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  1. ❉✐✈✐s✐❜✐❧✐t② ❛♥❞ ✐♥❢♦r♠❛t✐♦♥ ✢♦✇ ❢♦r ♥♦♥✲✐♥✈❡rt✐❜❧❡ q✉❜✐t ❞②♥❛♠✐❝❛❧ ♠❛♣s ✶ ✴ ✹✶ ❉✐✈✐s✐❜✐❧✐t② ❛♥❞ ✐♥❢♦r♠❛t✐♦♥ ✢♦✇ ❢♦r ♥♦♥✲✐♥✈❡rt✐❜❧❡ q✉❜✐t ❞②♥❛♠✐❝❛❧ ♠❛♣s ❉❛r✐✉s③ ❈❤r✉➧❝✐➠s❦✐ ◆✐❝♦❧❛✉s ❈♦♣❡r♥✐❝✉s ❯♥✐✈❡rs✐t②✱ ❚♦r✉➠✱ P♦❧❛♥❞ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ➪✳ ❘✐✈❛s✱ ❛♥❞ ❊✳ ❙tør♠❡r✱ ❛♥❞ ❙✳ ❈❤❛❦r❛❜♦rt②

  2. ❉✐✈✐s✐❜✐❧✐t② ❛♥❞ ✐♥❢♦r♠❛t✐♦♥ ✢♦✇ ❢♦r ♥♦♥✲✐♥✈❡rt✐❜❧❡ q✉❜✐t ❞②♥❛♠✐❝❛❧ ♠❛♣s ✷ ✴ ✹✶

  3. ❉✐✈✐s✐❜✐❧✐t② ❛♥❞ ✐♥❢♦r♠❛t✐♦♥ ✢♦✇ ❢♦r ♥♦♥✲✐♥✈❡rt✐❜❧❡ q✉❜✐t ❞②♥❛♠✐❝❛❧ ♠❛♣s ✸ ✴ ✹✶ ❉②♥❛♠✐❝❛❧ ♠❛♣ Quantum evolution ← → ρ → ρ t := Λ t ( ρ ) Λ t : B ( H ) → B ( H ) ; t ≥ 0 ❝♦♠♣❧❡t❡❧② ♣♦s✐t✐✈❡ tr❛❝❡✲♣r❡s❡r✈✐♥❣ Λ t =0 = id

  4. ❉✐✈✐s✐❜✐❧✐t② ❛♥❞ ✐♥❢♦r♠❛t✐♦♥ ✢♦✇ ❢♦r ♥♦♥✲✐♥✈❡rt✐❜❧❡ q✉❜✐t ❞②♥❛♠✐❝❛❧ ♠❛♣s ✹ ✴ ✹✶ ▼❛r❦♦✈✐❛♥ s❡♠✐❣r♦✉♣ ∂ t Λ t = L Λ t ❚❤❡♦r❡♠ ✭●♦r✐♥✐✲❑♦ss❛❦♦✇s❦✐✲❙✉❞❛rs❤❛♥✲▲✐♥❞❜❧❛❞ ✭✶✾✼✻✮✮ Λ t = e t L ✐s ❈P❚P ✐❢ ❛♥❞ ♦♥❧② ✐❢ � �� � � α − 1 V α ρV † V † α V α ρ + ρV † L ( ρ ) = − i [ H, ρ ]+ γ α α V α ; γ α > 0 2 α

  5. ❉✐✈✐s✐❜✐❧✐t② ❛♥❞ ✐♥❢♦r♠❛t✐♦♥ ✢♦✇ ❢♦r ♥♦♥✲✐♥✈❡rt✐❜❧❡ q✉❜✐t ❞②♥❛♠✐❝❛❧ ♠❛♣s ✺ ✴ ✹✶

