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❙♣❡❝tr❛❧ ♣r♦♣❡rt✐❡s ♦❢ q✉❛♥t✉♠ ♦♣❡r❛t✐♦♥s ✲ ❛❧❣❡❜r❛✐❝ ♠❡t❤♦❞s ▼✐❝❤❛➟ ❇✐❛➟♦➠❝③②❦✱ ✐♥ ❝♦♦♣❡r❛t✐♦♥ ✇✐t❤ ♣r♦❢✳ ❑❛r♦❧ ➏②❝③❦♦✇s❦✐ ❏❛❣✐❡❧❧♦♥✐❛♥ ❯♥✐✈❡rs✐t② ✹✾ ❙②♠♣♦s✐✉♠ ♦♥ ▼❛t❤❡♠❛t✐❝❛❧ P❤②s✐❝s✱ ❚♦r✉➠ ✶✼✳✵✻✳✷✵✶✼

◗✉❛♥t✉♠ st❛t❡s ❛♥❞ ♦♣❡r❛t✐♦♥s ✶ P❡rr♦♥ ✲ ❋r♦❜❡♥✐✉s t❤❡♦r② ✷ ❇❧♦❝❦ ❞❡❝♦♠♣♦s✐t✐♦♥s ✸

◗✉❛♥t✉♠ st❛t❡s ❛♥❞ ♦♣❡r❛t✐♦♥s P❡rr♦♥ ✲ ❋r♦❜❡♥✐✉s t❤❡♦r② ❇❧♦❝❦ ❞❡❝♦♠♣♦s✐t✐♦♥s ❈♦♥❡s ❛♥❞ ❢❛❝❡s P♦s✐t✐✈❡ ❝♦♥❡s ✐♥ ✈❡❝t♦r s♣❛❝❡s ▲❡t ( ❱ , R ) ❜❡ ❛ r❡❛❧ ✈❡❝t♦r s♣❛❝❡✳ ❆ s❡t ❑ ⊂ ❱ ✐s ❝❛❧❧❡❞ ❛ ❝♦♥❡ ✐❢ ❋♦r ① , ② ∈ ❑ ❛♥❞ α, β ≥ ✵ t❤❡♥ α ① + β ② ≥ ✵✱ ❑ ✐s ❝❧♦s❡❞ ✐♥ t❤❡ ♥❛t✉r❛❧ t♦♣♦❧♦❣② ♦❢ ❱ ❛♥❞ ✐♥t ❑ � = ∅ ✱ ❑ ∩ ( − ❑ ) = { ✵ } ✳ ❱❡❝t♦rs ① ∈ ✐♥t ❑ ❛r❡ ❝❛❧❧❡❞ str✐❝t❧② ♣♦s✐t✐✈❡ ❛♥❞ ❞❡♥♦t❡❞ ① > ✵✳ ❊①❛♠♣❧❡ ♦❢ ❛ ❝♦♥❡ ✿ ♣♦s✐t✐✈❡ ❤②♣❡r♦❝t❛♥t ❑ = R ♥ + ✳ ❋❛❝❡ ♦❢ ❛ ❝♦♥❡ ❚❤❡ ❢❛❝❡ ❋ ⊂ ❑ ✐s ❛ ❝♦♥❡ s✉❝❤ t❤❛t ✿ ② ≤ ① ❛♥❞ ① ∈ ❋ ⇒ ② ∈ ❋ ✭✶✮ ❖♥❡ ❞✐♠❡♥s✐♦♥❛❧ ❢❛❝❡ ✐s ❝❛❧❧❡❞ ❛♥ ❡①tr❡♠❛❧ r❛② ✳ ❊①❛♠♣❧❡ ✿ ❋♦r ❑ = R ♥ + ❢❛❝❡s ❛r❡ t❤❡ s❡ts R ❦ + ✱ ❦ = ✵ , . . . ♥ ✳ ▼✐❝❤❛➟ ❇✐❛➟♦➠❝③②❦✱ ✐♥ ❝♦♦♣❡r❛t✐♦♥ ✇✐t❤ ♣r♦❢✳ ❑❛r♦❧ ➏②❝③❦♦✇s❦✐ ❙♣❡❝tr❛❧ ♣r♦♣❡rt✐❡s ♦❢ q✉❛♥t✉♠ ♦♣❡r❛t✐♦♥s ✲ ❛❧❣❡❜r❛✐❝ ♠❡t❤♦❞s

