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❉✐✈✐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

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

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

❉②♥❛♠✐❝❛❧ ♠❛♣

Quantum evolution ← → ρ → ρt := Λt(ρ) Λt : B(H) → B(H) ; t ≥ 0 ❝♦♠♣❧❡t❡❧② ♣♦s✐t✐✈❡ tr❛❝❡✲♣r❡s❡r✈✐♥❣ Λt=0 = id

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

▼❛r❦♦✈✐❛♥ s❡♠✐❣r♦✉♣

∂tΛt = LΛt ❚❤❡♦r❡♠ ✭●♦r✐♥✐✲❑♦ss❛❦♦✇s❦✐✲❙✉❞❛rs❤❛♥✲▲✐♥❞❜❧❛❞ ✭✶✾✼✻✮✮ Λt = etL ✐s ❈P❚P ✐❢ ❛♥❞ ♦♥❧② ✐❢ L(ρ) = −i[H, ρ]+

• α

γα

• VαρV †

α − 1

2

• V †

αVαρ + ρV † αVα

• ; γα > 0

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

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

❇❡②♦♥❞ ▼❛r❦♦✈✐❛♥ s❡♠✐❣r♦✉♣

♥♦♥✲❧♦❝❛❧ ♠❛st❡r ❡q✉❛t✐♦♥ ✭◆❛❦❛❥✐♠❛✲❩✇❛♥③✐❣ ❡q✉❛t✐♦♥✮ ∂tΛt = t Kt−τΛτ dτ

s❡♠✐✲▼❛r❦♦✈ ✭❇r❡✉❡r✲❱❛❝❝❤✐♥✐✱ ❑♦ss❛❦♦✇s❦✐✱ ❉❈✮ ❝♦❧❧✐s✐♦♥ ♠♦❞❡❧s ✭●✐♦✈❛♥❡tt✐✱ P❛❧♠❛✱ ❈✐❝❛r❡❧❧♦✱ ▲♦r❡♥③♦ ✳✳✳ ✮ ❊♥❣✐♥❡❡r✐♥❣ q✉❛♥t✉♠ ❝❤❛♥♥❡❧s ✭▼❛rs❤❛❧❧✱ ❩❛♥❛r❞✐✱ ✳✳✳ ✮

❧♦❝❛❧ ✐♥ t✐♠❡ ♠❛st❡r ❡q✉❛t✐♦♥ ✭❚❈▲✮ ∂tΛt = LtΛt

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

■♥✈❡rt✐❜❧❡ ♠❛♣s

Φ : B(H) → B(H) ✭❈P❚P✮ 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

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

■♥✈❡rt✐❜❧❡ ♠❛♣s

∂tΛt = LtΛt , Λ0 = id Lt = [∂tΛt] Λ−1

t

Lt ✖ r❡❣✉❧❛r ❣❡♥❡r❛t♦r

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

❊①❛♠♣❧❡✿ q✉❜✐t ♠❛♣

Λt(ρ) =

• ρ11

ρ12 f(t) ρ21 f∗(t) ρ22

• ;

f(t) ∈ C Λt ✐s ❈P❚P ✐✛ |f(t)| ≤ 1 Λt ✐s ✐♥✈❡rt✐❜❧❡ ✐✛ f(t) = 0 Lt(ρ) = −iω(t)[σ3, ρ] + γ(t)[σ3ρσ3 − ρ] γ(t) = −Re ˙ f(t) f(t) ; ω(t) = −Im ˙ f(t) f(t)

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

❉✐✈✐s✐❜✐❧✐t② ✈s ▼❛r❦♦✈✐❛♥✐t②

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

❉✐✈✐s✐❜✐❧✐t②

Λt = Vt,s Λs ; t ≥ s Vt,s : B(H) → B(H) P✲❞✐✈✐s✐❜❧❡ ✐✛ Vt,s ✐s P❚P ❈P✲❞✐✈✐s✐❜❧❡ ✐✛ Vt,s ✐s ❈P❚P ❈P✲❞✐✈✐s✐❜❧❡ = ⇒ P✲❞✐✈✐s✐❜❧❡ ❚❤❡♦r❡♠ ✭❇❡♥❛tt✐✱❉❈✱❋✐❧❧✐♣♦✈ ✭✷✵✶✼✮✮ Λt ✐s ❈P✲❞✐✈✐s✐❜❧❡ ✐✛ Λt ⊗ Λt ✐s P✲❞✐✈✐s✐❜❧❡

