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▲♦❣❛r✐t❤♠✐❝ ❙♦❜♦❧❡✈ ■♥❡q✉❛❧✐t✐❡s ❢♦r ❊♥tr♦♣② Pr♦❞✉❝t✐♦♥ ❉❛♥✐❡❧ ❙t✐❧❝❦ ❋r❛♥ç❛ ❚❡❝❤♥✐s❝❤❡ ❯♥✐✈❡rs✐tät ▼ü♥❝❤❡♥ ❈❊◗❯■P ✷✵✶✺ ❏♦✐♥t ✇♦r❦ ✇✐t❤ ❆❧❡①❛♥❞❡r ▼ü❧❧❡r✲❍❡r♠❡s ❛♥❞ ▼✐❝❤❛❡❧ ▼✳ ❲♦❧❢ ❛r❳✐✈✿✶✺✵✺✳✵✹✻✼✽ ❉❛♥✐❡❧ ❙t✐❧❝❦ ❋r❛♥ç❛ ▲♦❣❛r✐t❤♠✐❝ ❙♦❜♦❧❡✈ ■♥❡q✉❛❧✐t✐❡s ❢♦r ❊♥tr♦♣② Pr♦❞✉❝t✐♦♥

❚❤❡ ▲♦❣❛r✐t❤♠✐❝ ❙♦❜♦❧❡✈ ✶ ❝♦♥st❛♥t ❢♦r ❞❡♣♦❧❛r✐③✐♥❣ s❡♠✐❣r♦✉♣s ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s t♦ t❤❡ ❝♦♥❝❛✈✐t② ♦❢ t❤❡ ✈♦♥ ◆❡✉♠❛♥♥ ❡♥tr♦♣②✳ ❚❤❡ ▲♦❣❛r✐t❤♠✐❝ ❙♦❜♦❧❡✈ ✷ ❝♦♥st❛♥t✱ ❤②♣❡r❝♦♥tr❛❝t✐✈✐t② ❛♥❞ ▲❙ ✐♥❡q✉❛❧✐t✐❡s t❤❛t t❡♥s♦r✐③❡ ✇✐t❤ ❛♣♣❧✐❝❛t✐♦♥s t♦ t❤❡ ❡♥tr♦♣② ♣r♦❞✉❝t✐♦♥✳ ❖✉t❧✐♥❡ ❉❡✜♥✐t✐♦♥ ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s ♦❢ t❤❡ ▲♦❣❛r✐t❤♠✐❝ ❙♦❜♦❧❡✈ ✶ ❝♦♥st❛♥t✳ ❉❛♥✐❡❧ ❙t✐❧❝❦ ❋r❛♥ç❛ ▲♦❣❛r✐t❤♠✐❝ ❙♦❜♦❧❡✈ ■♥❡q✉❛❧✐t✐❡s ❢♦r ❊♥tr♦♣② Pr♦❞✉❝t✐♦♥

❚❤❡ ▲♦❣❛r✐t❤♠✐❝ ❙♦❜♦❧❡✈ ✷ ❝♦♥st❛♥t✱ ❤②♣❡r❝♦♥tr❛❝t✐✈✐t② ❛♥❞ ▲❙ ✐♥❡q✉❛❧✐t✐❡s t❤❛t t❡♥s♦r✐③❡ ✇✐t❤ ❛♣♣❧✐❝❛t✐♦♥s t♦ t❤❡ ❡♥tr♦♣② ♣r♦❞✉❝t✐♦♥✳ ❖✉t❧✐♥❡ ❉❡✜♥✐t✐♦♥ ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s ♦❢ t❤❡ ▲♦❣❛r✐t❤♠✐❝ ❙♦❜♦❧❡✈ ✶ ❝♦♥st❛♥t✳ ❚❤❡ ▲♦❣❛r✐t❤♠✐❝ ❙♦❜♦❧❡✈ ✶ ❝♦♥st❛♥t ❢♦r ❞❡♣♦❧❛r✐③✐♥❣ s❡♠✐❣r♦✉♣s ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s t♦ t❤❡ ❝♦♥❝❛✈✐t② ♦❢ t❤❡ ✈♦♥ ◆❡✉♠❛♥♥ ❡♥tr♦♣②✳ ❉❛♥✐❡❧ ❙t✐❧❝❦ ❋r❛♥ç❛ ▲♦❣❛r✐t❤♠✐❝ ❙♦❜♦❧❡✈ ■♥❡q✉❛❧✐t✐❡s ❢♦r ❊♥tr♦♣② Pr♦❞✉❝t✐♦♥

