❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦
■♥st✐t✉t♦ ❞❡ ❋ís✐❝❛
❊st❛❞♦s ❡♠❛r❛♥❤❛❞♦s q✉â♥t✐❝♦s tr✐✲♣❛rt✐❞♦s ❝♦♠
✉♠ q✉❜✐t
▼❛r❝✐♦ ❋❡r♥❛♥❞♦ ❈♦r♥❡❧✐♦
❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❆♥t♦♥✐♦ ❋❡r♥❛♥❞♦ ❘✐❜❡✐r♦ ❞❡ ❚♦❧❡❞♦ P✐③❛
❚❡s❡ ❞❡ ❞♦✉t♦r❛❞♦ s✉❜♠❡t✐❞❛ ❛♦ ■♥st✐t✉t♦ ❞❡ ❋ís✐❝❛ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ❉♦✉t♦r ❡♠ ❈✐ê♥❝✐❛s
❈♦♠✐ssã♦ ❡①❛♠✐♥❛❞♦r❛✿
Pr♦❢✳ ❉r✳ ❆♥t♦♥✐♦ ❋❡r♥❛♥❞♦ ❘✐❜❡✐r♦ ❞❡ ❚♦❧❡❞♦ P✐③❛ ✭❯❙P✮ Pr♦❢✳ ❉r✳ ❆r♥❛❧❞♦ ●❛♠♠❛❧ ✭❯❙P✮
Pr♦❢✳ ❉r✳ ▼❛r❝♦s ❈❡s❛r ❞❡ ❖❧✐✈❡✐r❛ ✭❯♥✐❝❛♠♣✮ Pr♦❢✳ ❉r✳ P❛✉❧♦ ❆❧❜❡rt♦ ◆✉ss❡♥③✈❡✐❣ ✭❯❙P✮ Pr♦❢✳ ❉r✳ ❘♦❜❡rt♦ ▼❡♥❡③❡s ❙❡rr❛ ✭❯❋❆❇❈✮
❙ã♦ P❛✉❧♦
FICHA CATALOGRÁFICA
Preparada pelo Serviço de Biblioteca e Informação
do Instituto de Física da Universidade de São Paulo
Cornelio, Marcio Fernando
Estados emaranhados quânticos tri-partidos com um
qubit - 2008
Tese (Doutorado) – Universidade de São Paulo.
Instituto de Física - Depto. de Física Matemática
Orientador: Prof. Dr. Antonio Fernando Ribeiro de
Toledo Piza
Área de Concentração: Física
Unitermos: 1. Física; 2. Mecânica Quântica;
3.Informação Quântica.
✈
❘❡s✉♠♦
✈✐✐
❆❜str❛❝t
❙✉♠ár✐♦
❆❣r❛❞❡❝✐♠❡♥t♦s ①✐
■♥tr♦❞✉çã♦ ✶
✶ ❖ ❡♠❛r❛♥❤❛♠❡♥t♦ q✉â♥t✐❝♦ ✸
✶✳✶ ❊st❛❞♦s ❡♠❛r❛♥❤❛❞♦s ♣✉r♦s ❜✐✲♣❛rt✐❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✶✳✶ ❖ q✉❡ sã♦ ❡st❛❞♦s ❡♠❛r❛♥❤❛❞♦s❄ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✶✳✷ ❈♦rr❡❧❛çõ❡s ◗✉â♥t✐❝❛s ❡♠ ❡st❛❞♦s ❡♠❛r❛♥❤❛❞♦s✿ ❡st❛❞♦s r❡❧❛t✐✈♦s ✳ ✳ ✽ ✶✳✶✳✸ ❖ ❡st❛❞♦ ❞♦s s✉❜s✐st❡♠❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✶✳✹ ❆ ❘❡♣r❡s❡♥t❛çã♦ ❞❡ ❙❝❤♠✐❞t ♦✉ ❜✐✲♦rt♦❣♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✶✳✺ ◗✉❛♥t✐✜❝❛çã♦ ❞♦ ❡♠❛r❛♥❤❛♠❡♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✷ ❊st❛❞♦s ❡♠❛r❛♥❤❛❞♦s ♠✉❧t✐✲♣❛rt✐❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✶✳✸ ■♥❢♦r♠❛çã♦ q✉â♥t✐❝❛ ❡ ❡♠❛r❛♥❤❛♠❡♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✶✳✸✳✶ ❈ó❞✐❣♦ ❞❡♥s♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✶✳✸✳✷ ❚❡❧❡tr❛♥s♣♦rt❡ q✉â♥t✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾
✷ ❚r❛♥s❢♦r♠❛çõ❡s ❧♦❝❛✐s ❞♦s ❡st❛❞♦s ❡♠❛r❛♥❤❛❞♦s ✷✶ ✷✳✶ ❚r❛♥s❢♦r♠❛çõ❡s ✉♥✐tár✐❛s ❧♦❝❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✷ ❖♣❡r❛çõ❡s ❧♦❝❛✐s ❡st♦❝ást✐❝❛s ❝♦♠ ❝♦♠✉♥✐❝❛çã♦ ❝❧áss✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✷✳✶ ❖♣❡r❛çõ❡s ◗✉â♥t✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✷✳✷ ❊①❡♠♣❧♦s ❝♦♠ ❞♦✐s q✉❜✐ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✷✳✸ ❈r✐tér✐♦ ❞❡ ❝♦♥✈❡rt✐❜✐❧✐❞❛❞❡ ♣❛r❛ ❙▲❖❈❈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✷✳✷✳✹ ❊①❡♠♣❧♦✿ ❈❧❛ss✐✜❝❛çã♦ ❙▲❖❈❈ ♣❛r❛ ♦ ❝❛s♦ ❜✐✲♣❛rt✐❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✷✳✷✳✺ ❖♣❡r❛çõ❡s ❙▲❖❈❈ ❡ s✐st❡♠❛s ♠✉❧t✐✲♣❛rt✐❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺
✸ ❊♠❛r❛♥❤❛♠❡♥t♦ tr✐✲♣❛rt✐❞♦ ❝♦♠ ✉♠ q✉❜✐t ✸✼
✸✳✶ ❉❡❝♦♠♣♦s✐çõ❡s ❞❡ ❡st❛❞♦s ❡♠❛r❛♥❤❛❞♦s ❝♦♠ ✉♠ q✉❜✐t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✸✳✷ ❊①❡♠♣❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✸✳✷✳✶ ❊①❡♠♣❧♦s✿ ❚rês q✉❜✐ts✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✸✳✷✳✷ ❊①❡♠♣❧♦s✿ ❞♦✐s s✐st❡♠❛s ❞❡ ✸ ♥í✈❡✐s ❡ ✉♠ q✉❜✐t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼
① ❙❯▼➪❘■❖
✸✳✷✳✸ ❊①❡♠♣❧♦s✿ ❞♦✐s s✉❜s✐st❡♠❛s ❞❡ q✉❛tr♦ ♥í✈❡✐s ❡ ✉♠ q✉❜✐t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✸✳✷✳✹ ❈♦♥tr❛✲❡①❡♠♣❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✻ ✸✳✸ ❉❡❝♦♠♣♦s✐çõ❡s s✉❜✲❙❝❤♠✐❞t ❡ ♦♣❡r❛çõ❡s ❞♦ t✐♣♦ ❙▲❖❈❈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✽ ✸✳✹ ❉✐s❝✉ssã♦✿ ❊st❛❞♦s ❡♠❛r❛♥❤❛❞♦s tr✐✲♣❛rt✐❞♦s ♠❛✐s ❣❡r❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✷
❈♦♥❝❧✉sã♦ ✽✺
❆❣r❛❞❡❝✐♠❡♥t♦s
Pr✐♠❡✐r❛♠❡♥t❡✱ ❣♦st❛r✐❛ ❞❡ ❛❣r❛❞❡❝❡r ❛ ♠❡✉s ♣❛✐s✱ ▲✉✐③ ❋❡r♥❛♥❞♦ ❈♦r♥❡❧✐♦ ❡ ❱❛♥✐❝❡ ▼❛r✐❛ ❈♦r♥❡❧✐♦✱ ♣❡❧♦ ❣r❛♥❞❡ ❛♣♦✐♦ ❡ ❝♦♥✜❛♥ç❛ q✉❡ s❡♠♣r❡ ❞❡♣♦s✐t❛r❛♠ ❡♠ ♠✐♠✳ ●♦st❛r✐❛ ❞❡ ❛❣r❛❞❡❝❡r ❛✐♥❞❛ ♠❡✉s ❝♦❧❡❣❛s ❞♦ ❉❡♣✳ ❞❡ ❋ís✐❝❛ ▼❛t❡♠át✐❝❛ ❞❡♥tr❡ ♦s q✉❛✐s ✜③ ❣r❛♥❞❡s ❛♠✐❣♦s✳
●♦st❛r✐❛ ❞❡ ❛❣r❛❞❡❝❡r t❛♠❜é♠ ❛♦ Pr♦❢✳ ❉r✳ ❆♥t♦♥✐♦ ❋❡r♥❛♥❞♦ ❘✐❜❡✐r♦ ❞❡ ❚♦❧❡❞♦ P✐③❛ ♣❡❧❛ ♦r✐❡♥t❛çã♦✱ ♣❡❧♦s ❝♦♥s❡❧❤♦s ❡ s✉❣❡stõ❡s s❡♠♣r❡ ❜❛st❛♥t❡ r❡❧❡✈❛♥t❡s✳ ◆ã♦ ♣♦ss♦ ❞❡✐①❛r ❞❡ ❛❣r❛❞❡❝❡r t❛♠❜é♠ ❛♦ Pr♦❢✳ ❉r✳ ❉❛♥✐❡❧ ❚❛✉s❦ ♣❡❧❛ ❛❥✉❞❛ ♥❛ ❞❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ✸✳✶ ❡ ❛♦ ❉r✳ ❏❛❝♦❜ ❉✉♥♥✐♥❣❤❛♠ ❡ ❛♦ Pr♦❢✳ ❉r✳ ❱❧❛t❦♦ ❱❡❞r❛❧ ♣❡❧❛ s✉♣❡r✈✐sã♦ ❞♦ ♠❡✉ ❡stá❣✐♦ ♥❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ▲❡❡❞s✳
