ) L 3 = 1 2 ( σ 3 w eq I ) ,
Λ 0 = 1 , Λ 1 = Λ 2 = exp ( t T 2 ) , Λ 3 = exp ( t T 1 ) .
ρ = [ a d d * c ]
Φ ( ρ ) = [ A D D * C ]
A = 1 2 ( a + c ) ( 1 + w eq ) + 1 2 Λ 3 ( a c w eq ( a + c ) ) ,
C = 1 2 ( a + c ) ( 1 w eq ) 1 2 Λ 3 ( a c w eq ( a + c ) ) ,
D = Λ 2 d , D * = Λ 2 d * .
Λ 2 2 u 2 + Λ 2 2 v 2 + ( Λ 3 w + ( 1 Λ 3 ) w eq ) 2 1 .
( u Λ 1 ) 2 + ( v Λ 1 ) 2 + ( w Λ 3 + w eq ( 1 1 Λ 3 ) ) 2 = 1 .

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