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❖♥ t❤❡ ❣❡♦♠❡tr② ♦❢ ♦♥❡✲♠♦❞❡ ●❛✉ss✐❛♥ ❝❤❛♥♥❡❧s

❑❛t❛r③②♥❛ ❙✐✉❞③✐➠s❦❛✶ ❑✐♠♠♦ ▲✉♦♠❛✷ ❲❛❧t❡r ❚✳ ❙tr✉♥③✷

✶■♥st✐t✉t❡ ♦❢ P❤②s✐❝s✱ ❋❛❝✉❧t② ♦❢ P❤②s✐❝s✱ ❆str♦♥♦♠② ❛♥❞ ■♥❢♦r♠❛t✐❝s

◆✐❝♦❧❛✉s ❈♦♣❡r♥✐❝✉s ❯♥✐✈❡rs✐t②✱ ●r✉❞③✐→❞③❦❛ ✺✴✼✱ ✽✼✕✶✵✵ ❚♦r✉➠✱ P♦❧❛♥❞

✷■♥st✐t✉t ❢ür ❚❤❡♦r❡t✐s❝❤❡ P❤②s✐❦✱ ❚❡❝❤♥✐s❝❤❡ ❯♥✐✈❡rs✐tät ❉r❡s❞❡♥✱

❉✲✵✶✵✻✷✱ ❉r❡s❞❡♥✱ ●❡r♠❛♥②

❏✉♥❡ ✶✻✱ ✷✵✶✾

❈♦♥t❡♥ts

✶

• ❛✉ss✐❛♥ st❛t❡s ❛♥❞ ●❛✉ss✐❛♥ ❝❤❛♥♥❡❧s

❈♦✈❛r✐❛♥❝❡ ♠❛tr✐① ❛♥❞ ❞✐s♣❧❛❝❡♠❡♥t ✈❡❝t♦r ❈♦♠♣❧❡t❡ ♣♦s✐t✐✈✐t② ❝♦♥❞✐t✐♦♥

✷

❈❤♦✐✲❏❛♠✐♦➟❦♦✇s❦✐ ✐s♦♠♦r♣❤✐s♠

✸

❍✐❧❜❡rt✲❙❝❤♠✐❞t ❞✐st❛♥❝❡ ❛♥❞ ✈♦❧✉♠❡ ❙t❛t❡ ❝♦♦r❞✐♥❛t❡s ❚♦t❛❧ ✈♦❧✉♠❡ ♦❢ ●❛✉ss✐❛♥ ❝❤❛♥♥❡❧s

✹

❊♥t❛♥❣❧❡♠❡♥t ❛♥❞ ✐♥❝♦♠♣❛t✐❜✐❧✐t② ❜r❡❛❦✐♥❣ ❝❤❛♥♥❡❧s ❈r✐t❡r✐❛ ✐♥ t❡r♠s ♦❢ t❤❡ ❈❏ st❛t❡s ❱♦❧✉♠❡ r❛t✐♦s

✺

❙✉♠♠❛r②

❙✐✉❞③✐➠s❦❛✱ ▲✉♦♠❛✱ ❙tr✉♥③ ❖♥ t❤❡ ❣❡♦♠❡tr② ♦❢ ♦♥❡✲♠♦❞❡ ●❛✉ss✐❛♥ ❝❤❛♥♥❡❧s ✶✴ ✶✼

• ❛✉ss✐❛♥ st❛t❡s

❈♦♥s✐❞❡r t❤❡ n✲♣❛rt✐❝❧❡ ❝♦♥t✐♥✉♦✉s ✈❛r✐❛❜❧❡ s②st❡♠✳ ■♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❍✐❧❜❡rt s♣❛❝❡ H = n

k=✶ L✷(R)✳

❱❡❝t♦r st❛t❡s R = (q✶, p✶, . . . , qn, pn)T s❛t✐s❢②✐♥❣ t❤❡ ❝♦♠♠✉t❛t✐♦♥ r❡❧❛t✐♦♥s [Ri, R†

j ] = ✷iΩij,

Ω =

n

• k=✶

✵ ✶ −✶ ✵

• ,

✭✶✮ ✇✐t❤ t❤❡ s②♠♣❧❡❝t✐❝ ❢♦r♠ Ω✳ ❉❡♥s✐t② ♦♣❡r❛t♦rs ❛r❡ ❣✐✈❡♥ ❜② ρ =

• R✷n

d✷n ξ πn χ(ξ)D(−ξ) ✭✷✮ ✇✐t❤ t❤❡ ❞✐s♣❧❛❝❡♠❡♥t ✭❲❡②❧✮ ♦♣❡r❛t♦rs D(ξ) = eiRT Ωξ. ✭✸✮