P♦r ✜♠✱ ❣♦st❛r✐❛ ❞❡ ❛❣r❛❞❡❝❡r à ❋❆P❊❙P ✭❋✉♥❞❛çã♦ ❞❡ ❆♠♣❛r♦ ❛ P❡sq✉✐s❛ ❞♦ ❊s✲ t❛❞♦ ❞❡ ❙ã♦ P❛✉❧♦✮ ♣❡❧♦ s✉♣♦rt❡ ✜♥❛♥❝❡✐r♦ ❡ t❛♠❜é♠ ❛♦ ■❋❯❙P ✭■♥st✐t✉t♦ ❞❡ ❋ís✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦✮ ♣❡❧♦ ✉s✉❢r✉t♦ ❞❛s ❡①❝❡❧❡♥t❡s ✐♥st❛❧❛çõ❡s ❡ r❡❝✉rs♦✳
■♥tr♦❞✉çã♦
❆t✉❛❧♠❡♥t❡✱ ♦ ✐♥t❡r❡ss❡ ❡♠ ✐♥❢♦r♠❛çã♦ q✉â♥t✐❝❛ t❡♠ ♠♦t✐✈❛❞♦ ❢♦rt❡♠❡♥t❡ ♦ ❡st✉❞♦ ❞♦ ❡♠❛r❛♥❤❛♠❡♥t♦ q✉â♥t✐❝♦✳ ❊♠❜♦r❛ ♦ ❡♠❛r❛♥❤❛♠❡♥t♦ q✉â♥t✐❝♦ ♠✉❧t✐✲♣❛rt✐❞♦ ❛✐♥❞❛ s❡❥❛ ♣♦✉❝♦ ❝♦♥❤❡❝✐❞♦✱ ♥ã♦ ❤á ❞ú✈✐❞❛s s♦❜r❡ ❛ s✉❛ ❢✉♥❞❛♠❡♥t❛❧ ✐♠♣♦rtâ♥❝✐❛ ♣❛r❛ ❛ ✐♥❢♦r♠❛çã♦ q✉â♥t✐❝❛✳ ■ss♦ ❞❡✈❡✲s❡ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❛ ❛❧❣✉♠❛s ❛♣❧✐❝❛çõ❡s s✐♠♣❧❡s ❞♦s ❡st❛❞♦s ❡♠❛r❛♥❤❛✲ ❞♦s ❜✐✲♣❛rt✐❞♦s ❛ t❛r❡❢❛s ❞❡ ✐♥❢♦r♠❛çã♦ q✉â♥t✐❝❛ ❝♦♠♦ ♦ ❝ó❞✐❣♦ ❞❡♥s♦ ❬✶❪✱ ♦ t❡❧❡tr❛♥s♣♦rt❡ q✉â♥t✐❝♦ ❬✷❪ ❡ ❛ ❞✐str✐❜✉✐çã♦ ❞❡ ❝❤❛✈❡ s❡❝r❡t❛ ❬✸❪✳ ❆❧é♠ ❞✐ss♦✱ ❛ ❝♦❡rê♥❝✐❛ ❞❡ ✉♠ ❡st❛❞♦ ❡♠❛r❛♥❤❛❞♦ ♠✉❧t✐✲♣❛rt✐❞♦ é ❢✉♥❞❛♠❡♥t❛❧ ♣❛r❛ ❛ ✐♠♣❧❡♠❡♥t❛çã♦ ❞❡ ❛❧❣♦r✐t♠♦s q✉â♥t✐❝♦s ❝♦♠♦ ♦s ❛❧❣♦r✐t♠♦s ❞❡ ❙❤♦r ❡ ❞❡ ●r♦✈❡r ❬✹✱ ✺❪✳ ◆❡ss❡ s❡♥t✐❞♦✱ ✉♠ ♠❡❧❤♦r ❡♥t❡♥❞✐♠❡♥t♦ ❞♦ ❡♠❛r❛♥❤❛♠❡♥t♦ q✉â♥t✐❝♦ é ❛❧❣♦ ❜❛st❛♥t❡ ❞❡s❡❥á✈❡❧✳
P❛r❛ ❛t✐♥❣✐r♠♦s t❛❧ ♦❜❥❡t✐✈♦✱ ♣r✐♠❡✐r♦ ❣♦st❛rí❛♠♦s ❞❡ ♦❜s❡r✈❛r q✉❡ ♦ ❡♥t❡♥❞✐♠❡♥t♦ ❛t✉❛❧ ❞♦s ❡st❛❞♦s ❡♠❛r❛♥❤❛❞♦s ♣✉r♦s ❞❡ s✐st❡♠❛s ❝♦♠♣♦st♦s ❜✐✲♣❛rt✐❞♦s é ❜❛st❛♥t❡ ❡♠✲ ❜❛s❛❞♦ ♥❛ ❞❡❝♦♠♣♦s✐çã♦ ❞❡ ❙❝❤♠✐❞t ❬✻❪ ❡ t❛♠❜é♠ ♥❛ ❞❡❝♦♠♣♦s✐çã♦ ❡♠ ❡st❛❞♦s r❡❧❛t✐✈♦s ❬✼✱ ✽✱ ✾❪✳ ❊ss❛s r❡♣r❡s❡♥t❛çõ❡s sã♦ ❝❡♥tr❛✐s ♥♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ❞✐✈❡rs♦s tr❛❜❛❧❤♦s ❢✉♥✲ ❞❛♠❡♥t❛✐s ❡♠ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛ t❛✐s ❝♦♠♦ ❛ t❡♦r✐❛ ❞❡ ♠❡❞✐çã♦ ❞❡ ✈♦♥ ◆❡✉♠❛♥♥ ❬✶✵❪✱ ♦ ❛r❣✉♠❡♥t♦ ❞❡ ✐♥❝♦♠♣❧❡t✉❞❡ ❞❡ ❊✐♥st❡✐♥✱ P♦❞♦❧s❦✐ ❡ ❘♦s❡♥ ❬✶✶❪ ❡ ♦ ♣r♦❜❧❡♠❛ ❞♦ ❣❛t♦ ❞❡ ❙❝❤rö❞✐♥❣❡r ❬✼✱ ✽✱ ✶✷❪✳ ❍♦❥❡ ❡♠ ❞✐❛✱ ❡ss❛s ❞❡❝♦♠♣♦s✐çõ❡s tê♠ s✐❞♦ ❜❛st❛♥t❡ ✉t✐❧✐③❛❞❛s ❡♠ ✈ár✐♦s r❡s✉❧t❛❞♦s ❞❛ t❡♦r✐❛ ❞♦ ❡♠❛r❛♥❤❛♠❡♥t♦ ❜✐✲♣❛rt✐❞♦ ♣❛r❛ ❡st❛❞♦s ♣✉r♦s ❬✶✸✱ ✶✹✱ ✶✺❪✳ ❯♠❛ ♣r♦♣r✐❡❞❛❞❡ ✐♠♣♦rt❛♥t❡ ❞❡ss❛s ❞❡❝♦♠♣♦s✐çõ❡s é ❛ s✉❛ s✐♠♣❧✐❝✐❞❛❞❡✱ ♣♦✐s ♦ ❡st❛❞♦ ❡♠❛r❛♥❤❛❞♦ é ❡s❝r✐t♦ ❝♦♠ ♦ ♠❡♥♦r ♥ú♠❡r♦ ♣♦ssí✈❡❧ ❞❡ ❡st❛❞♦s ♣r♦❞✉t♦✳
P❛r❛ s✐st❡♠❛s ♠✉❧t✐✲♣❛rt✐❞♦s✱ ♣♦❞❡rí❛♠♦s ✉t✐❧✐③❛r ❡ss❛s ❞❡❝♦♠♣♦s✐çõ❡s r❡❝✉rs✐✈❛♠❡♥t❡ ❢❛③❡♥❞♦ s✉❝❡ss✐✈❛s ❜✐✲♣❛rt✐çõ❡s ❞♦ s✐st❡♠❛ ❝♦♠♣♦st♦ ❬✶✻❪✳ ◆♦ ❝❛s♦ ❞❡ ✉♠ s✐st❡♠❛ ❝♦♠✲ ♣♦st♦ ❞❡ três q✉❜✐ts✱ ♣♦r ❡①❡♠♣❧♦✱ ❝♦♥s❡❣✉✐rí❛♠♦s s❡♠♣r❡ ♦❜t❡r ✉♠❛ ❞❡❝♦♠♣♦s✐çã♦ ❡♠ ♥♦ ♠á①✐♠♦ q✉❛tr♦ ❡st❛❞♦s ♣r♦❞✉t♦✳ ◆♦ ❡♥t❛♥t♦✱ ❉ür ❡t ❛❧ ❬✶✼❪ ❞❡♠♦♥str❛r❛♠ q✉❡ s❡♠♣r❡ ❡①✐st❡♠ ❞❡❝♦♠♣♦s✐çõ❡s ♠❛✐s s✐♠♣❧❡s ❝♦♠ ❞♦✐s ♦✉ três ❡st❛❞♦s ♣r♦❞✉t♦✳
❆❧é♠ ❞✐ss♦✱ ❡♠ ❣❡r❛❧✱ ❛s ❛♣❧✐❝❛çõ❡s ❞♦ ❡♠❛r❛♥❤❛♠❡♥t♦ q✉â♥t✐❝♦ à ✐♥❢♦r♠❛çã♦ q✉â♥t✐❝❛ ❡stã♦ r❡❧❛❝✐♦♥❛❞❛s ❛♦s ❛s♣❡❝t♦s ♥ã♦ ❧♦❝❛✐s ❞♦s ❡st❛❞♦s ❡♠❛r❛♥❤❛❞♦s✳ ◆❡ss❡s ❝❛s♦s✱ ♥♦s q✉❛✐s ❞❡✈❡♠♦s ❝♦♥s✐❞❡r❛r q✉❡ ♦s s✉❜s✐st❡♠❛s ❡stã♦ ❡s♣❛❝✐❛❧♠❡♥t❡ s❡♣❛r❛❞♦s✱ é ✐♠♣♦rt❛♥t❡ s❛❜❡r♠♦s ❡♠ q✉❛✐s ❡st❛❞♦s ✉♠ ❞❡t❡r♠✐♥❛❞♦ ❡st❛❞♦ ❡♠❛r❛♥❤❛❞♦ ♣♦❞❡ s❡r tr❛♥s❢♦r♠❛❞♦ ♣♦r ♦♣❡r❛çõ❡s ❧♦❝❛✐s✳ ◆❡ss❡ s❡♥t✐❞♦✱ ❉ür ❡t ❛❧ ❬✶✼❪ t❛♠❜é♠ ❞❡♠♦♥str❛r❛♠ q✉❡ ❛s ❞❡❝♦♠♣♦s✐çõ❡s
✷ ■♥tr♦❞✉çã♦
♠❛✐s s✐♠♣❧❡s✱ ✐st♦ é✱ ♥♦ ♠❡♥♦r ♥ú♠❡r♦ ❞❡ ❡st❛❞♦s ♣r♦❞✉t♦✱ sã♦ út❡✐s ♣❛r❛ ❡♥t❡♥❞❡r♠♦s ❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦s ❡st❛❞♦s ❡♠❛r❛♥❤❛❞♦s ♠✉❧t✐✲♣❛rt✐❞♦s ❝♦♠ r❡❧❛çã♦ às ♦♣❡r❛çõ❡s ❧♦❝❛✐s ❡st♦❝ást✐❝❛s ❝♦♠ ❝♦♠✉♥✐❝❛çã♦ ❝❧áss✐❝❛ ✭❙▲❖❈❈✮ ❬✶✽❪✳
◆♦ ❡♥t❛♥t♦✱ ♥ã♦ ❡①✐st❡ ♥❡♥❤✉♠ ♠ét♦❞♦ ♣❛r❛ ❡♥❝♦♥tr❛r t❛✐s ❞❡❝♦♠♣♦s✐çõ❡s ❡ ✐ss♦ ❞✐✲ ✜❝✉❧t❛ ❜❛st❛♥t❡ ♦ ❡♥t❡♥❞✐♠❡♥t♦ ❞♦s ❡st❛❞♦s ❡♠❛r❛♥❤❛❞♦s ♠✉❧t✐✲♣❛rt✐❞♦s✳ ◆❡ss❡ s❡♥t✐❞♦✱ ❞❡s❡♥✈♦❧✈❡r❡♠♦s ✉♠ ♠ét♦❞♦ ❝❛♣❛③ ❞❡ ❡♥❝♦♥tr❛r ❡ss❛s ❞❡❝♦♠♣♦s✐çõ❡s ♣❛r❛ ❝❡rt♦s ❡st❛❞♦s ❡♠❛r❛♥❤❛❞♦s tr✐✲♣❛rt✐❞♦s s❡♥❞♦ q✉❡ ✉♠ ❞♦s s✉❜s✐st❡♠❛s é ✉♠ q✉❜✐t✳ ❆s ❞❡❝♦♠♣♦s✐çõ❡s ❡♥❝♦♥tr❛❞❛s s❡rã♦ út❡✐s ♣❛r❛ ♦❜t❡r♠♦s ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ss❡s ❡st❛❞♦s ❜❡♠ ❝♦♠♦ ❡st❛❜❡❧❡❝❡r q✉❛✐s ❞❡❧❡s sã♦ ❡q✉✐✈❛❧❡♥t❡s ♣♦r ♦♣❡r❛çõ❡s ❧♦❝❛✐s ❞♦ t✐♣♦ ❙▲❖❈❈✳
❈❛♣ít✉❧♦ ✶
❖ ❡♠❛r❛♥❤❛♠❡♥t♦ q✉â♥t✐❝♦
◆❡st❡ ❈❛♣ít✉❧♦✱ ❢❛r❡♠♦s ✉♠❛ ❜r❡✈❡ ✐♥tr♦❞✉çã♦ ❛♦ ❡♠❛r❛♥❤❛♠❡♥t♦ q✉â♥t✐❝♦✳ ❆t✉❛❧✲ ♠❡♥t❡✱ ♦ ❛ss✉♥t♦ ❡♠❛r❛♥❤❛♠❡♥t♦ q✉â♥t✐❝♦ ❥á s❡ t♦r♥♦✉ ❜❛st❛♥t❡ ✈❛st♦✳ ❯♠ ❜♦❛ r❡✈✐sã♦ ❞♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❛t✉❛❧ ❢♦✐ r❡❞✐❣✐❞❛ r❡❝❡♥t❡♠❡♥t❡ ♣♦r ❍♦r♦❞❡❝❦✐ ❡t ❛❧ ❬✶✺❪✳ ❆q✉✐✱ ❡♥❢❛t✐✲ ③❛r❡♠♦s ♠❛✐s ❛ ❞❡s❝r✐çã♦ ❞❛s ❞❡❝♦♠♣♦s✐çõ❡s ❞♦s ❡st❛❞♦s ❡♠❛r❛♥❤❛❞♦s ♣✉r♦s ❞❡ s✐st❡♠❛s ❝♦♠♣♦st♦s ❜✐✲♣❛rt✐❞♦s ❞❡ ❞✐♠❡♥sã♦ ✜♥✐t❛✳ ❚❛♠❜é♠ ❞✐s❝✉t✐r❡♠♦s ♦ ❡♠❛r❛♥❤❛♠❡♥t♦ ♠✉❧t✐✲ ♣❛rt✐❞♦ ❡ ❛❧❣✉♠❛s ❛♣❧✐❝❛çõ❡s à ✐♥❢♦r♠❛çã♦ q✉â♥t✐❝❛✳
✶✳✶ ❊st❛❞♦s ❡♠❛r❛♥❤❛❞♦s ♣✉r♦s ❜✐✲♣❛rt✐❞♦s
✶✳✶✳✶ ❖ q✉❡ sã♦ ❡st❛❞♦s ❡♠❛r❛♥❤❛❞♦s❄
