Quantum Computing verstehen: Grundlagen - Anwendungen - by Matthias Homeister

By Matthias Homeister

Quantenrechner versprechen ungeahnte Rechenleistung. Quantenverschlüsselungs-Systeme sind sicherer als alle Alternativen. Theoretische Informatiker überdenken die Grundlagen des Rechnens neu - die Forschungen zum Quantencomputer haben die Informatik verändert. Dieses Buch führt anschaulich und auf Beispiele gestützt in die Grundlagen des Quantum Computing ein. Alle wesentlichen Ergebnisse werden detailliert und intuitiv dargestellt. Der Leser erfährt, used to be er von dieser Technologie erwarten kann und wo ihre Grenzen liegen. Das Besondere an dieser Einführung ist, dass keine speziellen Kenntnisse vorausgesetzt werden. Alle wichtigen Begriffe werden umfassend eingeführt, ein mathematischer Anhang stellt benötigte Grundlagen kurz dar.

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Example text

S❡ ❙❝❤r❡✐❜✇❡✐s❡ ❡r✐♥♥❡rt ♥✐❝❤t ♦❤♥❡ ●r✉♥❞ ❛♥ ❡✐♥ Pr♦❞✉❦t✳ ❙❡✐ ❞❛s ❡rst❡ ◗✉❛♥t❡♥❜✐t ✐♠ ❩✉st❛♥❞ |x0 = γ0 |0 + γ1 |1 ✉♥❞ ❞❛s ③✇❡✐t❡ ✐♠ ❩✉st❛♥❞ |x1 = β0 |0 + β1 |1 . ❉❛♥♥ ❡r❤❛❧t❡♥ ✇✐r ❞❡♥ ❩✉st❛♥❞ ❞❡s ◗✉❛♥t❡♥r❡❣✐st❡rs ❞✉r❝❤ ❆✉s♠✉❧t✐♣❧✐③✐❡✲ r❡♥❀ ✐♠ ❆❜s❝❤♥✐tt ✷✳✼ ✇❡r❞❡♥ ✇✐r ❞✐❡s❡s ❱♦r❣❡❤❡♥ ♠❛t❤❡♠❛t✐s❝❤ ❢✉♥❞✐❡r❡♥✳ R = |x1 |x0 = (β0 |0 + β1 |1 ) · (γ0 |0 + γ1 |1 ) = β0 γ0 |0 |0 + β0 γ1 |0 |1 + β1 γ0 |1 |0 + β1 γ1 |1 |1 . ❉❛s ✐st ✉♥ü❜❡rs✐❝❤t❧✐❝❤✿ ❲✐r ❜❡③❡✐❝❤♥❡♥ ❞✐❡ ❆♠♣❧✐t✉❞❡♥ ❞❡r ❩✉stä♥❞❡ ❞❡s ◗✉❛♥t❡♥r❡❣✐st❡rs ♠✐t αi j = βi γ j ✉♥❞ ❡r❤❛❧t❡♥✿ R = α00 |0 |0 + α01 |0 |1 + α10 |1 |0 + α11 |1 |1 .

