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❆♥❛❧②t✐❝ q✉❛♥t✉♠ ✇❡❛❦ ❝♦✐♥ ✢✐♣♣✐♥❣ ♣r♦t♦❝♦❧s ✇✐t❤ ❛r❜✐tr❛r✐❧② s♠❛❧❧ ❜✐❛s ❆t✉❧ ❙✳ ❆r♦r❛✱ ❏éré♠✐❡ ❘♦❧❛♥❞✱ ❈❤r②s♦✉❧❛ ❱❧❛❝❤♦✉ ❛r❳✐✈✿✶✾✶✶✳✶✸✷✽✸ ◗❈r②♣t ✷✵✷✵

❙❡❝✉r❡ t✇♦✲♣❛rt② ❝♦♠♣✉t❛t✐♦♥ ❚✇♦ ♣❛rt✐❡s ❥♦✐♥t❧② ❝♦♠♣✉t❡ ❛♥ ❛r❜✐tr❛r② ❢✉♥❝t✐♦♥ ♦♥ t❤❡✐r ✐♥♣✉ts ✇✐t❤♦✉t s❤❛r✐♥❣ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡✐r ✐♥♣✉ts ✇✐t❤ t❤❡ ♦t❤❡r ❈❧❛ss✐❝❛❧ ❖❜❧✐✈✐♦✉s ❚r❛♥s❢❡r ⇒ ❇✐t ❈♦♠♠✐t♠❡♥t ⇒ ❈♦✐♥ ❋❧✐♣♣✐♥❣ P❡r❢❡❝t s❡❝✉r✐t② ✐♠♣♦ss✐❜❧❡ ✇✐t❤♦✉t ❡①tr❛ ❛ss✉♠♣t✐♦♥s ✭❡✳❣✳ ❝♦♠♣✉t❛t✐♦♥❛❧ ❤❛r❞♥❡ss✮ ◗✉❛♥t✉♠ ❖❜❧✐✈✐♦✉s ❚r❛♥s❢❡r ⇔ ❇✐t ❈♦♠♠✐t♠❡♥t ⇒ ❈♦✐♥ ❋❧✐♣♣✐♥❣ P❡r❢❡❝t s❡❝✉r✐t② ✐s ✐♠♣♦ss✐❜❧❡ ✭♥♦♥✲r❡❧❛t✐✈✐st✐❝✮ ◗✉❛♥t✉♠ ✇❡❛❦ ❝♦✐♥ ✢✐♣♣✐♥❣ ✐s t❤❡ str♦♥❣❡st ❦♥♦✇♥ ♣r✐♠✐t✐✈❡ ✇✐t❤ ❛r❜✐tr❛r✐❧② ♣❡r❢❡❝t s❡❝✉r✐t②

❈♦✐♥ ✢✐♣♣✐♥❣ 1 ♦✈❡r t❤❡ t❡❧❡♣❤♦♥❡ ❚✇♦ ❞✐str✉st❢✉❧ ♣❛rt✐❡s✱ ❆❧✐❝❡ ❛♥❞ ❇♦❜✱ ✇✐s❤ t♦ r❡♠♦t❡❧② ❣❡♥❡r❛t❡ ❛♥ ✉♥❜✐❛s❡❞ r❛♥❞♦♠ ❜✐t✳ ◮ ❙tr♦♥❣ ❈♦✐♥ ❋❧✐♣♣✐♥❣ ✭❙❈❋✮ ❚❤❡ ♣❛rt✐❡s ❞♦ ♥♦t ❦♥♦✇ ❛ ♣r✐♦r✐ t❤❡ ♣r❡❢❡rr❡❞ ♦✉t❝♦♠❡ ♦❢ t❤❡ ♦t❤❡r ◮ ❲❡❛❦ ❈♦✐♥ ❋❧✐♣♣✐♥❣ ✭❲❈❋✮ ❚❤❡ ♣❛rt✐❡s ❤❛✈❡ ❛ ♣r✐♦r✐ ❦♥♦✇♥ ♦♣♣♦s✐t❡ ♣r❡❢❡rr❡❞ ♦✉t❝♦♠❡s 1 ▼✳ ❇❧✉♠✱ ❙■●❆❈❚ ◆❡✇s ✶✺✳✶✱ ♣♣✳✷✸✲✷✼ ✭✶✾✽✸✮✳