✧❲❤❡♥ t✇♦ s②st❡♠s✱ ✳✳✳ ❡♥t❡r ✐♥t♦ t❡♠♣♦r❛r② ♣❤②s✐❝❛❧ ✐♥t❡r❛❝t✐♦♥ ✳✳✳ ❛♥❞ ✳✳✳ s❡♣❛r❛t❡ ❛❣❛✐♥✱ t❤❡♥ t❤❡② ❝❛♥ ♥♦ ❧♦♥❣❡r ❜❡ ❞❡s❝r✐❜❡❞ ✳✳✳ ❜② ❡♥❞♦✇✐♥❣ ❡❛❝❤ ♦❢ t❤❡♠ ✇✐t❤ ❛ r❡♣r❡s❡♥t❛t✐✈❡ ♦❢ ✐ts ♦✇♥✳ ■ ✇♦✉❧❞ ♥♦t ❝❛❧❧ t❤❛t ♦♥❡ ❜✉t r❛t❤❡r t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ tr❛✐t ♦❢ q✉❛♥t✉♠ ♠❡❝❤❛♥✐❝s✱ t❤❡ ♦♥❡ t❤❛t ❡♥❢♦r❝❡s ✐ts ❡♥t✐r❡ ❞❡♣❛rt✉r❡ ❢r♦♠ ❝❧❛ss✐❝❛❧ ❧✐♥❡s ♦❢ t❤♦✉❣❤t✳ ❇② t❤❡ ✐♥t❡r❛❝t✐♦♥ t❤❡ t✇♦ r❡♣r❡s❡♥✲ t❛t✐✈❡s ❤❛✈❡ ❜❡❝♦♠❡ ❡♥t❛♥❣❧❡❞✳✧
❙❝❤rö❞✐♥❣❡r✱ ✶✾✸✺ ❬✼❪
❊♠❛r❛♥❤❛♠❡♥t♦ é ✉♠ ❢❡♥ô♠❡♥♦ q✉â♥t✐❝♦ ❜❛st❛♥t❡ ❝♦♠✉♠ q✉❛♥❞♦ ❝♦♥s✐❞❡r❛♠♦s ❞✐✲ ✈❡rs♦s s✐st❡♠❛s q✉â♥t✐❝♦s ❡♠ ❝♦♥❥✉♥t♦✳ ❊ss❡ ❝♦♥❥✉♥t♦ ❞❡ s✐st❡♠❛s q✉â♥t✐❝♦s✱ ❡♠ ❣❡r❛❧✱ é ❝❤❛♠❛❞♦ ❞❡ s✐st❡♠❛ ❝♦♠♣♦st♦ ❡ ❝❛❞❛ ✉♠ ❞♦s s✐st❡♠❛s q✉❡ ♦ ❝♦♥st✐t✉✐ ❞❡ s✉❜s✐st❡♠❛ ♦✉ ♣❛rt❡ ❞♦ s✐st❡♠❛ ❝♦♠♣♦st♦✳ ❈♦♠♦ é ❜❡♠ ❝♦♥❤❡❝✐❞♦✱ ❝❛❞❛ ❡st❛❞♦ ❞❡ ✉♠ s✐st❡♠❛ ❢ís✐❝♦ é r❡♣r❡s❡♥t❛❞♦ ♣♦r ✉♠ r❛✐♦ ❡♠ ✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt✳ ❆ss✐♠✱ ❝♦♠♦ ♦ s✐st❡♠❛ ❝♦♠♣♦st♦ ❡ ✉♠❛ ♣❛rt❡ ❞❡st❡ sã♦ s✐st❡♠❛s ❢ís✐❝♦s ❞✐st✐♥t♦s✱ ❡❧❡s s❡rã♦ ❛ss♦❝✐❛❞♦s ❛ r❡♣r❡s❡♥t❛♥t❡s ❞✐s✲ t✐♥t♦s ❡♠ ❡s♣❛ç♦s ❞❡ ❍✐❧❜❡rt ❞✐st✐♥t♦s✳ ❖ ❝♦♥❝❡✐t♦ ❞❡ ❡♠❛r❛♥❤❛♠❡♥t♦ ❡stá ✐♥t✐♠❛♠❡♥t❡
✹ ❈❆P❮❚❯▲❖ ✶✳ ❖ ❊▼❆❘❆◆❍❆▼❊◆❚❖ ◗❯➶◆❚■❈❖
r❡❧❛❝✐♦♥❛❞♦ ❝♦♠ ❛ r❡❧❛çã♦ ❡♥tr❡ ❡ss❡s r❡♣r❡s❡♥t❛♥t❡s ❞♦ ❡st❛❞♦ q✉â♥t✐❝♦✳ P❛r❛ ❝❡rt♦s ❡s✲ t❛❞♦s ❞♦ s✐st❡♠❛s ❝♦♠♣♦st♦✱ ♣♦❞❡♠♦s ❞✐③❡r q✉❡ ♦s ✧r❡♣r❡s❡♥t❛♥t❡s ❞♦s s✉❜s✐st❡♠❛s ❡stã♦ ❡♠❛r❛♥❤❛❞♦s✧ ❝♦♠♦ ❞✐ss❡ ❙❝❤rö❞✐♥❣❡r ❬✽❪✳ ❍♦❥❡ ❡♠ ❞✐❛✱ ❡ss❡s ❡st❛❞♦s sã♦ ❝❤❛♠❛❞♦s ❞❡ ❡♠❛r❛♥❤❛❞♦s ♦✉ ❞✐③❡♠♦s q✉❡ ♦ s✐st❡♠❛ ♣♦ss✉✐ ✉♠ ❝❡rt♦ ❡♠❛r❛♥❤❛♠❡♥t♦✳
●❡r❛❧♠❡♥t❡✱ ✉♠ s✐st❡♠❛ ❢ís✐❝♦ ❞✐t♦ ❝♦♠♣♦st♦ ♣♦ss✉✐ ✉♠❛ ❣r❛♥❞❡ q✉❛♥t✐❞❛❞❡ ❞❡ ♦❜s❡r✲ ✈á✈❡✐s ❢ís✐❝♦s ❡ ♣♦❞❡ s❡r ❞✐✈✐❞✐❞♦ ❡♠ ✉♠❛ ❣r❛♥❞❡ q✉❛♥t✐❞❛❞❡ ❞❡ s✉❜s✐st❡♠❛s✳ ❊ss❛ ❞✐✈✐sã♦ ❡♠ ♠✉✐t❛s ♣❛rt❡s ♣♦❞❡ ❞❛r ♦r✐❣❡♠ ❛ ❡st❛❞♦s ❡♠❛r❛♥❤❛❞♦s ❜❛st❛♥t❡ ❝♦♠♣❧❡①♦s✱ ♦s q✉❛✐s✱ ❛✐♥❞❛ ❛❣✉❛r❞❛♠ ♣♦r ✉♠❛ ❜♦❛ t❡♦r✐❛ q✉❡ ♣♦ss❛ ❞❡s❝r❡✈❡r s✉❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ♠❛♥❡✐r❛ s❛t✐s❢❛tór✐❛ ❬✶✺❪✳ ◆❡st❡ ❝❛♣ít✉❧♦ ✐♥tr♦❞✉tór✐♦ ♥♦s r❡str✐♥❣✐r❡♠♦s ❛ ❡st✉❞❛r ♦ ❝❛s♦ ❞♦s ❡s✲ t❛❞♦s ♣✉r♦s ❜✐♣❛rt✐❞♦s✱ ♦s q✉❛✐s✱ sã♦ ❜❡♠ ❝♦♥❤❡❝✐❞♦s✳ ❈♦♠♦ ✉♠ ❡①❡♠♣❧♦✱ ✈❛♠♦s ❡♥tã♦ ❝♦♠❡ç❛r ❝♦♠ ♦ s✐st❡♠❛ ❝♦♠♣♦st♦ ♠❛✐s s✐♠♣❧❡s q✉❡ ❡①✐st❡✿ ❞♦✐s s✐st❡♠❛s ❞❡ ❞♦✐s ♥í✈❡✐s ♦✉ q✉❜✐ts✱ ♦s q✉❛✐s ♣♦❞❡♠ s❡r r❡❛❧✐③❛❞♦s ❛tr❛✈és ❞♦s s♣✐♥s ❞❡ ❞✉❛s ♣❛rtí❝✉❧❛s ❞❡ s♣✐♥1/2♦✉ ❞❛ ♣♦❧❛r✐③❛çã♦ ❞❡ ❞♦✐s ❢ót♦♥s✳
P❛r❛ ❡♥t❡♥❞❡r♠♦s ❝♦♠♦ ❛ r❡♣r❡s❡♥t❛çã♦ ❞♦ s✐st❡♠❛ ❝♦♠♣♦st♦ ❞❡ ❞♦✐s q✉❜✐ts s❡ r❡❧❛❝✐♦♥❛ ❝♦♠ ❛ r❡♣r❡s❡♥t❛çã♦ ❞❡ ❝❛❞❛ ✉♠ ❞♦s q✉❜✐ts✱ é ❝♦♥✈❡♥✐❡♥t❡ ❧❡♠❜r❛r♠♦s ❝♦♠♦ ❝♦♥str✉✐r ❛ r❡♣r❡s❡♥t❛çã♦ ❞♦ s♣✐♥ ❞❡ ✉♠❛ ♣❛rtí❝✉❧❛✳ ◆❡st❡ ❝❛s♦✱ s❛❜❡♠♦s ❡①♣❡r✐♠❡♥t❛❧♠❡♥t❡ q✉❡ ♦ s♣✐♥ ♣♦❞❡ ❛ss✉♠✐r ❞♦✐s ✈❛❧♦r❡s✳ ❆ ❝❛❞❛ ✉♠ ❞❡st❡s ✈❛❧♦r❡s ♣♦❞❡♠♦s ❛ss♦❝✐❛r ✉♠ ❡st❛❞♦ q✉â♥t✐❝♦✱ ♦s q✉❛✐s ❞❡♥♦t❛r❡♠♦s ♣♦r |0i ❡ |1i✳ ❆ ♣❛rt✐r ❞♦ ♣r✐♥❝í♣✐♦ ❞❛ s✉♣❡r♣♦s✐çã♦✱ s❛❜❡♠♦s q✉❡
q✉❛❧q✉❡r ❡st❛❞♦ ❞❛ ❢♦r♠❛
|ψi=α|0i+β|1i,
♦♥❞❡α ❡β sã♦ ❝♦❡✜❝✐❡♥t❡s ❝♦♠♣❧❡①♦s t❛✐s q✉❡|α|2+|β|2 = 1✱ é t❛♠❜é♠ ✉♠ ❡st❛❞♦ ✈á❧✐❞♦✳
❉❡ss❛ ❢♦r♠❛✱ ♦s ❡st❛❞♦s |0i ❡ |1i sã♦ ❛ss♦❝✐❛❞♦s ❛ ✉♠❛ ❜❛s❡ ♦rt♦♥♦r♠❛❧ ♣❛r❛ ✉♠ ❡s♣❛ç♦
❧✐♥❡❛r ❞❡ ❞✐♠❡♥sã♦ ✷ ❡ q✉❛❧q✉❡r ❡st❛❞♦ ❞♦ s♣✐♥ ❞❡ ✉♠❛ ♣❛rtí❝✉❧❛ ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞♦ ♣♦r ✉♠ r❛✐♦ ♥❡st❡ ❡s♣❛ç♦✳
P♦❞❡♠♦s ❛❣♦r❛ ❞✐s❝✉t✐r ❛ r❡♣r❡s❡♥t❛çã♦ ❞❡ ✉♠ s✐st❡♠❛ ❝♦♠♣♦st♦ ❞❡ ❞♦✐s q✉❜✐ts✳ ❱❛♠♦s ❝❤❛♠❛r ✉♠ ❞♦s q✉❜✐ts ❞❡ s✉❜s✐st❡♠❛a✱Sa✱ ❡ ♦ ♦✉tr♦ ❞❡ s✉❜s✐st❡♠❛b✱Sb✳ ❉✐r❡♠♦s t❛♠❜é♠
q✉❡ Sa ❡stá ❝♦♠ ❛ ♦❜s❡r✈❛❞♦r❛ ❆❧✐❝❡ ❡ Sb ❝♦♠ ♦ ♦❜s❡r✈❛❞♦r ❇♦❜✳ ❖s ❞♦✐s ♥í✈❡✐s ❞❡st❡s
q✉❜✐ts ♣♦❞❡♠ s❡r ❞❡✜♥✐❞♦s ❝♦♠♦ ❛✉t♦❡st❛❞♦s ❞❡ ❝❡rt♦s ♦❜s❡r✈á✈❡✐s✱ Za ♣❛r❛ ♦ q✉❜✐t Sa
❡ Zb ♣❛r❛ ♦ q✉❜✐t Sb✳ P♦❞❡♠♦s ❡s❝♦❧❤❡r ♦s ❞♦✐s ❛✉t♦✈❛❧♦r❡s ❞❡st❡s ♦❜s❡r✈á✈❡✐s ❝♦♠♦ ±1✳
❆ss✐♠ ❞❡✜♥✐♠♦s✿
Za|0ai = − |0ai Zb|0bi = − |0bi Za|1ai = +|1ai Zb|1bi = +|1bi.
❈♦♠ ❡ss❛s ❝♦♥✈❡♥çõ❡s✱ Za ♥❛ ❜❛s❡ {|0ai,|1ai} é ❛ ✉s✉❛❧ ♠❛tr✐③ ❞❡ P❛✉❧✐ ✉t✐❧✐③❛❞❛ ♣❛r❛
r❡♣r❡s❡♥t❛r ♦ s♣✐♥ ❞♦ q✉❜✐tSa ♥❛ ❞✐r❡çã♦ z✳ ❖ ♠❡s♠♦ ✈❛❧❡ ♣❛r❛ Zb ♥❛ ❜❛s❡ {|0bi,|1bi}✳
❯♠❛ ✈❡③ q✉❡Za ❡ Zb ❝♦rr❡s♣♦♥❞❡♠ ❛ ♦❜s❡r✈❛çõ❡s q✉❡ ♣♦❞❡♠ s❡r ❢❡✐t❛s ✐♥❞❡♣❡♥❞❡♥t❡✲
✶✳✶✳ ❊❙❚❆❉❖❙ ❊▼❆❘❆◆❍❆❉❖❙ P❯❘❖❙ ❇■✲P❆❘❚■❉❖❙ ✺
✐❞❡♥t✐✜❝❛r q✉❛tr♦ ❡st❛❞♦s s✐♠✉❧t❛♥❡❛♠❡♥t❡ ❞✐st✐♥❣✉í✈❡✐s ♣❛r❛ ♦ s✐st❡♠❛ ❝♦♠♣♦st♦ ❞❡ ❞♦✐s q✉❜✐ts✱ Sab✱ ♦s q✉❛✐s ♣♦❞❡♠ s❡r ❞❡♥♦t❛❞♦s ♣♦r
|0ai ⊗ |0bi, |0ai ⊗ |1bi, |1ai ⊗ |0bi ❡ |1ai ⊗ |1bi. ✭✶✳✶✮
❆❧é♠ ❞✐ss♦✱ ♣❡❧♦ ♣r✐♥❝í♣✐♦ ❞❛ s✉♣❡r♣♦s✐çã♦✱ s❛❜❡♠♦s q✉❡ q✉❛❧q✉❡r ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞❡ss❡s ❡st❛❞♦s ❡♠ ✭✶✳✶✮ é t❛♠❜é♠ ✉♠ ❡st❛❞♦ ✈á❧✐❞♦ ❞♦ s✐st❡♠❛ ❝♦♠♣♦st♦✳ ❆ss✐♠✱ ♠❛t❡♠❛t✐❝❛✲ ♠❡♥t❡✱ ❛ r❡♣r❡s❡♥t❛çã♦ ❞♦ s✐st❡♠❛ ❝♦♠♣♦st♦ ❛♣❛r❡❝❡ ♥❛t✉r❛❧♠❡♥t❡ ❝♦♠♦ s❡♥❞♦ ♦ ❡s♣❛ç♦ ❧✐♥❡❛r ❝♦♠♣❧❡①♦ ❞❡✜♥✐❞♦ ♣❡❧♦ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧ ❞♦s ❡s♣❛ç♦s ❧✐♥❡❛r❡s ❞♦s s✉❜s✐st❡♠❛s✳
