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❈♦♥❞✐t✐♦♥s ❢♦r ❧❡❣✐t✐♠❛t❡ ♠❡♠♦r② ❦❡r♥❡❧ ♠❛st❡r ❡q✉❛t✐♦♥ ✶ ✴ ✺✺ ❈♦♥❞✐t✐♦♥s ❢♦r ❧❡❣✐t✐♠❛t❡ ♠❡♠♦r② ❦❡r♥❡❧ ♠❛st❡r ❡q✉❛t✐♦♥ ❉❛r✐✉s③ ❈❤r✉➧❝✐➠s❦✐ ◆✐❝♦❧❛✉s ❈♦♣❡r♥✐❝✉s ❯♥✐✈❡rs✐t②✱ ❚♦r✉➠✱ P❖▲❆◆❉ ✻✵ ❨❡❛rs ❆❧❜❡rt♦ ■❜♦rt ❋❡st

❈♦♥❞✐t✐♦♥s ❢♦r ❧❡❣✐t✐♠❛t❡ ♠❡♠♦r② ❦❡r♥❡❧ ♠❛st❡r ❡q✉❛t✐♦♥ ✷ ✴ ✺✺ ❚❤❡ ♣r♦❜❧❡♠ ❍♦✇ t♦ ❞❡s❝r✐❜❡ ✭♥♦♥✲r❡❧❛t✐✈✐st✐❝✮ ❡✈♦❧✉t✐♦♥ ♦❢ t❤❡ ❞❡♥s✐t② ♠❛tr✐① ρ − → ρ t ❜❡②♦♥❞ ▼❛r❦♦✈✐❛♥ s❡♠✐✲❣r♦✉♣ ♥♦♥✲▼❛r❦♦✈✐❛♥ ✭♦r ❡✈♦❧✉t✐♦♥ ✇✐t❤ ♠❡♠♦r②✮✿ ❤♦✇ t♦ ❞❡✜♥❡✱ ❤♦✇ t♦ ❝❤❛r❛❝t❡r✐③❡❄

❈♦♥❞✐t✐♦♥s ❢♦r ❧❡❣✐t✐♠❛t❡ ♠❡♠♦r② ❦❡r♥❡❧ ♠❛st❡r ❡q✉❛t✐♦♥ ✸ ✴ ✺✺ ◗✉❛♥t✉♠ ❡✈♦❧✉t✐♦♥ ← → ❞②♥❛♠✐❝❛❧ ♠❛♣ Λ t : D ( H ) − → D ( H ) ; ( t ≥ 0) D ( H ) = { ρ ≥ 0 ; Tr ρ = 1 } ❝♦♠♣❧❡t❡❧② ♣♦s✐t✐✈❡ tr❛❝❡✲♣r❡s❡r✈✐♥❣ Λ 0 = 1 l

❈♦♥❞✐t✐♦♥s ❢♦r ❧❡❣✐t✐♠❛t❡ ♠❡♠♦r② ❦❡r♥❡❧ ♠❛st❡r ❡q✉❛t✐♦♥ ✹ ✴ ✺✺ ❈♦♠♣❧❡t❡❧② ♣♦s✐t✐✈❡ ♠❛♣s Φ : A − → B ( H ) ❙t✐♥❡s♣r✐♥❣ ✶✾✺✺ Φ ✐s ❝♦♠♣❧❡t❡❧② ♣♦s✐t✐✈❡ ✐✛ t❤❡r❡ ❡①✐sts ❛ ❍✐❧❜❡rt s♣❛❝❡ K t❤❡r❡ ❡①✐sts ⋆ ✲❤♦♠♦♠♦r❤✐s♠ π : A − → B ( K ) t❤❡r❡ ❡①✐sts V : K − → H Φ[ a ] = V π ( a ) V ∗

❈♦♥❞✐t✐♦♥s ❢♦r ❧❡❣✐t✐♠❛t❡ ♠❡♠♦r② ❦❡r♥❡❧ ♠❛st❡r ❡q✉❛t✐♦♥ ✺ ✴ ✺✺ dim H = d < ∞ ❑r❛✉s r❡♣r❡s❡♥t❛t✐♦♥ � K α XK † Φ[ X ] = α α � K † α K α = I α → Φ[ X ] = UXU † ✉♥✐t❛r② ♠❛♣ −