  6. ❉✐✈✐s✐❜✐❧✐t② ❛♥❞ ✐♥❢♦r♠❛t✐♦♥ ✢♦✇ ❢♦r ♥♦♥✲✐♥✈❡rt✐❜❧❡ q✉❜✐t ❞②♥❛♠✐❝❛❧ ♠❛♣s ✻ ✴ ✹✶ ❇❡②♦♥❞ ▼❛r❦♦✈✐❛♥ s❡♠✐❣r♦✉♣ ♥♦♥✲❧♦❝❛❧ ♠❛st❡r ❡q✉❛t✐♦♥ ✭◆❛❦❛❥✐♠❛✲❩✇❛♥③✐❣ ❡q✉❛t✐♦♥✮ � t ∂ t Λ t = K t − τ Λ τ dτ 0 s❡♠✐✲▼❛r❦♦✈ ✭❇r❡✉❡r✲❱❛❝❝❤✐♥✐✱ ❑♦ss❛❦♦✇s❦✐✱ ❉❈✮ ❝♦❧❧✐s✐♦♥ ♠♦❞❡❧s ✭●✐♦✈❛♥❡tt✐✱ P❛❧♠❛✱ ❈✐❝❛r❡❧❧♦✱ ▲♦r❡♥③♦ ✳✳✳ ✮ ❊♥❣✐♥❡❡r✐♥❣ q✉❛♥t✉♠ ❝❤❛♥♥❡❧s ✭▼❛rs❤❛❧❧✱ ❩❛♥❛r❞✐✱ ✳✳✳ ✮ ❧♦❝❛❧ ✐♥ t✐♠❡ ♠❛st❡r ❡q✉❛t✐♦♥ ✭❚❈▲✮ ∂ t Λ t = L t Λ t

  7. ❉✐✈✐s✐❜✐❧✐t② ❛♥❞ ✐♥❢♦r♠❛t✐♦♥ ✢♦✇ ❢♦r ♥♦♥✲✐♥✈❡rt✐❜❧❡ q✉❜✐t ❞②♥❛♠✐❝❛❧ ♠❛♣s ✼ ✴ ✹✶ ■♥✈❡rt✐❜❧❡ ♠❛♣s ✭❈P❚P✮ Φ : B ( H ) → B ( H ) Φ − 1 : B ( H ) → B ( H ) Φ − 1 ✐s tr❛❝❡✲♣r❡s❡r✈✐♥❣ ❜✉t ♥♦t ❈P ❚❤❡♦r❡♠ Φ − 1 ✐s ❈P❚P ✐✛ Φ( ρ ) = Uρ U † Φ − 1 ( X ) = U † XU − →

  8. ❉✐✈✐s✐❜✐❧✐t② ❛♥❞ ✐♥❢♦r♠❛t✐♦♥ ✢♦✇ ❢♦r ♥♦♥✲✐♥✈❡rt✐❜❧❡ q✉❜✐t ❞②♥❛♠✐❝❛❧ ♠❛♣s ✽ ✴ ✹✶ ■♥✈❡rt✐❜❧❡ ♠❛♣s ∂ t Λ t = L t Λ t , Λ 0 = id L t = [ ∂ t Λ t ] Λ − 1 t L t ✖ r❡❣✉❧❛r ❣❡♥❡r❛t♦r

  9. ❉✐✈✐s✐❜✐❧✐t② ❛♥❞ ✐♥❢♦r♠❛t✐♦♥ ✢♦✇ ❢♦r ♥♦♥✲✐♥✈❡rt✐❜❧❡ q✉❜✐t ❞②♥❛♠✐❝❛❧ ♠❛♣s ✾ ✴ ✹✶ ❊①❛♠♣❧❡✿ q✉❜✐t ♠❛♣ � � ρ 11 ρ 12 f ( t ) Λ t ( ρ ) = ; f ( t ) ∈ C ρ 21 f ∗ ( t ) ρ 22 Λ t ✐s ❈P❚P ✐✛ | f ( t ) | ≤ 1 Λ t ✐s ✐♥✈❡rt✐❜❧❡ ✐✛ f ( t ) � = 0 L t ( ρ ) = − iω ( t )[ σ 3 , ρ ] + γ ( t )[ σ 3 ρσ 3 − ρ ] ˙ ˙ f ( t ) f ( t ) γ ( t ) = − Re f ( t ) ; ω ( t ) = − Im f ( t )

  10. ❉✐✈✐s✐❜✐❧✐t② ❛♥❞ ✐♥❢♦r♠❛t✐♦♥ ✢♦✇ ❢♦r ♥♦♥✲✐♥✈❡rt✐❜❧❡ q✉❜✐t ❞②♥❛♠✐❝❛❧ ♠❛♣s ✶✵ ✴ ✹✶ ❉✐✈✐s✐❜✐❧✐t② ✈s ▼❛r❦♦✈✐❛♥✐t②