❖ ♣ró①✐♠♦ ♣❛ss♦ é ❡♥t❡♥❞❡r♠♦s ❝♦♠♦ ♦s ❡st❛❞♦s ❞♦s q✉❜✐ts ✐♥❞✐✈✐❞✉❛✐s ❡stã♦ r❡❧❛❝✐♦✲ ♥❛❞♦s ❝♦♠ ♦ ❡st❛❞♦ ❞♦ s✐st❡♠❛ ❝♦♠♣♦st♦ ❞❡ ❞♦✐s q✉❜✐ts✱ ♦✉ s❡❥❛✱ q✉❡r❡♠♦s r❡s♣♦♥❞❡r às s❡❣✉✐♥t❡s ♣❡r❣✉♥t❛s✿
✶✳ ❉❛❞♦ ♦ ❡st❛❞♦ ❞♦ s✐st❡♠❛ ❝♦♠♣♦st♦✱ q✉❛✐s sã♦ ♦s ❡st❛❞♦s ❝♦rr❡s♣♦♥❞❡♥t❡s ❞♦s s✉❜✲ s✐st❡♠❛s❄
✷✳ ❉❛❞♦s ♦s ❡st❛❞♦s ❞♦s s✉❜s✐st❡♠❛s✱ q✉❛❧ ♦ ❡st❛❞♦ ❝♦rr❡s♣♦♥❞❡♥t❡ ❞♦ s✐st❡♠❛ ❝♦♠✲ ♣♦st♦❄
◆❛ ❢ís✐❝❛ ❝❧áss✐❝❛✱ s❛❜❡♠♦s q✉❡ ✉♠ s✐st❡♠❛ ❢ís✐❝♦ ❝♦♠♣♦st♦ ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞♦ ♣❡✲ ❧❛s ✈❛r✐á✈❡✐s ❝❛♥ô♥✐❝❛s ❞♦s s✉❜s✐st❡♠❛s✱ ♦✉ s❡❥❛✱ s❡ ✉♠❛ ♣❛rt❡✱ Sa✱ é r❡♣r❡s❡♥t❛❞❛ ♣❡❧❛s
✈❛r✐á✈❡✐s{qa, pa}❡ ❛ ♦✉tr❛✱Sb✱ ♣❡❧❛s ✈❛r✐á✈❡✐s{qb, pb}✱ ♦ s✐st❡♠❛ ❝♦♠♣♦st♦✱Sab=Sa+Sb✱
é r❡♣r❡s❡♥t❛❞♦ ♣❡❧❛s ✈❛r✐á✈❡✐s {qa, qb, pa, pb}✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ q✉❛❧q✉❡r ❡st❛❞♦ ❞❡ ✉♠
s✐st❡♠❛ ❝♦♠♣♦st♦ ❝❧áss✐❝♦ ♣♦❞❡ s❡r ❞❡s❝r✐t♦ ♣❡❧❛s ✈❛r✐á✈❡✐s{qa, qb, pa, pb} ❡✱ ❞❛❞❛✱ ❛ r❡♣r❡✲
s❡♥t❛çã♦ ❞♦ s✐st❡♠❛ ❝♦♠♣♦st♦✱ ♣♦❞❡♠♦s ✐♥❢❡r✐r ✐♠❡❞✐❛t❛♠❡♥t❡ ❛ r❡♣r❡s❡♥t❛çã♦ ❞❛s ♣❛rt❡s✿
Sa ♣♦r{qa, pa} ❡Sb ♣♦r{qb, pb}✳
◆❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛✱ ❡ss❛ ❢❛❝✐❧✐❞❛❞❡ ❝♦♠ q✉❡ r❡s♣♦♥❞❡♠♦s às ♣❡r❣✉♥t❛s ✶ ❡ ✷ é ✉♠ ❝❛s♦ ❜❛st❛♥t❡ ♣❛rt✐❝✉❧❛r✳ ◗✉❛♥❞♦ ❛ ♣❛rt❡ Sa é r❡♣r❡s❡♥t❛❞❛ ♣❡❧♦ ❡st❛❞♦|ψai ❡ ❛ ♣❛rt❡Sb
♣♦r|ψbi✱ ♦ ❡st❛❞♦ ❞♦ s✐st❡♠❛ ❝♦♠♣♦st♦ |ψabi é r❡♣r❡s❡♥t❛❞♦ ♣♦r
|ψabi=|ψai ⊗ |ψbi ✭✶✳✷✮
❡ ✈✐❝❡✲✈❡rs❛✳ ❊ss❛ s✐t✉❛çã♦ ♣♦❞❡ s❡r ❡♥t❡♥❞✐❞❛ ❝♦♠♦ ✉♠ ❛♥á❧♦❣♦ ❞❛ s✐t✉❛çã♦ ❝❧áss✐❝❛ q✉❛♥t♦ à r❡♣r❡s❡♥t❛çã♦ ❞♦s s✐st❡♠❛s ❝♦♠♣♦st♦s✳ ◆♦ ❡♥t❛♥t♦✱ ✉♠❛ s✉♣❡r♣♦s✐çã♦ ❞❡ ❞♦✐s ❡st❛❞♦s ❞♦ t✐♣♦ ✭✶✳✷✮✱ ❝♦♠|ψ′
ai ❡ |ψ′bi ❞✐❢❡r❡♥t❡s ❞❡|ψai❡ |ψbi✱ r❡s✉❧t❛ ❡♠ ✉♠ ❡st❛❞♦
|ψemai=α|ψai ⊗ |ψbi+βψ′a
⊗ψb′
♣❛r❛ ♦ s✐st❡♠❛ ❝♦♠♣♦st♦ q✉❡ ♥ã♦ ♣♦❞❡ s❡r ❡s❝r✐t♦ ♥❛ ❢♦r♠❛ ❞❡ ✉♠ s✐♠♣❧❡s ♣r♦❞✉t♦ t❡♥s♦r✐❛❧ ❝♦♠♦ |ψabi ❡♠ ✭✶✳✷✮✳ ❆ss✐♠✱ ♥♦ ❝❛s♦ ❞♦ ❡st❛❞♦ |ψemai✱ ♥ã♦ é ♣♦ssí✈❡❧ ❞❡✜♥✐r ✉♠ ❡st❛❞♦
✻ ❈❆P❮❚❯▲❖ ✶✳ ❖ ❊▼❆❘❆◆❍❆▼❊◆❚❖ ◗❯➶◆❚■❈❖
❝♦♠♦ ✉♠ s✐♠♣❧❡s ♣r♦❞✉t♦ t❡♥s♦r✐❛❧ ❝♦♠♦ ❡♠ ✭✶✳✷✮ sã♦ ❝❤❛♠❛❞♦s ❞❡ ❡st❛❞♦s ❡♠❛r❛♥❤❛❞♦s✳ ❖s ❡st❛❞♦s q✉❡ ♣♦❞❡♠ s❡r r❡♣r❡s❡♥t❛❞♦s ♥❛ ❢♦r♠❛ ✭✶✳✷✮ sã♦ ❝❤❛♠❛❞♦s ❞❡ ❡st❛❞♦s ♣r♦❞✉t♦ ♦✉ s❡♣❛rá✈❡✐s✳ ❖s q✉❛tr♦ ❡st❛❞♦s ❡♠ ✭✶✳✶✮ sã♦ ❡st❛❞♦s ♣r♦❞✉t♦✳ ❙✉♣❡r♣♦♥❞♦ ❡ss❡s ❡st❛❞♦s✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r q✉❛tr♦ ❡①❡♠♣❧♦s ❞❡ ❡st❛❞♦s ❡♠❛r❛♥❤❛❞♦s q✉❡ s❡rã♦ ❜❛st❛♥t❡ út❡✐s✱
ψ−
= √1
2(|0ai ⊗ |1bi − |1ai ⊗ |0bi)
ψ+
= √1
2(|0ai ⊗ |1bi+|1ai ⊗ |0bi)
φ+
= √1
2(|0ai ⊗ |0bi+|1ai ⊗ |1bi)
φ−
= √1
2(|0ai ⊗ |0bi − |1ai ⊗ |1bi). ✭✶✳✸✮
❊ss❡s q✉❛tr♦ ❡st❛❞♦s sã♦ ❝❤❛♠❛❞♦s ❞❡ ❜❛s❡ ❞❡ ❇❡❧❧ ❡ ❛♣❛r❡❝❡♠ ❢r❡q✉❡♥t❡♠❡♥t❡ ♥❛ ❧✐t❡r❛t✉r❛✳ ❖ ❡st❛❞♦ |ψ−i ❢♦✐ ✉s❛❞♦ ♣♦r ❇♦❤♠ ❬✶✾❪ ♥❛ s✉❛ r❡❢♦r♠✉❧❛çã♦ ❞♦ ❢❛♠♦s♦ ❛r❣✉♠❡♥t♦ ❞❡
✐♥❝♦♠♣❧❡t✉❞❡ ❞❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛ ❞❡ ❊✐♥st❡✐♥✱ P♦❞♦❧s❦✐ ❡ ❘♦s❡♥ ❬✶✶❪✳ ❊ss❡s ❡st❛❞♦s t❛♠❜é♠ ♣♦❞❡♠ s❡r ✉s❛❞♦s ❝♦♠♦ ❡①❡♠♣❧♦s ❞❛ ✈✐♦❧❛çã♦ ❞❛s ❞❡s✐❣✉❛❧❞❛❞❡s ❞❡ ❇❡❧❧ ❬✷✵❪ ♣❡❧❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛✳ ◆♦ ❝❛♠♣♦ ❞❛ ✐♥❢♦r♠❛çã♦ q✉â♥t✐❝❛✱ ❡ss❡s ❡st❛❞♦s tê♠ ♣❛♣❡❧ ✐♠♣♦rt❛♥t❡ ♥♦ ❢❡♥ô♠❡♥♦ ❞♦ t❡❧❡tr❛♥s♣♦rt❡ ❞❡ ✉♠ ❡st❛❞♦ q✉â♥t✐❝♦ ❬✷✱ ✹✱ ✺❪ ❡ ❡♠ ♦✉tr❛s t❛r❡❢❛s ❞❡ ✐♥❢♦r♠❛çã♦ q✉â♥t✐❝❛✳
❉❡ss❛ ❢♦r♠❛✱ ♦❜t❡♠♦s ✉♠❛ ❞❡✜♥✐çã♦ ❜❛st❛♥t❡ ♠❛t❡♠át✐❝❛ ❞❡ ✉♠ ❡st❛❞♦ ❡♠❛r❛♥❤❛❞♦✳ ❉❛❞♦ ✉♠ s✐st❡♠❛ ❝♦♠♣♦st♦ ❝♦♠ ✉♠ ❝♦♥❥✉♥t♦ ❝♦♠♣❧❡t♦ ❞❡ ✈❛r✐á✈❡✐s ❞✐♥â♠✐❝❛s {A, B}✱
♣♦❞❡♠♦s ❞❡✜♥✐r ♦s ❡s♣❛ç♦s ❧✐♥❡❛r❡s ❝♦♠♣❧❡①♦s✱Ha❡Hb✱ ❣❡r❛❞♦s ♣❡❧❛s ❜❛s❡s ❞❡ ❛✉t♦✈❡t♦r❡s
❞❡A❡B✱{|aii}i=1,2,3...❡{|bii}i=1,2,3...✱ ❞♦s s✉❜s✐st❡♠❛sSa❡Sb✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ P♦❞❡♠♦s
❡♥tã♦ ❞❡✜♥✐r ♦ ❡s♣❛ç♦ ❞♦ s✐st❡♠❛ ❝♦♠♣♦st♦ ❝♦♠♦Ha⊗ Hb✳ ◆❡ss❛s ❝♦♥❞✐çõ❡s✱ ❞✐③❡♠♦s q✉❡
✉♠ ❡st❛❞♦ |ψi❞♦ s✐st❡♠❛ ❝♦♠♣♦st♦ é ❡♠❛r❛♥❤❛❞♦ s❡ ♥ã♦ ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞♦ ♥❛ ❢♦r♠❛
❞❡ ✉♠ ♣r♦❞✉t♦ t❡♥s♦r✐❛❧ ❞❡ ✉♠ ❡st❛❞♦ ❞❡ Sa ❡ ♦✉tr♦ ❞❡ Sb✱
|ψi 6=|ψai ⊗ |ψbi,
♦♥❞❡|ψai ∈ Ha❡|ψbi ∈ Hb✳ ➱ ✐♠♣♦rt❛♥t❡ r❡ss❛❧t❛r♠♦s t❛♠❜é♠ q✉❡A❡Bt❛♠❜é♠ ♣♦❞❡♠
r❡♣r❡s❡♥t❛r ✉♠ ❝♦♥❥✉♥t♦ ❝♦♠♣❧❡t♦ ❞❡ ✈❛r✐á✈❡✐s ❞✐♥â♠✐❝❛s ♣❛r❛ ♦s r❡s♣❡❝t✐✈♦s s✉❜s✐st❡♠❛s ❛♦ ✐♥✈és ❞❡ ✉♠❛ ú♥✐❝❛ ✈❛r✐á✈❡❧ ❞✐♥â♠✐❝❛✳
✶✳✶✳ ❊❙❚❆❉❖❙ ❊▼❆❘❆◆❍❆❉❖❙ P❯❘❖❙ ❇■✲P❆❘❚■❉❖❙ ✼
❜❛s❡ ❞❡ ❇❡❧❧ ✭✶✳✸✮ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿
Aψ,φφ±
=φ±
❡ Aψ,φψ±
=−ψ±
;
B± x+
=
x+
❡ B±
x−
=−
x−
, ♦♥❞❡ x=ψ, φ.
❖✉ s❡❥❛✱ Aψ,φ ♥♦s ❞✐③ s❡ ♦ ❡st❛❞♦ ❞♦ s✐st❡♠❛ ❝♦♠♣♦st♦ é ❞♦ t✐♣♦ ψ ♦✉ φ ❡ B± ♥♦s ❞✐③ s❡
♦ ❡st❛❞♦ é ❞♦ t✐♣♦ ✰ ♦✉ ✲ ❡♠ ✭✶✳✷✮✳ ❉❡ss❛ ❢♦r♠❛✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ❞♦✐s s✐st❡♠❛s ❞❡ ❞♦✐s ♥í✈❡✐s ❝♦rr❡s♣♦♥❞❡♥t❡s ❛ ❝❛❞❛ ✉♠ ❞♦s ♦❜s❡r✈á✈❡✐sAψ,φ ❡ B±✳ ❖s ❡st❛❞♦s ❞❛ ❜❛s❡ ❞❡ ❇❡❧❧✱
♥❡st❛ ♣❛rt✐çã♦ ❞♦ s✐st❡♠❛ ❝♦♠♣♦st♦✱ sã♦ t♦❞♦s ❡st❛❞♦s ♣r♦❞✉t♦s
|φ+i=|φi ⊗ |+i, |φ−i=|φi ⊗ |−i,
|ψ+i=|ψi ⊗ |+i, ❡ |ψ−i=|ψi ⊗ |−i.
