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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 − → ρt ❜❡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♥❛♠✐❝❛❧ ♠❛♣ Λt : D(H) − → D(H) ; (t ≥ 0) D(H) = { ρ ≥ 0 ; Trρ = 1 } ❝♦♠♣❧❡t❡❧② ♣♦s✐t✐✈❡ tr❛❝❡✲♣r❡s❡r✈✐♥❣ Λ0 = 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✐♦♥ Φ[X] =

• α

KαXK†

α

• α

K†

αKα = I

✉♥✐t❛r② ♠❛♣ − → Φ[X] = UXU †

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

dim H = d < ∞ ❑r❛✉s r❡♣r❡s❡♥t❛t✐♦♥ Φ[X] =

• α

KαXK†

α

• α

K†

αKα = I

✉♥✐t❛r② ♠❛♣ − → Φ[X] = UXU †

❈♦♥❞✐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 Φ1 , Φ2 ✕ ♣♦s✐t✐✈❡ ♠❛♣s Φ1 ⊗ Φ2 ✕ ♥❡❡❞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 Φ1 , Φ2 ✕ ♣♦s✐t✐✈❡ ♠❛♣s Φ1 ⊗ Φ2 ✕ ♥❡❡❞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 Φ1 , Φ2 ✕ ❈P ♠❛♣s − → Φ1 ⊗ Φ2 ✕ ✐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 Φ1 , Φ2 ✕ ❈P ♠❛♣s − → Φ1 ⊗ Φ2 ✕ ✐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 dtΛt = LΛt ; − → Λt = etL ; t ≥ 0 ❲❤❛t ✐s t❤❡ ♠♦st ❣❡♥❡r❛❧ L ❄ ❚❤❡♦r❡♠ ✭●♦r✐♥✐✲❑♦ss❛❦♦✇s❦✐✲❙✉❞❛rs❤❛♥✲▲✐♥❞❜❧❛❞ ✭✶✾✼✻✮✮ Λt = etL ✐s ❈P❚P ✐❢ ❛♥❞ ♦♥❧② ✐❢ L[ρ] = −i[H, ρ] +

• kl

γkl

• VkρV †

l − 1

2{V †

l Vk, ρ}

• ; [γkl] ≥ 0

❈♦♥❞✐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 dtΛt = LΛt ; Λ0 = 1 l

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

L − → Lt d dtΛt = LtΛt ; Λ0 = 1 l −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− d dt Λt = t Kt−τΛτdτ ; Λ0 = 1 l Kt = δ(t) L − → ▼❛r❦♦✈✐❛♥ s❡♠✐❣r♦✉♣

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

L − → Lt d dtΛt = LtΛt ; Λ0 = 1 l −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− d dt Λt = t Kt−τΛτdτ ; Λ0 = 1 l Kt = δ(t) L − → ▼❛r❦♦✈✐❛♥ s❡♠✐❣r♦✉♣

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

❊①❛♠♣❧❡✿ q✉❜✐t ❞❡♣❤❛s✐♥❣

Λt[ρ] =

• ρ11

ρ12 cos t ρ21 cos t ρ22

• Lt[ρ] = γ(t)[σ3ρσ3 − ρ]

Kt[ρ] = k(t)[σ3ρσ3 − ρ] ❙♣❡❝tr✉♠✿ λ1(t) = λ2(t) = 1 ; λ3(t) = λ4(t) = cos t ■❢ cos t ❝r♦ss❡s ✵ t❤❡ ❣❡♥❡r❛t♦rs Lt ❜❡❝♦♠❡s s✐♥❣✉❧❛r✦✦✦

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

❊①❛♠♣❧❡✿ q✉❜✐t ❞❡♣❤❛s✐♥❣

Λt[ρ] =

• ρ11

ρ12 cos t ρ21 cos t ρ22

• Lt[ρ] = γ(t)[σ3ρσ3 − ρ]

Kt[ρ] = k(t)[σ3ρσ3 − ρ] ❙♣❡❝tr✉♠✿ λ1(t) = λ2(t) = 1 ; λ3(t) = λ4(t) = cos t ■❢ cos t ❝r♦ss❡s ✵ t❤❡ ❣❡♥❡r❛t♦rs Lt ❜❡❝♦♠❡s s✐♥❣✉❧❛r✦✦✦

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

❊①❛♠♣❧❡✿ q✉❜✐t ❞❡♣❤❛s✐♥❣

Λt[ρ] =

• ρ11

ρ12 cos t ρ21 cos t ρ22

• Lt[ρ] = γ(t)[σ3ρσ3 − ρ]

Kt[ρ] = k(t)[σ3ρσ3 − ρ] γ(t) = tan t ✭s✐♥❣✉❧❛r✮ k(t) = 1 ✭r❡❣✉❧❛r✮

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

d dtΛt = LtΛt ; Λ0 = 1 l Lt = ??? −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− d dt Λt = Kt ∗ Λt ; Λ0 = 1 l Kt = ???