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

■♥✈❡rt✐❜❧❡ ♠❛♣s

■♥✈❡rt✐❜❧❡ ♠❛♣ ✐s ❛❧✇❛②s ❞✐✈✐s✐❜❧❡ Vt,s := Λt Λ−1

s

❚❤❡♦r❡♠ ■❢ Λt ✐s ✐♥✈❡rt✐❜❧❡✱ t❤❡♥ ✐t ✐s ❈P✲❞✐✈✐s✐❜❧❡ ✐✛ Lt(ρ) = −i[H(t), ρ]+

• α

γα(t)

• [Vα(t), ρV †

α(t)] + [Vα(t)ρ, V † α(t)]

• ❛♥❞ γα(t) ≥ 0✳

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

▼❛r❦♦✈✐❛♥✐t② ✲ ❘❍P

❉❡✜♥✐t✐♦♥✿ ❬❘✐✈❛s✱❍✉❡❧❣❛✱P❧❡♥✐♦❪ ❊✈♦❧✉t✐♦♥ ✐s ▼❆❘❑❖❱■❆◆ ✐✛ Λt ✐s ❈P✲❞✐✈✐s✐❜❧❡

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

▼❛r❦♦✈✐❛♥✐t② ✲ ❇▲P

❉❡✜♥✐t✐♦♥✿ ❬❇r❡✉❡r✱▲❛✐♥❡✱P✐✐❧♦❪ ▼❆❘❑❖❱■❆◆■❚❨ ← → ♥♦ ❜❛❝❦✢♦✇ ♦❢ ✐♥❢♦r♠❛t✐♦♥ d dtΛt(σ − ρ)1 ≤ 0 X1 = Tr √ XX†

❉✐✈✐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❝❛❧❝♦✮ ✳✳✳

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

❉✐✈✐s✐❜✐❧✐t② ✈s✳ ♠♦♥♦t♦♥✐❝✐t② ♦❢ tr❛❝❡ ♥♦r♠

❚❤❡♦r❡♠ ■❢ Λt ✐s P✲❞✐✈✐s✐❜❧❡✱ t❤❡♥ d dt Λt(X) 1 ≤ 0 ❢♦r ❛♥② X ∈ B(H)✳ X = ρ − σ ❈P✲❞✐✈✐s✐❜✐❧✐t② = ⇒ P✲❞✐✈✐s✐❜✐❧✐t② = ⇒ ♥♦ ✐♥❢♦r♠❛t✐♦♥ ❜❛❝❦✢♦✇

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

{p1, ρ1; p2, ρ2} − → D(ρ1, ρ2) = p1ρ1 − p2ρ2 1 ❚❤❡♦r❡♠ ✭❉❈✱❑♦ss❛❦♦✇s❦✐✱❘✐✈❛s✮ ■❢ Λt ✐s ✐♥✈❡rt✐❜❧❡✱ t❤❡♥ ✐t ✐s P✲❞✐✈✐s✐❜❧❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ d dt Λt(p1ρ1 − p2ρ2) 1 ≤ 0 ❢♦r ❛❧❧ ♣❛✐r ρ1, ρ2 ❛♥❞ p1, p2✳ ■t ✐s ❈P✲❞✐✈✐s✐❜❧❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ d dt idd ⊗ Λt(p1 ρ1 − p2 ρ2) 1 ≤ 0 ❢♦r ❛❧❧ ♣❛✐r ρ1, ρ2 ❛♥❞ p1, p2✳

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

❚❤❡♦r❡♠ ✭❇②❧✐❝❦❛✱❏♦❤❛♥ss♦♥✱❆❝í♥✮ ■❢ Λt ✐s ✐♥✈❡rt✐❜❧❡✱ t❤❡♥ ✐t ✐s ❈P✲❞✐✈✐s✐❜❧❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ d dt idd+1 ⊗ Λt( ρ1 − ρ2) 1 ≤ 0 ❢♦r ❛❧❧ ♣❛✐r ρ1, ρ2✳

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

❊①❛♠♣❧❡✿ q✉❜✐t ❡✈♦❧✉t✐♦♥

Lt(ρ) = 1 2

3

• k=1

γk(t)[σkρσk − ρ] =:

• k

γk(t)Lk Λt ✐s ✐♥✈❡rt✐❜❧❡ ⇐ ⇒ Γk(t) = t

0 γk(τ)dτ < ∞

Λt ✐s ❈P✲❞✐✈✐s✐❜❧❡ ✐✛ γ1(t) ≥ 0 ; γ2(t) ≥ 0 ; γ3(t) ≥ 0 Λt ✐s P✲❞✐✈✐s✐❜❧❡ ✐✛ γ1(t) + γ2(t) ≥ 0 ; γ1(t) + γ3(t) ≥ 0 ; γ2(t) + γ3(t) ≥ 0

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

Λt = x1etL1 + x2etL2 + x3etL3

Λt ✐s ❛ ▼❛r❦♦✈✐❛♥ s❡♠✐✲❣r♦✉♣ ✐❢ ♦♥❧② ♦♥❡ xk = 1 Λt ✐s P✲❞✐✈✐s✐❜❧❡ ❢♦r ❛❧❧ xk Λt ✐s ❈P✲❞✐✈✐s✐❜❧❡ ❢♦r ✏s♠❛❧❧ r❡❣✐♦♥✑

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

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

Λt = 1

2

• etL1 + etL2 )

Lt(ρ) = 1 2

3

• k=1

γk(t)[σkρσk − ρ] γ1(t) = γ2(t) = 1 , γ3(t) = −tanh t ❵❡t❡r♥❛❧❧② ♥♦♥✲▼❛r❦♦✈✐❛♥✬

1 2 3 4 5

• 1.0
• 0.5

0.5 1.0

❍❛❧❧✱ ❈r❡ss❡r✱ ❆♥❞❡rss♦♥✱ P❘❆ ✷✵✶✹

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

❇✉t ✇❤❛t ✐❢ t❤❡ ♠❛♣ ✐s ♥♦t ✐♥✈❡rt✐❜❧❡❄ ❋✳ ❇✉s❝❡♠✐ ❛♥❞ ◆✳ ❉❛tt❛✱ P❤②s✳ ❘❡✈✳ ❆ ✭✷✵✶✻✮ ❉❈✱ ❆✳ ❘✐✈❛s✱ ❛♥❞ ❊✳ ❙tør♠❡r✱ P❤②s✳ ❘❡✈✳ ▲❡tt✳ ✭✷✵✶✽✮✳ ❙✳ ❈❤❛❦r❛❜♦rt②✱ ❉❈✱ P❤②s✳ ❘❡✈ ❆ ✭✷✵✶✾✮

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

• ❡♥❡r❛❧ Λt

Λt = Vt,s Λs ; t ≥ s ❚❤❡♦r❡♠ Λt ✐s ❞✐✈✐s✐❜❧❡ ✐✛ ✐t ✐s ✏❦❡r♥❡❧ ♥♦♥✲❞❡❝r❡❛s✐♥❣✑ KerΛs ⊂ KerΛt Vt,s ✐s ◆❖❚ ✉♥✐q✉❡❧② ❞❡✜♥❡❞ ♦♥ B(H) Vt,s : Im Λs → Im Λt Extension : − → Vt,s : B(H) → B(H)

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

• ❡♥❡r❛❧ Λt

❚❤❡♦r❡♠ If d dt [1 l ⊗ Λt](p1ρ1 − p2ρ2) 1 ≤ 0 ❢♦r ❛❧❧ ρ1, ρ2 ∈ B(H ⊗ H)✱ ❛♥❞ p1, p2✱ t❤❡♥

✶ Λt ✐s ❞✐✈✐s✐❜❧❡ ✷ Vt,s ✐s ❈P❚P ♦♥ Im Λs✳

❈♦✉❧❞ ✇❡ ❡①t❡♥❞ Vt,s t♦ B(H)❄

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

❆r✈❡s♦♥ ❡①t❡♥s✐♦♥ t❤❡♦r❡♠

M ⊂ B(H) ✕ ♦♣❡r❛t♦r s②st❡♠ I ∈ M X ∈ M = ⇒ X† ∈ M ❚❤❡♦r❡♠ ✭❆r✈❡s♦♥✮ ▲❡t M ❜❡ ❛♥ ♦♣❡r❛t♦r s②st❡♠ ❛♥❞ Φ : M → B(H) ❛ ✉♥✐t❛❧ ❈P ♠❛♣✳ ❚❤❡♥ t❤❡r❡ ❡①✐sts ✉♥✐t❛❧ ❈P ❡①t❡♥s✐♦♥ Φ : B(H) → B(H)✳