Pr♦t♦❝♦❧ ❢❡❛t✉r❡s ❍♦♥❡st ✐s ❛ ♣❧❛②❡r ✇❤♦ ❢♦❧❧♦✇s t❤❡ ♣r♦t♦❝♦❧ ❡①❛❝t❧② ❛s ❞❡s❝r✐❜❡❞✳ ❆ ❇ ❋❡❛t✉r❡ Pr✭❆ ✇✐♥s✮ Pr✭❇ ✇✐♥s✮ ❍♦♥❡st ❍♦♥❡st ❈♦rr❡❝t♥❡ss P A = 1 / 2 P B = 1 / 2 P ∗ 1 − P ∗ ❈❤❡❛ts ❍♦♥❡st ❆ ❝❛♥ ❜✐❛s A A ❍♦♥❡st ❈❤❡❛ts ❇ ❝❛♥ ❜✐❛s 1 − P ∗ P ∗ B B ❈❤❡❛ts ❈❤❡❛ts ◆♦ ♣r♦t♦❝♦❧ ✕ ✕ ❆ ♣r♦t♦❝♦❧ ❤❛s ❜✐❛s ǫ ✐❢ ♥❡✐t❤❡r ♣❧❛②❡r ❝❛♥ ❢♦r❝❡ t❤❡✐r ❞❡s✐r❡❞ ♦✉t❝♦♠❡ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ❤✐❣❤❡r t❤❛♥ 1 2 + ǫ ✱ ✐✳❡✳ t❤❡ ❜✐❛s ✐s t❤❡ B ≤ 1 s♠❛❧❧❡st ǫ s✉❝❤ t❤❛t P ∗ A , P ∗ 2 + ǫ ✳

❇♦✉♥❞s ❛♥❞ ❜❡st ❡①♣❧✐❝✐t ♣r♦t♦❝♦❧s ❈❧❛ss✐❝❛❧ ❈♦♠♣❧❡t❡❧② ✐♥s❡❝✉r❡ ǫ = 1 2 ✱ ✉♥❧❡ss ❡①tr❛ ❛ss✉♠♣t✐♦♥s ❛r❡ ♠❛❞❡ ◗✉❛♥t✉♠ ❇♦✉♥❞ Pr♦t♦❝♦❧ 2 ❛♥❞ ǫ = 1 1 3 1 2 − 1 1 2 − 1 ❙❈❋ ǫ ≥ ǫ → √ √ 2 2 4 6 ✱ ♥✉♠❡r✐❝❛❧❧② ǫ → 0 6 ǫ → 0 4 , 5 1 ❲❈❋ ǫ = 10 1 ❆✳ ❨✳ ❑✐t❛❡✈✱ ◗■P ✇♦r❦s❤♦♣ ✭✷✵✵✸✮✳ 2 ❆✳ ❈❤❛✐❧❧♦✉① ❛♥❞ ■✳ ❑❡r❡♥✐❞✐s✱ ✺✵t❤ ❋❖❈❙✱ ♣♣✳ ✺✷✼✲✺✸✸ ✭✷✵✵✾✮✳ 3 ❆✳ ❆♠❜❛✐♥✐s✱ ❏ ❈♦♠♣ ❛♥❞ ❙②s ❙❝✐ ✻✽✳✷✱ ♣♣✳ ✸✾✽✲✹✶✻ ✭✷✵✵✹✮✳ 4 ❈✳ ▼♦❝❤♦♥✱ ❛r❳✐✈✿✵✼✶✶✳✹✶✶✹ ✭✷✵✵✼✮✳ 5 ❉✳ ❆❤❛r♦♥♦✈✱ ❆✳ ❈❤❛✐❧❧♦✉①✱ ▼✳ ●❛♥③✱ ■✳ ❑❡r❡♥✐❞✐s ❛♥❞ ▲✳ ▼❛❣♥✐♥✱ ❙■❆▼ ❏ ❈♦♠♣ ✹✺✳✸✱ ♣♣✳ ✻✸✸✲✻✼✾ ✭✷✵✶✻✮✳ 6 ❆✳ ❙✳ ❆r♦r❛✱ ❏✳ ❘♦❧❛♥❞ ❛♥❞ ❙✳ ❲❡✐s✱ ✺✶st ❆❈▼ ❙■●❆❈❚ ❙❚❖❈✱ ♣♣✳ ✷✵✺✲✷✶✻ ✭✷✵✶✾✮✳