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

d dtΛt = LtΛt ; Λ0 = 1 l t❡❝❤♥✐❝❛❧❧② s✐♠♣❧❡r −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− d dt Λt = Kt ∗ Λt ; Λ0 = 1 l ♠♦r❡ ❢✉♥❞❛♠❡♥t❛❧

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

◆❛❦❛❥✐♠❛✲❩✇❛♥③✐❣ ♣r♦❥❡❝t✐♦♥ ♠❡t❤♦❞

HS ⊗ HE H = HS ⊗ IE + IS ⊗ HE + HI Λt[ρS] := TrE

• e−iHt[ρS ⊗ ρE]eiHt

d dt Λt = t Kt−τΛτ dτ ♠❡♠♦r② ❦❡r♥❡❧ ♠❛st❡r ❡q✉❛t✐♦♥ ✐s ✉♥✐✈❡rs❛❧

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

▼❛r❦♦✈✐❛♥ s❡♠✐❣r♦✉♣

d dtΛt = LΛt ; Λ0 = 1 l ❇♦r♥✲▼❛r❦♦✈ ❛♣♣r♦①✐♠❛t✐♦♥ ✇❡❛❦ ❝♦✉♣❧✐♥❣ s✐♥❣✉❧❛r ❝♦✉♣❧✐♥❣ ✭r❡s❡r✈♦✐r ❝♦rr❡❧❛t✐♦♥ ❢✉♥❝t✐♦♥s ∼ δ(t)✮ ✳✳✳ ❈✉rr❡♥t ❡①♣❡r✐♠❡♥ts ❝❛❧❧ ❢♦r ♠♦r❡ r❡✜♥❡ ❛♣♣r♦❛❝❤

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

▼❛r❦♦✈✐❛♥ ✈s✳ ♥♦♥✲▼❛r❦♦✈✐❛♥

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

❉✐✈✐s✐❜✐❧✐t②

Λt = Vt,sΛs ; t ≥ s ❈P✲❞✐✈✐s✐❜❧❡ ✐❢ Vt,s ✐s ❈P P✲❞✐✈✐s✐❜❧❡ ✐❢ Vt,s ✐s ♣♦s✐t✐✈❡ ❈P✲❞✐✈✐s✐❜❧❡ − → P✲❞✐✈✐s✐❜❧❡ ▼❛r❦♦✈✐❛♥ s❡♠✐❣r♦✉♣ − → Vt,s = e(t−s)L ✭❈P✲❞✐✈✐s✐❜❧❡✮ ❚❤❡♦r❡♠ ✭❇❡♥❛tt✐✱❉❈✱❋✐❧❧✐♣♦✈ ✭✷✵✶✼✮✮ Λt ✐s ❈P✲❞✐✈✐s✐❜❧❡ ✐✛ Λt ⊗ Λt ✐s P✲❞✐✈✐s✐❜❧❡

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

❉✐✈✐s✐❜✐❧✐t②

Λt = Vt,sΛs ; t ≥ s ❈P✲❞✐✈✐s✐❜❧❡ ✐❢ Vt,s ✐s ❈P P✲❞✐✈✐s✐❜❧❡ ✐❢ Vt,s ✐s ♣♦s✐t✐✈❡ ❈P✲❞✐✈✐s✐❜❧❡ − → P✲❞✐✈✐s✐❜❧❡ ▼❛r❦♦✈✐❛♥ s❡♠✐❣r♦✉♣ − → Vt,s = e(t−s)L ✭❈P✲❞✐✈✐s✐❜❧❡✮ ❚❤❡♦r❡♠ ✭❇❡♥❛tt✐✱❉❈✱❋✐❧❧✐♣♦✈ ✭✷✵✶✼✮✮ Λt ✐s ❈P✲❞✐✈✐s✐❜❧❡ ✐✛ Λt ⊗ Λt ✐s P✲❞✐✈✐s✐❜❧❡

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

❉✐✈✐s✐❜✐❧✐t②

Λt = Vt,sΛs ; t ≥ s ❈P✲❞✐✈✐s✐❜❧❡ ✐❢ Vt,s ✐s ❈P P✲❞✐✈✐s✐❜❧❡ ✐❢ Vt,s ✐s ♣♦s✐t✐✈❡ ❈P✲❞✐✈✐s✐❜❧❡ − → P✲❞✐✈✐s✐❜❧❡ ▼❛r❦♦✈✐❛♥ s❡♠✐❣r♦✉♣ − → Vt,s = e(t−s)L ✭❈P✲❞✐✈✐s✐❜❧❡✮ ❚❤❡♦r❡♠ ✭❇❡♥❛tt✐✱❉❈✱❋✐❧❧✐♣♦✈ ✭✷✵✶✼✮✮ Λt ✐s ❈P✲❞✐✈✐s✐❜❧❡ ✐✛ Λt ⊗ Λt ✐s P✲❞✐✈✐s✐❜❧❡