◗✉❛♥t✉♠ st❛t❡s ❛♥❞ ♦♣❡r❛t✐♦♥s P❡rr♦♥ ✲ ❋r♦❜❡♥✐✉s t❤❡♦r② ❇❧♦❝❦ ❞❡❝♦♠♣♦s✐t✐♦♥s ❚❤❡ ❝♦♥❡ ♦❢ ♣♦s✐t✐✈❡ ✲ s❡♠✐❞❡✜♥✐t❡ ♠❛tr✐❝❡s ❉❡♥♦t❡ t❤❡ s❡t ♦❢ ♥ × ♥ ♠❛tr✐❝❡s ❛s M ♥ ( C ) ✳ P♦s✐t✐✈❡ ✲ s❡♠✐❞❡✜♥✐t❡ ♠❛tr✐❝❡s ✿ M + ♥ ( C ) = { ρ ∈ M ♥ ( C ) : ∀ | ψ � � ψ | ρ | ψ � ≥ ✵ } ⊂ M R ♥ ( C ) ✭✷✮ ❢♦r♠ ❛ ♣♦s✐t✐✈❡ ❝♦♥❡ ✐♥ t❤❡ s♣❛❝❡ ♦❢ ❤❡r♠✐t❡❛♥ ♠❛tr✐❝❡s ( M R ♥ ( C ) , R ) ✳ ❖♥❡ ❤❛s ρ > ✵ ⇔ ❑❡r ρ = { ✵ } ✳ ❙❡t ♦❢ ♠✐①❡❞ st❛t❡s ♦❢ s✐③❡ ◆ ❙t❛t❡s ♦❢ ✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ♦♣❡♥ q✉❛♥t✉♠ s②st❡♠ ❛r❡ ♠♦❞❡❧❧❡❞ ❜② t❤❡ s❡t ♦❢ ❞❡♥s✐t② ♠❛tr✐❝❡s ✿ M + , ✶ ( C ) = { ρ ∈ M ♥ ( C ) : ρ ≥ ✵ , ❚r ρ = ✶ } ⊂ M R ( C ) . ✭✸✮ ♥ P✉r❡ st❛t❡s P✉r❡ st❛t❡s ❛r❡ ♦♥❡ ❞✐♠❡♥s✐♦♥❛❧ ♣r♦❥❡❝t♦rs ✱✐✳❡✳ ❞❡♥s✐t② ♠❛tr✐❝❡s ♦❢ t❤❡ ❢♦r♠ ρ = | ψ � � ψ | ✳ ▼✐❝❤❛➟ ❇✐❛➟♦➠❝③②❦✱ ✐♥ ❝♦♦♣❡r❛t✐♦♥ ✇✐t❤ ♣r♦❢✳ ❑❛r♦❧ ➏②❝③❦♦✇s❦✐ ❙♣❡❝tr❛❧ ♣r♦♣❡rt✐❡s ♦❢ q✉❛♥t✉♠ ♦♣❡r❛t✐♦♥s ✲ ❛❧❣❡❜r❛✐❝ ♠❡t❤♦❞s

◗✉❛♥t✉♠ st❛t❡s ❛♥❞ ♦♣❡r❛t✐♦♥s P❡rr♦♥ ✲ ❋r♦❜❡♥✐✉s t❤❡♦r② ❇❧♦❝❦ ❞❡❝♦♠♣♦s✐t✐♦♥s P♦s✐t✐✈❡ ♠❛♣s P♦s✐t✐✈❡ ♠❛♣s ❆ ♠❛♣ Φ : M ♥ ( C ) → M ♥ ( C ) ✐s ❝❛❧❧❡❞ ♣♦s✐t✐✈❡ ✐❢ Φ( M + ♥ ( C )) ⊆ M + ♥ ( C ) ✳ ❚❤❛t ✐s ♣♦s✐t✐✈❡ ♠❛♣ ❧❡❛✈❡s t❤❡ ❝♦♥❡ ✐♥✈❛r✐❛♥t ✳ ❊①❛♠♣❧❡ ✿ ◆♦♥♥❡❣❛t✐✈❡ ♠❛tr✐① ✐s t❤❡ ♣♦s✐t✐✈❡ ♠❛♣ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❝♦♥❡ R ♥ + ✳ ❊✈♦❧✉t✐♦♥ ♦❢ t❤❡ q✉❛♥t✉♠ s②st❡♠ ❲❤❡♥ Φ : M ♥ ( C ) → M ♥ ( C ) ❝❛♥ ♠♦❞❡❧ ❞✐s❝r❡t❡ ❡✈♦❧✉t✐♦♥ ♦❢ ❛♥ ♦♣❡♥ q✉❛♥t✉♠ s②st❡♠ ✿ ρ → Φ( ρ )? ✭✹✮ ❈♦♠♣❧❡t❡❧② ♣♦s✐t✐✈❡ ♠❛♣ P♦s✐t✐✈❡ ♠❛♣ Φ : M ♥ ( C ) → M ♥ ( C ) ✐s ❝♦♠♣❧❡t❡❧② ♣♦s✐t✐✈❡ ✭❈P ♠❛♣✮ ✐❢ t❤❡ ♠❛♣ Φ ⊗ I ❦ : M ♥❦ ( C ) → M ♥❦ ( C ) ✐s ♣♦s✐t✐✈❡ ❢♦r ❦ = ✶ , . . . ✳ ▼✐❝❤❛➟ ❇✐❛➟♦➠❝③②❦✱ ✐♥ ❝♦♦♣❡r❛t✐♦♥ ✇✐t❤ ♣r♦❢✳ ❑❛r♦❧ ➏②❝③❦♦✇s❦✐ ❙♣❡❝tr❛❧ ♣r♦♣❡rt✐❡s ♦❢ q✉❛♥t✉♠ ♦♣❡r❛t✐♦♥s ✲ ❛❧❣❡❜r❛✐❝ ♠❡t❤♦❞s