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

• ❡♥❡r❛❧✐③✐♥❣ ❆r✈❡s♦♥ t❤❡♦r❡♠

M ⊂ B(H) ✕ s♣❛♥♥❡❞ ❜② ♣♦s✐t✐✈❡ ♦♣❡r❛t♦rs ❚❤❡♦r❡♠ ✭❏❡♥❝♦✈❛✮ ■❢ Φ : M → B(H) ✐s ❛ ❈P ♠❛♣✱ t❤❡♥ t❤❡r❡ ❡①✐sts ❈P ❡①t❡♥s✐♦♥

• Φ : B(H) → B(H)✳

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

❚❤❡♦r❡♠ ✭❉❈✱❘✐✈❛s✱❙tør♠❡r✮ If d dt [1 l ⊗ Λt](p1ρ1 − p2ρ2) 1 ≤ 0 ❢♦r ❛❧❧ ρ1, ρ2 ∈ B(H ⊗ H)✱ ❛♥❞ p1, p2✱ t❤❡♥ Vt,s ✐s ❈P❚P ♦♥ Im Λs✱ ❛♥❞ ✐t ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ ❈P ♠❛♣

• Vt,s : B(H) → B(H)
• Vt,s ✐s ❛❧✇❛②s tr❛❝❡✲♣r❡s❡r✈✐♥❣ ♦♥ Im Λs
• Vt,s ♥❡❡❞s ◆❖❚ ❜❡ tr❛❝❡✲♣r❡s❡r✈✐♥❣ ✦✦✦

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

❈♦✉❧❞ ✇❡ ❤❛✈❡ ❜♦t❤ ❈P ❛♥❞ tr❛❝❡✲♣r❡s❡r✈❛t✐♦♥ ❄

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

Λt = Vt,s Λs ; t ≥ s Λt ✐s ✏✐♠❛❣❡ ♥♦♥✲✐♥❝r❡❛s✐♥❣✑ ⇐ ⇒ ImΛt ⊂ ImΛs ❚❤❡♦r❡♠ ✭❉❈✱❘✐✈❛s✱❙tør♠❡r✮ ■❢ Λt ✐s ✐♠❛❣❡ ♥♦♥✲✐♥❝r❡❛s✐♥❣✱ t❤❡♥ ✐t ✐s ❈P✲❞✐✈✐s✐❜❧❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ d dt [1 l ⊗ Λt](p1ρ1 − p2ρ2) 1 ≤ 0 ❢♦r ❛❧❧ ρ1, ρ2 ∈ B(H ⊗ H)✱ ❛♥❞ p1, p2✱

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

❊①❛♠♣❧❡ ✕ ✐♠❛❣❡ ♥♦♥✲✐♥❝r❡❛s✐♥❣

Λt1 Λt2 = Λt2 Λt1 Λt ✐s ❞✐❛❣♦♥❛❧✐③❛❜❧❡ ❚❤❡♦r❡♠ ■❢ Λt ✐s ❦❡r♥❡❧ ♥♦♥✲❞❡❝r❡❛s✐♥❣✱ t❤❡♥ ✐t ✐s ✐♠❛❣❡ ♥♦♥✲✐♥❝r❡❛s✐♥❣✳ ❆ ❧♦t ♦❢ st✉❞✐❡❞ ❡①❛♠♣❧❡s ✜t t❤✐s ❝❧❛ss

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

◗✉❜✐t

❚❤❡♦r❡♠ ✭❉❈✱ ❙✳ ❈❤❛❦r❛❜♦rt②✮ Λt ✐s ❈P✲❞✐✈✐s✐❜❧❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ d dt [1 l ⊗ Λt](p1ρ1 − p2ρ2) 1 ≤ 0 ❢♦r ❛❧❧ ρ1, ρ2 ❛♥❞ p1, p2✳

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

◗✉❜✐t ❝❤❛♥♥❡❧s

B = ❇❧♦❝❤ ❜❛❧❧ ; S = ❇❧♦❝❤ s♣❤❡r❡ Φ(B) ⊂ B P✉r❡ ❖✉t♣✉t − → PO = Φ(B) ∩ S

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

◗✉❜✐t ❝❤❛♥♥❡❧s

P✉r❡ ❖✉t♣✉t − → PO = Φ(B) ∩ S ❚❤❡♦r❡♠ ✭❇r❛✉♥✱ ●✐r❛✉❞✱ ◆❡❝❤✐t❛✱ P❡❧❧❡❣r✐♥✐✱ ➎♥✐❞❛r✐↔ ✭✷✵✶✹✮✮ ❋♦r ❛♥② q✉❜✐t ❝❤❛♥♥❡❧ PO = {∅} |PO| = 1 |PO| = 2 |PO| > 2 = ⇒ PO = S ; (Φ(ρ) = Uρ U†)