❈♦♥❞✐t✐♦♥s ❢♦r ❧❡❣✐t✐♠❛t❡ ♠❡♠♦r② ❦❡r♥❡❧ ♠❛st❡r ❡q✉❛t✐♦♥ ✺ ✴ ✺✺ dim H = d < ∞ ❑r❛✉s r❡♣r❡s❡♥t❛t✐♦♥ � K α XK † Φ[ X ] = α α � K † α K α = I α → Φ[ X ] = UXU † ✉♥✐t❛r② ♠❛♣ −

❈♦♥❞✐t✐♦♥s ❢♦r ❧❡❣✐t✐♠❛t❡ ♠❡♠♦r② ❦❡r♥❡❧ ♠❛st❡r ❡q✉❛t✐♦♥ ✻ ✴ ✺✺ ❲❤② ❝♦♠♣❧❡t❡ ♣♦s✐t✐✈✐t② P♦s✐t✐✈❡ ♠❛♣s X ≥ 0 − → Φ[ X ] ≥ 0 Φ 1 , Φ 2 ✕ ♣♦s✐t✐✈❡ ♠❛♣s Φ 1 ⊗ Φ 2 ✕ ♥❡❡❞s ◆❖❚ ❜❡ ❛ ♣♦s✐t✐✈❡ ♠❛♣✦✦✦

❈♦♥❞✐t✐♦♥s ❢♦r ❧❡❣✐t✐♠❛t❡ ♠❡♠♦r② ❦❡r♥❡❧ ♠❛st❡r ❡q✉❛t✐♦♥ ✻ ✴ ✺✺ ❲❤② ❝♦♠♣❧❡t❡ ♣♦s✐t✐✈✐t② P♦s✐t✐✈❡ ♠❛♣s X ≥ 0 − → Φ[ X ] ≥ 0 Φ 1 , Φ 2 ✕ ♣♦s✐t✐✈❡ ♠❛♣s Φ 1 ⊗ Φ 2 ✕ ♥❡❡❞s ◆❖❚ ❜❡ ❛ ♣♦s✐t✐✈❡ ♠❛♣✦✦✦

❈♦♥❞✐t✐♦♥s ❢♦r ❧❡❣✐t✐♠❛t❡ ♠❡♠♦r② ❦❡r♥❡❧ ♠❛st❡r ❡q✉❛t✐♦♥ ✼ ✴ ✺✺ ❲❤② ❝♦♠♣❧❡t❡ ♣♦s✐t✐✈✐t② ❝♦♠♣❧❡t❡❧② ♣♦s✐t✐✈❡ ♠❛♣s ⊂ ♣♦s✐t✐✈❡ ♠❛♣s Φ 1 , Φ 2 ✕ ❈P ♠❛♣s − → Φ 1 ⊗ Φ 2 ✕ ✐s ❛ ❈P ♠❛♣

❈♦♥❞✐t✐♦♥s ❢♦r ❧❡❣✐t✐♠❛t❡ ♠❡♠♦r② ❦❡r♥❡❧ ♠❛st❡r ❡q✉❛t✐♦♥ ✼ ✴ ✺✺ ❲❤② ❝♦♠♣❧❡t❡ ♣♦s✐t✐✈✐t② ❝♦♠♣❧❡t❡❧② ♣♦s✐t✐✈❡ ♠❛♣s ⊂ ♣♦s✐t✐✈❡ ♠❛♣s Φ 1 , Φ 2 ✕ ❈P ♠❛♣s − → Φ 1 ⊗ Φ 2 ✕ ✐s ❛ ❈P ♠❛♣

❈♦♥❞✐t✐♦♥s ❢♦r ❧❡❣✐t✐♠❛t❡ ♠❡♠♦r② ❦❡r♥❡❧ ♠❛st❡r ❡q✉❛t✐♦♥ ✽ ✴ ✺✺ ✶ ▼❛r❦♦✈✐❛♥ s❡♠✐❣r♦✉♣ ✷ ❛♥❞ ❜❡②♦♥❞