■ss♦ s✐❣♥✐✜❝❛ q✉❡ ♦ ❢❛t♦ ❞❡ q✉❡ ✉♠ ❡st❛❞♦ ❞♦ s✐st❡♠❛ ❝♦♠♣♦st♦ ❡stá ♦✉ ♥ã♦ ❡♠❛r❛♥❤❛❞♦ ❞❡♣❡♥❞❡ ❞❡ ❝♦♠♦ ❞✐✈✐❞✐♠♦s ♦ s✐st❡♠❛ ❝♦♠♣♦st♦ ❡♠ s✉❜s✐st❡♠❛s✱ ✐st♦ é✱ q✉❛✐s ✈❛r✐á✈❡✐s ❞✐✲ ♥â♠✐❝❛s ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ❝♦♠♣❧❡t♦ ❡s❝♦❧❤❡♠♦s ❝♦♠♦ ♣❡rt❡♥❝❡♥t❡s às ♣❛rt❡s Sa ♦✉ Sb✳ ❊✱
♠❛t❡♠❛t✐❝❛♠❡♥t❡✱ ❡①✐st❡ ✉♠❛ ✐♥✜♥✐❞❛❞❡ ❞❡ ❝♦♥❥✉♥t♦s ❝♦♠♣❧❡t♦s ♣❛r❛ ♦ s✐st❡♠❛ ❝♦♠♣♦st♦✳ ❊♥tã♦ q✉❛❧ ❝♦♥❥✉♥t♦ ❞❡✈❡♠♦s ♣r❡❢❡r✐r ♣❛r❛ ❞✐③❡r♠♦s q✉❡ ✉♠ ❡st❛❞♦ é ♦✉ ♥ã♦ ❡♠❛r❛♥❤❛❞♦❄ ❊ss❛ q✉❡stã♦ é ❡①❛t❛♠❡♥t❡ ❛ ♣❛rt❡ ❢ís✐❝❛ ❞❛ ❞❡✜♥✐çã♦ ❞❡ ❡♠❛r❛♥❤❛♠❡♥t♦✳ ➱ ✉s✉❛❧ ❝♦♥s✐❞❡✲ r❛r♠♦s ❡♠❛r❛♥❤❛♠❡♥t♦ ❡♥tr❡ ♦ s♣✐♥ ❞❡ ❞✉❛s ♣❛rtí❝✉❧❛s ♣♦rq✉❡ ♣♦❞❡♠♦s ❞❛r ❝❛❞❛ ✉♠❛ ❞❛s ♣❛rtí❝✉❧❛s ♣❛r❛ ♦❜s❡r✈❛❞♦r❡s ❞✐st✐♥t♦s✱ ❞✐❣❛♠♦s ❆❧✐❝❡ ❡ ❇♦❜✱ ♦s q✉❛✐s ♣♦❞❡♠ s❡♣❛r❛r✲s❡ ❡ ❡♥tã♦ r❡❛❧✐③❛r ♠❡❞✐❞❛s ❞♦ s♣✐♥ ❞❡ s✉❛s ♣❛rtí❝✉❧❛s✳ ◆❡ss❡ s❡♥t✐❞♦✱ s❡ ❆❧✐❝❡ ❡ ❇♦❜ ❝♦♠♣❛r✲ t✐❧❤❛♠ ✉♠ ❡st❛❞♦ ❞❛ ❜❛s❡ ❞❡ ❇❡❧❧ t❛❧ q✉❡ ❆❧✐❝❡ ❡stá ❝♦♠ ✉♠❛ ♣❛rtí❝✉❧❛ ❡ ❇♦❜ ❡stá ❝♦♠ ❛ ♦✉tr❛✱ ❡❧❡s sã♦ ❝❛♣❛③❡s ❞❡ ✉s❛r ❡ss❡ ❡♠❛r❛♥❤❛♠❡♥t♦ ♣❛r❛ ✈❡r✐✜❝❛r ❛s ❞❡s✐❣✉❛❧❞❛❞❡s ❞❡ ❇❡❧❧✱ t❡❧❡tr❛♥s♣♦rt❛r ✉♠ ❡st❛❞♦ q✉â♥t✐❝♦ ❡♥tr❡ ♦✉tr❛s t❛r❡❢❛s q✉❡ ♥ã♦ s❡r✐❛♠ ♣♦ssí✈❡✐s ❝❧❛ss✐❝❛✲ ♠❡♥t❡✳ ◆♦ ❡♥t❛♥t♦✱ ♣❛r❛ q✉❡ ❛❧❣✉♠ ♦❜s❡r✈❛❞♦r ♣♦ss❛ ♠❡❞✐r ♦s ♦❜s❡r✈á✈❡✐sAψ,φ ❡ B±✱ ❡❧❡
♣r❡❝✐s❛ t❡r ❛s ❞✉❛s ♣❛rtí❝✉❧❛s ❝♦♥s✐❣♦ ♦✉ ♣r❡❝✐s❛ ❢❛③❡r ❛s ♣❛rtí❝✉❧❛s ✐♥t❡r❛❣✐r❡♠ ❛♥t❡s ❞❛ ♦❜s❡r✈❛çã♦✳ ❆ss✐♠✱ ♥ã♦ é ♣♦ssí✈❡❧ s❡♣❛r❛r ❡s♣❛❝✐❛❧♠❡♥t❡ ♦s ✧s✉❜s✐st❡♠❛s✧ ❞❡✜♥✐❞♦s ♣❡❧♦s ♦❜s❡r✈á✈❡✐s Aψ,φ ❡ B±✳ ❚❛♠❜é♠ é ❞✐❢í❝✐❧ ❡♥t❡♥❞❡r♠♦s ✜s✐❝❛♠❡♥t❡ ♦s ♦❜s❡r✈á✈❡✐s Aψ,φ ❡ B± ✐♥❞❡♣❡♥❞❡♥t❡♠❡♥t❡✳ ➱ ❝♦♠✉♠ ✉t✐❧✐③❛r❛r♠♦s ✉♠❛ ♠❡❞✐❞❛ ♥❛ ❜❛s❡ ❞❡ ❇❡❧❧✱ ❛ q✉❛❧ ♣♦❞❡
s❡r ❡♥t❡♥❞✐❞❛ ❝♦♠ ❡♥✈♦❧✈❡♥❞♦ ✉♠❛ ♠❡❞✐❞❛ ❡♠ ❝♦♥❥✉♥t♦ ❞♦s ♦❜s❡r✈á✈❡✐s Aψ,φ ❡ B±✳ ❯♠
❡①❡♠♣❧♦ é ♦ t❡❧❡tr❛♥s♣♦rt❡ ❬✷❪ ❡ ✉♠ ❝❛s♦ ♠❛✐s ❝♦♠♣❧✐❝❛❞♦ é ♦ ❞❛ ❝♦♠♣✉t❛çã♦ ❜❛s❡❛❞❛ ❡♠ ♠❡❞✐❞❛s ❬✷✶❪✳ ◆♦ ❡♥t❛♥t♦✱ ♥ã♦ ❝♦♥❤❡❝❡♠♦s ♥❡♥❤✉♠ ❝❛s♦ ❡♠ q✉❡ ❛ ♠❡❞✐❞❛ ❞❡ ❛♣❡♥❛sAψ,φ
♦✉B± t❡♠ ❛❧❣✉♠ s✐❣♥✐✜❝❛❞♦ ♦✉ ✐♠♣♦rtâ♥❝✐❛✳
❉❡ss❛ ❢♦r♠❛✱ ❞❡✈❡♠♦s ❡s❝♦❧❤❡r ♦s ♦❜s❡r✈á✈❡✐s A ❡ B ❞❡ ❢♦r♠❛ q✉❡ ❡st❡s ♣♦ss❛♠ s❡r
♠❡❞✐❞♦s ♣♦r ♦❜s❡r✈❛❞♦r❡s ❡s♣❛❝✐❛❧♠❡♥t❡ s❡♣❛r❛❞♦s✳ ➱ ❜❛st❛♥t❡ ❝♦♠✉♠ q✉❡ A ❡ B s❡❥❛♠
✽ ❈❆P❮❚❯▲❖ ✶✳ ❖ ❊▼❆❘❆◆❍❆▼❊◆❚❖ ◗❯➶◆❚■❈❖
❡ s✐♠ ❛♦ ♥ú♠❡r♦ ❞❡ ♣❛rtí❝✉❧❛s ❡♠ ❞❡t❡r♠✐♥❛❞♦ ❡st❛❞♦✳
✶✳✶✳✷ ❈♦rr❡❧❛çõ❡s ◗✉â♥t✐❝❛s ❡♠ ❡st❛❞♦s ❡♠❛r❛♥❤❛❞♦s✿ ❡st❛❞♦s r❡❧❛t✐✈♦s
❯♠❛ ❝❛r❛❝t❡ríst✐❝❛ ❞♦s ❡st❛❞♦s ❡♠❛r❛♥❤❛❞♦s é ❛ ♣r❡s❡♥ç❛ ❞❡ ❝♦rr❡❧❛çõ❡s ❡♥tr❡ ❛s ♦❜✲ s❡r✈❛çõ❡s r❡❛❧✐③❛❞❛s ♣♦r ❆❧✐❝❡ ❡ ❇♦❜ ♥♦s s❡✉s r❡s♣❡❝t✐✈♦s s✉❜s✐st❡♠❛s✳ ◗✉❛♥❞♦ t❡♠♦s ✉♠ s✐st❡♠❛ ❝♦♠♣♦st♦ ❝❧áss✐❝♦ ❡ t❡♠♦s ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❝♦♠♣❧❡t♦ ❞❡st❡✱ ♥ã♦ ❡①✐st❡♠ ❝♦rr❡❧❛✲ çõ❡s ❡♥tr❡ ♦s s✉❜s✐st❡♠❛s✳ ◆♦ ❡♥t❛♥t♦✱ ♥❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛✱ q✉❛♥❞♦ ♦ ❡st❛❞♦ s✐st❡♠❛ ❝♦♠♣♦st♦ é ♣✉r♦ ❡ ❡♠❛r❛♥❤❛❞♦✱ ♦s ♦❜s❡r✈á✈❡✐s ❞♦s s✉❜s✐st❡♠❛s ❛♣r❡s❡♥t❛♠ ❝♦rr❡❧❛çõ❡s✳
P❛r❛ ❝♦♠❡ç❛r♠♦s ❛ ❡♥t❡♥❞❡r ❡ss❛s ❝♦rr❡❧❛çõ❡s✱ ✈❛♠♦s s✉♣♦r q✉❡ ❛ ❆❧✐❝❡ ❡ ❇♦❜ ❝♦♠♣❛r✲ t✐❧❤❡♠ ✉♠ s✐st❡♠❛ q✉â♥t✐❝♦ ❜✐✲♣❛rt✐❞♦✱ ✐st♦ é✱ ❆❧✐❝❡ ♣♦ss✉✐ ✉♠❛ ♣❛rt❡ ❞♦ s✐st❡♠❛ q✉â♥t✐❝♦ q✉❡ ♣♦❞❡ s❡r ❞❡s❝r✐t♦ ♣♦r ✉♠ ♦❜s❡r✈á✈❡❧ A ❝♦♠ ♣♦ssí✈❡✐s ❛✉t♦✲✈❛❧♦r❡s ai ❡✱ s✐♠✐❧❛r♠❡♥t❡✱
❇♦❜ ♣♦ss✉✐ ❛ ♦✉tr❛ ♣❛rt❡ ❞♦ s✐st❡♠❛ ❞❡s❝r✐t♦ ♣♦r ✉♠ ♦❜s❡r✈á✈❡❧ B ❡ ♣♦ssí✈❡✐s ✈❛❧♦r❡s bi✳
❆ss✐♠✱ ✉♠ ❡st❛❞♦ q✉❛❧q✉❡r|ψi❞♦ s✐st❡♠❛ ❝♦♠♣♦st♦ ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞♦ ♥❛ ❜❛s❡ ♣r♦❞✉t♦
{|aii ⊗ |bji}✱
|ψi=X
ij
cij|aii ⊗ |bji,
♦♥❞❡ |aii é ♦ ❛✉t♦✈❡t♦r ❞❡ A ❝♦♠ ❛✉t♦✈❛❧♦r ai✱|bii é ♦ ❛✉t♦✈❡t♦r ❞❡ B ❝♦♠ ❛✉t♦✲✈❛❧♦r bi
❡ cij sã♦ ♦s ❝♦❡✜❝✐❡♥t❡s ❝♦♠♣❧❡①♦s ❞❛ ❡①♣❛♥sã♦ ❞❡|ψi ♥❡st❛ ❜❛s❡✳ P♦r t❡r♠♦s ❡s❝♦❧❤✐❞♦ ❛
❜❛s❡ ❞♦s ❛✉t♦✈❡t♦r❡s ❞❡ A✱ ♣♦❞❡♠♦s r❡❡s❝r❡✈❡r ❢❛❝✐❧♠❡♥t❡|ψi❞❡ ♠❛♥❡✐r❛ ❛ ❡①♣❧✐❝✐t❛r♠♦s
❛s ❝♦rr❡❧❛çõ❡s q✉❡ ❆❧✐❝❡ ♦❜té♠ ♦❜s❡r✈❛♥❞♦ ♦ ✈❛❧♦r ❞❡A✱
|ψi = X
i
√pi |aii ⊗
1
√p
i
X
j
cij|bji
= X
i
√pi
|aii ⊗ |raii ✭✶✳✹✮
♦♥❞❡
|raii= 1
√pi X
j
cij|bji ❡ pi=
X
j
|cij|2
é ✉♠❛ ❝♦♥st❛♥t❡ ❞❡ ♥♦r♠❛❧✐③❛çã♦✳ ❆ss✐♠✱ s❡ ❆❧✐❝❡ ♦❜té♠ ♦ ❛✉t♦✈❛❧♦rai ♥❛ s✉❛ ♦❜s❡r✈❛çã♦
❝♦rr❡s♣♦♥❞❡♥t❡ ❛♦ ❛✉t♦✲❡st❛❞♦ |aii✱ ❡❧❛ ♣♦❞❡ ✐♥❢❡r✐r q✉❡ ♦ ❡st❛❞♦ ❞♦ s✉❜s✐st❡♠❛ ❞♦ ❇♦❜ é