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

❉✐✈✐s✐❜✐❧✐t②

Λt = Vt,sΛs ; t ≥ s ❈P✲❞✐✈✐s✐❜❧❡ ✐❢ Vt,s ✐s ❈P P✲❞✐✈✐s✐❜❧❡ ✐❢ Vt,s ✐s ♣♦s✐t✐✈❡ ❈P✲❞✐✈✐s✐❜❧❡ − → P✲❞✐✈✐s✐❜❧❡ ▼❛r❦♦✈✐❛♥ s❡♠✐❣r♦✉♣ − → Vt,s = e(t−s)L ✭❈P✲❞✐✈✐s✐❜❧❡✮ ❚❤❡♦r❡♠ ✭❇❡♥❛tt✐✱❉❈✱❋✐❧❧✐♣♦✈ ✭✷✵✶✼✮✮ Λt ✐s ❈P✲❞✐✈✐s✐❜❧❡ ✐✛ Λt ⊗ Λt ✐s P✲❞✐✈✐s✐❜❧❡

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

▼❛r❦♦✈✐❛♥ ✈s✳ ♥♦♥✲▼❛r❦♦✈✐❛♥

▼❛r❦♦✈✐❛♥✐t② ✐s ❞❡✜♥❡❞ ❢♦r ❝❧❛ss✐❝❛❧ st♦❝❤❛st✐❝ ♣r♦❝❡ss❡s ▼❛r❦♦✈✐❛♥✐t② ❂ s❡♠✐❣r♦✉♣ ❞②♥❛♠✐❝s✿ Λt = etL ▼❛r❦♦✈✐❛♥✐t② ❂ ❈P✲❞✐✈✐s✐❜✐❧✐t② ✭❘✐✈❛s✱ ❍✉❡❧❣❛✱ P❧❡♥✐♦✮✿ Λt = Vt,sΛs ▼❛r❦♦✈✐❛♥✐t② ❂ ♥❡❣❛t✐✈❡ ✐♥❢♦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 ✈✐❛ ❝❤❛♥♥❡❧ ❝❛♣❛❝✐t② ✭❇②❧✐❝❦❛✱ ❉❈✱ ▼❛♥✐s❝❛❧❝♦✮ ♥♦♥✲▼❛r❦♦✈✐❛♥✐t② ✈✐❛ ❝❤❛♥♥❡❧ ❞✐s❝r✐♠✐♥❛t✐♦♥ ✭❇❛❡✱ ❉❈✮ ✳✳✳

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

❙✉♣♣♦s❡ t❤❛t Λ−1

t

❡①✐sts ❚❤❡♦r❡♠ ✭❉❈✱ ❑♦ss❛❦♦✇s❦✐✱ ❘✐✈❛s ✭✷✵✶✶✮✮ Λt ✐s ❈P✲❞✐✈✐s✐❜❧❡ ✐✛ d dt || [1 l ⊗ Λt]X ||1 ≤ 0 ❢♦r ❛❧❧ X = X† ∈ B(H) ⊗ B(H)✳ Λt ✐s P✲❞✐✈✐s✐❜❧❡ ✐✛ d dt || ΛtX ||1 ≤ 0 ❢♦r ❛❧❧ X = X† ∈ B(H)✳

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

❙✉♣♣♦s❡ t❤❛t Λ−1

t

❡①✐sts ❚❤❡♦r❡♠ ✭❉❈✱ ❑♦ss❛❦♦✇s❦✐✱ ❘✐✈❛s ✭✷✵✶✶✮✮ Λt ✐s ❈P✲❞✐✈✐s✐❜❧❡ ✐✛ d dt || [1 l ⊗ Λt]X ||1 ≤ 0 ❢♦r ❛❧❧ X = X† ∈ B(H) ⊗ B(H)✳ Λt ✐s P✲❞✐✈✐s✐❜❧❡ ✐✛ d dt || ΛtX ||1 ≤ 0 ❢♦r ❛❧❧ X = X† ∈ B(H)✳

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

❇r❡✉❡r✲▲❛✐♥❡✲P✐✐❧♦ ✭❇▲P✮ ❝♦♥❞✐t✐♦♥ ✕ P❘▲ ✷✵✶✵

❊✈♦❧✉t✐♦♥ ✐s ▼❛r❦♦✈✐❛♥ ✐❢ σ(ρ1, ρ2; t) := d dt ||Λt(ρ1 − ρ2)||1 ≤ 0 ❢♦r ❛❧❧ ♣❛✐rs ρ1 ❛♥❞ ρ2✳ d dt || ΛtX ||1 ≤ 0 ; X = ρ1 − ρ2 ❈P✲❞✐✈✐s✐❜✐❧✐t② = ⇒ P✲❞✐✈✐s✐❜✐❧✐t② = ⇒ ❇▲P ❝♦♥❞✐t✐♦♥