❙✐✉❞③✐➠s❦❛✱ ▲✉♦♠❛✱ ❙tr✉♥③ ❖♥ t❤❡ ❣❡♦♠❡tr② ♦❢ ♦♥❡✲♠♦❞❡ ●❛✉ss✐❛♥ ❝❤❛♥♥❡❧s ✷✴ ✶✼

• ❛✉ss✐❛♥ st❛t❡s

❉❡✜♥✐t✐♦♥ ✶

❆ ❞❡♥s✐t② ♦♣❡r❛t♦r ρ =

• R✷n

d✷n ξ πn χ(ξ)D(−ξ) ✭✹✮ ✐s ❛ ●❛✉ss✐❛♥ st❛t❡ ✐❢ ✐ts ❝❤❛r❛❝t❡r✐st✐❝ ❢✉♥❝t✐♦♥ χ(ξ) ✐s ❛ ●❛✉ss✐❛♥ ❢✉♥❝t✐♦♥✳ ❆ ●❛✉ss✐❛♥ ❝❤❛r❛❝t❡r✐st✐❝ ❢✉♥❝t✐♦♥ ✐s r❡♣r❡s❡♥t❡❞ ❜② χ(ξ) = exp

• −✶

✷ξTΩΣΩTξ + iℓTΩξ

• ,

✭✺✮ ✇❤❡r❡ ℓk = Tr[ρRk] ✐s t❤❡ ❞✐s♣❧❛❝❡♠❡♥t ✈❡❝t♦r❀ Σij = ✶

✷ Tr[ρ(RiRj + RjRi)] − ℓiℓj ✐s t❤❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐①✳

Σ ✐s t❤❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐① ♦❢ ❛ ●❛✉ss✐❛♥ st❛t❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ Σ + iΩ ≥ ✵. ✭✻✮

❙✐✉❞③✐➠s❦❛✱ ▲✉♦♠❛✱ ❙tr✉♥③ ❖♥ t❤❡ ❣❡♦♠❡tr② ♦❢ ♦♥❡✲♠♦❞❡ ●❛✉ss✐❛♥ ❝❤❛♥♥❡❧s ✸✴ ✶✼

• ❛✉ss✐❛♥ ❝❤❛♥♥❡❧s

❉❡✜♥✐t✐♦♥ ✷

❆ ●❛✉ss✐❛♥ ❝❤❛♥♥❡❧ Λ ✐s ❛ q✉❛♥t✉♠ ❝❤❛♥♥❡❧ t❤❛t tr❛♥s❢♦r♠s ●❛✉ss✐❛♥ st❛t❡s ✐♥t♦

• ❛✉ss✐❛♥ st❛t❡s✳

❘❡♣r❡s❡♥t❛t✐♦♥ ✐♥ t❤❡ ❍❡✐s❡♥❜❡r❣ ♣✐❝t✉r❡✿ Λ∗[D(ξ)] = D(Mξ) exp

• −✶

✷ξTNξ + icTξ

• .

✭✼✮ ❊❛❝❤ ❝❤❛♥♥❡❧ ✐s ❝♦♠♣❧❡t❡❧② ❝❤❛r❛❝t❡r✐③❡❞ ❜② ❛ tr✐♣❧❡ (M, N, c)✱ ❛♥❞ ✐t ❛❝ts ♦♥ t❤❡ ●❛✉ss✐❛♥ st❛t❡ ρ(Σ, ℓ) ❛s ❢♦❧❧♦✇s✱ Σ → MTΣM + N, ℓ → MTℓ + c. ✭✽✮ ❚❤❡ ❝♦♠♣❧❡t❡ ♣♦s✐t✐✈✐t② ❝♦♥❞✐t✐♦♥✿ N − iMTΩM + iΩ ≥ ✵. ✭✾✮