|raii✳ ❖s ❡st❛❞♦s{|raii}❢♦r❛♠ ✐♥tr♦❞✉③✐❞♦s ♣r✐♠❡✐r❛♠❡♥t❡ ♣♦r ❙❝❤rö❞✐♥❣❡r ❬✽❪ ❡✱ t❛♠❜é♠✱
♣♦st❡r✐♦r♠❡♥t❡ ♥❛ t❡s❡ ❞❡ ❊✈❡r❡tt ❬✾❪✱ ♦ q✉❛❧ ♦s ❝❤❛♠♦✉ ❞❡ ❡st❛❞♦s r❡❧❛t✐✈♦s✳ ➱ ✐♠♣♦rt❛♥t❡ ♥♦t❛r♠♦s q✉❡ ♦s ❡st❛❞♦s{|raii}♥ã♦ sã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ♦rt♦❣♦♥❛✐s ❡♥tr❡ s✐✱ ♣♦❞❡♥❞♦ t❛♥t♦
s❡r❡♠ t♦❞♦s ✐❣✉❛✐s✱ ♦ q✉❡ ❝♦rr❡s♣♦♥❞❡♥t❡ ❛♦ ❝❛s♦ ❡♠ q✉❡ ♥ã♦ ❡①✐st❡♠ ❝♦rr❡❧❛çõ❡s ❡♥tr❡ ♦s s✉❜s✐st❡♠❛sSa ❡ Sb✱ ✐✳ ❡✳✱ |ψi é ✉♠ ❡st❛❞♦ ♣r♦❞✉t♦✱ q✉❛♥t♦ ♦rt♦❣♦♥❛✐s ♦ q✉❡ ❝♦rr❡s♣♦♥❞❡
❛ ✉♠ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❜❛st❛♥t❡ ✐♥t❡r❡ss❛♥t❡ q✉❡ s❡rá ❞✐s❝✉t✐❞♦ ♥❛ s❡çã♦ ✶✳✶✳✹✳
✶✳✶✳ ❊❙❚❆❉❖❙ ❊▼❆❘❆◆❍❆❉❖❙ P❯❘❖❙ ❇■✲P❆❘❚■❉❖❙ ✾
❛♣❡♥❛s ❞♦ ❡st❛❞♦ |aii✱ s❡♥❞♦ ✐♥❞❡♣❡♥❞❡♥t❡ ❞♦s ♦✉tr♦s ❡❧❡♠❡♥t♦s ❞❛ ❜❛s❡ {|aj6=ii} ❡ ❞♦
♦❜s❡r✈á✈❡❧A✳ P❛rt✐❝✉❧❛r♠❡♥t❡✱ ♣♦❞❡♠♦s ❞❡✜♥✐r|raii ❞❡ ❢♦r♠❛ ✐♥❞❡♣❡♥❞❡♥t❡ ❞❡st❡s ♦✉tr♦s
❡❧❡♠❡♥t♦s ❝♦♠♦
|raii= 1
√p
i h ai|ψiH
Sa,
♦♥❞❡ hai|ψiHsar❡♣r❡s❡♥t❛ ♦ ♣r♦❞✉t♦ ❡s❝❛❧❛r ♣❛r❝✐❛❧ ♥♦ s✉❜s✐st❡♠❛ ❞❡ ❆❧✐❝❡✱ ♦ q✉❛❧ r❡s✉❧t❛
❡♠ ✉♠ ❡st❛❞♦ ♥♦ s✉❜s✐st❡♠❛ ❞❡ ❇♦❜✳ ▼❛t❡♠❛t✐❝❛♠❡♥t❡✱ ♣♦❞❡♠♦s ❡♥t❡♥❞❡r ❡ss❡ ♣r♦❞✉t♦ ❡s❝❛❧❛r ♣❛r❝✐❛❧ ❝♦♠♦ ✉♠❛ ❛♣❧✐❝❛çã♦ ❧✐♥❡❛r ❞❡✜♥✐❞❛ ♣♦r |ψi ❞♦ ❡s♣❛ç♦ ❞✉❛❧ ❞♦ s✉❜s✐st❡♠❛ Sa✱H∗Sa✱ ❡♠HSb✳ ❊ss❛ ✐♥t❡r♣r❡t❛çã♦ s❡rá ❜❛st❛♥t❡ út✐❧ ♥♦ ❡st✉❞♦ ❞♦s s✐st❡♠❛s ❝♦♠♣♦st♦s
tr✐✲♣❛rt✐❞♦s ❞❡s❡♥✈♦❧✈✐❞♦ ♥♦ ❈❛♣ít✉❧♦ ✸✳ ❆ss✐♠✱ ❡ss❛ ❛♣❧✐❝❛çã♦ ❧✐♥❡❛r ♠❛♣❡✐❛ ♦ r❡s✉❧t❛❞♦ ❞❛ ♠❡❞✐❞❛ ♦❜t✐❞❛ ♣♦r ❆❧✐❝❡ ❝♦♠ ♦ r❡s♣❡❝t✐✈♦ ❡st❛❞♦ ❞♦ s✉❜s✐st❡♠❛ ❞❡ ❇♦❜✳
❖✉tr❛ ♦❜s❡r✈❛çã♦ r❡❧❡✈❛♥t❡ é q✉❡ ❡ss❛s ❝♦rr❡❧❛çõ❡s ♥ã♦ sã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ r❡❝í♣r♦❝❛s✱ ✐st♦ é✱ ♦ ❡st❛❞♦ r❡❧❛t✐✈♦ ❞❡ |raii ❡♠ ❣❡r❛❧ ♥ã♦ é ♦ ❡st❛❞♦ |aii✳ P❛r❛ ✈❡r♠♦s ✐ss♦✱ ❜❛st❛
❝♦♥s✐❞❡r❛r♠♦s ♦ ♣r♦❞✉t♦ ❡s❝❛❧❛r ♣❛r❝✐❛❧ ♥♦ s✉❜s✐st❡♠❛ ❞❡ ❇♦❜
hral|ψiHsb =|ali+ X
i6=l
√p
i|aii hrai|rali.
P♦rt❛♥t♦✱|ali❡|ralis❡rã♦ r❡❝✐♣r♦❝❛♠❡♥t❡ r❡❧❛t✐✈♦s s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ♦ ❡st❛❞♦ r❡❧❛t✐✈♦|rali
❢♦r ♦rt♦❣♦♥❛❧ ❛♦s ♦✉tr♦s ❡st❛❞♦s r❡❧❛t✐✈♦s |raii ♣❛r❛ i6=l✳ ❈♦♠♦ ✈❡r❡♠♦s ♥❛ s❡çã♦ ✶✳✶✳✹✱
❡ss❛ s✐t✉❛çã♦ ❝♦rr❡s♣♦♥❞❡ ✉♠❛ ❡s❝♦❧❤❛ ♣❛rt✐❝✉❧❛r ❞♦ ❡st❛❞♦|ali✱ ♥❛ q✉❛❧ ❡❧❡ é ✉♠ ❡❧❡♠❡♥t♦
❞❛ ❝❤❛♠❛❞❛ ❜❛s❡ ❙❝❤♠✐❞t✳
✶✳✶✳✸ ❖ ❡st❛❞♦ ❞♦s s✉❜s✐st❡♠❛s
◆❛ s❡çã♦ ✶✳✶✳✶✱ ❞❡✐①❛♠♦s ❡♠ ❛❜❡rt♦ ❛ q✉❡stã♦ ❞♦ ❡st❛❞♦ ❞♦s s✉❜s✐st❡♠❛s q✉❛♥❞♦ ♦ s✐st❡♠❛ ❝♦♠♣♦st♦ ❡stá ❡♠❛r❛♥❤❛❞♦✳ ◆❡st❛ s❡çã♦ ✈❛♠♦s ❞✐s❝✉t✐r ❡st❛ q✉❡stã♦✳ ❈♦♠♦ ✈✐♠♦s ♥❛q✉❡❧❛ s❡çã♦✱ ♦s s✐st❡♠❛s ❢ís✐❝♦s Sb ❡ Sab sã♦ s✐st❡♠❛s ❢ís✐❝♦s ❞✐st✐♥t♦s ❡ ♣♦ss✉❡♠
❞♦✐s r❡♣r❡s❡♥t❛♥t❡s ❞✐st✐♥t♦s✳ ◆♦ ❡♥t❛♥t♦✱ ❞❛❞♦ ✉♠ ♦❜s❡r✈á✈❡❧ B ❞♦ s✉❜s✐st❡♠❛ Sb✱ ❡❧❡ é
t❛♠❜é♠ ✉♠ ♦❜s❡r✈á✈❡❧ ❞♦ s✐st❡♠❛ ❝♦♠♣♦st♦Sab✳ ❆ss✐♠ s❡♥❞♦✱ t❡♠♦s ❞✉❛s r❡♣r❡s❡♥t❛çõ❡s
♣❛r❛ ✉♠ ♠❡s♠♦ ♦❜s❡r✈á✈❡❧✱ ♦✉ s❡❥❛✱ ❞✉❛s ♠❛♥❡✐r❛s ❞❡ ❝❛❧❝✉❧❛r♠♦s ♦ ✈❛❧♦r ❡s♣❡r❛❞♦ ❞❡B✳
❈♦♠♦ ♦ ✈❛❧♦r ❡s♣❡r❛❞♦ é ✉♠ ❞❛❞♦ ❡♠♣ír✐❝♦✱ ❡❧❡ ❞❡✈❡ s❡r ✐♥❞❡♣❡♥❞❡♥t❡ ❞❛ r❡♣r❡s❡♥t❛çã♦ ❡s❝♦❧❤✐❞❛ ❡✱ ♣♦rt❛♥t♦✱ ❡ss❛s ❞✉❛s ♠❛♥❡✐r❛s ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❞❡✈❡♠ ❝♦✐♥❝✐❞✐r✳ ❊ss❡ é ♦ ♥♦ss♦ ♣♦♥t♦ ❞❡ ♣❛rt✐❞❛ ♣❛r❛ ❡♥t❡♥❞❡r♠♦s ❝♦♠♦ ❞❡s❝r❡✈❡r ♦ ❡st❛❞♦ ❞❡ ✉♠ s✉❜s✐st❡♠❛ ❞❡ ✉♠ s✐st❡♠❛ ❢ís✐❝♦ ❡♠❛r❛♥❤❛❞♦✳
❱❛♠♦s ❡s❝♦❧❤❡r ✉♠ ♦❜s❡r✈á✈❡❧ q✉❛❧q✉❡r ❞♦ s✉❜s✐st❡♠❛ ❞♦ ❇♦❜ B✳ ❊ss❡ ♦❜s❡r✈á✈❡❧ é
r❡♣r❡s❡♥t❛❞♦ ♥♦ s✉❜s✐st❡♠❛ ♣❡❧♦ ♦♣❡r❛❞♦r B✳ ◆♦ s✐st❡♠❛ ❝♦♠♣♦st♦✱ ❡ss❡ ♦❜s❡r✈á✈❡❧ ❞❡✈❡
✶✵ ❈❆P❮❚❯▲❖ ✶✳ ❖ ❊▼❆❘❆◆❍❆▼❊◆❚❖ ◗❯➶◆❚■❈❖
✈❛❧♦r ♠é❞✐♦ ❞❡ B ❞❡✈❡ s❡r ✐♥❞❡♣❡♥❞❡♥t❡ ❞♦ ❡st❛❞♦ ❞♦ s✉❜s✐st❡♠❛ ❞❡ ❆❧✐❝❡|ψai✱ ✐st♦ é✱
hBi=hψa| ⊗ hψb|Ia⊗B|ψai ⊗ |ψbi=hψb|B|ψbi.
❯t✐❧✐③❛♥❞♦ ❛ ❞❡❝♦♠♣♦s✐çã♦ ❡♠ ❡st❛❞♦s r❡❧❛t✐✈♦s ✭✶✳✹✮✱ ✜❝❛ ❢á❝✐❧ ❝❛❧❝✉❧❛r♠♦s ✈❛❧♦r ♠é❞✐♦ ❞❡ B ♣❛r❛ ✉♠ ❡st❛❞♦ ❛r❜✐trár✐♦|ψi ❞♦ s✐st❡♠❛ ❝♦♠♣♦st♦✱
hBi = ❚r[|ψi hψ|Ia⊗B]
= X
nm
han| ⊗ hbm| X i′
√pi
′|ai′i ⊗ |ra
i′i !
X
i
√pi
hai| ⊗ hrai| !
Ia⊗B|ani ⊗ |bmi
❆ s♦♠❛ ❡♠ n✱ ❞❛s ❝♦♠♣♦♥❡♥t❡s |ani✱ é ❝❤❛♠❛❞❛ ❞❡ tr❛ç♦ ♣❛r❝✐❛❧ s♦❜r❡ ♦ s✉❜s✐st❡♠❛ ❡
r❡s✉❧t❛ ♥♦ ♣r♦❞✉t♦δn,iδn,i′✳ ❖ ✈❛❧♦r ♠é❞✐♦ ❞❡ B ✜❝❛
hBi=X
m
hbm|
X
i
pi|raii hrai|B|bmi
q✉❡ r❡♣r❡s❡♥t❛ ✉♠ tr❛ç♦ ♥♦ s✉❜s✐st❡♠❛ ❞❡ ❇♦❜✳ P♦❞❡♠♦s ❡♥tã♦ ❞❡✜♥✐r ♦ ♦♣❡r❛❞♦r
ρb =X
i
pi|raii hrai|=❚rHa|ψi hψ|, ✭✶✳✺✮
♦♥❞❡ ❚rHa ❞❡♥♦t❛ ♦ tr❛ç♦ ♣❛r❝✐❛❧ s♦❜r❡ ♦ ❡s♣❛ç♦ ❞♦ s✉❜s✐st❡♠❛ Sa✱ ❡ r❡❡s❝r❡✈❡r ♦ ✈❛❧♦r
♠é❞✐♦
hBi=❚r[ρbB].