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

❇r❡✉❡r✲▲❛✐♥❡✲P✐✐❧♦ ✭❇▲P✮ ❝♦♥❞✐t✐♦♥ ✕ P❘▲ ✷✵✶✵

❊✈♦❧✉t✐♦♥ ✐s ▼❛r❦♦✈✐❛♥ ✐❢ σ(ρ1, ρ2; t) := d dt ||Λt(ρ1 − ρ2)||1 ≤ 0 ❢♦r ❛❧❧ ♣❛✐rs ρ1 ❛♥❞ ρ2✳ d dt || ΛtX ||1 ≤ 0 ; X = ρ1 − ρ2 ❈P✲❞✐✈✐s✐❜✐❧✐t② = ⇒ P✲❞✐✈✐s✐❜✐❧✐t② = ⇒ ❇▲P ❝♦♥❞✐t✐♦♥

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

❊①❛♠♣❧❡✿ r❛♥❞♦♠ ✉♥✐t❛r②

Ltρ = 1 2

3

• k=1

γk(t)[σkρσk − ρ] =

• k

γk(t)Lk Λt ✐s ❈P✲❞✐✈✐s✐❜❧❡ ✐✛ γ1(t) ≥ 0 ; γ2(t) ≥ 0 ; γ3(t) ≥ 0 Λt ✐s P✲❞✐✈✐s✐❜❧❡ ✐✛ ✭P✲❞✐✈✐s✐❜❧❡ ≡ ❇▲P ❝♦♥❞✐t✐♦♥✮ γ1(t) + γ2(t) ≥ 0 ; γ1(t) + γ3(t) ≥ 0 ; γ2(t) + γ3(t) ≥ 0

❉❈✱ ❋✳ ❲✉❞❛rs❦✐✱ P▲❆ ✷✵✶✸✳

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

Λt = x1etL1 + x2etL2 + x3etL3

Λt ✐s ❛ ▼❛r❦♦✈✐❛♥ s❡♠✐✲❣r♦✉♣ ✐❢ ♦♥❧② ♦♥❡ xk = 1 Λt ✐s ❈P✲❞✐✈✐s✐❜❧❡ Λt ✐s P✲❞✐✈✐s✐❜❧❡ ❢♦r ❛❧❧ xk ✭≡ ❇▲P ❝♦♥❞✐t✐♦♥✮

◆✳▼❡❣✐❡r✱ ❉❈✱ ❏✳P✐✐❧♦✱ ❲✳❙tr✉♥③✱ ❙❝✳ ❘❡♣✳ ✷✵✶✼

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

▼❡♠♦r② ❡✛❡❝ts ✈✐❛ ❛ ♠❡♠♦r② ❦❡r♥❡❧

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

Pr♦❜❧❡♠✿ ♣❤②s✐❝❛❧❧② ❛❞♠✐ss✐❜❧❡ ❦❡r♥❡❧s

d dt Λt = t Kt−τΛτdτ Kt =???

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

d dt Λt = t Kt−τΛτdτ ❋✐♥❞ t❤❡ str✉❝t✉r❡ ♦❢ Kt s✉❝❤ t❤❛t Λt ✐s ❈P❚P ❇❛r♥❡tt ❛♥❞ ❙t❡♥❤♦❧♠ ✭✷✵✵✶✮✿ Kt = k(t)L ▲✐❞❛r ❛♥❞ ❙❤❛❜❛♥✐ ✭✷✵✵✺✮✿ Kt = k(t)LetL ✳✳✳ k(t) =???

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

❍♦✇ t♦ ❝♦♥str✉❝t ❛ ❧❡❣✐t✐♠❛t❡ Kt ❄

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

▼❛r❦♦✈✐❛♥ s❡♠✐❣r♦✉♣ ✖ q✉❛♥t✉♠ ❥✉♠♣s r❡♣r❡s❡♥t❛t✐♦♥

L[ρ] = −i[H, ρ] +

• α
• VαρV †

α − 1

2{V †

αVα, ρ}

• L = B − Z

B[ρ] =

• α

VαρV †

α

Z[ρ] = i(Cρ − ρC†) ; C = H − i 2

• α

V †

αVα

✭❲✐❣♥❡r✕❲❡✐ss❦♦♣❢ t❤❡♦r② ✮

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

B[ρ] =

• α

VαρV †

α ;

C = H − i 2

• α

V †

αVα

˙ Λt = (B − Z)Λt ; Λ0 = 1 l ˙ Nt = −ZNt ; N0 = 1 l Nt[ρ] = e−iCtρeiC†t ❉②s♦♥ ♣❡rt✉r❜❛t✐♦♥ s❡r✐❡s Λt = Nt + Nt ∗ BNt + Nt ∗ BNt ∗ BNt + . . .