❙✐✉❞③✐➠s❦❛✱ ▲✉♦♠❛✱ ❙tr✉♥③ ❖♥ t❤❡ ❣❡♦♠❡tr② ♦❢ ♦♥❡✲♠♦❞❡ ●❛✉ss✐❛♥ ❝❤❛♥♥❡❧s ✹✴ ✶✼

❈❤♦✐✲❏❛♠✐♦➟❦♦✇s❦✐ ✐s♦♠♦r♣❤✐s♠

❚❤❡♦r❡♠ ✸

❚❤❡r❡ ❡①✐sts ❛ ♦♥❡✲t♦✲♦♥❡ ❝♦rr❡s♣♦♥❞❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❜✐♣❛rt✐t❡ ●❛✉ss✐❛♥ st❛t❡s ρAB ✇✐t❤ ❛ ❝♦♠♠♦♥ ♠❛r❣✐♥❛❧ σ = TrA ρAB ❛♥❞ t❤❡ ●❛✉ss✐❛♥ ❝❤❛♥♥❡❧s Λ : HB → HA✱ s✉❝❤ t❤❛t ρAB = (Λ ⊗ IB)(ρΩ), ✭✶✵✮ ✇❤❡r❡ t❤❡ ●❛✉ss✐❛♥ st❛t❡ ρΩ ✐s ❝❤❛r❛❝t❡r✐③❡❞ ❜② ΣΩ =

• Σσ

ST

σ ZσSσ

ST

σ ZσSσ

Σσ

• ,

ℓΩ = ℓσ ⊕ ℓσ. ✭✶✶✮

❏✳ ❑✐✉❦❛s✱ ❈✳ ❇✉❞r♦♥✐✱ ❘✳ ❯♦❧❛✱ ❛♥❞ ❏✳✲P✳ P❡❧❧♦♥♣ää✱ P❤②s✳ ❘❡✈✳ ❆ ✾✻✱ ✵✹✷✸✸✶ ✭✷✵✶✼✮✳

Sσ ✐s t❤❡ s②♠♣❧❡❝t✐❝ ♠❛tr✐① ❞✐❛❣♦♥❛❧✐③✐♥❣ Σσ❀ Zσ = n

k=✶ σ✸

• ν✷

σ,k − ✶❀

νσ,k ❛r❡ t❤❡ s②♠♣❧❡❝t✐❝ ❡✐❣❡♥✈❛❧✉❡s ♦❢ Σσ✳

❙✐✉❞③✐➠s❦❛✱ ▲✉♦♠❛✱ ❙tr✉♥③ ❖♥ t❤❡ ❣❡♦♠❡tr② ♦❢ ♦♥❡✲♠♦❞❡ ●❛✉ss✐❛♥ ❝❤❛♥♥❡❧s ✺✴ ✶✼

▲✐♥❡ ❛♥❞ ✈♦❧✉♠❡ ❡❧❡♠❡♥t

❚❤❡ ❍✐❧❜❡rt✲❙❝❤♠✐❞t ❞✐st❛♥❝❡ ✐s ❞❡✜♥❡❞ ❜② d s✷ = Tr(d ρ✷)✳ ❋♦r t❤❡ ●❛✉ss✐❛♥ st❛t❡s ρ(Σ, ℓ = ✵)✱ ♦♥❡ ❤❛s d s✷ = ✶ ✶✻ √ det Σ