❖✉t❧✐♥❡ ❉❡✜♥✐t✐♦♥ ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s ♦❢ t❤❡ ▲♦❣❛r✐t❤♠✐❝ ❙♦❜♦❧❡✈ ✶ ❝♦♥st❛♥t✳ ❚❤❡ ▲♦❣❛r✐t❤♠✐❝ ❙♦❜♦❧❡✈ ✶ ❝♦♥st❛♥t ❢♦r ❞❡♣♦❧❛r✐③✐♥❣ s❡♠✐❣r♦✉♣s ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s t♦ t❤❡ ❝♦♥❝❛✈✐t② ♦❢ t❤❡ ✈♦♥ ◆❡✉♠❛♥♥ ❡♥tr♦♣②✳ ❚❤❡ ▲♦❣❛r✐t❤♠✐❝ ❙♦❜♦❧❡✈ ✷ ❝♦♥st❛♥t✱ ❤②♣❡r❝♦♥tr❛❝t✐✈✐t② ❛♥❞ ▲❙ ✐♥❡q✉❛❧✐t✐❡s t❤❛t t❡♥s♦r✐③❡ ✇✐t❤ ❛♣♣❧✐❝❛t✐♦♥s t♦ t❤❡ ❡♥tr♦♣② ♣r♦❞✉❝t✐♦♥✳ ❉❛♥✐❡❧ ❙t✐❧❝❦ ❋r❛♥ç❛ ▲♦❣❛r✐t❤♠✐❝ ❙♦❜♦❧❡✈ ■♥❡q✉❛❧✐t✐❡s ❢♦r ❊♥tr♦♣② Pr♦❞✉❝t✐♦♥

❉ ❡ t ✷ ✶ t ❉ ❡ ✇✐t❤ ❉ tr ❧♦❣ ❧♦❣ ✳ ▲♦❣❛r✐t❤♠✐❝ ❙♦❜♦❧❡✈ ✶ ❈♦♥st❛♥t ●✐✈❡♥ ❛ ▲✐♦✉✈✐❧❧✐❛♥ L : M ❞ → M ❞ ✇✐t❤ st❛t✐♦♥❛r② st❛t❡ σ ∈ D + ❞ ✇❡ ✇❛♥t t♦ ❡st✐♠❛t❡ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ✐♥ t❤❡ r❡❧❛t✐✈❡ ❡♥tr♦♣②✿ ❉❛♥✐❡❧ ❙t✐❧❝❦ ❋r❛♥ç❛ ▲♦❣❛r✐t❤♠✐❝ ❙♦❜♦❧❡✈ ■♥❡q✉❛❧✐t✐❡s ❢♦r ❊♥tr♦♣② Pr♦❞✉❝t✐♦♥

▲♦❣❛r✐t❤♠✐❝ ❙♦❜♦❧❡✈ ✶ ❈♦♥st❛♥t ●✐✈❡♥ ❛ ▲✐♦✉✈✐❧❧✐❛♥ L : M ❞ → M ❞ ✇✐t❤ st❛t✐♦♥❛r② st❛t❡ σ ∈ D + ❞ ✇❡ ✇❛♥t t♦ ❡st✐♠❛t❡ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ✐♥ t❤❡ r❡❧❛t✐✈❡ ❡♥tr♦♣②✿ ❡ t L ρ || σ ≤ ❡ − ✷ α ✶ t ❉ ( ρ || σ ) � � ❉ ✇✐t❤ ❉ ( ρ || σ ) = tr [ ρ ( ❧♦❣ ( ρ ) − ❧♦❣ ( σ ))] ✳ ❉❛♥✐❡❧ ❙t✐❧❝❦ ❋r❛♥ç❛ ▲♦❣❛r✐t❤♠✐❝ ❙♦❜♦❧❡✈ ■♥❡q✉❛❧✐t✐❡s ❢♦r ❊♥tr♦♣② Pr♦❞✉❝t✐♦♥

