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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 ✶ , . . . , q n , p n ) T s❛t✐s❢②✐♥❣ t❤❡ ❝♦♠♠✉t❛t✐♦♥ r❡❧❛t✐♦♥s � ✵ n � ✶ [ R i , R † � j ] = ✷ i Ω ij , Ω = , ✭✶✮ − ✶ ✵ k = ✶ ✇✐t❤ t❤❡ s②♠♣❧❡❝t✐❝ ❢♦r♠ Ω ✳ ❉❡♥s✐t② ♦♣❡r❛t♦rs ❛r❡ ❣✐✈❡♥ ❜② d ✷ n ξ � π n χ ( ξ ) D ( − ξ ) ρ = ✭✷✮ R ✷ n ✇✐t❤ t❤❡ ❞✐s♣❧❛❝❡♠❡♥t ✭❲❡②❧✮ ♦♣❡r❛t♦rs D ( ξ ) = e iR T Ω ξ . ✭✸✮ ❙✐✉❞③✐➠s❦❛✱ ▲✉♦♠❛✱ ❙tr✉♥③ ❖♥ t❤❡ ❣❡♦♠❡tr② ♦❢ ♦♥❡✲♠♦❞❡ ●❛✉ss✐❛♥ ❝❤❛♥♥❡❧s ✷✴ ✶✼

●❛✉ss✐❛♥ st❛t❡s ❉❡✜♥✐t✐♦♥ ✶ ❆ ❞❡♥s✐t② ♦♣❡r❛t♦r d ✷ n ξ � π n χ ( ξ ) D ( − ξ ) ρ = ✭✹✮ R ✷ n ✐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 � − ✶ � ✷ ξ T ΩΣΩ T ξ + i ℓ T Ω ξ χ ( ξ ) = exp , ✭✺✮ ✇❤❡r❡ ℓ k = Tr[ ρ R k ] ✐s t❤❡ ❞✐s♣❧❛❝❡♠❡♥t ✈❡❝t♦r❀ Σ ij = ✶ ✷ Tr[ ρ ( R i R j + R j R i )] − ℓ i ℓ j ✐s t❤❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐①✳ Σ ✐s t❤❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐① ♦❢ ❛ ●❛✉ss✐❛♥ st❛t❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ Σ + i Ω ≥ ✵ . ✭✻✮ ❙✐✉❞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 ( M ξ ) exp ✷ ξ T N ξ + ic T ξ . ✭✼✮ ❊❛❝❤ ❝❤❛♥♥❡❧ ✐s ❝♦♠♣❧❡t❡❧② ❝❤❛r❛❝t❡r✐③❡❞ ❜② ❛ tr✐♣❧❡ ( M , N , c ) ✱ ❛♥❞ ✐t ❛❝ts ♦♥ t❤❡ ●❛✉ss✐❛♥ st❛t❡ ρ (Σ , ℓ ) ❛s ❢♦❧❧♦✇s✱ Σ �→ M T Σ M + N , ℓ �→ M T ℓ + c . ✭✽✮ ❚❤❡ ❝♦♠♣❧❡t❡ ♣♦s✐t✐✈✐t② ❝♦♥❞✐t✐♦♥✿ N − iM T Ω M + i Ω ≥ ✵ . ✭✾✮ ❙✐✉❞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 ρ AB ✇✐t❤ ❛ ❝♦♠♠♦♥ ♠❛r❣✐♥❛❧ σ = Tr A ρ AB ❛♥❞ t❤❡ ●❛✉ss✐❛♥ ❝❤❛♥♥❡❧s Λ : H B → H A ✱ s✉❝❤ t❤❛t ρ AB = (Λ ⊗ I B )( ρ Ω ) , ✭✶✵✮ ✇❤❡r❡ t❤❡ ●❛✉ss✐❛♥ st❛t❡ ρ Ω ✐s ❝❤❛r❛❝t❡r✐③❡❞ ❜② � S T � Σ σ σ Z σ S σ Σ Ω = ℓ Ω = ℓ σ ⊕ ℓ σ . ✭✶✶✮ , S T Σ σ σ Z σ S σ ❏✳ ❑✐✉❦❛s✱ ❈✳ ❇✉❞r♦♥✐✱ ❘✳ ❯♦❧❛✱ ❛♥❞ ❏✳✲P✳ P❡❧❧♦♥♣ää✱ P❤②s✳ ❘❡✈✳ ❆ ✾✻ ✱ ✵✹✷✸✸✶ ✭✷✵✶✼✮✳ S σ ✐s t❤❡ s②♠♣❧❡❝t✐❝ ♠❛tr✐① ❞✐❛❣♦♥❛❧✐③✐♥❣ Σ σ ❀ � Z σ = � n ν ✷ σ, k − ✶❀ k = ✶ σ ✸ ν σ, k ❛r❡ t❤❡ s②♠♣❧❡❝t✐❝ ❡✐❣❡♥✈❛❧✉❡s ♦❢ Σ σ ✳ ❙✐✉❞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 ✷ = � ✷ Tr(Σ − ✶ d Σ) ✷ + [Tr(Σ − ✶ d Σ)] ✷ � √ . ✭✶✷✮ ✶✻ det Σ ❚❤❡ ✈♦❧✉♠❡ ❡❧❡♠❡♥t ❝♦rr❡s♣♦♥❞✐♥❣ t♦ d s ✷ = d Σ T G d Σ r❡❛❞s ✹ n ✷ √ � d V = det G d Σ k , ✭✶✸✮ k = ✶ ✇❤❡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❤ � �   Γ T = N + M T Σ σ M , Σ A Σ A  Σ =  ,   = S T ✭✶✹✮ Γ Σ σ Γ σ Z σ S σ M ,   = c + M T ℓ σ . = ℓ A ⊕ ℓ σ , ℓ ℓ A   ❚❤❡ ♣✉r✐t②✲s❡r❛❧✐❛♥ ❝♦♦r❞✐♥❛t❡s✿ ✶ ✶ √ µ = , µ A /σ = , ∆ = det Σ A + det Σ σ + ✷ det Γ . ✭✶✺✮ � det Σ A /σ det Σ ❚❤❡ ✈♦❧✉♠❡ ❡❧❡♠❡♥t✿ µ ✶✶ / ✷ √ d V = d µ A d µ d ∆ d θ d m ( S A ) , ✭✶✻✮ ✷ µ ✸ A µ ✷ ✻✹ σ ✇❤❡r❡ d m ( S A ) ✐s t❤❡ ♠❡❛s✉r❡ ♦❢ t❤❡ ♥♦♥✲❝♦♠♣❛❝t s②♠♣❧❡❝t✐❝ ❣r♦✉♣ Sp ( ✷ ) ✳ ●✳ ❆❞❡ss♦✱ ❆✳ ❙❡r❛✜♥✐✱ ❛♥❞ ❋✳ ■❧❧✉♠✐♥❛t✐✱ P❤②s✳ ❘❡✈✳ ▲❡tt✳ ✾✷ ✱ ✵✽✼✾✵✶ ✭✷✵✵✹✮✳ ❙✐✉❞③✐➠s❦❛✱ ▲✉♦♠❛✱ ❙tr✉♥③ ❖♥ t❤❡ ❣❡♦♠❡tr② ♦❢ ♦♥❡✲♠♦❞❡ ●❛✉ss✐❛♥ ❝❤❛♥♥❡❧s ✼✴ ✶✼