Pr♦t♦❝♦❧ ❞❡s❝r✐♣t✐♦♥ ❆ ♥❡✇ ❢r❛♠❡✇♦r❦ ✐s ♥❡❡❞❡❞ ♣❡r♠✐tt✐♥❣ ✉s t♦ ✜♥❞ ❜♦t❤ t❤❡ ♣r♦t♦❝♦❧ ❛♥❞ ✐ts ❜✐❛s✳

❚✐♠❡✲❞❡♣❡♥❞❡♥t ♣♦✐♥t ❣❛♠❡s ∗ ✭❚❉P●✮ ❙❡q✉❡♥❝❡ ♦❢ ❢r❛♠❡s ✐♥❝❧✉❞✐♥❣ ♣♦✐♥ts ♦♥ x − y ♣❧❛♥❡ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ✇❡✐❣❤ts ❛ss✐❣♥❡❞ ◮ ❙t❛rt✐♥❣ ♣♦✐♥ts✿ (0 , 1) ❛♥❞ (1 , 0) ✇✐t❤ p = 1 / 2 ✳ ◮ ❚r❛♥s✐t✐♦♥s ❜❡t✇❡❡♥ ❢r❛♠❡s✿ � � p z = p z ′ z z ′ λz ′ λz � � λ + z p z ≤ λ + z ′ p z ′ , ∀ λ ≥ 0 z z ′ ◮ ❋✐♥❛❧ ♣♦✐♥t ( β, α ) ✇✐t❤ p = 1 ✳ ∗ ▼♦❝❤♦♥ ✐♥ ❛r❳✐✈✿✵✼✶✶✳✹✶✶✹ ❛ttr✐❜✉t❡s t❤❡ ♣♦✐♥t✲❣❛♠❡ ❢♦r♠❛❧✐s♠ t♦ ❆✳ ❨✳ ❑✐t❛❡✈✳

❊①❛♠♣❧❡s ♦❢ ❛❧❧♦✇❡❞ ♠♦✈❡s

❚r❛♥s✐t✐♦♥s ❡①♣r❡ss✐❜❧❡ ❜② ♠❛tr✐❝❡s ✭❊❇▼✮ ❈♦♥s✐❞❡r ❛ ❍❡r♠✐t✐❛♥ ♠❛tr✐① Z ≥ 0 ❛♥❞ ❧❡t Π [ z ] ❜❡ t❤❡ ♣r♦❥❡❝t♦r ♦♥ t❤❡ z z Π [ z ] ✳ ▲❡t | ψ � ❜❡ ❛ ✈❡❝t♦r ❡✐❣❡♥s♣❛❝❡ ♦❢ t❤❡ ❡✐❣❡♥✈❛❧✉❡ z ✳ ❚❤❡♥ Z = � ✭♥♦t ♥❡❝❡ss❛r✐❧② ♥♦r♠❛❧✐s❡❞✮✳ ❲❡ ❞❡✜♥❡ t❤❡ ❢✉♥❝t✐♦♥ Pr♦❜ [ Z, | ψ � ] : [0 , ∞ ) → [0 , ∞ ) ✇✐t❤ ✜♥✐t❡ s✉♣♣♦rt ❛s � � ψ | Π [ z ] | ψ � ✐❢ z ∈ s♣❡❝tr✉♠ ( Z ) Pr♦❜ [ Z, | ψ � ]( z ) = 0 ♦t❤❡r✇✐s❡ . ▲❡t g, h : [0 , ∞ ) → [0 , ∞ ) ❜❡ t✇♦ ❢✉♥❝t✐♦♥s ✇✐t❤ ✜♥✐t❡ s✉♣♣♦rts✳ ❚❤❡ ❧✐♥❡ tr❛♥s✐t✐♦♥ g → h ✐s ❝❛❧❧❡❞ ❊❇▼ ✐❢ t❤❡r❡ ❡①✐st t✇♦ ♠❛tr✐❝❡s 0 ≤ G ≤ H ❛♥❞ ❛ ✈❡❝t♦r | ψ � s✉❝❤ t❤❛t✿ g = Pr♦❜ [ G, | ψ � ] ❛♥❞ h = Pr♦❜ [ H, | ψ � ] . ❋♦r ❡❛❝❤ ❊❇▼ ❚❉P● t❤❡r❡ ❡①✐sts ❛ ❲❈❋ ♣r♦t♦❝♦❧ ✇✐t❤ P ∗ A ≤ α, P ∗ B ≤ β ✳