▲♦❣❛r✐t❤♠✐❝ ❙♦❜♦❧❡✈ ✶ ❈♦♥st❛♥t ●✐✈❡♥ ❛ ♣r✐♠✐t✐✈❡ ▲✐♦✉✈✐❧❧✐❛♥ L : M ❞ → M ❞ ✇✐t❤ st❛t✐♦♥❛r② st❛t❡ σ ∈ D + ❞ ✇❡ ✇❛♥t t♦ ❡st✐♠❛t❡ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ✐♥ t❤❡ r❡❧❛t✐✈❡ ❡♥tr♦♣②✿ ❡ t L ρ || σ ≤ ❡ − ✷ α ✶ t ❉ ( ρ || σ ) � � ✭✶✮ ❉ ✇✐t❤ ❉ ( ρ || σ ) = tr [ ρ ( ❧♦❣ ( ρ ) − ❧♦❣ ( σ ))] ✳ ❚❤❡ ❧❛r❣❡st α ✶ s✳t✳ ✭✶✮ ❤♦❧❞s ❢♦r ❛❧❧ t > ✵ ✐s t❤❡ ▲♦❣❛r✐t❤♠✐❝ ❙♦❜♦❧❡✈ ✶ ❝♦♥st❛♥t✳ ❉❛♥✐❡❧ ❙t✐❧❝❦ ❋r❛♥ç❛ ▲♦❣❛r✐t❤♠✐❝ ❙♦❜♦❧❡✈ ■♥❡q✉❛❧✐t✐❡s ❢♦r ❊♥tr♦♣② Pr♦❞✉❝t✐♦♥

❙ ❡ t ✷ t ❙ ✶ ❡ ❧♦❣ ❞ ❙ Pr♦✈✐❞❡s ❛ ✇❛② ♦❢ q✉❛♥t✐❢②✐♥❣ t❤❡ ♣r♦❞✉❝t✐♦♥ ♦❢ ❡♥tr♦♣② ❜② t❤❡ s❡♠✐❣r♦✉♣✳ ❊♥tr♦♣② Pr♦❞✉❝t✐♦♥ ❋♦r ❙ ( ρ ) = − tr [ ρ ❧♦❣ ( ρ )] t❤❡ ✈♦♥ ◆❡✉♠❛♥♥ ❡♥tr♦♣② ❛♥❞ ❞♦✉❜❧② st♦❝❤❛st✐❝ ▲✐♦✉✈✐❧❧✐❛♥s ✭ L ( 1 ) = L ∗ ( 1 ) = ✵✮✱ ❛ ▲❙✲✶ ✐♥❡q✉❛❧✐t② ✐s ❡q✉✐✈❛❧❡♥t t♦✿ ❉❛♥✐❡❧ ❙t✐❧❝❦ ❋r❛♥ç❛ ▲♦❣❛r✐t❤♠✐❝ ❙♦❜♦❧❡✈ ■♥❡q✉❛❧✐t✐❡s ❢♦r ❊♥tr♦♣② Pr♦❞✉❝t✐♦♥

Pr♦✈✐❞❡s ❛ ✇❛② ♦❢ q✉❛♥t✐❢②✐♥❣ t❤❡ ♣r♦❞✉❝t✐♦♥ ♦❢ ❡♥tr♦♣② ❜② t❤❡ s❡♠✐❣r♦✉♣✳ ❊♥tr♦♣② Pr♦❞✉❝t✐♦♥ ❋♦r ❙ ( ρ ) = − tr [ ρ ❧♦❣ ( ρ )] t❤❡ ✈♦♥ ◆❡✉♠❛♥♥ ❡♥tr♦♣② ❛♥❞ ❞♦✉❜❧② st♦❝❤❛st✐❝ ▲✐♦✉✈✐❧❧✐❛♥s ✭ L ( 1 ) = L ∗ ( 1 ) = ✵✮✱ ❛ ▲❙✲✶ ✐♥❡q✉❛❧✐t② ✐s ❡q✉✐✈❛❧❡♥t t♦✿ ❙ ( ❡ t L ρ ) − ❙ ( ρ ) ≥ ( ✶ − ❡ − ✷ α t )( ❧♦❣ ( ❞ ) − ❙ ( ρ )) ❉❛♥✐❡❧ ❙t✐❧❝❦ ❋r❛♥ç❛ ▲♦❣❛r✐t❤♠✐❝ ❙♦❜♦❧❡✈ ■♥❡q✉❛❧✐t✐❡s ❢♦r ❊♥tr♦♣② Pr♦❞✉❝t✐♦♥