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

Λt = Nt + Nt ∗ BNt + Nt ∗ BNt ∗ BNt + . . . Qt := BNt Λt = Nt + Nt ∗ (Qt + Qt ∗ Qt + Qt ∗ Qt ∗ Qt + . . .) Nt, Qt ✕❝♦♠♣❧❡t❡❧② ♣♦s✐t✐✈❡ ♠❛♣s q✉❛♥t✉♠ ❥✉♠♣ r❡♣r❡s❡♥t❛t✐♦♥

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

˙ Λt = LΛt Λt = etL = 1 l + tL + (tL)2 2! + (tL)3 3! + . . . L = B − Z Λt = Nt + Nt ∗ Qt + Nt ∗ Qt ∗ Qt + . . . tr❛❝❡ ♣r❡s❡r✈❛t✐♦♥ ✈s✳ ❝♦♠♣❧❡t❡ ♣♦s✐t✐✈✐t②

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

L − → Kt

˙ Λt = LΛt L = B − Z Λt = Nt + Nt ∗ BNt + Nt ∗ BNt ∗ BNt + . . . ˙ Λt = Kt ∗ Λt Kt = Bt − Zt Λt = Nt + Nt ∗ Bt ∗ Nt + Nt ∗ Bt ∗ Nt ∗ Bt ∗ Nt + . . .

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

Λt = Nt + Nt ∗ Bt ∗ Nt + Nt ∗ Bt ∗ Nt ∗ Bt ∗ Nt + . . . Qt := Bt ∗ Nt Λt = Nt + Nt ∗ (Qt + Qt ∗ Qt + Qt ∗ Qt ∗ Qt + . . .)

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

▲❡❣✐t✐♠❛t❡ ♣❛✐r

❲❡ ❝❛❧❧ {Nt, Qt} ❛ ❧❡❣✐t✐♠❛t❡ ♣❛✐r ✐✛

✶ Nt, Qt ❛r❡ ❈P✱ ❛♥❞ N0 = 1

l

✷ Tr[(Qt + ˙

Nt)ρ] = 0 Λt = Nt + Nt ∗ (Qt + Qt ∗ Qt + Qt ∗ Qt ∗ Qt + . . .) Kt = Bt − Zt Qt = Bt ∗ Nt ; ˙ Nt = −Zt ∗ Nt

❉❈✱ ❆✳ ❑♦ss❛❦♦✇s❦✐✱ P❘❆ ✭✷✵✶✻✮

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

❇❛r♥❡tt ❛♥❞ ❙t❡♥❤♦❧♠ ❣❡♥❡r❛t♦r✿ Kt = k(t)L

Nt =

• 1 −

t f(τ)dτ

• 1

l f(t) ≥ 0 ; ∞ f(τ)dτ ≤ 1 Qt = f(t)E ; E ✕ ❛r❜✐tr❛r② q✉❛♥t✉♠ ❝❤❛♥♥❡❧ Kt = k(t)(E − 1 l) = k(t)L

• k(s) =

s f(s) 1 − f(s) f(t) = γe−γt = ⇒ k(t) = γδ(t) − → ▼❛r❦♦✈✐❛♥ s❡♠✐✲❣r♦✉♣

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

❇❛r♥❡tt ❛♥❞ ❙t❡♥❤♦❧♠ ❣❡♥❡r❛t♦r✿ Kt = k(t)L

Nt =

• 1 −

t f(τ)dτ

• 1

l f(t) ≥ 0 ; ∞ f(τ)dτ ≤ 1 Qt = f(t)E ; E ✕ ❛r❜✐tr❛r② q✉❛♥t✉♠ ❝❤❛♥♥❡❧ Kt = k(t)(E − 1 l) = k(t)L

• k(s) =

s f(s) 1 − f(s) f(t) = γe−γt = ⇒ k(t) = γδ(t) − → ▼❛r❦♦✈✐❛♥ s❡♠✐✲❣r♦✉♣

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

Pr♦♣❡rt✐❡s ✖ ✶

❈♦♥✈❡①✐t② ■❢ {N(k)

t

, Q(k)

t } ❛r❡ ❧❡❣✐t✐♠❛t❡ ♣❛✐rs t❤❡♥ ❛ ❝♦♥✈❡① ❝♦♠❜✐♥❛t✐♦♥

Nt =

• k

pkN(k)

t

; Qt =

• k

pkQ(k)

t

♣r♦✈✐❞❡ ❛ ❧❡❣✐t✐♠❛t❡ ♣❛✐r✳

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

Pr♦♣❡rt✐❡s ✖ ✷

❘❡❞✉❝❡❞ ♣❛✐r ❙✉♣♣♦s❡ t❤❛t {Nt, Qt} ❞❡✜♥❡s ❛ ❧❡❣✐t✐♠❛t❡ ♣❛✐r ❢♦r t❤❡ ❡✈♦❧✉t✐♦♥ ✐♥ H ⊗ HE✳ ❚❤❡♥ ❢♦r ❛r❜✐tr❛r② st❛t❡ ω ✐♥ HE Nt[ρ] = TrE(Nt[ρ ⊗ ω]), Qt[ρ] = TrE(Qt[ρ ⊗ ω]), ♣r♦✈✐❞❡ ❛ ❧❡❣✐t✐♠❛t❡ ♣❛✐r ❢♦r t❤❡ ❡✈♦❧✉t✐♦♥ ✐♥ H✳