• ✷ Tr(Σ−✶ d Σ)✷ + [Tr(Σ−✶ d Σ)]✷

. ✭✶✷✮ ❚❤❡ ✈♦❧✉♠❡ ❡❧❡♠❡♥t ❝♦rr❡s♣♦♥❞✐♥❣ t♦ d s✷ = d ΣTG d Σ r❡❛❞s d V = √ det G

✹n✷

• k=✶

d Σk, ✭✶✸✮ ✇❤❡r❡ d Σ = ✈❡❝ d Σ✱ ❛♥❞ G ✐s t❤❡ ♠❡tr✐❝✳

▲✐♥❦ ❛♥❞ ❲✳ ❚✳ ❙tr✉♥③✱ ❏✳ P❤②s✳ ❆✿ ▼❛t❤✳ ❚❤❡♦r✳ ✹✽✱ ✷✼✺✸✵✶ ✭✷✵✶✺✮✳

❙✐✉❞③✐➠s❦❛✱ ▲✉♦♠❛✱ ❙tr✉♥③ ❖♥ t❤❡ ❣❡♦♠❡tr② ♦❢ ♦♥❡✲♠♦❞❡ ●❛✉ss✐❛♥ ❝❤❛♥♥❡❧s ✻✴ ✶✼

▲✐♥❡ ❛♥❞ ✈♦❧✉♠❡ ❡❧❡♠❡♥t

❈♦♥s✐❞❡r t❤❡ ♦♥❡✲♠♦❞❡ ●❛✉ss✐❛♥ ❝❤❛♥♥❡❧s ✭n = ✶✮✱ ✇❤✐❝❤ ❝♦rr❡s♣♦♥❞ t♦ t❤❡ t✇♦✲♠♦❞❡ ❈❏ ●❛✉ss✐❛♥ st❛t❡s ✇✐t❤      Σ =

• ΣA

ΓT Γ Σσ

• ,

ℓ = ℓA ⊕ ℓσ,      ΣA = N + MTΣσM, Γ = ST

σ ZσSσM,

ℓA = c + MTℓσ. ✭✶✹✮ ❚❤❡ ♣✉r✐t②✲s❡r❛❧✐❛♥ ❝♦♦r❞✐♥❛t❡s✿ µ = ✶ √ det Σ , µA/σ = ✶ det ΣA/σ , ∆ = det ΣA + det Σσ + ✷ det Γ. ✭✶✺✮ ❚❤❡ ✈♦❧✉♠❡ ❡❧❡♠❡♥t✿ d V = µ✶✶/✷ ✻✹ √ ✷µ✸

Aµ✷ σ

d µA d µ d ∆ d θ d m(SA), ✭✶✻✮ ✇❤❡r❡ d m(SA) ✐s t❤❡ ♠❡❛s✉r❡ ♦❢ t❤❡ ♥♦♥✲❝♦♠♣❛❝t s②♠♣❧❡❝t✐❝ ❣r♦✉♣ Sp(✷)✳

• ✳ ❆❞❡ss♦✱ ❆✳ ❙❡r❛✜♥✐✱ ❛♥❞ ❋✳ ■❧❧✉♠✐♥❛t✐✱ P❤②s✳ ❘❡✈✳ ▲❡tt✳ ✾✷✱ ✵✽✼✾✵✶ ✭✷✵✵✹✮✳

❙✐✉❞③✐➠s❦❛✱ ▲✉♦♠❛✱ ❙tr✉♥③ ❖♥ t❤❡ ❣❡♦♠❡tr② ♦❢ ♦♥❡✲♠♦❞❡ ●❛✉ss✐❛♥ ❝❤❛♥♥❡❧s ✼✴ ✶✼

❱♦❧✉♠❡ ♦❢ ●❛✉ss✐❛♥ ❝❤❛♥♥❡❧s

❚❤❡ r❛♥❣❡ ♦❢ ❝♦♦r❞✐♥❛t❡s t❤❛t ❞❡✜♥❡ ❛ ♣❤②s✐❝❛❧ ●❛✉ss✐❛♥ st❛t❡✿        ✵ ≤ µA/σ ≤ ✶, µAµσ ≤ µ ≤ µAµσ µAµσ + |µA − µσ|, ✷ µ + (µA − µσ)✷ µ✷

Aµ✷ σ

≤ ∆ ≤ min

• − ✷

µ + (µA + µσ)✷ µ✷

Aµ✷ σ

, ✶ + ✶ µ✷

• .