  11. ❉✐✈✐s✐❜✐❧✐t② ❛♥❞ ✐♥❢♦r♠❛t✐♦♥ ✢♦✇ ❢♦r ♥♦♥✲✐♥✈❡rt✐❜❧❡ q✉❜✐t ❞②♥❛♠✐❝❛❧ ♠❛♣s ✶✶ ✴ ✹✶ ❉✐✈✐s✐❜✐❧✐t② Λ t = V t,s Λ s ; t ≥ s V t,s : B ( H ) → B ( H ) P✲❞✐✈✐s✐❜❧❡ ✐✛ V t,s ✐s P❚P ❈P✲❞✐✈✐s✐❜❧❡ ✐✛ V t,s ✐s ❈P❚P ❈P✲❞✐✈✐s✐❜❧❡ = ⇒ P✲❞✐✈✐s✐❜❧❡ ❚❤❡♦r❡♠ ✭❇❡♥❛tt✐✱❉❈✱❋✐❧❧✐♣♦✈ ✭✷✵✶✼✮✮ Λ t ✐s ❈P✲❞✐✈✐s✐❜❧❡ ✐✛ Λ t ⊗ Λ t ✐s P✲❞✐✈✐s✐❜❧❡

  12. ❉✐✈✐s✐❜✐❧✐t② ❛♥❞ ✐♥❢♦r♠❛t✐♦♥ ✢♦✇ ❢♦r ♥♦♥✲✐♥✈❡rt✐❜❧❡ q✉❜✐t ❞②♥❛♠✐❝❛❧ ♠❛♣s ✶✷ ✴ ✹✶ ■♥✈❡rt✐❜❧❡ ♠❛♣s ■♥✈❡rt✐❜❧❡ ♠❛♣ ✐s ❛❧✇❛②s ❞✐✈✐s✐❜❧❡ V t,s := Λ t Λ − 1 s ❚❤❡♦r❡♠ ■❢ Λ t ✐s ✐♥✈❡rt✐❜❧❡ ✱ t❤❡♥ ✐t ✐s ❈P✲❞✐✈✐s✐❜❧❡ ✐✛ � � � [ V α ( t ) , ρV † α ( t )] + [ V α ( t ) ρ, V † L t ( ρ ) = − i [ H ( t ) , ρ ]+ γ α ( t ) α ( t )] α ❛♥❞ γ α ( t ) ≥ 0 ✳

  13. ❉✐✈✐s✐❜✐❧✐t② ❛♥❞ ✐♥❢♦r♠❛t✐♦♥ ✢♦✇ ❢♦r ♥♦♥✲✐♥✈❡rt✐❜❧❡ q✉❜✐t ❞②♥❛♠✐❝❛❧ ♠❛♣s ✶✸ ✴ ✹✶ ▼❛r❦♦✈✐❛♥✐t② ✲ ❘❍P ❉❡✜♥✐t✐♦♥✿ ❬❘✐✈❛s✱❍✉❡❧❣❛✱P❧❡♥✐♦❪ ❊✈♦❧✉t✐♦♥ ✐s ▼❆❘❑❖❱■❆◆ ✐✛ Λ t ✐s ❈P✲❞✐✈✐s✐❜❧❡

  14. ❉✐✈✐s✐❜✐❧✐t② ❛♥❞ ✐♥❢♦r♠❛t✐♦♥ ✢♦✇ ❢♦r ♥♦♥✲✐♥✈❡rt✐❜❧❡ q✉❜✐t ❞②♥❛♠✐❝❛❧ ♠❛♣s ✶✹ ✴ ✹✶ ▼❛r❦♦✈✐❛♥✐t② ✲ ❇▲P ❉❡✜♥✐t✐♦♥✿ ❬❇r❡✉❡r✱▲❛✐♥❡✱P✐✐❧♦❪ ▼❆❘❑❖❱■❆◆■❚❨ ← → ♥♦ ❜❛❝❦✢♦✇ ♦❢ ✐♥❢♦r♠❛t✐♦♥ d dt � Λ t ( σ − ρ ) � 1 ≤ 0 √ � X � 1 = Tr XX †