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

Pr♦♣❡rt✐❡s ✖ ✸

• ❛✉❣❡ tr❛♥s❢♦r♠❛t✐♦♥s

■❢ {Nt, Qt} ✐s ❛ ❧❡❣✐t✐♠❛t❡ ♣❛✐r ❛♥❞ Ft ✐s ❛ ❞②♥❛♠✐❝❛❧ ♠❛♣✱ t❤❡♥ N′

t = FtNt ;

Q′

t = FtQt,

♣r♦✈✐❞❡ ❛ ❧❡❣✐t✐♠❛t❡ ♣❛✐r ❛s ✇❡❧❧✳

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

■♥❤♦♠♦❣❡♥❡♦✉s ♠❛st❡r ❡q✉❛t✐♦♥

d dtρt = t Kt−τρτdτ {Nt, Qt} d dtρt = t Kt−τρτdτ + d dtNtρ0 ρ0 ✕ ✐♥✐t✐❛❧ st❛t❡ Kt = d dtQt + δ(t)Q0

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

■♥❤♦♠♦❣❡♥❡♦✉s ♠❛st❡r ❡q✉❛t✐♦♥

{Nt, Qt} d dtρt = Q0ρt + t d dtQt−τ

• ρτdτ + d

dtNtρ0

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

■♥❤♦♠♦❣❡♥❡♦✉s ♠❛st❡r ❡q✉❛t✐♦♥

d dtρt = Q0ρt + t d dtQt−τ

• ρτdτ + d

dtNtρ0 ❊①❛♠♣❧❡ Qt = f(t)E ; Nt = (1 − t f(u)du)1 l ˙ ρt = γEρt + t ˙ f(t − τ)E[ρτ]dτ − f(t)ρ0 γ := f(0)

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

❲❤❛t ❞♦❡s ✜t ♦✉r ❝❧❛ss ❄ q✉❛♥t✉♠ s❡♠✐✲▼❛r❦♦✈ ❡✈♦❧✉t✐♦♥ ✭❇r❡✉❡r✱ ❱❛❝❝❤✐♥✐✮ ❝♦❧❧✐s✐♦♥ ♠♦❞❡❧s ✭P❛❧♠❛✱ ●✐♦✈❛♥❡tt✐✱ ▲♦r❡♥③♦✱ ❈✐❝❛r❡❧❧♦✱ ❱❛❝❝❤✐♥✐✱✳✳✳✮ ✳✳✳✳✳

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

◗✉❛♥t✉♠ s❡♠✐✲▼❛r❦♦✈ ❡✈♦❧✉t✐♦♥

Qt ✖ q✉❛♥t✉♠ s❡♠✐✲▼❛r❦♦✈ ♠❛♣ ft = Q†

t[I]

✖ q✉❛♥t✉♠ ✇❛✐t✐♥❣ t✐♠❡ ♦♣❡r❛t♦r gt = I − t fτdτ ✖ q✉❛♥t✉♠ s✉r✈✐✈❛❧ ♦♣❡r❛t♦r Nt[ρ] = √gt ρ √gt {Qt, Nt} ; ✖ ❧❡❣✐t✐♠❛t❡ ♣❛✐r ft = Γe−Γt ; Γ ≥ 0 = ⇒ ▼❛r❦♦✈✐❛♥ s❡♠✐❣r♦✉♣

❉❈ ✫ ❆✳ ❑♦ss❛❦♦✇s❦✐ ✭✷✵✶✼✮

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

◗✉❛♥t✉♠ s❡♠✐✲▼❛r❦♦✈ ❡✈♦❧✉t✐♦♥

Qt ✖ q✉❛♥t✉♠ s❡♠✐✲▼❛r❦♦✈ ♠❛♣ ft = Q†

t[I]

✖ q✉❛♥t✉♠ ✇❛✐t✐♥❣ t✐♠❡ ♦♣❡r❛t♦r gt = I − t fτdτ ✖ q✉❛♥t✉♠ s✉r✈✐✈❛❧ ♦♣❡r❛t♦r Nt[ρ] = √gt ρ √gt {Qt, Nt} ; ✖ ❧❡❣✐t✐♠❛t❡ ♣❛✐r ft = Γe−Γt ; Γ ≥ 0 = ⇒ ▼❛r❦♦✈✐❛♥ s❡♠✐❣r♦✉♣

❉❈ ✫ ❆✳ ❑♦ss❛❦♦✇s❦✐ ✭✷✵✶✼✮

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

❚❤✐s ❝♦♥str✉❝t✐♦♥ ✈✐❛ {Qt, Nt} ❝♦✈❡rs ▼❆◆❨ ❡①❛♠♣❧❡s ❇✉t st✐❧❧ t❤❡r❡ ❛r❡ ❡①❛♠♣❧❡s ♦✉ts✐❞❡ t❤✐s ❝❧❛ss