✭✶✼✮ ❚❤❡ ✈♦❧✉♠❡ ♦❢ ❛❧❧ ♦♥❡✲♠♦❞❡ ●❛✉ss✐❛♥ ❝❤❛♥♥❡❧s ✐s ❡q✉❛❧ t♦ VGC = C

• CP

µ✶✶/✷ ✻✹ √ ✷µ✸

Aµ✷ σ

d µA d µ d ∆ = C ✹ + µ✾/✷

σ (✾µ✷ σ − ✶✸)

✶✽✵✶✽ √ ✷µ✸

σ

, ✭✶✽✮ ✇❤❡r❡ CP ✐s t❤❡ r❡❣✐♦♥ ❞❡t❡r♠✐♥❡❞ ❜② ❝♦♥❞✐t✐♦♥s ✭✶✼✮✱ ❛♥❞ t❤❡ ✐♥✜♥✐t❡ ❝♦♥st❛♥t C =

• M

d m(SA) ✷π

✵

d θ. ✭✶✾✮

❙✐✉❞③✐➠s❦❛✱ ▲✉♦♠❛✱ ❙tr✉♥③ ❖♥ t❤❡ ❣❡♦♠❡tr② ♦❢ ♦♥❡✲♠♦❞❡ ●❛✉ss✐❛♥ ❝❤❛♥♥❡❧s ✽✴ ✶✼

❱♦❧✉♠❡ ♦❢ ❡♥t❛♥❣❧❡♠❡♥t ❜r❡❛❦✐♥❣ ❝❤❛♥♥❡❧s

❉❡✜♥✐t✐♦♥ ✹

❆ q✉❛♥t✉♠ ❝❤❛♥♥❡❧ Λ : HB → HA ✐s ❡♥t❛♥❣❧❡♠❡♥t ❜r❡❛❦✐♥❣ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ρAB = (Λ ⊗ IB)(ρ) ✭✷✵✮ ✐s s❡♣❛r❛❜❧❡ ❢♦r ❛❧❧ st❛t❡s ρ✳ ❋♦r t❤❡ ●❛✉ss✐❛♥ ❝❤❛♥♥❡❧s✱ ✐t ✐s ❡♥♦✉❣❤ t❤❛t ρAB ✐s s❡♣❛r❛❜❧❡ ❢♦r ρΩ ✇✐t❤ ❛ ♠❛r❣✐♥❛❧ σ = TrA ρAB✳

❚❤❡♦r❡♠ ✺

❚❤❡ t✇♦✲♠♦❞❡ ●❛✉ss✐❛♥ st❛t❡ ρAB ✐s s❡♣❛r❛❜❧❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t s❛t✐s✜❡s t❤❡ P❡r❡s✲❍♦r♦❞❡❝❦✐ ❝r✐t❡r✐♦♥ det(ΣPPT + iΩ) ≥ ✵, ✭✷✶✮ ✇❤❡r❡ ΣPPT = ΘΣΘ ❛♥❞ Θ = ❞✐❛❣(−✶, ✶, ✶, ✶)✳

❘✳ ❙✐♠♦♥✱ P❤②s✳ ❘❡✈✳ ▲❡tt✳ ✽✹✱ ✷✼✷✻ ✭✷✵✵✵✮✳

❙✐✉❞③✐➠s❦❛✱ ▲✉♦♠❛✱ ❙tr✉♥③ ❖♥ t❤❡ ❣❡♦♠❡tr② ♦❢ ♦♥❡✲♠♦❞❡ ●❛✉ss✐❛♥ ❝❤❛♥♥❡❧s ✾✴ ✶✼

❱♦❧✉♠❡ ♦❢ ❡♥t❛♥❣❧❡♠❡♥t ❜r❡❛❦✐♥❣ ❝❤❛♥♥❡❧s

■♥ t❤❡ s❡r❛❧✐❛♥✲♣✉r✐t② ❝♦♦r❞✐♥❛t❡s✱ ❝♦♥❞✐t✐♦♥ det(ΣPPT + iΩ) ≥ ✵ r❡❛❞s ✶ + ✶ µ✷ + ∆ − ✷ µ✷

A

− ✷ µ✷

σ

≥ ✵. ✭✷✷✮ ❚❤❡ ✈♦❧✉♠❡ ♦❢ ❛❧❧ ❡♥t❛♥❣❧❡♠❡♥t ❜r❡❛❦✐♥❣ ♦♥❡✲♠♦❞❡ ●❛✉ss✐❛♥ ❝❤❛♥♥❡❧s ✐s ❡q✉❛❧ t♦ VEBC = C