◗✉❛♥t✉♠ st❛t❡s ❛♥❞ ♦♣❡r❛t✐♦♥s P❡rr♦♥ ✲ ❋r♦❜❡♥✐✉s t❤❡♦r② ❇❧♦❝❦ ❞❡❝♦♠♣♦s✐t✐♦♥s Φ ❛s ❛ s✉♣❡r♦♣❡r❛t♦r ❋♦r ✜♥✐t❡ ✲ ❞✐♠❡♥s✐♦♥❛❧ s②st❡♠s Φ ❛❝ts ❛s ❛ ♠❛tr✐① ✿ ρ ✐❥ → Φ ✐❥ ❦❧ ρ ❦❧ . ✭✺✮ ❉②♥❛♠✐❝❛❧ ♠❛tr✐① ✭❙✉❞❛rs❤❛♥ ❛t ❛❧✳✱ ✶✾✻✶✮ ❉❡✜♥❡ r❡s❤✉✤❡❞ ♠❛tr✐① ❉ ✿ ❉ ✐ ❥ ❦ ❧ = Φ ✐ ❦ ❥ ❧ . ❈❤♦✐✬s t❤❡♦r❡♠ ✭✶✾✼✺✮ Φ ✐s ❝♦♠♣❧❡t❡❧② ♣♦s✐t✐✈❡ ⇔ ❉ ≥ ✵✳ ❑r❛✉s ❢♦r♠ Φ : M ♥ ( C ) → M ♥ ( C ) ✲ ❈P ♠❛♣✱ t❤❡♥ ❑ � ❆ ✐ ❳❆ † ✐ , ❑ ≤ ◆ ✷ . Φ( ❳ ) = ✭✻✮ ✐ = ✶ ▼✐❝❤❛➟ ❇✐❛➟♦➠❝③②❦✱ ✐♥ ❝♦♦♣❡r❛t✐♦♥ ✇✐t❤ ♣r♦❢✳ ❑❛r♦❧ ➏②❝③❦♦✇s❦✐ ❙♣❡❝tr❛❧ ♣r♦♣❡rt✐❡s ♦❢ q✉❛♥t✉♠ ♦♣❡r❛t✐♦♥s ✲ ❛❧❣❡❜r❛✐❝ ♠❡t❤♦❞s

◗✉❛♥t✉♠ st❛t❡s ❛♥❞ ♦♣❡r❛t✐♦♥s P❡rr♦♥ ✲ ❋r♦❜❡♥✐✉s t❤❡♦r② ❇❧♦❝❦ ❞❡❝♦♠♣♦s✐t✐♦♥s ❚r❛❝❡ ♣r❡s❡r✈✐♥❣ ❛♥❞ ✉♥✐t❛❧ ♠❛♣s ❚r❛❝❡ ♣r❡s❡r✈❛t✐♦♥✿ ❑ � ❆ † ❚r Φ( ρ ) = ❚r ρ ⇔ ✐ ❆ ✐ = I ✭✼✮ ✐ = ✶ ❯♥✐t❛❧✐t②✿ ❑ � ❆ ✐ ❆ † Φ( I ) = I ⇔ ✐ = I ✭✽✮ ✐ = ✶ ❉✉❛❧ ♠❛♣ ❋♦r ♣♦s✐t✐✈❡ Φ : M ♥ ( C ) → M ♥ ( C ) t❤❡ ❞✉❛❧ ♠❛♣ Φ ◦ ✐s ❞❡✜♥❡❞✿ � ❳ † , Φ( ❨ ) � ❍❙ = � Φ ◦ ( ❳ ) , ❨ � ❍❙ Φ ❤❛s t❤❡ s❛♠❡ s♣❡❝tr✉♠ ❛s Φ ◦ ✳ ▼✐❝❤❛➟ ❇✐❛➟♦➠❝③②❦✱ ✐♥ ❝♦♦♣❡r❛t✐♦♥ ✇✐t❤ ♣r♦❢✳ ❑❛r♦❧ ➏②❝③❦♦✇s❦✐ ❙♣❡❝tr❛❧ ♣r♦♣❡rt✐❡s ♦❢ q✉❛♥t✉♠ ♦♣❡r❛t✐♦♥s ✲ ❛❧❣❡❜r❛✐❝ ♠❡t❤♦❞s