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

Λt = x1etL1 + x2etL2 + x3etL3

Λt ✐s ❛ ▼❛r❦♦✈✐❛♥ s❡♠✐✲❣r♦✉♣ ✐❢ ♦♥❧② ♦♥❡ xk = 1 Λt ✐s s❡♠✐✲▼❛r❦♦✈ ✐❢ x1 = x2 = x3 = 1

3

Λt ✐s ❈P✲❞✐✈✐s✐❜❧❡ Λt ✐s P✲❞✐✈✐s✐❜❧❡ ❢♦r ❛❧❧ xk

◆✳▼❡❣✐❡r✱ ❉❈✱ ❏✳P✐✐❧♦✱ ❲✳❙tr✉♥③✱ ❙❝✳ ❘❡♣✳ ✷✵✶✼

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

◆❡✇ ❢❛♠✐❧② ♦❢ ❦❡r♥❡❧s

✭❲✉❞❛rs❦✐✱ ◆❛❧❡③②t②✱ ❙❛r❜✐❝❦✐✱ ❛♥❞ ❉❈✱ P❘❆ ✷✵✶✺✮

d dt Λt = t Kt−τΛτ dτ Kt[ρ] = 1 2

3

• i=1

ki(t)(σiρσi − ρ) ; Kt[σi] = κi(t)σi ˜ λi(s) = 1 s − ˜ κi(s) i = 1, 2, 3

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

◆❡✇ ❢❛♠✐❧② ♦❢ ❦❡r♥❡❧s

✭❲✉❞❛rs❦✐✱ ◆❛❧❡③②t②✱ ❙❛r❜✐❝❦✐✱ ❛♥❞ ❉❈✱ P❘❆ ✷✵✶✺✮

d dt Λt = t Kt−τΛτ dτ Kt[ρ] = 1 2

3

• i=1

ki(t)(σiρσi − ρ) ; Kt[σi] = κi(t)σi ˜ λi(s) = 1 s − ˜ κi(s) i = 1, 2, 3

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

❚❤❡♦r❡♠ ▲❡t a1, a2, a2 > 0 ❛♥❞ f : [0, ∞) → R s✉❝❤ t❤❛t 1 a1 + 1 a2 ≥ 1 a3 + ❝②❝❧✐❝ ♣❡r♠✳ 0 ≤

t

• f(τ)dτ ≤ 4

1 a1 + 1 a2 + 1 a3 −1 t❤❡♥

• κi(s) = −

s f(s) ai − f(s) ❞❡✜♥❡ ❛ ❧❡❣✐t✐♠❛t❡ ♠❡♠♦r② ❦❡r♥❡❧ Kt✳

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

❇❧♦❝❤ ❡q✉❛t✐♦♥

L[ρ] = 1 2

3

• k=1

γk(σkρσk − ρ) ❇❧♦❝❤ ✈❡❝t♦r → xk = Tr(ρσk) d dtxk = 1 Tk xk ; k = 1, 2, 3 1 T1 = γ2 + γ3 ; 1 T2 = γ3 + γ1 ; 1 T3 = γ1 + γ2 ❈P ⇐ ⇒ 1 T1 + 1 T2 ≥ 1 T3 ❡t❝

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

❇❧♦❝❤ ❡q✉❛t✐♦♥

L[ρ] = 1 2

3

• k=1

γk(σkρσk − ρ) ❇❧♦❝❤ ✈❡❝t♦r → xk = Tr(ρσk) d dtxk = 1 Tk xk ; k = 1, 2, 3 1 T1 = γ2 + γ3 ; 1 T2 = γ3 + γ1 ; 1 T3 = γ1 + γ2 ❈P ⇐ ⇒ 1 T1 + 1 T2 ≥ 1 T3 ❡t❝

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

▼✉❧t✐✲❡①♣♦♥❡♥t✐❛❧ ❞❡❝❛② ♦❢ ♠❡♠♦r② ❡✛❡❝ts

▲❡t W(s) ❜❡ ❛ ♣♦❧②♥♦♠✐❛❧ ♦❢ t❤❡ ❢♦r♠ W(s) = (s + z1) . . . (s + zn) ❛♥❞ ❧❡t ❛❧❧ zi > 0✳ ■❢

n

• i=1

zi ≥ 1 4 1 a1 + 1 a2 + 1 a3

• t❤❡♥

f(s) = 1/W(s) s❛t✐s✜❡s t❤❡ ❚❤❡♦r❡♠✳ zk = zl = ⇒ κi(t) = Ai0δ(t) + Ai1e−wi1t + . . . + Aire−wirt ✭❇❡r♥st❡✐♥ t❤❡♦r❡♠