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

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

◗✉❜✐t ❝❤❛♥♥❡❧s

Pr♦♣♦s✐t✐♦♥ ✭❈❤❛❦r❛❜♦rt②✱ ❉❈ ✭✷✵✶✾✮✮ ❚❤❡r❡ ✐s ◆❖ q✉❜✐t ❈P❚P ♣r♦❥❡❝t♦r s✉❝❤ t❤❛t dim Im(Φ) = 3✳

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

◗✉❛♥t✉♠ ❝❤❛♥♥❡❧ ✇✐t❤ ✸✲❞✐♠✳ ✐♠❛❣❡

❊①❛♠♣❧❡ Φ(ρ) = 1 2ρ + 1 4

• σ1ρσ1 + σ2ρσ2
• .

Φ(1 l) = 1 l, Φ(σ1) = 1 2σ1, Φ(σ2) = 1 2σ2, Φ(σ3) = 0, dim Im(Φ) = 3✳ Φ ✐s ♥♦t ❛ ♣r♦❥❡❝t♦r✦

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

❈P❚P ✈s✳ P❚P

Sz(ρ) = 1 2

• ρ + σ3ρσ3
• ❈P❚P ♣r♦❥❡❝t♦r

Πxy(ρ) = 1 4

• 3ρ + σ1ρσ1 + σ2ρσ2 − σ3ρσ3
• P❚P ♣r♦❥❡❝t♦r

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

■❢ t❤❡ ✐♠❛❣❡ ✐s ✷ ❞✐♠✳

{ρ1, ρ2} ; {σ1, σ2} ❉♦❡s t❤❡r❡ ❡①✐st ❛ q✉❛♥t✉♠ ❝❤❛♥❡❧ Φ s✉❝❤ t❤❛t σk = Φ(ρk) ❄ ❚❤❡♦r❡♠ ✭❆❧❜❡rt✐ ❛♥❞ ❯❤❧♠❛♥♥✮ ρ1 − tρ21 ≥ σ1 − tσ21 ❢♦r ❛❧❧ t > 0✳ ❊q✉✐✈❛❧❡♥t❧② p1ρ1 − p2ρ21 ≥ p1σ1 − p2σ21 ❢♦r ❛❧❧ p1 + p2 = 1✳ ❚❤❡ t❤❡♦r❡♠ ✐s ◆❖❚ tr✉❡ ❢♦r ❤✐❣❤❡r ❞✐♠❡♥s✐♦♥s✦

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

❈♦♥❝❧✉s✐♦♥s

▼❛r❦♦✈✐❛♥✐t② ❂ ❈P✲❞✐✈✐s✐❜❧✐t② ← → Λt = Vt,s ◦ Λs ❢♦r ✐♥✈❡rt✐❜❧❡ ♠❛♣s✿ ❈P✲❞✐✈✐s✐❜✐❧✐t② ← → ♠♦♥♦t♦♥✐❝✐t② ♦❢ t❤❡ tr❛❝❡ ♥♦r♠ ✇❡ ❣❡♥❡r❛❧✐③❡ ✐t ❢♦r ♥♦♥✲✐♥✈❡rt✐❜❧❡ ♠❛♣s ❜✉t ❢♦r ♥♦♥✲✐♥✈❡rt✐❜❧❡ ♠❛♣s✿ Vt,s ♠✐❣❤t ❜❡ ❈P ♦♥ B(H) ❜✉t tr❛❝❡✲♣r❡s❡r✈✐♥❣ ♦♥❧② ♦♥ ImΛs ❢♦r ✐♠❛❣❡ ❞❡❝r❡❛s✐♥❣ ♠❛♣s ✐t ✐s ❛❧s♦ tr❛❝❡✲♣r❡s❡r✈✐♥❣ ♦♥ B(H) ❢♦r t❤❡ q✉❜✐t ❡✈♦❧✉t✐♦♥✿ ❈P✲❞✐✈✐s✐❜✐❧✐t② ← → ♠♦♥♦t♦♥✐❝✐t② ♦❢ t❤❡ tr❛❝❡ ♥♦r♠