◗✉❛♥t✉♠ st❛t❡s ❛♥❞ ♦♣❡r❛t✐♦♥s P❡rr♦♥ ✲ ❋r♦❜❡♥✐✉s t❤❡♦r② ❇❧♦❝❦ ❞❡❝♦♠♣♦s✐t✐♦♥s ❉✉❛❧ ❈P ♠❛♣ ❑ ❑ � � ❆ ✐ ❳❆ † ❆ † ✐ ⇔ Φ ◦ ( ❳ ) = Φ( ❳ ) = ✐ ❳❆ ✐ . ✐ = ✶ ✐ = ✶ Φ tr❛❝❡ ♣r❡s❡r✈✐♥❣ ⇔ Φ ◦ ✉♥✐t❛❧✳ ❙♣❡❝tr❛❧ ♣r♦♣❡rt✐❡s ♦❢ ♣♦s✐t✐✈❡ ♠❛♣s ■❢ Φ : M ♥ ( C ) → M ♥ ( C ) ✐s ♣♦s✐t✐✈❡✱ tr❛❝❡ ♣r❡s❡r✈✐♥❣ ♦r ✉♥✐t❛❧✱ t❤❡♥ ❆❧❧ ❡✐❣❡♥✈❛❧✉❡s ❜❡❧♦♥❣ t♦ t❤❡ ✉♥✐t ❞✐s❦✱ ❚❤❡ ❧❡❛❞✐♥❣ ❡✐❣❡♥✈❛❧✉❡ ❡q✉❛❧s ✶ ✭✐✳❡✳ t❤❡r❡ ❛❧✇❛②s ❡①✐sts t❤❡ ✜①❡❞ ♣♦✐♥t✱ ♣♦ss✐❜❧② ❞❡❣❡♥❡r❛t❡❞✮✱ ❋✐①❡❞ ♣♦✐♥t ❝❛♥ ❜❡ ❝❤♦s❡♥ t♦ ❜❡ ❛ ❞❡♥s✐t② ♠❛tr✐①✳ ▼✐❝❤❛➟ ❇✐❛➟♦➠❝③②❦✱ ✐♥ ❝♦♦♣❡r❛t✐♦♥ ✇✐t❤ ♣r♦❢✳ ❑❛r♦❧ ➏②❝③❦♦✇s❦✐ ❙♣❡❝tr❛❧ ♣r♦♣❡rt✐❡s ♦❢ q✉❛♥t✉♠ ♦♣❡r❛t✐♦♥s ✲ ❛❧❣❡❜r❛✐❝ ♠❡t❤♦❞s

◗✉❛♥t✉♠ st❛t❡s ❛♥❞ ♦♣❡r❛t✐♦♥s P❡rr♦♥ ✲ ❋r♦❜❡♥✐✉s t❤❡♦r② ❇❧♦❝❦ ❞❡❝♦♠♣♦s✐t✐♦♥s 1.0 0.5 0.0 - 0.5 - 1.0 - 1.0 - 0.5 0.0 0.5 1.0 ▼✐❝❤❛➟ ❇✐❛➟♦➠❝③②❦✱ ✐♥ ❝♦♦♣❡r❛t✐♦♥ ✇✐t❤ ♣r♦❢✳ ❑❛r♦❧ ➏②❝③❦♦✇s❦✐ ❙♣❡❝tr❛❧ ♣r♦♣❡rt✐❡s ♦❢ q✉❛♥t✉♠ ♦♣❡r❛t✐♦♥s ✲ ❛❧❣❡❜r❛✐❝ ♠❡t❤♦❞s