❚✐♠❡✲✐♥❞❡♣❡♥❞❡♥t ♣♦✐♥t ❣❛♠❡s ✭❚■P●✮ ❋♦r ❛♥ ❊❇▼ tr❛♥s✐t✐♦♥ g → h ✱ ✇❡ ❞❡✜♥❡ t❤❡ ❊❇▼ ❢✉♥❝t✐♦♥ g − h ✳ ❚❤❡ s❡t ♦❢ ❊❇▼ ❢✉♥❝t✐♦♥s ✐s t❤❡ s❛♠❡ ✭✉♣ t♦ ❝❧♦s✉r❡s✮ ❛s t❤❡ s❡t ♦❢ ✈❛❧✐❞ ❢✉♥❝t✐♦♥s✳ ❆ ❢✉♥❝t✐♦♥ f ( x ) ✐s ✈❛❧✐❞ ✐❢ � x f ( x ) = 0 ❛♥❞ � f ( x ) λ + x ≤ 0 , ∀ λ ≥ 0 . x ❋♦r ❡❛❝❤ ❚■P● t❤❡r❡ ❡①✐sts ❛♥ ❊❇▼ ❚❉P● ✇✐t❤ t❤❡ s❛♠❡ ✜♥❛❧ ❢r❛♠❡

❊①✐st❡♥❝❡ ♦❢ ❛ ❲❈❋ ♣r♦t♦❝♦❧ ✇✐t❤ ǫ → 0 1 ❋❛♠✐❧② ♦❢ ❚■P● 2 ❛♣♣r♦❛❝❤✐♥❣ ❜✐❛s 1 ǫ = 4 k + 2 , ✇❤❡r❡ 2 k ✐s t❤❡ ♥✉♠❜❡r ♦❢ ♣♦✐♥ts ✐♥✈♦❧✈❡❞ ✐♥ t❤❡ ♠❛✐♥ ♠♦✈❡ ♦❢ t❤❡ ♣♦✐♥t ❣❛♠❡ 1 ❈✳ ▼♦❝❤♦♥✱ ❛r❳✐✈✿✵✼✶✶✳✹✶✶✹ ✭✷✵✵✼✮✳ 2 P✐❝t✉r❡ ❢r♦♠ P✳ ❍ø②❡r ❛♥❞ ❊✳ P❡❧❝❤❛t✱ ▼❆ t❤❡s✐s✱ ❯♥✐✈❡rs✐t② ♦❢ ❈❛❧❣❛r② ✭✷✵✶✸✮✳

❊q✉✐✈❛❧❡♥t ❢r❛♠❡✇♦r❦s ❛♥❞ t❤❡ ♣r♦♦❢ ♦❢ ❡①✐st❡♥❝❡ 1 , 2 1 ❈✳ ▼♦❝❤♦♥✱ ❛r❳✐✈✿✵✼✶✶✳✹✶✶✹ ✭✷✵✵✼✮✳ 2 ❉✳ ❆❤❛r♦♥♦✈✱ ❆✳ ❈❤❛✐❧❧♦✉①✱ ▼✳ ●❛♥③✱ ■✳ ❑❡r❡♥✐❞✐s ❛♥❞ ▲✳ ▼❛❣♥✐♥✱ ❙■❆▼ ❏ ❈♦♠♣ ✹✺✳✸✱ ♣♣✳ ✻✸✸✲✻✼✾ ✭✷✵✶✻✮✳

❚❉P●✲t♦✲❡①♣❧✐❝✐t✲♣r♦t♦❝♦❧ ❢r❛♠❡✇♦r❦ ✭❚❊❋✮ 1 ❈♦♥✈❡rs✐♦♥ ♦❢ ❛ ❚❉P● t♦ ❛♥ ❡①♣❧✐❝✐t ❲❈❋ ♣r♦t♦❝♦❧ ✇✐t❤ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❜✐❛s✱ ❣✐✈❡♥ t❤❛t ❢♦r ❡✈❡r② tr❛♥s✐t✐♦♥ ♦❢ t❤❡ ❚❉P●✱ ❛ ✉♥✐t❛r② s❛t✐s❢②✐♥❣ ❝❡rt❛✐♥ ❝♦♥str❛✐♥ts ❝❛♥ ❜❡ ❢♦✉♥❞ 1 ❆✳ ❙✳ ❆r♦r❛✱ ❏✳ ❘♦❧❛♥❞ ❛♥❞ ❙✳ ❲❡✐s✱ ✺✶st ❆❈▼ ❙■●❆❈❚ ❙❚❖❈✱ ♣♣✳ ✷✵✺✲✷✶✻ ✭✷✵✶✾✮✳