Nội dung chính của bài viết trình bày tính chất đan rối và viễn tải lượng tử với trạng thái thêm và bớt một photon lên hai mode kết hợp SU(2). Để hiểu rõ hơn, mời các bạn tham khảo chi tiết nội dung bài viết này.
❑❍❷❖ ❙⑩❚ ❚➑◆❍ ❈❍❻❚ ✣❆◆ ❘➮■ ❱⑨ ❱■➍◆ ❚❷■ ▲×Đ◆● ❚Û ❱❰■ ❚❘❸◆● ❚❍⑩■ ❚❍➊▼ ❱⑨ ❇❰❚ ▼❐❚ P❍❖❚❖◆ ▲➊◆ ❍❆■ ▼❖❉❊ ❑➌❚ ❍ÑP ❙❯✭✷✮ 2,∗ ✱ P❍❆◆ ◆●➴❈ ❉❯❨ ❚➚◆❍2 ❚❘❺◆ ❚❍➚ ❚❍❆◆❍ ❇➐◆❍ ✱ ❚❘×❒◆● ▼■◆❍ ✣Ù❈ ❍å❝ ✈✐➯♥ ❈❛♦ ❤å❝✱ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❙÷ P❤↕♠✱ t ỵ r t ✈➔ ❱▲❚❚✱ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❙÷ P❤↕♠✱ ✣↕✐ ❤å❝ ❍✉➳ ∗ ❊♠❛✐❧✿ tr✉♦♥❣♠✐♥❤❞✉❝❅❞❤s♣❤✉❡✳❡❞✉✳✈♥ ❚r♦♥❣ ❜➔✐ ❜→♦ ♥➔②✱ ❝❤ó♥❣ tỉ✐ ❦❤↔♦ s→t t➼♥❤ ❝❤➜t ✤❛♥ rè✐ ✈➔ ✈✐➵♥ t↔✐ ❧÷đ♥❣ tû ✈ỵ✐ tr↕♥❣ t❤→✐ t❤➯♠ ✈➔ ❜ỵt ♠ët ♣❤♦t♦♥ ❧➯♥ ❤❛✐ ♠♦❞❡ ❦➳t ❤đ♣ ❙❯✭✷✮✳ ❑➳t q✉↔ t❤✉ ✤÷đ❝ ❧➔ tr↕♥❣ t❤→✐ t❤➯♠ ✈➔ ❜ỵt ♠ët ♣❤♦t♦♥ ❧➯♥ ❤❛✐ ♠♦❞❡ ❦➳t ❤ñ♣ ❙❯✭✷✮ t❤➸ ❤✐➺♥ t➼♥❤ ✤❛♥ rè✐ t❤❡♦ t✐➯✉ ❝❤✉➞♥ ✤❛♥ rè✐ ❍✐❧❧❡r②✲❩✉❜❛✐r②✳ ❍ì♥ ♥ú❛✱ tr↕♥❣ t❤→✐ ♥➔② ❝á♥ t❤➸ ❤✐➺♥ t➼♥❤ ❝❤➜t ✤❛♥ rè✐ ❝❛♦ t❤❡♦ t✐➯✉ ❝❤✉➞♥ ✤à♥❤ ❧÷đ♥❣ ✤ë rè✐ ❊♥tr♦♣② t✉②➳♥ t➼♥❤✳ ❙❛✉ ✤â sû ❞ư♥❣ tr↕♥❣ t❤→✐ t❤➯♠ ✈➔ ❜ỵt ♠ët ♣❤♦t♦♥ ❧➯♥ ❤❛✐ ♠♦❞❡ ❦➳t ❤ñ♣ ❙❯✭✷✮ ♥➔② ✤➸ t❤ü❝ ❤✐➺♥ ♥❣❤✐➯♥ ❝ù✉ q✉→ tr➻♥❤ ✈✐➵♥ t↔✐✱ ❝❤ó♥❣ tỉ✐ ♥❤➟♥ t❤➜② r➡♥❣ q✉→ tr➻♥❤ ✈✐➵♥ t↔✐ t❤➔♥❤ ❝ỉ♥❣ ✈ỵ✐ ✤ë tr✉♥❣ ❜➻♥❤ Fav ♥➡♠ tr♦♥❣ ❦❤♦↔♥❣ 0, ≤ Fav ≤ 1✳ ❚ø ❦❤â❛✿ ❚r↕♥❣ t❤→✐ ❤❛✐ ♠♦❞❡ ❦➳t ❤ñ♣✱ t✐➯✉ ❝❤✉➞♥ ✤❛♥ rè✐ ❍✐❧❧❡r②✲❩✉❜❛✐r②✱ t✐➯✉ ❝❤✉➞♥ ✤❛♥ rè✐ ❊♥tr♦♣② t✉②➳♥ t➼♥❤✱ ✤ë tr✉♥❣ t❤ü❝ tr✉♥❣ ❜➻♥❤✳ ❚â♠ t➢t✿ ✶ ▼Ð ✣❺❯ ◆❣➔② ♥❛② ❦❤♦❛ ❤å❝✱ ❦➽ t❤✉➟t ✈➔ ❝æ♥❣ ♥❣❤➺ ❦❤æ♥❣ ♥❣ø♥❣ ♣❤→t tr✐➸♥✱ ❝♦♥ ♥❣÷í✐ t➻♠ ❝→❝❤ sè ❤â❛ t➜t ❝↔ ❝→❝ ❞ú ❧✐➺✉ t❤ỉ♥❣ t✐♥✱ ❦➳t ♥è✐ t➜t ❝↔ ❝❤ó♥❣ t❛ ❧↕✐ ✈ỵ✐ ♥❤❛✉✳ ❍å ❦❤ỉ♥❣ ♥❣ø♥❣ ❝↔✐ t✐➳♥ ❝→❝❤ t❤ù❝ ❧✐➯♥ ❧↕❝ tr♦♥❣ ❝✉ë❝ sè♥❣✳ ❚r♦♥❣ t❤ỉ♥❣ t✐♥ ❧÷đ♥❣ tû✱ ❧➔♠ t❤➳ ♥➔♦ ✤➸ tr✉②➲♥ t➼♥ ❤✐➺✉ ✤✐ ①❛ ♠➔ ✈➝♥ ✤↔♠ ❜↔♦ t➼♥❤ ❧å❝ ❧ü❛ ❝❛♦ ✈➔ ❣✐↔♠ ✤÷đ❝ t❤➠♥❣ ❣✐→♥❣ ✤➳♥ ♠ù❝ t❤➜♣ ♥❤➜t ❧➔ ✈➜♥ ✤➲ ❝➜♣ tt t ỵ ỵ tt ụ ữ t❤ü❝ ♥❣❤✐➺♠✳ ❚r♦♥❣ ✤â✱ ❝→❝ tr↕♥❣ t❤→✐ ♣❤✐ ❝ê ✤✐➸♥ ỗ t õ tr ù♥❣ ❝❤♦ ②➯✉ ❝➛✉ ♥➔②✳ ◆➠♠ ✶✾✻✸✱ ●❧❛✉❜❡r ✈➔ ❙✉❞❛rs❤❛♥ ữ r tr t t ủ ỵ ❧➔ |α ✱ ✤➙② ❧➔ tr↕♥❣ t❤→✐ ù♥❣ ✈ỵ✐ t❤➠♥❣ ❣✐→♥❣ ❧÷đ♥❣ tû ♥❤ä ♥❤➜t s✉② r❛ tø ❤➺ t❤ù❝ ❜➜t ✤à♥❤ ❍❡✐s❡♥❜❡r❣✳ ❙❛✉ ✤â ❆❣r❛✇❛❧ ✈➔ ❚❛r❛ ✤➣ ✤➲ t ỵ tữ tr t t ủ t t ❬✸❪ ✈➔ ❝ơ♥❣ ✤➣ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ♥â ❧➔ ♠ët tr↕♥❣ t❤→✐ ♣❤✐ ❝ê ✤✐➸♥ ♥➠♠ ✶✾✾✶✳ ❚r↕♥❣ t❤→✐ ❤❛✐ ♠♦❞❡ ❦➳t ❤ñ♣ ❙❯✭✷✮ ❬✹❪ N |ϕ ab = (1 + |ξ|2 )−N/2 n 1/2 n (CN ) ξ |n, N − n , ✭✶✮ n=0 tr♦♥❣ ✤â ξ = − (θ/2) exp (−iϕ) ; (θ/2) = r ✈ỵ✐ θ r➜t ❜➨✱ ✈➔ ❤➺ sè ❦❤❛✐ tr✐➸♥ CNn = (N −n!n)!n! ✳ ❚ø tr↕♥❣ t❤→✐ ❤❛✐ ♠♦❞❡ ❦➳t ❤ñ♣ ú tổ ữ r ởt tr t ợ ❝→❝❤ t❤➯♠ ✈➔ ❚↕♣ ❝❤➼ ❑❤♦❛ ❤å❝✱ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠✱ ✣↕✐ ❤å❝ ❍✉➳ ■❙❙◆ ✶✽✺✾✲✶✻✶✷✱ ❙è ✸✭✺✾✮✴✷✵✷✶✿ tr✳✹✻✲✺✹ ◆❣➔② ♥❤➟♥ ❜➔✐✿ ✷✼✴✶✶✴✷✵✷✵❀ ❍♦➔♥ t❤➔♥❤ ♣❤↔♥ ❜✐➺♥✿ ✶✺✴✶✷✴✷✵✷✵❀ ◆❣➔② ♥❤➟♥ ✤➠♥❣✿ ✶✻✴✶✷✴✷✵✷✵ ❑❍❷❖ ❙⑩❚ ❚➑◆❍ ❈❍❻❚ ✣❆◆ ❘➮■ ❱⑨ ìẹ ợt ởt t tr t❤→✐ ❤❛✐ ♠♦❞❡ ❦➳t ❤đ♣ ❙❯✭✷✮ ♥❤÷ s❛✉✿ |ψ ab = N (ˆ a+ + ˆb)|ϕ ✭✷✮ ab , tr♦♥❣ ✤â aˆ+, aˆ ✈➔ ˆb+, ˆb ❧➔ t♦→♥ tû s✐♥❤ ✭❤õ②✮ ♣❤♦t♦♥ ❝õ❛ ♠♦❞❡ ❛ ✈➔ ♠♦❞❡ ❜✱ N ❧➔ ❤➺ sè ❝❤✉➞♥ ❤â❛ ❝â ❞↕♥❣ + |ξ|2 N = N N N! |ξ|2n (N + 1) n=0 (N − n)!n! ✭✸✮ ❑❤✐ tr✐➸♥ ❦❤❛✐ tr♦♥❣ ❝→❝ tr↕♥❣ t❤→✐ ❋♦❝❦✱ tr↕♥❣ t❤→✐ t❤➯♠ ✈➔ ❜ỵt ♠ët ♣❤♦t♦♥ ❧➯♥ ❤❛✐ ♠♦❞❡ ❦➳t ❤đ♣ ❙❯✭✷✮ tr♦♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✮ ✤÷đ❝ ✤÷❛ r❛ ♥❤÷ s❛✉✿ + |ξ|2 |ψ ab = N N N (1 + |ξ|2 )−N/2 N! n=0 |ξ|2n (N + 1) (N − n)!n! n=0 √ √ × ξ n ( n + 1|n + 1, N − n ab + N − n|n, N − n − N! n!(N − n)! 1/2 ✭✹✮ ab ) ❚r♦♥❣ ❜➔✐ ❜→♦ ♥➔②✱ ❝❤ó♥❣ tỉ✐ t✐➳♥ ❤➔♥❤ ❦❤↔♦ s→t t➼♥❤ ✤❛♥ rè✐ ❝õ❛ tr↕♥❣ t❤→✐ t❤➯♠ ✈➔ ❜ỵt ♠ët ♣❤♦t♦♥ ❧➯♥ ❤❛✐ ♠♦❞❡ ❦➳t ❤ñ♣ ❙❯✭✷✮ ❜➡♥❣ t✐➯✉ ❝❤✉➞♥ ✤❛♥ rè✐ ❍✐❧❧❡r②✲❩✉❜❛✐r② ✈➔ ✤à♥❤ ❧÷đ♥❣ ✤ë rè✐ t❤❡♦ t✐➯✉ ❝❤✉➞♥ ❊♥tr♦♣② t✉②➳♥ t➼♥❤✳ ❙❛✉ ✤â✱ ❝❤ó♥❣ tỉ✐ t✐➳♥ ❤➔♥❤ ✈✐➵♥ t↔✐ ❧÷đ♥❣ tû ♠ët tr↕♥❣ t❤→✐ ❦➳t ❤đ♣ t❤ỉ♥❣ q✉❛ tr↕♥❣ t❤→✐ ♥➔② ✈➔ ✤→♥❤ ❣✐→ sü t❤➔♥❤ ❝æ♥❣ ❝õ❛ q✉→ tr➻♥❤ ✈✐➵♥ t↔✐ t❤æ♥❣ q✉❛ ✤ë tr✉♥❣ t❤ü❝ tr✉♥❣ ❜➻♥❤✳ ❈→❝ ❦➳t q✉↔ t❤✉ ✤÷đ❝ s➩ ✤÷đ❝ ❝❤ó♥❣ tỉ✐ ❜✐➺♥ ❧✉➟♥ ❝❤✐ t✐➳t ð ♣❤➛♥ ❦➳t ❧✉➟♥✳ ✷ ❑❍❷❖ ❙⑩❚ ❚➑◆❍ ❈❍❻❚ ✣❆◆ ❘➮■ ❱⑨ ✣➚◆❍ ▲×Đ◆● ✣❐ ❘➮■ ❚➼♥❤ ❝❤➜t ✤❛♥ rè✐ ❝õ❛ tr↕♥❣ t❤→✐ t❤➯♠ ✈➔ ❜ỵt ♠ët ♣❤♦t♦♥ ❧➯♥ ❤❛✐ ♠♦❞❡ ❦➳t ❤đ♣ ❙❯✭✷✮ ✤÷đ❝ ❦❤↔♦ s→t t❤❡♦ t✐➯✉ ❝❤✉➞♥ ✤❛♥ rè✐ ❍✐❧❧❡r②✲❩✉❜❛✐r② ❬✺❪✳ ✣✐➲✉ ❦✐➺♥ ✤❛♥ rè✐ tê♥❣ q✉→t ✤÷đ❝ ✤÷❛ r❛ ❜➡♥❣ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ s❛✉ a ˆ+m a ˆm ˆb+nˆbn < a ˆmˆb+n ✭✺✮ ❚❤❡♦ ❍✐❧❧❡r②✲❩✉❜❛✐r② ❬✻❪✱ ♠ët tr↕♥❣ t❤→✐ ❜à ✤❛♥ rè✐ ♥➳✉ ♠ët tr♦♥❣ ❤❛✐ ♠♦❞❡ t❤ä❛ ♠➣♥ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ tr➯♥✳ ✣➸ ✤ì♥ ❣✐↔♥✱ t❛ ✤➦t m = n✱ ✈ỵ✐ n = 2k(k > 0)✳ ❈❤å♥ k = ữ r t số rố R1 ữợ ❞↕♥❣ R1 = a ˆ+2 a ˆ2 ˆb+2ˆb2 − a ˆ2ˆb+2 ✭✻✮ ▼ët tr↕♥❣ t❤→✐ ❧➔ ✤❛♥ rè✐ ♥➳✉ t❤❛♠ sè ✤❛♥ rè✐ R1 < 0✳ ❚❤❛♠ sè R1 ❝➔♥❣ ➙♠ t❤➻ ♠ù❝ ✤ë ✤❛♥ rè✐ ❝➔♥❣ t➠♥❣✱ ♥❣÷đ❝ ❧↕✐ ♥➳✉ R1 ≥ t❤➻ tr↕♥❣ t❤→✐ ✤â ❦❤æ♥❣ ✤❛♥ rè✐✳ ❚❤ü❝ ❤✐➺♥ t➼♥❤ t♦→♥ ❝→❝ sè ❤↕♥❣ tr♦♥❣ R1 ✈➔ ✤➦t ϕ = 0, θ = 2r ợ r t ữủ = tanhr t ❦➳t q✉↔ ✈➔♦ ❜✐➸✉ t❤ù❝ ❚❘❺◆ ❚❍➚ ❚❍❆◆❍ ❇➐◆❍ ✹✽ ✈➔ ❝s✳ ✭✻✮ t❛ t❤✉ ✤÷đ❝ R1 = N N4 (1 + |ξ|2 )2N n=0 N! (N − n)!n! |ξ|2n × (n + 1)3 + n(n − 1)(N − n) × [(n + 1)(N − n)(N − n − 1) + (N − n)(N − n − 1)(N − n − 2)] − N × n=0 ❍➻♥❤ ✶✿ (1 + |ξ|2 )2N ✭✼✮ N! (N − n)!n! 2n |ξ| ξ × [n(n − 1)(n + 1) + n(n − 1)(N − n)] ❙ü ♣❤ö t❤✉ë❝ ❝õ❛ t❤❛♠ sè ✤❛♥ rè✐ ✈➔♦ ❝õ❛ tr↕♥❣ t❤→✐ t❤➯♠ ❜ỵt ♠ët ♣❤♦t♦♥ ❧➯♥ ❤❛✐ ♠♦❞❡ ❦➳t ❤đ♣ ❙❯✭✷✮ ✈ỵ✐ ❍➻♥❤ ✷✿ N4 N = 3, 6, ❙ü ♣❤ö t❤✉ë❝ ❝õ❛ t❤❛♠ sè ✤❛♥ rè✐ R1 ✈➔♦ r ✈ỵ✐ N =5 ❝õ❛ tr↕♥❣ t❤→✐ t❤➯♠ ❜ỵt ♠ët ♣❤♦t♦♥ ❧➯♥ ❤❛✐ ♠♦❞❡ ❦➳t ❤ñ♣ ❙❯✭✷✮ ✈➔ tr↕♥❣ t❤→✐ ❤❛✐ ♠♦❞❡ ❦➳t ủ ỗ t tr t t q s→t ♠ù❝ ✤ë ✤❛♥ rè✐ R t❤❡♦ r ✈➔ N ✳ ❈→❝ ❣✐→ trà N t÷ì♥❣ ù♥❣ ð ❤➻♥❤ ✶ ❧➔ N = ✱ N = ✱ N = ✤➸ ❦❤↔♦ s→t ❝❤♦ tr↕♥❣ t❤→✐ t❤➯♠ ✈➔ ❜ỵt ♠ët ♣❤♦t♦♥ ❧➯♥ ❤❛✐ ♠♦❞❡ ❦➳t ❤đ♣ ❙❯✭✷✮✳ ❍➻♥❤ st ợ N = tữỡ ự ợ tr↕♥❣ t❤→✐ t❤➯♠ ✈➔ ❜ỵt ♠ët ♣❤♦t♦♥ ❧➯♥ ❑❍❷❖ ❙⑩❚ ❚➑◆❍ ❈❍❻❚ ✣❆◆ ❘➮■ ❱⑨ ❱■➍◆ ❚❷■ ▲×Đ◆● ❚Û✳✳✳ ✹✾ ❤❛✐ ♠♦❞❡ ❦➳t ❤đ♣ ❙❯✭✷✮ ✭✤÷í♥❣ ✤ùt ♥➨t✮✱ ✈➔ tr↕♥❣ t❤→✐ ❤❛✐ ♠♦❞❡ ❦➳t ❤đ♣ ❙❯✭✷✮ ✭✤÷í♥❣ ❧✐➲♥ ♥➨t✮✳ ❚ø ỗ t t t rố R < t❤ä❛ ♠➣♥ ✈ỵ✐ ❝→❝ ❣✐→ trà r tr♦♥❣ ❦❤♦↔♥❣ 0.8 ≤ r ≤ 1✳ ❈→❝ ✤÷í♥❣ ❝♦♥❣ ✤✐ ①✉è♥❣ ❝❤♦ t❤➜② ✤ë ✤❛♥ rè✐ t➠♥❣ ♠↕♥❤ ❦❤✐ r t➠♥❣✳ ỗ t t t rố ❝õ❛ tr↕♥❣ t❤→✐ ❤❛✐ ♠♦❞❡ ❦➳t ❤ñ♣ ❙❯✭✷✮ ❧➔ ♠↕♥❤ ❤ì♥ ❝õ❛ tr↕♥❣ t❤→✐ t❤➯♠ ✈➔ ❜ỵt ♠ët ♣❤♦t♦♥ ❧➯♥ t ủ ữ ợ ự ổ ✤→♥❣ ❦➸✳ ❱➟② tr↕♥❣ t❤→✐ t❤➯♠ ✈➔ ❜ỵt ♠ët ♣❤♦t♦♥ ❧➯♥ ❤❛✐ ♠♦❞❡ ❦➳t ❤ñ♣ ❙❯✭✷✮ ❧➔ tr↕♥❣ t❤→✐ ✤❛♥ rè✐ ❦❤æ♥❣ ❤♦➔♥ t♦➔♥✳ ❚✐➯✉ ❝❤✉➞♥ ✤❛♥ rè✐ ❍✐❧❧❡r②✲❩✉❜❛✐r② ❝❤➾ ♥❤÷ ❧➔ ✤✐➲✉ ❦✐➺♥ ✤õ ❦❤✐ ✤→♥❤ ❣✐→ ✤ë rè✐✱ ❦❤✐ ✤â ❝❤ó♥❣ t❛ ❝➛♥ ❦✐➸♠ tr❛ ❧↕✐ ❝→❝ ❦➳t q tr ởt ữỡ ợ ❝→❝❤ tr➯♥✳ ❱➻ ✈➟② ✤➸ ✤→♥❤ ❣✐→ ❝➜♣ ✤ë ✤❛♥ rè✐ ❝õ❛ tr↕♥❣ t❤→✐ t❤➯♠ ✈➔ ❜ỵt ♠ët ♣❤♦t♦♥ ❧➯♥ ❤❛✐ ♠♦❞❡ ❦➳t ❤đ♣ ❙❯✭✷✮✱ t❛ sû ❞ư♥❣ ❡♥tr♦♣② t✉②➳♥ t➼♥❤ M = − T r ρˆ2a , ✭✽✮ tr♦♥❣ ✤â T r ρˆ2a ❧➔ ♣❤➨♣ ❧➜② ✈➳t ♠❛ tr➟♥ ♠➟t ✤ë rót ❣å♥ ❜➻♥❤ ♣❤÷ì♥❣✳ ▼ët tr↕♥❣ t❤→✐ ✤❛♥ rè✐ ❝➔♥❣ ♠↕♥❤ ♥➳✉ M = ✱ tr÷í♥❣ ủ M = tữỡ ự ợ tr t ổ ✤❛♥ rè✐✳ ❳➨t tr♦♥❣ tr÷í♥❣ ❤đ♣ tê♥❣ q✉→t✱ ♠❛ tr➟♥ ♠➟t ✤ë ρˆ ❝õ❛ tr↕♥❣ t❤→✐ t❤➯♠ ✈➔ ❜ỵt ♠ët ♣❤♦t♦♥ ❧➯♥ ❤❛✐ ♠♦❞❡ ❦➳t ❤ñ♣ ❙❯✭✷✮ ❝â ❞↕♥❣ ρˆ = |ψ = ❛❜ ❜❛ ψ| N N2 |ξ|2 )N (1 + n=0 N! (N − n)!n! ✭✾✮ |ξ|2n × {(n + 1) ×b N − n | N − m b + (N − n) ×b N − n − | N − m − b } , tr♦♥❣ ✤â✱ T rb (ˆρ) ❧➔ ♣❤➨♣ ❧➜② ✈➳t ♠❛ tr➟♥ ♠➟t ✤ë ρˆ ❝õ❛ tr↕♥❣ t❤→✐ t❤➯♠ ✈➔ ❜ỵt ♠ët ♣❤♦t♦♥ ❧➯♥ ❤❛✐ ♠♦❞❡ ❦➳t ❤đ♣ ❙❯✭✷✮ ❧➯♥ ♠♦❞❡ ❜✳ ❚ø ✤â✱ t❛ ❝â t❤➸ t➼♥❤ ✤÷đ❝ ρˆ2a = N N4 (1 + |ξ|2 )2N n=0 N! (N − n)!n! |ξ|4n (n + 1)2 ×b N − n | N − n b + 2(n + 1)(N − n)b N − n | N − n ✭✶✵✮ b ×b N − n − | N − n − b + (N − n)2 ×b N − n − | N − n − b ❱➟② t❛ t❤❛② ❣✐→ trà ❝õ❛ ρˆ2a ✈➔♦ ✭✽✮ ✈➔ t❤✉ ✤÷đ❝ ❦➳t q✉↔ ❡♥tr♦♣② t✉②➳♥ t➼♥❤ ❝õ❛ tr↕♥❣ t❤→✐ t❤➯♠ ✈➔ ❜ỵt ♠ët ♣❤♦t♦♥ ❧➯♥ ❤❛✐ ♠♦❞❡ ❦➳t ❤ñ♣ ❙❯✭✷✮ ❧➔ M = − T r(ˆ ρ2a ) N4 =1− (1 + |ξ|2 )2N N n=0 N! (N − n)!n! |ξ|4n ×{(n + 1)2 + 2(n + 1)(N − n) + (N − n)2 } ✭✶✶✮ ❚❘❺◆ ❚❍➚ ❚❍❆◆❍ ❇➐◆❍ ✺✵ ✈➔ ❝s✳ ❍➻♥❤ ✸✿ ❙ü ♣❤ö t❤✉ë❝ ❝õ❛ ❡♥tr♦♣② t✉②➳♥ t➼♥❤ M ✈➔♦ r ợ tr N = ữớ ựt t N = ✭✤÷í♥❣ ❧✐➲♥ ♥➨t✮✱ N =6 ✭✤÷í♥❣ ❝❤➜♠ ❣↕❝❤✮✳ ❍➻♥❤ ✸ t❤➸ ❤✐➺♥ sü ♣❤ö t❤✉ë❝ ❡♥tr♦♣② t✉②➳♥ t➼♥❤ M ✈➔♦ ❜✐➯♥ ✤ë ❦➳t ❤đ♣ r ✈ỵ✐ ❣✐→ trà N = 1, ✈➔ 6✳ ❈→❝ ❣✐→ trà N ữủ t tự tỹ tữỡ ự ợ ữớ ✤ùt ♥➨t✱ ✤÷í♥❣ ❧✐➲♥ ♥➨t ✈➔ ✤÷í♥❣ ❝❤➜♠ ❣↕❝❤✳ ❙❛✉ ❦❤✐ ❦❤↔♦ s→t ❡♥tr♦♣② t✉②➳♥ t➼♥❤ ❝õ❛ tr↕♥❣ t❤→✐ t❤➯♠ ✈➔ ❜ỵt ♠ët ♣❤♦t♦♥ ❧➯♥ ❤❛✐ ♠♦❞❡ ❦➳t ❤đ♣ ❙❯✭✷✮✱ ỗ t ú t t tr t ❧➔ tr↕♥❣ t❤→✐ ✤❛♥ rè✐✳ ❑❤✐ ❜✐➯♥ ✤ë r ✤õ ❧ỵ♥✱ ❝➜♣ ✤ë ✤❛♥ rè✐ t➠♥❣ t❤❡♦ ❣✐→ trà ❝õ❛ r tr♦♥❣ ❦❤♦↔♥❣ ≤ r ≤ ✳ ◆❣÷đ❝ ❧↕✐✱ ❝➜♣ ✤ë ✤❛♥ rè✐ ❣✐↔♠ t❤❡♦ ❣✐→ trà ❝õ❛ r tr♦♥❣ ❦❤♦↔♥❣ ≤ r ≤ ✳ ▼➦t ❦❤→❝✱ ❦❤✐ t➠♥❣ tê♥❣ ❧➯♥ t❤➻ ❣✐→ trà M ❝➔♥❣ t✐➳♥ ✤➳♥ ✶✱ ❝❤ù♥❣ tä tr↕♥❣ t❤→✐ ♥➔② ❝➔♥❣ ✤❛♥ rè✐✳ ◆❤÷ ✈➟② tr↕♥❣ t❤→✐ t❤➯♠ ✈➔ ❜ỵt ♠ët ♣❤♦t♦♥ ❧➯♥ ❤❛✐ ♠♦❞❡ ❦➳t ❤đ♣ ❙❯✭✷✮ ✤↕t ✤➳♥ ♠ù❝ ✤ë ✤❛♥ rè✐ ❝ü❝ ✤↕✐ ❦❤✐ t❛ ❝❤å♥ t❤ỉ♥❣ sè t❤➼❝❤ ❤đ♣ ✈➔ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ✤❛♥ rè✐ ✤➸ t❤ü❝ ❤✐➺♥ ♥❤✐➺♠ ✈ö q✉→ tr➻♥❤ ✈✐➵♥ t↔✐ ❧÷đ♥❣ tû✳ ◆❤÷ ✈➟②✱ ð ♣❤➛♥ ♥➔② ❝❤ó♥❣ tỉ✐ t❤➜② r➡♥❣ tr↕♥❣ t❤→✐ t❤➯♠ ✈➔ ❜ỵt ♠ët ♣❤♦t♦♥ ❧➯♥ ❤❛✐ ♠♦❞❡ ❦➳t ❤ñ♣ ❙❯✭✷✮ ✤❛♥ rè✐ t❤❡♦ t✐➯✉ ❝❤✉➞♥ ✤❛♥ rè✐ ❍✐❧❧❡r②✲❩✉❜❛✐r② ❜➟❝ ❝❛♦✳ ▼➦t ❦❤→❝✱ ❦❤✐ t✐➳♥ ❤➔♥❤ ✤à♥❤ ữủ rố tr t t ợt ởt t ❧➯♥ ❤❛✐ ♠♦❞❡ ❦➳t ❤ñ♣ ❙❯✭✷✮ ❜➡♥❣ t✐➯✉ ❝❤✉➞♥ ❊♥tr♦♣② t✉②➳♥ t➼♥❤ t❤➻ ♥â ❤♦➔♥ t♦➔♥ ♣❤ị ❤đ♣ t➼♥❤ ✤❛♥ rè✐ ♥❤➡♠ ❦❤➥♥❣ ✤à♥❤ t❤➯♠ tr↕♥❣ t❤→✐ t❤➯♠ ✈➔ ❜ỵt ♠ët ♣❤♦t♦♥ ❧➯♥ ❤❛✐ ♠♦❞❡ ❦➳t ❤ñ♣ ❙❯✭✷✮ ❧➔ tr↕♥❣ t rố õ t ỗ rố q tr➻♥❤ ✈✐➵♥ t↔✐ ❧÷đ♥❣ tû ð ♣❤➛♥ s❛✉✳ ✸ ❑❍❷❖ ❙⑩❚ ◗❯⑩ ❚❘➐◆❍ ❱■➍◆ ❚❷■ ▲×Đ◆● ❚Û ✣➸ t❤ü❝ ❤✐➺♥ q✉→ tr➻♥❤ ✈✐➵♥ t↔✐ ❧÷đ♥❣ tû✱ ❝❤ó♥❣ tỉ✐ sû ❞ư♥❣ ỗ rố tr t t ợt ởt t ❧➯♥ ❤❛✐ ♠♦❞❡ ❦➳t ❤ñ♣ ❙❯✭✷✮ |ψ ab = N N N/2 N! (N − n)!n! n=0 + |ξ|2 √ × { n + 1|n + 1, N − n ab + √ ξn N − n|n, N − n − ✭✶✷✮ ab } ❚❤❡♦ ♠æ ❤➻♥❤ ✈✐➵♥ t↔✐ ❝õ❛ ❆❣❛r✇❛❧ ✈➔ ●❛s❜r✐s ❬✼❪✱ ❬✽❪✱ ❣✐↔ sỷ ữủ ữ tợ ữủ ữ tợ tổ t ữủ õ tr tr↕♥❣ t❤→✐ ❦➳t ❤đ♣ ✤÷đ❝ ✈✐➵♥ t↔✐ ❧➔ |γ c✳ ỡ ỷ tổ t trữợ tỹ ✤♦ ❇❡❧❧ ❬✾❪ ❆❧✐❝❡ s➩ t❤ü❝ ❤✐➺♥ ✈✐➺❝ tê ❤ñ♣ tr↕♥❣ t❤→✐ ❑❍❷❖ ❙⑩❚ ❚➑◆❍ ❈❍❻❚ ✣❆◆ ❘➮■ ❱⑨ ❱■➍◆ ❚❷■ ▲×Đ◆● ❚Û✳✳✳ |γ c ✈➔ |ψ ab ✱ t❛ ✤÷đ❝ |B (X, P ) ac ac =√ π B (X, P )| = √ π ✺✶ ∞ ˆ c (β) |k, k D ac , ✭✶✸✮ k=0 ∞ ˆ+ ac k, k| Dc (β) k=0 ❑❤✐ ♣❤➨♣ ✤♦ tê ❤đ♣ ❤♦➔♥ t❤➔♥❤✱ tr↕♥❣ t❤→✐ t➼❝❤ sư♣ ✤ê ❞♦ ❇♦❜ ✈➔ ❆❧✐❝❡ ❝ò♥❣ ❝❤✐❛ s➫ tr↕♥❣ t❤→✐ rè✐ ♥➯♥ ❇♦❜ ❝â tr↕♥❣ t❤→✐ ♥❤÷ s❛✉ |ψ abc |ψ ❛❜❝✱ ❇ =ac B(X, P ) | ψ abc N =√ × π (1 + |ξ|2 )N/2 ∞ N N! (N − n)!n! k=0 n=0 ˆ c+ (β) n + 1, N − n × k, k D ab √ + ˆ + N − nαc k, k Dc (β) n, N − n − ab ξn √ n + 1ac ✭✶✹✮ |γ c ú tỗ t tr t ự ợ ♠♦❞❡ ❜ ❝❤ù❛ ❝→❝ t❤æ♥❣ t✐♥ ✈➲ ♠♦❞❡ ❝✳ ❇♦❜ t❤ü❝ ❤✐➺♥ ❞à❝❤ ❝❤✉②➸♥ D (gβ) ✤➸ ①➙② ❞ü♥❣ ❧↕✐ tr↕♥❣ t❤→✐ ✤÷đ❝ ✈✐➵♥ t↔✐ ❜❛♥ ✤➛✉ |γ c ✱ tr↕♥❣ t❤→✐ ❝✉è✐ ❝ị♥❣ t❤✉ ✤÷đ❝ tr♦♥❣ q✉→ tr➻♥❤ ✈✐➵♥ t↔✐ ✤â ❧➔ |ψ abc, ♦✉t ˆ = D(gβ)|ψ ❛❜❝✱ ❇ N =√ × exp π (1 + |ξ|2 )N/2 N × n=0 N! (N − n)!n! ξn × β ∗ γ − βγ ∗ exp − 21 |γ − β|2 (γ − β)n+1 √ ˆ √ n + 1D(gβ)|N −n n + 1! (γ − β)n √ ˆ + √ N − nD(gβ)|N −n−1 n! b ✭✶✺✮ b , ✈ỵ✐ ❤➺ sè ❝❤✉➞♥ ❤â❛ ✤➣ t➻♠ ✤÷đ❝ ❧➔ + |ξ|2 N = N N N! |ξ|2n (N + 1) n=0 (N − n)!n! ✭✶✻✮ ◗✉→ tr➻♥❤ ✈✐➵♥ t↔✐ ❧÷đ♥❣ tû ❧ó❝ ♥➔② ✤➣ ❤♦➔♥ t❤➔♥❤✳ ✣➸ ✤→♥❤ ❣✐→ ♠ù❝ ✤ë t❤➔♥❤ ❝æ♥❣ ❝õ❛ q✉→ tr➻♥❤ ✈✐➵♥ t↔✐✱ t❛ ❞ü❛ ✈➔♦ ✤ë tr✉♥❣ t❤ü❝ tr✉♥❣ ❜➻♥❤ Fav ♠➔ ❝❤ó♥❣ tỉ✐ ✤÷❛ r❛ ð ♣❤➛♥ t✐➳♣ t❤❡♦✳ ✹ ✣❐ ❚❘❯◆● ❚❍Ü❈ ❚❘❯◆● ❇➐◆❍ ❈Õ❆ ◗❯⑩ ❚❘➐◆❍ ❱■➍◆ ❚❷■ ▲×Đ◆● ❚Û ✣ë tr✉♥❣ t❤ü❝ tr✉♥❣ ❜➻♥❤ tr♦♥❣ q✉→ tr➻♥❤ ✈✐➵♥ t↔✐ ✤÷đ❝ ①→❝ ✤à♥❤ q✉❛ ❜✐➸✉ t❤ù❝ Fav = ❂ |in ψ|ψ | γ|ψ 2 out | d β out | 2 d β ✭✶✼✮ ❚❘❺◆ ❚❍➚ ❚❍❆◆❍ ❇➐◆❍ ✺✷ ✈➔ ❝s✳ ◗✉→ tr➻♥❤ ✈✐➵♥ t↔✐ ✤÷đ❝ ①❡♠ ❧➔ t❤➔♥❤ ❝ỉ♥❣ ♥➳✉ ✤ë tr✉♥❣ t❤ü❝ tr✉♥❣ ❜➻♥❤ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ≤ Fav ≤ 1✳ ✣➸ t✐➳♥ ❤➔♥❤ t➼♥❤ t♦→♥ Fav ✱ ❝❤ó♥❣ tỉ✐ sû ❞ư♥❣ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ t♦→♥ tû ❞à❝❤ ❝❤✉②➸♥ ✈➔ ❦❤❛✐ tr✐➸♥ ❝→❝ tr↕♥❣ t❤→✐ tr♦♥❣ tr↕♥❣ t ữợ Fav = N 4N e|| (1 + |ξ|2 )N n=0 N! (N − n)!n! |ξ|2n × γ 2(N −n) (n + 1) (N − n)! γ 2(N −n) n γ 2(N −n) ξ γ 2(N −n−1) (N − n)2 + + + (N − n + 1)(N − n − 1)!ξ γ (N − n − 2)! (N − n)! ✭✶✽✮ ❍➻♥❤ ✹✿ ❙ü ♣❤ö t❤✉ë❝ ❝õ❛ ✤ë tr✉♥❣ ❜➻♥❤ Fav ✈➔♦ ❜✐➯♥ ✤ë ❦➳t ❤đ♣ r ù♥❣ ✈ỵ✐ ❣✐→ trà γ = 2.35 ✈➔ N = 11 ✭✤÷í♥❣ ✤ùt ♥➨t✮✱ N = 13 ✭✤÷í♥❣ ❧✐➲♥ ♥➨t✮✱ N = 16 ✭✤÷í♥❣ ❝❤➜♠ ❣↕❝❤✮ ❉ü❛ ✈➔♦ ❤➻♥❤ ✹✱ ❦❤✐ t❛ ❝❤å♥ ❣✐→ trà = 2.35 ợ N = 11 tữỡ ự ợ ✤÷í♥❣ ✤ùt ♥➨t t❤➻ ✤ë tr✉♥❣ t❤ü❝ tr✉♥❣ ❜➻♥❤ Fav ❝â ❣✐→ trà ♥➡♠ tr♦♥❣ ❦❤♦↔♥❣ tø 0.5 ≤ Fav ≤ ❦❤✐ 0.4 ≤ r ≤ 1.1✳ ❑❤✐ t➠♥❣ tr N = 13 ự ợ ữớ t t❤➻ ✤ë tr✉♥❣ t❤ü❝ tr✉♥❣ ❜➻♥❤ Fav ❝â ❣✐→ trà ♥➡♠ tr♦♥❣ ❦❤♦↔♥❣ tû 0.5 ≤ Fav ≤ ❦❤✐ 0.6 ≤ r ≤ 1.15✳ ❑❤✐ N = 16 ù♥❣ ợ ữớ t tr tỹ tr Fav ❝â ❣✐→ trà ♥➡♠ tr♦♥❣ ❦❤♦↔♥❣ tø 0.5 ≤ Fav ≤ ❦❤✐ 0.7 ≤ r ≤ 1.2✳ ◆❤➻♥ ỗ t t t q tr t t ❝æ♥❣ ①↔② r❛ tr♦♥❣ ❦❤♦↔♥❣ r ❤➭♣ ❞➛♥✳ ✣✐➲✉ ♥➔② ❝❤ù♥❣ tä N ❝➔♥❣ ❜➨ t❤➻ q✉→ tr➻♥❤ ✈✐➵♥ t↔✐ ❝➔♥❣ t❤➔♥❤ ❝ỉ♥❣✳ ❱ỵ✐ ❝→❝ ❣✐→ trà ❝õ❛ N = 11, 13 ✈➔ N = 16 t❤➻ q✉→ tr➻♥❤ ✈✐➵♥ t ữủ tỷ r t ổ ợ tr t❤ü❝ tr✉♥❣ ❜➻♥❤ 0.5 ≤ Fav ≤ ✈ỵ✐ 0.4 ≤ r ≤ 1.1✳ ❑❤✐ N ❜➨ t❤➻ q✉→ tr➻♥❤ ✈✐➵♥ t↔✐ ①↔② r❛ tèt ♥❤➜t ♥➯♥ t❛ ①➨t ❦❤✐ N = 11 ù♥❣ ✈ỵ✐ ❝→❝ ❣✐→ trà γ ❦❤→❝ ♥❤❛✉ ✤➸ ❦❤↔♦ s→t ✤ë tr✉♥❣ t❤ü❝ tr✉♥❣ ❜➻♥❤ Fav ♥❤÷ ❤➻♥❤ ✺✳ ❑❍❷❖ ❙⑩❚ ❚➑◆❍ ❈❍❻❚ ✣❆◆ ❘➮■ ❱⑨ ❱■➍◆ ❚❷■ ▲×Đ◆● ❚Û✳✳✳ ❍➻♥❤ ✺✿ ❙ü ♣❤ư t❤✉ë❝ ❝õ❛ ✤ë tr✉♥❣ ❜➻♥❤ γ = 2.35 ✭✤÷í♥❣ ✤ùt ♥➨t✮✱ γ = 2.90 Fav ✈➔♦ ❜✐➯♥ ✤ë ❦➳t ❤ñ♣ r ù♥❣ ợ tr ữớ t = 3.47 N = 11 ✈➔ ✭✤÷í♥❣ ❝❤➜♠ ❣↕❝❤✮✳ ❑❤✐ ❝❤å♥ ❣✐→ trà N = 11 ✈ỵ✐ γ = 2.35 ù♥❣ ✈ỵ✐ ✤÷í♥❣ ✤ùt ♥➨t ✤ë tr✉♥❣ t❤ü❝ tr✉♥❣ ❜➻♥❤ ✤↕t ❣✐→ trà ❝ü❝ ✤↕✐ ❣➛♥ ❜➡♥❣ ✶✳ ❑❤✐ γ t➠♥❣ ù♥❣ ợ tr = 2.90 ự ợ ữớ ♥➨t t❤➻ ✤ë tr✉♥❣ t❤ü❝ tr✉♥❣ ❜➻♥❤ ✤↕t ❝ü❝ ✤↕✐ 0.8 = 3.47 ự ợ ữớ t❤➻ ✤ë tr✉♥❣ t❤ü❝ tr✉♥❣ ❜➻♥❤ ✤↕t ❝ü❝ ✤↕✐ 0.50✳ ❚✐➳♣ tö❝ t➠♥❣ γ t❤➻ ✤ë tr✉♥❣ t❤ü❝ tr✉♥❣ ❜➻♥❤ t ữợ 0.5 q tr t ổ t ❝ỉ♥❣ ♥ú❛✳ ❱➟② ♥➯♥ ❦❤✐ ❝❤ó♥❣ t❛ ❝❤å♥ γ ❝➔♥❣ ❜➨ t❤➻ q✉→ tr➻♥❤ ✈✐➵♥ t↔✐ ❝➔♥❣ t❤➔♥❤ ❝ỉ♥❣ ✈ỵ✐ ✤ë tr✉♥❣ t❤ü❝ tr✉♥❣ ❜➻♥❤ 0.5 ≤ Fav ≤ ợ 2.35 3.47 ữ ❝→❝❤ ❝❤å♥ ❝→❝ t❤❛♠ sè ♠ët ❝→❝❤ ♣❤ị ❤đ♣✱ ❝❤ó♥❣ tỉ✐ ✤➣ t❤ü❝ ❤✐➺♥ ✈✐➵♥ t↔✐ ❧÷đ♥❣ tû t❤➔♥❤ ❝ỉ♥❣ ởt tr t t ủ tổ q ỗ rố ❧➔ tr↕♥❣ t❤→✐ t❤➯♠ ✈➔ ❜ỵt ♠ët ♣❤♦t♦♥ ❧➯♥ ❤❛✐ ♠♦❞❡ ❦➳t ❤ñ♣ ❙❯✭✷✮✳ ✺ ❑➌❚ ▲❯❾◆ ❚r♦♥❣ ❜➔✐ ❜→♦ ♥➔②✱ ❝❤ó♥❣ tỉ✐ ✤➣ ❦❤↔♦ s→t t➼♥❤ ❝❤➜t ✤❛♥ rè✐ ✈➔ ✤à♥❤ ❧÷đ♥❣ ✤ë rè✐ ❝õ❛ tr↕♥❣ t❤→✐ t❤➯♠ ✈➔ ❜ỵt ♠ët ♣❤♦t♦♥ ❧➯♥ ❤❛✐ ♠♦❞❡ ❦➳t ❤đ♣ ❙❯✭✷✮ ❜➡♥❣ t✐➯✉ ❝❤✉➞♥ ✤❛♥ rè✐ ❍✐❧❧❡r②✲❩✉❜❛✐r② ❜➟❝ ❝❛♦ ✈➔ t✐➯✉ ❝❤✉➞♥ ✤❛♥ rè✐ ❊♥tr♦♣② t✉②➳♥ t➼♥❤✳ ❑➳t q✉↔ t➼♥❤ t♦→♥ ❝❤♦ t❤➜② tr↕♥❣ t❤→✐ t❤➯♠ ✈➔ ❜ỵt ♠ët ♣❤♦t♦♥ ❧➯♥ ❤❛✐ ♠♦❞❡ ❦➳t ❤ñ♣ ❙❯✭✷✮ ❧➔ tr↕♥❣ t❤→✐ ✤❛♥ rè✐✳ ❚✉② ♥❤✐➯♥✱ t➼♥❤ ❝❤➜t ✤❛♥ rè✐ ❧➔ ❦❤æ♥❣ ❤♦➔♥ t♦➔♥✱ ♥❣❤➽❛ ❧➔ ❝â ✈ị♥❣ ✤❛♥ rè✐ ♠↕♥❤ ♥❤÷♥❣ ❝ơ♥❣ ❝â ✈ị♥❣ ❦❤ỉ♥❣ t❤ü❝ sü ✤❛♥ rè✐✳ ❙❛✉ ✤â✱ ❝❤ó♥❣ tỉ✐ sû tr t ỗ rố tỹ ✈✐➵♥ t↔✐ ❧÷đ♥❣ tû ♠ët tr↕♥❣ t❤→✐ ❦➳t ❤đ♣ ✈➔ ✤→♥❤ ❣✐→ ♠ù❝ ✤ë t❤➔♥❤ ❝æ♥❣ ❝õ❛ q✉→ tr➻♥❤ ✈✐➵♥ t↔✐ t↔✐ t❤æ♥❣ q✉❛ ✤ë tr✉♥❣ t❤ü❝ tr✉♥❣ ❜➻♥❤ Fav ✳ ❈→❝ ❦➳t q✉↔ t❤✉ ✤÷đ❝ ❝❤♦ t❤➜② q✉→ tr➻♥❤ ✈✐➵♥ t↔✐ ❧➔ t❤➔♥❤ ❝ỉ♥❣ tr♦♥❣ ✈ị♥❣ t❤❛♠ sè ✤÷đ❝ ❧ü❛ ❝❤å♥ ♠ët ❝→❝❤ ♣❤ị ❤đ♣ ✈ỵ✐ tr↕♥❣ t❤→✐ ❝â ❜✐➯♥ ✤ë ❜➨✱ ✈➔ ✤ë tr✉♥❣ t❤ü❝ tr✉♥❣ ❜➻♥❤ ❝õ❛ q✉→ tr➻♥❤ ✈✐➵♥ t↔✐ ♥➡♠ tr♦♥❣ ❦❤♦↔♥❣ 0.5 ≤ Fav ≤ ✳ ❚✉② ♥❤✐➯♥✱ ✤ë tr✉♥❣ t❤ü❝ tr✉♥❣ ❜➻♥❤ ❝õ❛ q✉→ tr➻♥❤ ✈✐➵♥ t↔✐ ❧➔ ❝❤÷❛ ê♥ ✤à♥❤ ✈➔ ♣❤ö t❤✉ë❝ ♥❤✐➲✉ ✈➔♦ ❝→❝ t❤❛♠ sè ✤➛✉ ✈➔♦ ❝õ❛ q✉→ tr➻♥❤ ✈✐➵♥ t↔✐ ❧÷đ♥❣ tû✳ ▲❮■ ❈❷▼ ❒◆ P❤❛♥ ◆❣å❝ ❉✉② ❚à♥❤ ❝↔♠ ì♥ t➔✐ trđ ❝õ❛ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠✱ ✣↕✐ ❤å❝ ❍✉➳ tr♦♥❣ ✤➲ t➔✐ ♠➣ sè ❚✳✷✵✲❚◆✳❙❱✲✵✸✳ ❚❘❺◆ ❚❍➚ ❚❍❆◆❍ ❇➐◆❍ ✺✹ ✈➔ ❝s✳ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ❬✶❪ ●❧❛✉❜❡r ❘✳ ❏✳ ✭✶✾✻✸✮✱ ✧❈♦❤❡r❡♥t ❛♥❞ ■♥❝♦❤❡r❡♥t ❙t❛t❡s ♦❢ t❤❡ ❘❛❞✐❛t✐♦♥ ❋✐❡❧❞✧✱ P❤②s✐❝❛❧ ❘❡✈✐❡✇ ▲❡tt❡rs✱ ✶✸✶✱ ✷✼✻✻✳ ❬✷❪ ❙✉❞❛rs❤❛♥ ❊✳ ❈✳ ●✳ ✭✶✾✻✸✮✱✧❊q✉✐✈❛❧❡♥❝❡ ♦❢ ❙❡♠✐❝❧❛ss✐❝❛❧ ❛♥❞ ◗✉❛♥t✉♠ ▼❡❝❤❛♥✐❝❛❧ ❉❡s❝r✐♣✲ t✐♦♥s ♦❢ ❙t❛t✐st✐❝❛❧ ▲✐❣❤t ❇❡❛♠s✧✱ P❤②s✐❝❛❧ ❘❡✈✐❡✇ ▲❡tt❡rs✱ ✶✵✱ ✷✼✼✳ ❬✸❪ ❆❣❛r✇❛❧ ●✳ ❙✳ ❛♥❞ ❚❛r❛ ❑✳ ✭✶✾✾✶✮✱ ✧◆♦♥❝❧❛ss✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ st❛t❡s ❣❡♥❡r❛t❡❞ ❜② t❤❡ ❡①❝✐✲ t❛t✐♦♥s ♦♥ ❛ ❝♦❤❡r❡♥t st❛t❡✧✱ P❤②s✐❝❛❧ ❘❡✈✐❡✇ ❆✱ ✹✸✱ ✹✾✷✳ ❬✹❪ ●❡rr✉ ❈✳ ❈✳✱ ●r♦❜❡ ❘✳ ✭✶✾✾✻✮✱ ✏ ❚✇♦✲♠♦❞❡ ❙❯✭✷✮ ❛♥❞ ❙❯✭✶✱✶✮ ❙❝❤r♦❞✐♥❣❡r ❝❛t st❛t❡s✑ ✱ ❏♦✉r♥❛❧ ♦❢ ♠♦❞❡r♥ ♦♣t✐❝s✱ ✹✹ ✭✶✮✱ ✹✶✳ ❬✺❪ ❍✐❧❧❡r② ▼✳ ❛♥❞ ❩✉❜❛✐r② ▼✳ ❙✳ ✭✷✵✵✻✮✱ ✏❊♥t❛♥❣❧❡♠❡♥t ❝♦♥❞✐t✐♦♥s ❢♦r t✇♦ ✲ ♠♦❞❡ st❛t❡s✑ ✱ P❤②s✐❝❛❧ ❘❡✈✐❡✇ ▲❡tt❡rs ✾✻✭✺✮✱ ✵✺✵✺✵✸✳ ❬✻❪ ❍✐❧❧❡r② ▼✳ ❛♥❞ ❩✉❜❛✐r② ▼✳ ❙✳ ✭✷✵✵✻✮✱ ✏❊♥t❛♥❣❧❡♠❡♥t ❝♦♥❞✐t✐♦♥s ❢♦r t✇♦ ✲ ♠♦❞❡ st❛t❡s✿ ❆♣♣❧✐❝❛t✐♦♥s✑ ✱ P❤②s✐❝❛❧ ❘❡✈✐❡✇ ❆ ✼✹✭✸✮✱ ✵✸✷✸✸✸✳ ❬✼❪ ❆❣❛r✇❛❧ ●✳ ❙✳ ❛♥❞ ❇✐s✇❛s ❆✳ ✭✷✵✵✺✮✱ ✧■♥s❡♣❛r❛❜✐❧✐t② ✐♥❡q✉❛❧✐t✐❡s ❢♦r ❤✐❣❤❡r ♦❞❡r ♠♦♠❡♥ts ❢♦r ❜✐♣❛rt✐t❡ s②st❡♠s✧✱ ◆❡✇ ❏♦✉r♥❛❧ ♦❢ P❤②s✐❝s✱ ✼✭✶✮✱ ✷✶✶✳ ❬✽❪ ❆❣❛r✇❛❧ ●✳ ❙✳✱ ●❛❜r✐s ❆✳ ✭✷✵✵✼✮✱✧◗✉❛♥t✉♠ t❡❧❡♣♦rt❛t✐♦♥ ✇✐t❤ ♣❛✐r✲❝♦❤❡r❡♥t st❛t❡s✧✱ ■♥t❡r✲ ♥❛t✐♦♥❛❧ ❏♦✉r♥❛❧ ♦❢ ◗✉❛♥t✉♠ ■♥❢♦r♠❛t✐♦♥✱ ✺✭✶✲✷✮✱ ✶✼✳ ❬✾❪ ❏❛♥s③❦② ❏✳✱ ❑♦♥✐♦r❝③②❦ ▼✳✱ ●❛❜r✐s ❆✳ ✭✷✵✵✶✮✱ ✧❖♥❡✲❝♦♠♣❧❡①✲♣❧❛♥❡ r❡♣r❡s❡♥t❛t✐♦♥ ❛♣♣r♦❛❝❤ t♦ ❝♦♥t✐♥✉♦✉s ✈❛r✐❛❜❧❡ q✉❛♥t✉♠ t❡❧❡♣♦rt❛t✐♦♥✧✱ P❤②s✐❝❛❧ ❘❡✈✐❡✇ ❆✱ ✻✹✱ ✵✸✹✸✵✷✳ ■◆❱❊❙❚■●❆❚■◆● ❚❍❊ ❊◆❚❆◆●▲❊▼❊◆❚ ❆◆❉ ❚❍❊ ◗❯❆◆❚❯▼ ❚❊▲❊P❖❘❚❆✲ ❚■❖◆ ❲■❚❍ ❚❍❊ ❖◆❊✲P❍❖❚❖◆✲❆❉❉❊❉ ❆◆❉ ❖◆❊✲P❍❖❚❖◆✲❙❯❇❚❘❆❈❚❊❉ ❚❲❖ ▼❖❉❊ ❙❯✭✷✮ ❈❖❍❊❘❊◆❚ ❙❚❆❚❊ ❚✐t❧❡✿ ■♥ t❤✐s ♣❛♣❡r✱ ✇❡ st✉❞② t❤❡ ❡♥t❛♥❣❧❡♠❡♥t ❛♥❞ t❤❡ q✉❛♥t✉♠ t❡❧❡♣♦rt❛t✐♦♥ ✐♥ t❤❡ ♦♥❡✲ ♣❤♦t♦♥✲❛❞❞❡❞ ❛♥❞ ♦♥❡✲♣❤♦t♦♥✲s✉❜tr❛❝t❡❞ t✇♦ ♠♦❞❡ ❙❯✭✷✮ ❝♦❤❡r❡♥t st❛t❡ ❜② ✉s✐♥❣ t❤❡ ❍✐❧❧❡r②✲ ❩✉❜❛✐r② ❛♥❞ t❤❡ ❧✐♥❡❛r ❡♥tr♦♣② ❝r✐t❡r✐❛✳ ❚❤❡ ✐♥✈❡st✐❣❛t✐♥❣ r❡s✉❧ts s❤♦✇ t❤❛t t❤✐s st❛t❡ ✐s ❛♥ ❡♥t❛♥❣❧❡❞ st❛t❡✳ ❚❤❡♥✱ ✉s✐♥❣ t❤❡ ♦♥❡✲♣❤♦t♦♥✲❛❞❞❡❞ ❛♥❞ ♦♥❡✲♣❤♦t♦♥✲s✉❜tr❛❝t❡❞ t✇♦ ♠♦❞❡ ❙❯✭✷✮ ❝♦❤❡r❡♥t st❛t❡ ❛s ❛♥ ❡♥t❛♥❣❧❡❞ r❡s♦✉r❝❡ t♦ t❡❧❡♣♦rt ❛ ❝♦❤❡r❡♥t st❛t❡✱ ✇❡ ❤❛✈❡ ❢♦✉♥❞ t❤❛t t❤❡ t❡❧❡♣♦rt❛t✐♦♥ ♣r♦❝❡ss ✐s s✉❝❝❡ss❢✉❧✱ ❛♥❞ t❤❡ ❛✈❡r❛❣❡ ❢✐❞❡❧✐t② Fav ♦❢ t❤❡ t❡❧❡♣♦rt❛t✐♦♥ ♣r♦❝❡ss ❝❛♥ ❣❡t t❤❡ ✈❛❧✉❡s ✐♥ t❤❡ r❛♥❣❡ 0, ≤ Fav ≤ ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ s✉✐t❛❜❧❡ ♣❛r❛♠❡t❡rs✳ ❑❡②✇♦r❞s✿ ❚✇♦✲♠♦❞❡ ❝♦❤❡r❡♥t st❛t❡✱ ❍✐❧❧❡r②✲❩✉❜❛✐r② ❝r✐t❡r✐♦♥✱ ▲✐♥❡❛r ❡♥tr♦♣② ❝r✐t❡r✐♦♥✱ ❆✈❡r✲ ❛❣❡ ❢✐❞❡❧✐t② ♦❢ t❤❡ t❡❧❡♣♦rt❛t✐♦♥✳ ❆❜str❛❝t✿