• SEP

µ✶✶/✷ ✻✹ √ ✷µ✸

Aµ✷ σ

d µA d µ d ∆ = C √µσ(✶ − µσ)✷(✶✶ + ✾µσ) ✶✽✵✶✽ √ ✷ , ✭✷✸✮ ✇❤❡r❡ SEP ✐s t❤❡ ♣❤②s✐❝❛❧✐t② r❡❣✐♦♥ CP ✇✐t❤ t❤❡ ❛❞❞✐t✐♦♥❛❧ ❝♦♥str❛✐♥t ✭✷✷✮✳

❙✐✉❞③✐➠s❦❛✱ ▲✉♦♠❛✱ ❙tr✉♥③ ❖♥ t❤❡ ❣❡♦♠❡tr② ♦❢ ♦♥❡✲♠♦❞❡ ●❛✉ss✐❛♥ ❝❤❛♥♥❡❧s ✶✵✴ ✶✼

■♥❝♦♠♣❛t✐❜✐❧✐t② ❜r❡❛❦✐♥❣ ❝❤❛♥♥❡❧s

❉❡✜♥✐t✐♦♥ ✻

❆ q✉❛♥t✉♠ ❝❤❛♥♥❡❧ Λ : HB → HA ✐s ✐♥❝♦♠♣❛t✐❜✐❧✐t② ❜r❡❛❦✐♥❣ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ρAB = (Λ ⊗ IB)(ρ) ✭✷✹✮ ✐s ♥♦♥✲st❡❡r❛❜❧❡ ❢♦r ❛❧❧ st❛t❡s ρ✳

❚❤❡♦r❡♠ ✼

❆ ♦♥❡✲♠♦❞❡ ●❛✉ss✐❛♥ ❝❤❛♥♥❡❧ Λ : HB → HA ✐s ✐♥❝♦♠♣❛t✐❜✐❧✐t② ❜r❡❛❦✐♥❣ ✐❢ ❛♥❞ ♦♥❧② ✐❢ Σ + i(✵ ⊕ ω) ≥ ✵, ω = ✵ ✶ −✶ ✵

• .

✭✷✺✮

❚✳ ❍❡✐♥♦s❛❛r✐✱ ❏✳ ❑✐✉❦❛s✱ ❛♥❞ ❏✳ ❙❝❤✉❧t③✱ ❏✳ ▼❛t❤✳ P❤②s✳ ✺✻✱ ✵✽✷✷✵✷ ✭✷✵✶✺✮✳

❙✐✉❞③✐➠s❦❛✱ ▲✉♦♠❛✱ ❙tr✉♥③ ❖♥ t❤❡ ❣❡♦♠❡tr② ♦❢ ♦♥❡✲♠♦❞❡ ●❛✉ss✐❛♥ ❝❤❛♥♥❡❧s ✶✶✴ ✶✼

■♥❝♦♠♣❛t✐❜✐❧✐t② ❜r❡❛❦✐♥❣ ❝❤❛♥♥❡❧s

■♥ t❤❡ ♣✉r✐t②✲s❡r❛❧✐❛♥ ❝♦♦r❞✐♥❛t❡s✱ ❝♦♥❞✐t✐♦♥ Σ + i(✵ ⊕ ω) ≥ ✵ ✐s ❡q✉✐✈❛❧❡♥t t♦ µ ≤ µA. ✭✷✻✮ ❚❤❡ ✈♦❧✉♠❡ ♦❢ ❛❧❧ ✐♥❝♦♠♣❛t✐❜✐❧✐t② ❜r❡❛❦✐♥❣ ♦♥❡✲♠♦❞❡ ●❛✉ss✐❛♥ ❝❤❛♥♥❡❧s ✐s ❡q✉❛❧ t♦ VICBC = C

• N S

µ✶✶/✷ ✻✹ √ ✷µ✸

Aµ✷ σ

d µA d µ d ∆ = C √µσ

• −✶✸µσ + ✾µ✸

σ − ✽ √ ✷(−✶✶+✼µσ) (✶+µσ)✼/✷

• ✶✽✵✶✽

√ ✷ , ✭✷✼✮ ✇❤❡r❡ NS ✐s t❤❡ ♣❤②s✐❝❛❧✐t② r❡❣✐♦♥ CP ✇✐t❤ t❤❡ ❛❞❞✐t✐♦♥❛❧ ❝♦♥str❛✐♥t ✭✷✻✮✳