❖ ♦♣❡r❛❞♦r ❞❡✜♥✐❞♦ ❡♠ ✭✶✳✺✮ é ❝❤❛♠❛❞♦ ❞❡ ♦♣❡r❛❞♦r ❞❡♥s✐❞❛❞❡ r❡❞✉③✐❞♦ ❡ t❡♠ ❛ ♠❡s♠❛ ✐♥t❡r♣r❡t❛çã♦ ❞❡ ✉♠❛ ♠✐st✉r❛ ❡st❛tíst✐❝❛✳ ❆ss✐♠✱ ♦s ❢❛t♦r❡s ❞❡ ♥♦r♠❛❧✐③❛çã♦pi ♣♦❞❡♠ s❡r
✐♥t❡r♣r❡t❛❞♦s ❝♦♠♦ ♦s ♣❡s♦s ❞❛ ♠✐st✉r❛ ❞♦s ❡st❛❞♦s |raii✳ ❈♦♠♦ ❡①✐st❡ ✉♠❛ ✐♥✜♥✐❞❛❞❡
❞❡ ♣♦ssí✈❡✐s ❞❡❝♦♠♣♦s✐çõ❡s ❡♠ ❡st❛❞♦s r❡❧❛t✐✈♦s✱ ♣♦❞❡♠♦s✱ ❡♥tã♦✱ ❞✐③❡r q✉❡ ♦s ❡st❛❞♦s r❡❧❛t✐✈♦s r❡♣r❡s❡♥t❛♠ ❞✐❢❡r❡♥t❡s ♠❛♥❡✐r❛s ❞❡ s❡ ❝♦♥str✉✐r ❛ ♠✐st✉r❛ρb✳ ❉❡ss❛ ✐♥t❡r♣r❡t❛çã♦✱
❝♦♥❝❧✉í♠♦s✱ ❛✐♥❞❛✱ q✉❡ ♦ ✈❛❧♦r ❞❡ q✉❛❧q✉❡r ❞♦s pi✬s ❞❡✈❡ ❡st❛r ❡♥tr❡ ♦ ♠❡♥♦r ❡ ♠❛✐♦r
❛✉t♦✈❛❧♦r ❞❡ ρb✱ ♣♦✐s✱ ❝❛s♦ ❝♦♥trár✐♦✱ ♣♦❞❡r✐❛ ❤❛✈❡r ✉♠ ❡st❛❞♦ ❝♦♠ ♣r♦❜❛❜✐❧✐❞❛❞❡ ♠❛✐♦r
✭♠❡♥♦r✮ q✉❡ ❛ ❞♦ ❛✉t♦❡st❛❞♦ ❝♦♠ ♠❛✐♦r ✭♠❡♥♦r✮ ❛✉t♦✈❛❧♦r✳
❊ss❡ r❡s✉❧t❛❞♦ r❡♣r❡s❡♥t❛ ✉♠❛ ❣r❛♥❞❡ q✉❡❜r❛ ❞♦ ♣❡♥s❛♠❡♥t♦ ❝❧áss✐❝♦✱ ♣♦✐s ❡♠❜♦r❛ t❡♥❤❛♠♦s t♦❞♦ ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ♣♦ssí✈❡❧ ❞♦ s✐st❡♠❛ ❝♦♠♣♦st♦ s❡❣✉♥❞♦ ❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛✱ r❡♣r❡s❡♥t❛❞♦ ♣❡❧♦ ❡st❛❞♦ ♣✉r♦|ψi✱ ♥ã♦ s♦♠♦s ❝❛♣❛③❡s ❞❡ ❞❡s❝r❡✈❡r ❝♦♠♣❧❡t❛♠❡♥t❡ ♦ ❡st❛❞♦
✶✳✶✳ ❊❙❚❆❉❖❙ ❊▼❆❘❆◆❍❆❉❖❙ P❯❘❖❙ ❇■✲P❆❘❚■❉❖❙ ✶✶
❇❡❧❧ ❬✷✵❪ ♠♦str♦✉ q✉❡ ❡ss❛s ❝♦rr❡❧❛çõ❡s sã♦ ♠❛✐♦r❡s ❞♦ q✉❡ s❡ ♣♦❞❡r✐❛ ❡♥❝♦♥tr❛r ❡♠ q✉❛❧q✉❡r s✐st❡♠❛ ❢ís✐❝♦ ❞❡s❝r✐t♦ ♣❡❧❛ ❢ís✐❝❛ ❝❧áss✐❝❛✶✳
✶✳✶✳✹ ❆ ❘❡♣r❡s❡♥t❛çã♦ ❞❡ ❙❝❤♠✐❞t ♦✉ ❜✐✲♦rt♦❣♦♥❛❧
◆❡st❛ s❡çã♦✱ ✈❛♠♦s ♠♦str❛r q✉❡ é s❡♠♣r❡ ♣♦ssí✈❡❧ ❡s❝♦❧❤❡r♠♦s ✉♠❛ ❞❡t❡r♠✐♥❛❞❛ ❜❛s❡
{|λa
ii}♣❛r❛ ♦ s✉❜s✐st❡♠❛ ❞❡ ❆❧✐❝❡ t❛❧ q✉❡ ♦s r❡s♣❡❝t✐✈♦s ❡st❛❞♦s r❡❧❛t✐✈♦s
rλa
i t❛♠❜é♠
s❡❥❛♠ ♦rt♦❣♦♥❛✐s ❡♥tr❡ s✐✳ ■ss♦ ♥♦s ♣❡r♠✐t✐rá ❡s❝r❡✈❡r ♦ ❡st❛❞♦ ❡♠❛r❛♥❤❛❞♦ ❞❡ ✉♠❛ ❢♦r♠❛ ❡♠ q✉❡ ❛s ❝♦rr❡❧❛çõ❡s ✜❝❛♠ ♠❛✐s ❡✈✐❞❡♥t❡s ❡ q✉❡✱ ❛❧é♠ ❞✐ss♦✱ é ❡①tr❡♠❛♠❡♥t❡ út✐❧✳
❯♠❛ ♠❛♥❡✐r❛ s✐♠♣❧❡s ❞❡ ♦❜t❡r♠♦s ❡ss❡ r❡s✉❧t❛❞♦ é ❝❛❧❝✉❧❛♥❞♦ ❛ ♠❛tr✐③ ❞❡♥s✐❞❛❞❡ ❞❡ ❆❧✐❝❡ ❡♠ ✉♠❛ ❜❛s❡ |aii✳ ❉❡ ✭✶✳✹✮ ♦❜t❡♠♦s
ρa = ❚rb[|ψi hψ|]
= X
j
hbj| X i
√pi
|aii ⊗ |raii !
X
i′
√pi
′hai′| ⊗ hra
i′|
!
|bji
= X
ii′
√pipi
′hra
i′|
X
j
|bji hbj|
|raii |aii hai′|
= X
ii′
√p
ipi′hra
i′|raii |aii hai′| ✭✶✳✻✮
♦♥❞❡ {|bii} é ✉♠❛ ❜❛s❡ ♣❛r❛ ♦ s✉❜s✐st❡♠❛ ❞♦ ❇♦❜✳ ❆❣♦r❛✱ ♣♦❞❡♠♦s s✉♣♦r q✉❡ ♦ ❡st❛❞♦s
|aii✬s sã♦ ♦s ❛✉t♦✈❡t♦r❡s ❞♦ ♦♣❡r❛❞♦r ❞❡♥s✐❞❛❞❡ r❡❞✉③✐❞♦ ρa✱ |aii = |λaii✳ ❈♦♠♦ s❛❜❡♠♦s
q✉❡ρa❞❡✈❡ s❡r ❞✐❛❣♦♥❛❧ ♥❡st❛ ❜❛s❡✱ é ♥❡❝❡ssár✐♦ q✉❡ ♦s ❡st❛❞♦s r❡❧❛t✐✈♦s ❞❡|λa
iis❡❥❛♠ t❛✐s
q✉❡
hrλa i′|rλ
a ii=δij
❡♠ ✭✶✳✻✮ ♣❛r❛ q✉❡ t❡♥❤❛♠♦s✷
ρa=X
i
pi|λaii hλai|.
❉❛ ❡q✉❛çã♦ ✭✶✳✺✮ s❛❜❡♠♦s t❛♠❜é♠ q✉❡ ♦ ♦♣❡r❛❞♦r ❞❡♥s✐❞❛❞❡ ❞❡ ❇♦❜✱ρb✱ ♣♦❞❡ s❡r r❡♣r❡✲
s❡♥t❛❞♦ ❝♦♠♦ ✉♠❛ ♠✐st✉r❛ ❞♦s ❡st❛❞♦s r❡❧❛t✐✈♦s ❡✱ ❝♦♠♦ s❛❜❡♠♦s q✉❡ ♦s rλa
i
✬s ❝♦♥st✐t✉❡♠ ✉♠❛ ❜❛s❡ ♦rt♦♥♦r♠❛❧ ♣❛r❛ ❡ss❡ s✉❜s✐st❡♠❛✱ t❡♠♦s q✉❡ ρb é ❞✐❛❣♦♥❛❧ ♥❡st❛ ❜❛s❡
ρb =
X
i pi
rλa i rλai
=
X
i
pi|λbiihλbi|,
✶❖✉ q✉❛❧q✉❡r t❡♦r✐❛ q✉❡ ♦❜❡❞❡❝❡ss❡ ♦s ♣r✐♥❝í♣✐♦s ❞❡ ❧♦❝❛❧✐❞❛❞❡ ❡ r❡❛❧✐❞❛❞❡ ❞❡ ❊✐♥st❡✐♥✱ P♦❞♦❧s❦✐ ❡ ❘♦s❡♥
❬✶✶❪✳
✷❊♠ ❛❧❣✉♥s ❝❛s♦✱ ♣♦❞❡♠♦s t❡r ❛❧❣✉♥spi✬s ✐❣✉❛❧ ❛ ③❡r♦✱ ♦ q✉❡ ❝♦rr❡s♣♦♥❞❡ ❛ρat❡r ✉♠ ♥ú❝❧❡♦ ♥ã♦ tr✐✈✐❛❧✳
◆❡st❡ ❝❛s♦✱ ♣♦❞❡♠♦s ❡s❝♦❧❤❡r ❛r❜✐tr❛r✐❛♠❡♥t❡ ♦s ❛✉t♦✈❡t♦r❡s ❞❡st❡ ♥ú❝❧❡♦ t❛♥t♦ ♣❛r❛ρa❡ρb✉♠❛ ✈❡③ q✉❡
✶✷ ❈❆P❮❚❯▲❖ ✶✳ ❖ ❊▼❆❘❆◆❍❆▼❊◆❚❖ ◗❯➶◆❚■❈❖
♦♥❞❡ ✜③❡♠♦s λbi
=
rλa i
✳ ❱♦❧t❛♥❞♦ ❛♦ ❡st❛❞♦ ❝♦♠♣❛rt✐❧❤❛❞♦|ψi✱ ♣♦❞❡♠♦s ♦❜t❡r ❞❡ ✭✶✳✹✮
❛ r❡♣r❡s❡♥t❛çã♦ ❞♦ ♠❡s♠♦ ♥❡st❛s ❜❛s❡s✱
|ψi=X
i
√p
i|λaii ⊗ |λbii. ✭✶✳✼✮
❊ss❛ ❢♦r♠❛ s✐♠♣❧❡s ✭✶✳✼✮ é ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❞❡❝♦♠♣♦s✐çã♦ ❞❡ ❙❝❤♠✐❞t ❬✻❪✳ ➱ t❛♠❜é♠ ❞✐t❛ ❜✐✲♦rt♦❣♦♥❛❧ ❞❡✈✐❞♦ ❛ ♦rt♦❣♦♥❛❧✐❞❛❞❡ ❞❛s ❜❛s❡s {|λa
ii} ❡
|λb
ii ✱ ❛s q✉❛✐s ❞❡♥♦♠✐♥❛✲
r❡♠♦s ❞❡ ❜❛s❡s ❞❡ ❙❝❤♠✐❞t✳ P♦ré♠✱ ❞❡✈❡♠♦s r❡ss❛❧t❛r ❛q✉✐ q✉❡ ❛s ❜❛s❡s ❞❡ ❙❝❤♠✐❞t ♥ã♦ sã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❜❛s❡s ❡♠Sa ❡Sb✳ ▼❛s✱ ❡❧❛s ♣♦❞❡♠ s❡r ❞✐t❛s ❜❛s❡s ❞♦s s✉♣♦rt❡s ❧♦❝❛✐s
❞❡ |ψi ❡♠ HSa ❡ HSb✱ ✐st♦ é✱ ❞♦s s✉❜❡s♣❛ç♦s ♦♥❞❡ ♦ ❡st❛❞♦ ❞♦s s✉❜s✐st❡♠❛s t❡♠ ❛❧❣✉♠❛
♣r♦❜❛❜✐❧✐❞❛❞❡ ❞✐❢❡r❡♥t❡ ❞❡ ③❡r♦ ❞❡ s❡r ❡♥❝♦♥tr❛❞♦✳ ❖❜✈✐❛♠❡♥t❡ s❡♠♣r❡ ♣♦❞❡♠♦s ❝♦♠♣❧❡t❛r ❛s ❜❛s❡s ❞♦s s✉♣♦rt❡s ❧♦❝❛✐s ♣❛r❛ ♦❜t❡r♠♦s ❜❛s❡s ♣❛r❛HSa ❡ HSb✳ ❉❡ss❛ ❢♦r♠❛✱ ♣♦❞❡♠♦s
♦❜s❡r✈❛r q✉❡✱ s❡ ❛s ❞✐♠❡♥sõ❡s ❞❡HSa ❡HSb ❢♦r❡♠ ❞✐❢❡r❡♥t❡s✱ s❡❣✉❡ ❞❡ ✭✶✳✼✮ q✉❡ ♦s s✉♣♦rt❡s
❧♦❝❛✐s ❞❡ |ψi ♥♦s ❞♦✐s s✉❜s✐st❡♠❛s tê♠ ❛ ♠❡s♠❛ ❞✐♠❡♥sã♦ ❡✱ ♣♦rt❛♥t♦✱ ❛ ♣❛rt✐r ❞❡ ❛❣♦r❛✱
s✉♣♦r❡♠♦s s❡♠♣r❡ q✉❡ ❛s ❞✐♠❡♥sõ❡s ❞❡ HSa✱HSb ❡ ❞❡ s❡✉s r❡s♣❡❝t✐✈♦s s✉♣♦rt❡s ❧♦❝❛✐s sã♦
s❡♠♣r❡ ✐❣✉❛✐s✳
❯♠❛ s✐t✉❛çã♦ ✐♥t❡r❡ss❛♥t❡ ❛❝♦♥t❡❝❡ q✉❛♥❞♦ ❆❧✐❝❡ r❡s♦❧✈❡ ♠❡❞✐r ✉♠ ♦❜s❡r✈á✈❡❧ ❞✐❛❣♦♥❛❧ ♥❛ ❜❛s❡ ❞❡ ❙❝❤♠✐❞t✱ Aλ✳ ◆❡ss❡ ❝❛s♦✱ s❡ ❡❧❛ ♦❜t✐✈❡r ♦ r❡s✉❧t❛❞♦ aλl✱ ❝♦rr❡s♣♦♥❞❡♥t❡ ❛♦
❡st❛❞♦ |λa
li✱ ♦ ❡st❛❞♦ ❞♦ s✐st❡♠❛ ❝♦♠♣♦st♦ r❡❞✉③✲s❡ ❛♦ ❡❧❡♠❡♥t♦ ❞❛ ❞❡❝♦♠♣♦s✐çã♦ ❞❡ ❙❝❤✲
♠✐❞t |λa
li ⊗ |λbli✳ ❆ss✐♠✱ ♦ r❡s✉❧t❛❞♦ ❞❡ ✉♠❛ ♠❡❞✐❞❛ r❡❛❧✐③❛❞❛ ♣♦r ❇♦❜ ❞❡ ✉♠ ♦❜s❡r✈á✈❡❧
t❛♠❜é♠ ❝♦♠♣❛tí✈❡❧ ❝♦♠ ❛ ❜❛s❡ ❞❡ ❙❝❤♠✐❞t✱ Bλ✱ r❡s✉❧t❛rá s❡♠♣r❡ ❡♠ bλl✱ ❝♦rr❡s♣♦♥❞❡♥t❡
❛♦ ❡st❛❞♦|λb
li✱ ✐st♦ é✱ ♦ s✐st❡♠❛ ❛♣r❡s❡♥t❛ ✉♠❛ ❝♦rr❡❧❛çã♦ ♣❡r❢❡✐t❛ ❡♥tr❡ ♦s ♦❜s❡r✈á✈❡✐s Aλ
❡ Bλ✳ ❉❡✈✐❞♦ ❛ ♥ã♦✲♦rt♦♥♦r♠❛❧✐❞❛❞❡ ❞♦s ❡st❛❞♦s r❡❧❛t✐✈♦s✱ ❡ss❛ ❝♦rr❡❧❛çã♦ ♣❡r❢❡✐t❛ ❡♥tr❡
♦❜s❡r✈á✈❡✐s ❡①✐st❡ ❛♣❡♥❛s ♣❛r❛ ♦❜s❡r✈á✈❡✐s ❝♦♠♣❛tí✈❡✐s ❝♦♠ ❛ ❜❛s❡ ❞❡ ❙❝❤♠✐❞t✳
❆ ✉♥✐❝✐❞❛❞❡ ❞❛ ❞❡❝♦♠♣♦s✐çã♦ ✭✶✳✼✮ ❞❡♣❡♥❞❡rá ❞❛ ❞❡❣❡♥❡r❡s❝ê♥❝✐❛ ❞♦s ❛✉t♦✈❛❧♦r❡s ❞❡
ρa♦✉ ρb✱ ♣♦✐s✱ ♥♦ ❝❛s♦ ♥ã♦ ❞❡❣❡♥❡r❛❞♦ ♦s ❛✉t♦✈❡t♦r❡s |λiai ❡|λbii sã♦ ❞❡✜♥✐❞♦s ✉♥✐❝❛♠❡♥t❡
❡✱ ♣♦rt❛♥t♦✱ t❛♠❜é♠ é ú♥✐❝❛ ❛ ❞❡❝♦♠♣♦s✐çã♦ ✭✶✳✼✮✳ ◆♦ ❡♥t❛♥t♦✱ s❡ ❛❧❣✉♥s ❞♦s ❛✉t♦✈❛❧♦r❡s ❢♦r❡♠ ❞❡❣❡♥❡r❛❞♦s✱ ❤❛✈❡rá ✉♠❛ ❛r❜✐tr❛r✐❡❞❛❞❡ ♥❛ ❡s❝♦❧❤❛ ❞❡ ❛✉t♦✈❡t♦r❡s ♦rt♦❣♦♥❛✐s ♣❡r✲ t❡♥❝❡♥t❡s ❛♦s s✉❜❡s♣❛ç♦s ✐♥✈❛r✐❛♥t❡s ❞❡ss❡s ❛✉t♦✈❛❧♦r❡s✱ ♠❛s✱ ❡s❝♦❧❤✐❞♦ ✉♠ ❛✉t♦✈❡t♦r|λa
li
❝♦♠ ❛✉t♦✈❛❧♦r ❞❡❣❡♥❡r❛❞♦ λl ♥♦ s✉❜s✐st❡♠❛ ❞❡ ❆❧✐❝❡✱ ❛✉t♦♠❛t✐❝❛♠❡♥t❡ ❡st❛rá ❞❡✜♥✐❞♦ ♦
❛✉t♦✈❡t♦r ❝♦rr❡s♣♦♥❞❡♥t❡|λbli❛tr❛✈és ❞♦ ❡st❛❞♦|ψi✱
|λbli=hψ|λali.