  15. ❉✐✈✐s✐❜✐❧✐t② ❛♥❞ ✐♥❢♦r♠❛t✐♦♥ ✢♦✇ ❢♦r ♥♦♥✲✐♥✈❡rt✐❜❧❡ q✉❜✐t ❞②♥❛♠✐❝❛❧ ♠❛♣s ✶✺ ✴ ✹✶ ▼❛r❦♦✈✐❛♥ ✈s✳ ♥♦♥✲▼❛r❦♦✈✐❛♥ ▼❛r❦♦✈✐❛♥✐t② ✐s ❞❡✜♥❡❞ ❢♦r ❝❧❛ss✐❝❛❧ st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s ◗✉❛♥t✉♠ ▼❛r❦♦✈✐❛♥ ♣r♦❝❡ss❡s ✭❆❝❝❛r❞✐✱ ❋r✐❣❡r♦✱ ▲❡✇✐s✱ ▲✐♥❞❜❧❛❞✮ ▼❛r❦♦✈✐❛♥✐t② ❂ ❈P✲❞✐✈✐s✐❜✐❧✐t② ✭❘✐✈❛s✱ ❍✉❡❧❣❛✱ P❧❡♥✐♦✮ ▼❛r❦♦✈✐❛♥✐t② ❂ ♥♦ ✐♥❢♦r♠❛t✐♦♥ ✢♦✇ ✭❇r❡✉❡r✱ ▲❛✐♥❡✱ P✐✐❧♦✮ ●❡♦♠❡tr✐❝❛❧ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ♥♦♥✲▼❛r❦♦✈✐❛♥✐t② ✭▲♦r❡♥③♦✱ P❧❛st✐♥❛✱ P❛t❡r♥♦str♦✮ ♥♦♥✲▼❛r❦♦✈✐❛♥✐t② ✈✐❛ ♠✉t✉❛❧ ✐♥❢♦r♠❛t✐♦♥ ✭▲✉♦✮ ♥♦♥✲▼❛r❦♦✈✐❛♥✐t② ✈✐❛ q✉❛♥t✉♠ r❡❣r❡ss✐♦♥ t❤❡♦r❡♠ ✭❱❛❝❝❤✐♥✐✱ ❙♠✐r♥❡✱ ❍✉❡❧❣❛✱ P❡tr✉❝❝✐♦♥❡✱✳✳✳✮ ♥♦♥✲▼❛r❦♦✈✐❛♥✐t② ✈✐❛ ❞✐s❝r✐♠✐♥❛t✐♦♥ ♦❢ st❛t❡s ✭❇✉s❝❡♠✐✱ ❉❛tt❛✮ ♥♦♥✲▼❛r❦♦✈✐❛♥✐t② ✈✐❛ ❝❤❛♥♥❡❧ ❝❛♣❛❝✐t② ✭❇②❧✐❝❦❛✱ ❉❈✱ ▼❛♥✐s❝❛❧❝♦✮ ✳✳✳

  16. ❉✐✈✐s✐❜✐❧✐t② ❛♥❞ ✐♥❢♦r♠❛t✐♦♥ ✢♦✇ ❢♦r ♥♦♥✲✐♥✈❡rt✐❜❧❡ q✉❜✐t ❞②♥❛♠✐❝❛❧ ♠❛♣s ✶✻ ✴ ✹✶ ❉✐✈✐s✐❜✐❧✐t② ✈s✳ ♠♦♥♦t♦♥✐❝✐t② ♦❢ tr❛❝❡ ♥♦r♠ ❚❤❡♦r❡♠ ■❢ Λ t ✐s P✲❞✐✈✐s✐❜❧❡✱ t❤❡♥ d dt � Λ t ( X ) � 1 ≤ 0 ❢♦r ❛♥② X ∈ B ( H ) ✳ X = ρ − σ ⇒ P✲❞✐✈✐s✐❜✐❧✐t② = ⇒ ♥♦ ✐♥❢♦r♠❛t✐♦♥ ❜❛❝❦✢♦✇ ❈P✲❞✐✈✐s✐❜✐❧✐t② =

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