❈♦♥❞✐t✐♦♥s ❢♦r ❧❡❣✐t✐♠❛t❡ ♠❡♠♦r② ❦❡r♥❡❧ ♠❛st❡r ❡q✉❛t✐♦♥ ✾ ✴ ✺✺ ▼❛r❦♦✈✐❛♥ s❡♠✐✲❣r♦✉♣ d Λ t = e tL ; t ≥ 0 dt Λ t = L Λ t ; − → ❲❤❛t ✐s t❤❡ ♠♦st ❣❡♥❡r❛❧ L ❄ ❚❤❡♦r❡♠ ✭●♦r✐♥✐✲❑♦ss❛❦♦✇s❦✐✲❙✉❞❛rs❤❛♥✲▲✐♥❞❜❧❛❞ ✭✶✾✼✻✮✮ Λ t = e tL ✐s ❈P❚P ✐❢ ❛♥❞ ♦♥❧② ✐❢ � � � l − 1 V k ρV † 2 { V † L [ ρ ] = − i [ H, ρ ] + γ kl l V k , ρ } ; [ γ kl ] ≥ 0 kl

❈♦♥❞✐t✐♦♥s ❢♦r ❧❡❣✐t✐♠❛t❡ ♠❡♠♦r② ❦❡r♥❡❧ ♠❛st❡r ❡q✉❛t✐♦♥ ✶✵ ✴ ✺✺

❈♦♥❞✐t✐♦♥s ❢♦r ❧❡❣✐t✐♠❛t❡ ♠❡♠♦r② ❦❡r♥❡❧ ♠❛st❡r ❡q✉❛t✐♦♥ ✶✶ ✴ ✺✺ ❍♦✇ t♦ ❣♦ ❜❡②♦♥❞ ▼❛r❦♦✈✐❛♥ s❡♠✐❣r♦✉♣ ❄ d dt Λ t = L Λ t ; Λ 0 = 1 l

❈♦♥❞✐t✐♦♥s ❢♦r ❧❡❣✐t✐♠❛t❡ ♠❡♠♦r② ❦❡r♥❡❧ ♠❛st❡r ❡q✉❛t✐♦♥ ✶✷ ✴ ✺✺ L − → L t d dt Λ t = L t Λ t ; Λ 0 = 1 l −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− � t d dt Λ t = K t − τ Λ τ dτ ; Λ 0 = 1 l 0 K t = δ ( t ) L − → ▼❛r❦♦✈✐❛♥ s❡♠✐❣r♦✉♣

❈♦♥❞✐t✐♦♥s ❢♦r ❧❡❣✐t✐♠❛t❡ ♠❡♠♦r② ❦❡r♥❡❧ ♠❛st❡r ❡q✉❛t✐♦♥ ✶✷ ✴ ✺✺ L − → L t d dt Λ t = L t Λ t ; Λ 0 = 1 l −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− � t d dt Λ t = K t − τ Λ τ dτ ; Λ 0 = 1 l 0 K t = δ ( t ) L − → ▼❛r❦♦✈✐❛♥ s❡♠✐❣r♦✉♣

❈♦♥❞✐t✐♦♥s ❢♦r ❧❡❣✐t✐♠❛t❡ ♠❡♠♦r② ❦❡r♥❡❧ ♠❛st❡r ❡q✉❛t✐♦♥ ✶✸ ✴ ✺✺ ❊①❛♠♣❧❡✿ q✉❜✐t ❞❡♣❤❛s✐♥❣ � � ρ 11 ρ 12 cos t Λ t [ ρ ] = ρ 21 cos t ρ 22 L t [ ρ ] = γ ( t )[ σ 3 ρσ 3 − ρ ] K t [ ρ ] = k ( t )[ σ 3 ρσ 3 − ρ ] ❙♣❡❝tr✉♠✿ λ 1 ( t ) = λ 2 ( t ) = 1 ; λ 3 ( t ) = λ 4 ( t ) = cos t ■❢ cos t ❝r♦ss❡s ✵ t❤❡ ❣❡♥❡r❛t♦rs L t ❜❡❝♦♠❡s s✐♥❣✉❧❛r✦✦✦