❙✐✉❞③✐➠s❦❛✱ ▲✉♦♠❛✱ ❙tr✉♥③ ❖♥ t❤❡ ❣❡♦♠❡tr② ♦❢ ♦♥❡✲♠♦❞❡ ●❛✉ss✐❛♥ ❝❤❛♥♥❡❧s ✶✷✴ ✶✼

❱♦❧✉♠❡ r❛t✐♦

❙✐✉❞③✐➠s❦❛✱ ▲✉♦♠❛✱ ❙tr✉♥③ ❖♥ t❤❡ ❣❡♦♠❡tr② ♦❢ ♦♥❡✲♠♦❞❡ ●❛✉ss✐❛♥ ❝❤❛♥♥❡❧s ✶✸✴ ✶✼

P❛rt✐❛❧ ❦♥♦✇❧❡❞❣❡ ❛❜♦✉t t❤❡ s②st❡♠

❆ss✉♠❡ t❤❛t ♦✉r ❦♥♦✇❧❡❞❣❡ ❛❜♦✉t ❛ t✇♦✲♠♦❞❡ ❈❏ ●❛✉ss✐❛♥ st❛t❡ ✐s ❧✐♠✐t❡❞ t♦ t❤❡ ✈❛❧✉❡s ♦❢ t♦t❛❧ µ ❛♥❞ ♠❛r❣✐♥❛❧ µA/σ ♣✉r✐t✐❡s✳ µAµσ ≤ µ ≤ µAµσ µA + µσ − µAµσ (s❡♣❛r❛❜❧❡ st❛t❡s) ✭✷✽✮ µAµσ µA + µσ − µAµσ ≤ µ ≤ µAµσ

• µ✷

A + µ✷ σ − µ✷ Aµ✷ σ

(❝♦❡①✐st❡♥❝❡ r❡❣✐♦♥) ✭✷✾✮ µAµσ

• µ✷

A + µ✷ σ − µ✷ Aµ✷ σ

≤ µ ≤ µAµσ µAµσ + |µA − µσ| (❡♥t❛♥❣❧❡❞ st❛t❡s) ✭✸✵✮ ■♥ t❤❡ ❝♦❡①✐st❡♥❝❡ r❡❣✐♦♥✱ ✐t ✐s ✐♠♣♦ss✐❜❧❡ t♦ ❞✐st✐♥❣✉✐s❤ ❜❡t✇❡❡♥ t❤❡ s❡♣❛r❛❜❧❡ ❛♥❞ ❡♥t❛♥❣❧❡❞ st❛t❡s ✇✐t❤♦✉t t❤❡ ❢✉❧❧ ❦♥♦✇❧❡❞❣❡ ❛❜♦✉t t❤❡ s②st❡♠✳

• ✳ ❆❞❡ss♦✱ ❆✳ ❙❡r❛✜♥✐✱ ❛♥❞ ❋✳ ■❧❧✉♠✐♥❛t✐✱ P❤②s✳ ❘❡✈✳ ▲❡tt✳ ✾✷✱ ✵✽✼✾✵✶ ✭✷✵✵✹✮✳

❙✐✉❞③✐➠s❦❛✱ ▲✉♦♠❛✱ ❙tr✉♥③ ❖♥ t❤❡ ❣❡♦♠❡tr② ♦❢ ♦♥❡✲♠♦❞❡ ●❛✉ss✐❛♥ ❝❤❛♥♥❡❧s ✶✹✴ ✶✼

P❛rt✐❛❧ ❦♥♦✇❧❡❞❣❡ ❛❜♦✉t t❤❡ s②st❡♠

❙✐✉❞③✐➠s❦❛✱ ▲✉♦♠❛✱ ❙tr✉♥③ ❖♥ t❤❡ ❣❡♦♠❡tr② ♦❢ ♦♥❡✲♠♦❞❡ ●❛✉ss✐❛♥ ❝❤❛♥♥❡❧s ✶✺✴ ✶✼

❙✉♠♠❛r②

✶ ❚❤❡ ❣❡♦♠❡tr✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ●❛✉ss✐❛♥ ❝❤❛♥♥❡❧s Λ ❝❛♥ ❜❡ ❞❡t❡r♠✐♥❡❞ ❜②