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

▼✉❧t✐✲❡①♣♦♥❡♥t✐❛❧ ❞❡❝❛② ♦❢ ♠❡♠♦r② ❡✛❡❝ts

▲❡t W(s) ❜❡ ❛ ♣♦❧②♥♦♠✐❛❧ ♦❢ t❤❡ ❢♦r♠ W(s) = (s + z1) . . . (s + zn) ❛♥❞ ❧❡t ❛❧❧ zi > 0✳ ■❢

n

• i=1

zi ≥ 1 4 1 a1 + 1 a2 + 1 a3

• t❤❡♥

f(s) = 1/W(s) s❛t✐s✜❡s t❤❡ ❚❤❡♦r❡♠✳ zk = zl = ⇒ κi(t) = Ai0δ(t) + Ai1e−wi1t + . . . + Aire−wirt ✭❇❡r♥st❡✐♥ t❤❡♦r❡♠

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

❊①❛♠♣❧❡ ✭▼✳ ❍❛❧❧✱ ❊✳ ❆♥❞❡rs♦♥✱ ✳✳✳ P❘❆ ✷✵✶✹✮

Lt[ρ] = 1 2

3

• i=1

γi(t)(σiρσi − ρ) γ1 = γ2 = 1 ; γ3(t) = −t❛♥❤ t < 0 Λt = 1 2(etL1 + etL2) W(s) = s + 2 Kt = 1 2

• δ(t)(L1 + L2) − e−tL3

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

❊①❛♠♣❧❡ ✭▼✳ ❍❛❧❧✱ ❊✳ ❆♥❞❡rs♦♥✱ ✳✳✳ P❘❆ ✷✵✶✹✮

Lt[ρ] = 1 2

3

• i=1

γi(t)(σiρσi − ρ) γ1 = γ2 = 1 ; γ3(t) = −t❛♥❤ t < 0 Λt = 1 2(etL1 + etL2) W(s) = s + 2 Kt = 1 2

• δ(t)(L1 + L2) − e−tL3

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

❙✉♠♠❛r②

■ ♣r♦✈✐❞❡❞ ❛ ❝♦♥str✉❝t✐♦♥ ❢♦r ❛ ❢❛♠✐❧② ♦❢ ❧❡❣✐t✐♠❛t❡ ❦❡r♥❡❧s ✐♥ t❡r♠s ♦❢ ❧❡❣✐t✐♠❛t❡ ♣❛✐rs {Nt, Qt} ♠❛♥② ❦♥♦✇♥ ❡①❛♠♣❧❡s ✜t t❤✐s ❝❧❛ss t❤✐s ❝❧❛ss ❞❡✜♥❡s ❛ ♥❛t✉r❛❧ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ❝❧❛ss✐❝❛❧ s❡♠✐✲▼❛r❦♦✈ ❡✈♦❧✉t✐♦♥ ♠❛② ❜❡ ✉s❡❞ t♦ ❡♥❣✐♥❡❡r✐♥❣ q✉❛♥t✉♠ ❡✈♦❧✉t✐♦♥ ✭s✉♣♣r❡ss✐♦♥ ♦❢ ❞❡❝♦❤❡r❡♥❝❡✴❞✐ss✐♣❛t✐♦♥✮

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

❘❡❢❡r❡♥❝❡s

❉✳❈✳ ❛♥❞ ❙✳ ▼❛♥✐s❝❛❧❝♦✱ P❘▲✱ ✶✶✷✱ ✶✷✵✹✵✹ ✭✷✵✶✹✮ ❋✳ ❲✉❞❛rs❦✐✱ P✳ ◆❛❧❡➺②t②✱ ●✳ ❙❛r❜✐❝❦✐✱ ❛♥❞ ❉✳❈✳ P❘❆ ✾✶✱ ✵✹✷✶✵✺ ✭✷✵✶✺✮✳ ❏✳ ❇❛❡ ❛♥❞ ❉✳❈✱ P❘▲ ✶✶✼✱ ✵✺✵✹✵✸ ✭✷✵✶✻✮ ❉✳❈✳ ❛♥❞ ❆✳ ❑♦ss❛❦♦✇s❦✐✱ P❘❆ ✾✹✱ ✵✷✵✶✵✸ ✭✷✵✶✻✮❀ P❘❆ ✾✺✱ ✵✹✷✶✸✶ ✭✷✵✶✼✮✳ ❉✳❈✳✱ ❈✳ ▼❛❝❝❤✐❛✈❡❧❧♦✱ ❛♥❞ ❙✳ ▼❛♥✐s❝❛❧❝♦✱ P❘▲ ✶✶✽✱ ✵✽✵✹✵✹ ✭✷✵✶✼✮✳ ◆✳ ▼❡❣✐❡r✱ ❉✳❈✳✱ ❏✳ P✐✐❧♦✱ ❛♥❞ ❲✳ ❙tr✉♥③✱ ❙❝✐❡♥t✐✜❝ ❘❡♣♦rts ✼✱ ✻✸✼✾ ✭✷✵✶✼✮✳

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

❆❧❜❡rt♦ ❤❛♣♣② ❜✐rt❤❞❛②✦✦✦