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.

© 2001 Optical Society of America

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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 Preserving Maps on M2,” preprint quantum-ph/0005004, http://xxx.lanl.gov/
  7. K. Wódkiewicz 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/

1998 (1)

C. H. Benett and P. W. Shore, “Quantum Information Theory,” IEEE Trans. Info. Theory 44, 2724–2748 (1998).
[Crossref]

1996 (1)

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]

1993 (1)

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]

1985 (1)

K. Wódkiewicz 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]

Agarwal, G. S.

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

Allen, L.

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

Benett, C. H.

C. H. Benett and P. W. Shore, “Quantum Information Theory,” IEEE Trans. Info. Theory 44, 2724–2748 (1998).
[Crossref]

Briegel, H. J.

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]

Eberly, J. H.

K. Wódkiewicz 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]

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

Englert, B. -G.

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]

Gardiner, C. W.

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

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

King, C.

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

Knight, P. L.

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]

Kraus, K.

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

Plenio, M. B.

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]

Ruskai, M. B.

M. B. Ruskai, S. Szarek, and E. Werner, “A Characterisation of Completely-Positive Trace Preserving Maps on M2,” preprint quantum-ph/0005004, http://xxx.lanl.gov/

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

Shore, P. W.

C. H. Benett and P. W. Shore, “Quantum Information Theory,” IEEE Trans. Info. Theory 44, 2724–2748 (1998).
[Crossref]

Szarek, S.

M. B. Ruskai, S. Szarek, and E. Werner, “A Characterisation of Completely-Positive Trace Preserving Maps on M2,” preprint quantum-ph/0005004, http://xxx.lanl.gov/

Werner, E.

M. B. Ruskai, S. Szarek, and E. Werner, “A Characterisation of Completely-Positive Trace Preserving Maps on M2,” preprint quantum-ph/0005004, http://xxx.lanl.gov/

Wódkiewicz, K.

K. Wódkiewicz 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]

IEEE Trans. Info. Theory (1)

C. H. Benett and P. W. Shore, “Quantum Information Theory,” IEEE Trans. Info. Theory 44, 2724–2748 (1998).
[Crossref]

Phys. Rev. A (3)

K. Wódkiewicz 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]

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]

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]

Other (7)

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

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).

M. B. Ruskai, S. Szarek, and E. Werner, “A Characterisation of Completely-Positive Trace Preserving Maps on M2,” preprint quantum-ph/0005004, http://xxx.lanl.gov/

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).

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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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