◆♦ ❝❛s♦ ❡①tr❡♠♦ ❞❡ ❞❡❣❡♥❡r❡s❝ê♥❝✐❛✱ ♦s ♦♣❡r❛❞♦r❡s ❞❡♥s✐❞❛❞❡sρa❡ρb sã♦ ♣r♦♣♦r❝✐♦♥❛✐s ❛♦
♦♣❡r❛❞♦r ✐❞❡♥t✐❞❛❞❡✱ ❛ss✐♠✱ q✉❛❧q✉❡r ❜❛s❡ ♦rt♦♥♦r♠❛❧ ❡♠ ✉♠ ❞♦s s✉❜s✐st❡♠❛s é ✉♠❛ ❜❛s❡ ❞❡ ❙❝❤♠✐❞t ❡ ❝❛❞❛ ✉♠ ❞❡ s❡✉s ❡❧❡♠❡♥t♦s ❡stá ❛ss♦❝✐❛❞♦✱ ❛tr❛✈és ❞♦ ❡st❛❞♦ |ψi✱ ❝♦♠ ✉♠
✶✳✶✳ ❊❙❚❆❉❖❙ ❊▼❆❘❆◆❍❆❉❖❙ P❯❘❖❙ ❇■✲P❆❘❚■❉❖❙ ✶✸
❆❧✐❝❡ ❡st❛rá ♣❡r❢❡✐t❛♠❡♥t❡ ❝♦rr❡❧❛❝✐♦♥❛❞♦ ❝♦♠ ✉♠ ♦❜s❡r✈á✈❡❧ ❝♦rr❡s♣♦♥❞❡♥t❡ ❞♦ s✉❜s✐st❡♠❛ ❞❡ ❇♦❜✳
❖✉tr♦ ❝❛s♦ ❜❛st❛♥t❡ ♣❛rt✐❝✉❧❛r ❡ ✐♠♣♦rt❛♥t❡ é q✉❛♥❞♦ ✉♠ ❞♦s ❛✉t♦✈❛❧♦r❡s ρa ❡ ρb é
✐❣✉❛❧ ❛ ✶✱ λl = 1 ❡✱ ♣♦rt❛♥t♦✱ t♦❞♦s ♦s ♦✉tr♦s ❛✉t♦✈❛❧♦r❡s sã♦ ♥✉❧♦s✱ λi6=l = 0✳ ◆❡ss❡
❝❛s♦✱ ♦ s✐st❡♠❛ ❝♦♠♣♦st♦ Sab r❡❞✉③✲s❡ ❛ ✉♠ ❡st❛❞♦ ♣r♦❞✉t♦ |ψi = |λali ⊗ |λbli ❡ ♥ã♦
❡①✐st❡♠ ❝♦rr❡❧❛çõ❡s ❡♥tr❡ ♦s s✉❜s✐st❡♠❛s✱ ♦s q✉❛✐s sã♦ ♣✉r♦s ❡ ♣♦❞❡♠ s❡r r❡♣r❡s❡♥t❛❞♦s ♣❡❧♦s r❡s♣❡❝t✐✈♦s ✈❡t♦r❡s ❞❡ ❡st❛❞♦|λali❡|λbl′i✳ ❉❡ss❛ ❢♦r♠❛✱ ♦ ❢❛t♦ ❞❡ ♦ ♦♣❡r❛❞♦r ❞❡♥s✐❞❛❞❡
r❡❞✉③✐❞♦ t❡r ❛♣❡♥❛s ✉♠ ♦✉ ✈ár✐♦s ❛✉t♦✈❛❧♦r❡s ♥ã♦ ♥✉❧♦ ❝♦♥st✐t✉✐ ✉♠ ❝r✐tér✐♦ s✐♠♣❧❡s ♣❛r❛ s❛❜❡r♠♦s s❡ ✉♠ ❡st❛❞♦|ψi❞❡ ✉♠ s✐st❡♠❛ ❝♦♠♣♦st♦ ❡stá ♦✉ ♥ã♦ ❡♠❛r❛♥❤❛❞♦✳
✶✳✶✳✺ ◗✉❛♥t✐✜❝❛çã♦ ❞♦ ❡♠❛r❛♥❤❛♠❡♥t♦
◆ós ✈✐♠♦s ♥❛ s❡çã♦ ❛♥t❡r✐♦r ✶✳✶✳✹ q✉❡✱ q✉❛♥❞♦ ♦ ❡st❛❞♦ ❞♦ s✉❜s✐st❡♠❛ é ♦ ❞❡ ❡♥tr♦✲ ♣✐❛ ♠á①✐♠❛✱ ❡①✐st❡ ✉♠❛ ❝♦rr❡❧❛çã♦ ♣❡r❢❡✐t❛ ❡♥tr❡ q✉❛❧q✉❡r ♦❜s❡r✈á✈❡❧ ❞❡ ✉♠ s✉❜s✐st❡♠❛ ❝♦♠ ✉♠ ❞♦ ♦✉tr♦✳ ❆❧é♠ ❞✐ss♦✱ ♥♦ ❝❛s♦ ❞♦ s✐st❡♠❛ ❝♦♠♣♦st♦ ❡♥❝♦♥tr❛r✲s❡ ❡♠ ✉♠ ❡st❛❞♦ ♣r♦❞✉t♦✱ ♦ ❡st❛❞♦ ❞♦s s✉❜s✐st❡♠❛s é ♣✉r♦ ❡ ♥ã♦ ❡①✐st❡♠ ❝♦rr❡❧❛çõ❡s ❡♥tr❡ ♦s ♦❜s❡r✈á✈❡✐s ❞♦s s✉❜s✐st❡♠❛s✳ ■♥t✉✐t✐✈❛♠❡♥t❡✱ ❡st❛ ✐♥t❡r♣r❡t❛çã♦ s✉❣❡r❡ q✉❡ ❛✱ ♠❡❞✐❞❛ ❡♠ q✉❡ ❛ ❡♥✲ tr♦♣✐❛ ❞♦ ❡st❛❞♦ ❞♦s s✉❜s✐st❡♠❛s ❛✉♠❡♥t❛✱ ❛✉♠❡♥t❛♠ t❛♠❜é♠ ❛s ❝♦rr❡❧❛çõ❡s q✉â♥t✐❝❛s ❡♥tr❡ ♦s s✉❜s✐st❡♠❛s✳ ❉❡ss❛ ❢♦r♠❛✱ ✉♠❛ ♠❛♥❡✐r❛ ❞❡ q✉❛♥t✐✜❝❛r♠♦s ❡ss❛s ❝♦rr❡❧❛çõ❡s ♦✉ ♦ ❡♠❛r❛♥❤❛♠❡♥t♦ ❞❡ ✉♠ ❡st❛❞♦ |ψi ❞♦ s✐st❡♠❛ ❝♦♠♣♦st♦ s❡r✐❛ ❛tr❛✈és ❞❛ ❡♥tr♦♣✐❛ ❞❡ ✈♦♥
◆❡✉♠❛♥♥ ❞❡ ✉♠ ❞♦s s✉❜s✐st❡♠❛s✱
ES(|ψi) =−❚r[ρrlogρr] =−
N
X
i=1
pi❧♦❣pi.
❊ss❛ ♥ã♦ é ❛ ú♥✐❝❛ ♠❛♥❡✐r❛ ❞❡ q✉❛♥t✐✜❝❛r♠♦s ♦ ❡♠❛r❛♥❤❛♠❡♥t♦ ❞❡ ✉♠ ❡st❛❞♦ ♣✉r♦ ❜✐✲♣❛rt✐❞♦✳ P♦ré♠✱ ES é ✉♠❛ ❞❛s ♠❡❞✐❞❛s ♠❛✐s ✉t✐❧✐③❛❞❛s✳ ❉❡ ✉♠ ♣♦♥t♦ ❞❡ ✈✐st❛ ♠❛✐s
r✐❣♦r♦s♦ ❡ ❛①✐♦♠át✐❝♦ ❬✶✺❪✱ ♣❛r❛ q✉❡ ✉♠❛ q✉❛♥t✐❞❛❞❡ s❡❥❛ ✉♠❛ ♠❡❞✐❞❛ ❞❡ ❡♠❛r❛♥❤❛♠❡♥t♦✱ ❜❛st❛ q✉❡ ❡❧❛ ♥ã♦ ❛✉♠❡♥t❡ ♣♦r ♦♣❡r❛çõ❡s ❧♦❝❛✐s ❝♦♠ ❝♦♠✉♥✐❝❛çã♦ ❝❧áss✐❝❛ ✭▲❖❈❈✮✸✳ ❈♦♠♦
♦✉tr❛s q✉❛♥t✐❞❛❞❡s q✉❡ ♣♦❞❡♠ s❡r ❝♦♥s✐❞❡r❛❞❛s ♠❡❞✐❞❛s ❞♦ ❡♠❛r❛♥❤❛♠❡♥t♦ ♣♦❞❡♠♦s ❝✐t❛r ♦ ♣♦st♦ ❞❡ ❙❝❤♠✐❞t ❞♦ ❡st❛❞♦✳ ❉❡ss❛ ❢♦r♠❛✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♥♦ ❝❛s♦ ♠✉❧t✐✲♣❛rt✐❞♦✱ é ♣♦ssí✈❡❧ ❞❡✜♥✐r♠♦s ✉♠❛ ❣r❛♥❞❡ q✉❛♥t✐❞❛❞❡ ❞❡ ♠❡❞✐❞❛s✳ ❊♥tr❡t❛♥t♦✱ é ❞❡s❡❥á✈❡❧ t❛♠❜é♠ q✉❡ ✉♠❛ ♠❡❞✐❞❛ ❞♦ ❡♠❛r❛♥❤❛♠❡♥t♦ t❡♥❤❛ ✉♠❛ ✐♥t❡r♣r❡t❛çã♦ ❢ís✐❝❛ ❡ é ♣♦r ✐ss♦ q✉❡ ❛ ❡♥tr♦♣✐❛ ❞❡ ✈♦♥ ◆❡✉♠❛♥♥ ❛♣r❡s❡♥t❛✲s❡ ❝♦♠♦ ✉♠❛ ❜♦❛ ♠❡❞✐❞❛✳ ■ss♦ ❞❡✈❡✲s❡ ❡♠ ❣r❛♥❞❡ ♣❛rt❡ ❛♦s ♣r♦t♦❝♦❧♦s ❞❡ ❞❡st✐❧❛çã♦ ❡ ❞✐❧✉✐çã♦ ❞♦ ❡♠❛r❛♥❤❛♠❡♥t♦ ✐♥tr♦❞✉③✐❞♦s ♣♦r ❇❡♥♥❡tt ❡t ❛❧ ❬✶✹❪✳
❊♠❜♦r❛ s❡❥❛ ✉♠ tó♣✐❝♦ ❞❡ ❜❛st❛♥t❡ ✐♠♣♦rtâ♥❝✐❛✱ ♥❡st❛ t❡s❡✱ ♥ã♦ ❞✐s❝✉t✐r❡♠♦s ♠✉✐t♦ ❛s ♠❡❞✐❞❛s ❞♦ ❡♠❛r❛♥❤❛♠❡♥t♦✳ ❯s❛r❡♠♦s ES ❝♦♠♦ ✉♠❛ r❡❢❡rê♥❝✐❛ ♣❛r❛ ❝♦♠♣❛r❛r♠♦s
✸❉❡✈❡♠♦s ❝♦♥s✐❞❡r❛r q✉❡ ❡ss❛ q✉❛♥t✐❞❛❞❡ ♥ã♦ ❛✉♠❡♥t❛ ❡♠ ♠é❞✐❛ ♣❛r❛ ♦ ❝❛s♦ ❞❡ ♦♣❡r❛çõ❡s ❧♦❝❛✐s