❈♦♥❞✐t✐♦♥s ❢♦r ❧❡❣✐t✐♠❛t❡ ♠❡♠♦r② ❦❡r♥❡❧ ♠❛st❡r ❡q✉❛t✐♦♥ ✶✸ ✴ ✺✺ ❊①❛♠♣❧❡✿ q✉❜✐t ❞❡♣❤❛s✐♥❣ � � ρ 11 ρ 12 cos t Λ t [ ρ ] = ρ 21 cos t ρ 22 L t [ ρ ] = γ ( t )[ σ 3 ρσ 3 − ρ ] K t [ ρ ] = k ( t )[ σ 3 ρσ 3 − ρ ] ❙♣❡❝tr✉♠✿ λ 1 ( t ) = λ 2 ( t ) = 1 ; λ 3 ( t ) = λ 4 ( t ) = cos t ■❢ cos t ❝r♦ss❡s ✵ t❤❡ ❣❡♥❡r❛t♦rs L t ❜❡❝♦♠❡s s✐♥❣✉❧❛r✦✦✦

❈♦♥❞✐t✐♦♥s ❢♦r ❧❡❣✐t✐♠❛t❡ ♠❡♠♦r② ❦❡r♥❡❧ ♠❛st❡r ❡q✉❛t✐♦♥ ✶✹ ✴ ✺✺ ❊①❛♠♣❧❡✿ q✉❜✐t ❞❡♣❤❛s✐♥❣ � � ρ 11 ρ 12 cos t Λ t [ ρ ] = ρ 21 cos t ρ 22 L t [ ρ ] = γ ( t )[ σ 3 ρσ 3 − ρ ] K t [ ρ ] = k ( t )[ σ 3 ρσ 3 − ρ ] γ ( t ) = tan t ✭s✐♥❣✉❧❛r✮ k ( t ) = 1 ✭r❡❣✉❧❛r✮

❈♦♥❞✐t✐♦♥s ❢♦r ❧❡❣✐t✐♠❛t❡ ♠❡♠♦r② ❦❡r♥❡❧ ♠❛st❡r ❡q✉❛t✐♦♥ ✶✺ ✴ ✺✺ d dt Λ t = L t Λ t ; Λ 0 = 1 l L t = ??? −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− d dt Λ t = K t ∗ Λ t ; Λ 0 = 1 l K t = ???

❈♦♥❞✐t✐♦♥s ❢♦r ❧❡❣✐t✐♠❛t❡ ♠❡♠♦r② ❦❡r♥❡❧ ♠❛st❡r ❡q✉❛t✐♦♥ ✶✻ ✴ ✺✺ d dt Λ t = L t Λ t ; Λ 0 = 1 l t❡❝❤♥✐❝❛❧❧② s✐♠♣❧❡r −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− d dt Λ t = K t ∗ Λ t ; Λ 0 = 1 l ♠♦r❡ ❢✉♥❞❛♠❡♥t❛❧

❈♦♥❞✐t✐♦♥s ❢♦r ❧❡❣✐t✐♠❛t❡ ♠❡♠♦r② ❦❡r♥❡❧ ♠❛st❡r ❡q✉❛t✐♦♥ ✶✼ ✴ ✺✺ ◆❛❦❛❥✐♠❛✲❩✇❛♥③✐❣ ♣r♦❥❡❝t✐♦♥ ♠❡t❤♦❞ H S ⊗ H E H = H S ⊗ I E + I S ⊗ H E + H I � e − iHt [ ρ S ⊗ ρ E ] e iHt � Λ t [ ρ S ] := Tr E � t d dt Λ t = K t − τ Λ τ dτ 0 ♠❡♠♦r② ❦❡r♥❡❧ ♠❛st❡r ❡q✉❛t✐♦♥ ✐s ✉♥✐✈❡rs❛❧