❊♥tr♦♣② Pr♦❞✉❝t✐♦♥ ❋♦r ❙ ( ρ ) = − tr [ ρ ❧♦❣ ( ρ )] t❤❡ ✈♦♥ ◆❡✉♠❛♥♥ ❡♥tr♦♣② ❛♥❞ ❞♦✉❜❧② st♦❝❤❛st✐❝ ▲✐♦✉✈✐❧❧✐❛♥s ✭ L ( 1 ) = L ∗ ( 1 ) = ✵✮✱ ❛ ▲❙✲✶ ✐♥❡q✉❛❧✐t② ✐s ❡q✉✐✈❛❧❡♥t t♦✿ ❙ ( ❡ t L ρ ) − ❙ ( ρ ) ≥ ( ✶ − ❡ − ✷ α t )( ❧♦❣ ( ❞ ) − ❙ ( ρ )) Pr♦✈✐❞❡s ❛ ✇❛② ♦❢ q✉❛♥t✐❢②✐♥❣ t❤❡ ♣r♦❞✉❝t✐♦♥ ♦❢ ❡♥tr♦♣② ❜② t❤❡ s❡♠✐❣r♦✉♣✳ ❉❛♥✐❡❧ ❙t✐❧❝❦ ❋r❛♥ç❛ ▲♦❣❛r✐t❤♠✐❝ ❙♦❜♦❧❡✈ ■♥❡q✉❛❧✐t✐❡s ❢♦r ❊♥tr♦♣② Pr♦❞✉❝t✐♦♥

❲❤② ②♦✉ s❤♦✉❧❞ ❝❛r❡ ▲❙ ✐♥❡q✉❛❧✐t✐❡s ❤❛✈❡ ❛❧r❡❛❞② ❢♦✉♥❞ ♠❛♥② ❛♣♣❧✐❝❛t✐♦♥s✱ s✉❝❤ ❛s✿ ✶ ■❢ ✇❡ ❤❛✈❡ ❛ ❢❛♠✐❧② ♦❢ ▲✐♦✉✈✐❧❧✐❛♥s ❞❡✜♥❡❞ ♦♥ ❛ ❧❛tt✐❝❡ t❤❛t ❤❛✈❡ ❛ ▲❙ ❝♦♥st❛♥t ✇❤✐❝❤ ❞♦❡s ♥♦t s❝❛❧❡ ✇✐t❤ s✐③❡ ♦❢ t❤❡ s②st❡♠✱ t❤✐s ✐♠♣❧✐❡s✿ ❙tr♦♥❣ ♥♦t✐♦♥ ♦❢ st❛❜✐❧✐t② ♦❢ ♦❜s❡r✈❛❜❧❡s ✇✳r✳t✳ ♣❡rt✉r❜❛t✐♦♥s ♦❢ t❤❡ ▲✐♦✉✈✐❧❧✐❛♥ ✶ ✳ ✶ ❚✳ ❙✳ ❈✉❜✐tt ❡t ❛❧✳ ✏❙t❛❜✐❧✐t② ♦❢ ❧♦❝❛❧ q✉❛♥t✉♠ ❞✐ss✐♣❛t✐✈❡ s②st❡♠s✑✳ ■♥✿ ❆r❳✐✈ ❡✲♣r✐♥ts ✭▼❛r✳ ✷✵✶✸✮✳ ❛r❳✐✈✿ ✶✸✵✸✳✹✼✹✹ ❬q✉❛♥t✲♣❤❪ ❉❛♥✐❡❧ ❙t✐❧❝❦ ❋r❛♥ç❛ ▲♦❣❛r✐t❤♠✐❝ ❙♦❜♦❧❡✈ ■♥❡q✉❛❧✐t✐❡s ❢♦r ❊♥tr♦♣② Pr♦❞✉❝t✐♦♥