❛♥❛❧②③✐♥❣ t❤❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❈❤♦✐✲❏❛♠✐♦➟❦♦✇s❦✐ st❛t❡s ρAB✳

✷ ❚❤❡ ✈♦❧✉♠❡ ❡❧❡♠❡♥t ♦❢ t❤❡ ♦♥❡✲♠♦❞❡ ●❛✉ss✐❛♥ ❝❤❛♥♥❡❧s ❞❡♣❡♥❞s ♦♥ t❤❡ t♦t❛❧

❛♥❞ ♠❛r❣✐♥❛❧ ♣✉r✐t✐❡s ♦❢ ρAB✱ ❛♥❞ ✐t ✐s ✢❛t ✐♥ t❤❡ s❡r❛❧✐❛♥✳

✸ ❚❤❡ ✈♦❧✉♠❡ r❛t✐♦s ♦❢ t❤❡ ♦♥❡✲♠♦❞❡ ❡♥t❛♥❣❧❡♠❡♥t ❛♥❞ ✐♥❝♦♠♣❛t✐❜✐❧✐t②

❜r❡❛❦✐♥❣ ●❛✉ss✐❛♥ ❝❤❛♥♥❡❧s ❛r❡ ♠♦♥♦t♦♥✐❝❛❧❧② ✐♥❝r❡❛s✐♥❣ ❢✉♥❝t✐♦♥s ♦❢ µσ✳

✹ ■♥ ♠❛♥② ❝❛s❡s✱ ✐t ✐s ♣♦ss✐❜❧❡ t♦ s❛② ✇❤❡t❤❡r ❛ ❣✐✈❡♥ ♦♥❡✲♠♦❞❡ ●❛✉ss✐❛♥

❝❤❛♥♥❡❧ ✐s ❡♥t❛♥❣❧❡♠❡♥t ♦r ✐♥❝♦♠♣❛t✐❜✐❧✐t② ❜r❡❛❦✐♥❣ ❢r♦♠ t❤❡ ♣✉r✐t✐❡s µ✱ µA/σ ❛❧♦♥❡✳

❙✐✉❞③✐➠s❦❛✱ ▲✉♦♠❛✱ ❙tr✉♥③ ❖♥ t❤❡ ❣❡♦♠❡tr② ♦❢ ♦♥❡✲♠♦❞❡ ●❛✉ss✐❛♥ ❝❤❛♥♥❡❧s ✶✻✴ ✶✼

❊♥❞ ♥♦t❡s

▼♦r❡ ♦♥ t❤❡ t♦♣✐❝ s♦♦♥ ♦♥ t❤❡ ❆r❳✐✈✿

❑✳ ❙✐✉❞③✐➠s❦❛✱ ❑✳ ▲✉♦♠❛✱ ❛♥❞ ❲✳ ❚✳ ❙tr✉♥③✱ ❖♥ t❤❡ ❣❡♦♠❡tr② ♦❢ ♦♥❡✲♠♦❞❡ ●❛✉ss✐❛♥ ❝❤❛♥♥❡❧s✳

❚❤✐s ✇♦r❦ ✇❛s s✉♣♣♦rt❡❞ ❜② t❤❡ P♦❧✐s❤ ◆❛t✐♦♥❛❧ ❙❝✐❡♥❝❡ ❈❡♥tr❡ ♣r♦❥❡❝t ◆♦✳ ✷✵✶✽✴✷✽✴❚✴❙❚✷✴✵✵✵✵✽✳

❙✐✉❞③✐➠s❦❛✱ ▲✉♦♠❛✱ ❙tr✉♥③ ❖♥ t❤❡ ❣❡♦♠❡tr② ♦❢ ♦♥❡✲♠♦❞❡ ●❛✉ss✐❛♥ ❝❤❛♥♥❡❧s ✶✼✴ ✶✼