A2n −1,i , s♦ ✇✐r❞ |i ❛✉❢ a0i |0 + a1i |1 + . . + a2n −1,i |2n − 1 ❛❜❣❡❜✐❧❞❡t✳ ❲✐r ❞❡♠♦♥str✐❡r❡♥ ❞✐❡ ◆✉♠♠❡r✐❡r✉♥❣ ✈♦♥ ❩❡✐❧❡♥ ✉♥❞ ❙♣❛❧t❡♥ ❛♥ ❞❡r ❖♣❡r❛t✐♦♥ CNOT✳ |00 |01 |10 |11 |00 1  0   0 0  |01 0 1 0 0 |10 |11  0 0 0 0   0 1  1 0 ✷✳✺ ◗✉❛♥t❡♥r❡❣✐st❡r ✸✶ ❙♦ s✐❡❤t ♠❛♥ s♦❢♦rt✱ ❞❛ss ❞✐❡s❡ ❖♣❡r❛t✐♦♥ ❞✐❡ ❆♠♣❧✐t✉❞❡♥ ✈♦♥ t❛✉s❝❤t✿    α0 α0  α1   α1   CNOT   α2  =  α3 α3 α2 |10 ✉♥❞ |11   .  ❊s ❤❛♥❞❡❧t s✐❝❤ ✉♠ ❡✐♥❡ P❡r♠✉t❛t✐♦♥s♠❛tr✐①✳ ❇❡✐s♣✐❡❧ ✷✳✻✿ ❲✐r ❜❡tr❛❝❤t❡♥ ▼❛tr✐③❡♥ ✈♦♥ ❞❡r ❢♦❧❣❡♥❞❡♥ ❋♦r♠✳ ■♥ ❥❡❞❡r ❙♣❛❧t❡ ✉♥❞ ✐♥ ❥❡❞❡r ❩❡✐❧❡ ❣✐❜t ❡s ❣❡♥❛✉ ❡✐♥❡ 1✱ ❛❧❧❡ ❛♥❞❡r❡♥ ❊✐♥trä❣❡ s✐♥❞ 0✳ ❉✐❡s❡ ▼❛tr✐③❡♥ ❤❡✐ÿ❡♥ P❡r♠✉t❛t✐♦♥s♠❛tr✐③❡♥✳ ❆♥❣❡✇❛♥❞t ❛✉❢ ❡✐♥❡♥ ❱❡❦t♦r✱ ✈❡rä♥❞❡rt ❡✐♥❡ s♦❧❝❤❡ ▼❛tr✐① ❞✐❡ ❘❡✐❤❡♥❢♦❧❣❡ ❞❡r ❑♦♠♣♦♥❡♥t❡♥✱ ä♥❞❡rt ❛❜❡r ✐❤r❡♥ ❲❡rt ♥✐❝❤t✳ ❆✉❢ ❡✐♥❡♥ ❩✉st❛♥❞s✈❡❦t♦r ❛♥❣❡✇❛♥❞t✱ ✇✐r❞ ❞✐❡ ❩✉♦r❞♥✉♥❣ ✈♦♥ ❆♠♣❧✐t✉❞❡♥ ③✉ ❇❛s✐s③✉stä♥❞❡♥ ❣❡ä♥❞❡rt✳ ❆✉❢❣❛❜❡ ✷✳✾✿ ❩❡✐❣❡♥ ❙✐❡✱ ❞❛ss P❡r♠✉t❛t✐♦♥s♠❛tr✐③❡♥ ✉♥✐tär s✐♥❞✳ ❉✐❡ ❢♦❧❣❡♥❞❡ ❉❡✜♥✐t✐♦♥ ✐st ✇❡s❡♥t❧✐❝❤✿ ❩✇❛r ✇✐ss❡♥ ✇✐r✱ ✇✐❡ s✐❝❤ ❡✐♥ ◗✉❛♥✲ t❡♥r❡❣✐st❡r ❡♥t✇✐❝❦❡❧t✳ ❆❜❡r ✇✐r ❤❛❜❡♥ ♥♦❝❤ ♥✐❝❤t ❣❡s❛❣t✱ ✇❛s ❡✐♥ ❡✐♥❢❛❝❤❡r ❘❡❝❤❡♥s❝❤r✐tt ✐st✳ ❊✐♥❢❛❝❤❡ ❘❡❝❤❡♥s❝❤r✐tt❡ ❛✉❢ ❡✐♥❡♠ ◗✉❛♥t❡♥r❡❣✐st❡r s✐♥❞ ❧♦❦❛❧❡ ✉♥✐tär❡ ❚r❛♥s❢♦r♠❛t✐♦♥❡♥✳ ▲♦❦❛❧ s✐♥❞ ❚r❛♥s❢♦r♠❛t✐♦♥❡♥✱ ❛♥ ❞❡♥❡♥ ❤ö❝❤st❡♥s ❞r❡✐ ❇✐ts ❜❡t❡✐❧✐❣t s✐♥❞✳ ❲❡♥♥ ✇✐r ❛❧s♦ ❡✐♥❡ ✉♥✐tär❡ ❚r❛♥s❢♦r♠❛t✐♦♥ ✈❡r✇❡♥❞❡♥✱ ❞✐❡ ❛✉❢ ❞❡♠ ❣❛♥③❡♥ ❘❡❣✐st❡r ♦♣❡r✐❡rt✱ ✐♥t❡r❡ss✐❡rt ✉♥s ❞✐❡ ❩❡r❧❡❣✉♥❣ ✐♥ ❧♦❦❛❧❡ ❚r❛♥s❢♦r♠❛t✐♦♥❡♥✳ ❉❡r ●r✉♥❞ ❞❛❢ür ✐st ❧❡✐❝❤t ❡✐♥③✉s❡❤❡♥✿ ❊✐♥❡ ❘❡❣✐st❡r♦♣❡r❛t✐♦♥✱ ❞✐❡ ✈♦♥ ❡✐♥❡r 2n × 2n ✲▼❛tr✐① ❜❡s❝❤r✐❡❜❡♥ ✇✐r❞✱ ❦❛♥♥ s❡❤r ❦♦♠♣❧✐③✐❡rt ✉♥❞ ♥✉r ♠✐t ❣r♦ÿ❡♠ ❆✉❢✇❛♥❞ ❛✉s❢ü❤r❜❛r s❡✐♥✳ ❉❛❣❡❣❡♥ ❦❛♥♥ ❡✐♥❡ ❉r❡✐✲❇✐t✲❚r❛♥s❢♦r♠❛t✐♦♥ ✭③✉✲ ♠✐♥❞❡st t❤❡♦r❡t✐s❝❤✮ ✈♦♥ ❡✐♥❢❛❝❤❡♥ ♣❤②s✐❦❛❧✐s❝❤❡♥ ❙②st❡♠❡♥ r❡❛❧✐s✐❡rt ✇❡r✲ ❞❡♥✳ ❉✐❡ ❩❛❤❧ ❞r❡✐ ❤❛❜❡♥ ✇✐r ♥❛❝❤ ♣r❛❦t✐s❝❤❡♥ ●❡s✐❝❤ts♣✉♥❦t❡♥ ❢❡st❣❡❧❡❣t✳ ■♠ Pr✐♥③✐♣ ✐st ❥❡❞❡ ✉♥✐tär❡ ❖♣❡r❛t✐♦♥ ❧♦❦❛❧✱ ❛♥ ❞❡r ❦♦♥st❛♥t ✈✐❡❧❡ ❇✐ts ❜❡t❡✐❧✐❣t s✐♥❞❀ ❞✐❡s❡ ❆♥③❛❤❧ ✇ä❝❤st ❛❧s♦ ♥✐❝❤t ♠✐t ❞❡r ❘❡❣✐st❡r❣röÿ❡✳ ●r✉♥❞sät③❧✐❝❤ ❦ä♠❡ ♠❛♥ ♠✐t ❚r❛♥s❢♦r♠❛t✐♦♥❡♥ ③✇❡✐❡r ❇✐ts ❛✉s✳ ❯♠ ❦❧❛ss✐s❝❤❡ ❘❡❝❤❡♥✈❡r✲ ❢❛❤r❡♥ ✐♥ ◗✉❛♥t❡♥❛❧❣♦r✐t❤♠❡♥ ③✉ ü❜❡r❢ü❤r❡♥✱ ✐st ❥❡❞♦❝❤ ❡✐♥ ❉r❡✐✲❇✐t✲●❛tt❡r ♥öt✐❣ ✭s✐❡❤❡ ❆❜s❝❤♥✐tt ✸✳✺✮✳ ❉❡r❛rt✐❣❡ ❩❡r❧❡❣✉♥❣❡♥ ✈♦♥ ❚r❛♥s❢♦r♠❛t✐♦♥❡♥ s✐♥❞ ❞❛s ❚❤❡♠❛ ❞❡s ❆❜✲ s❝❤♥✐tts ✷✳✼✳ ❲✐r ✉♥t❡rs✉❝❤❡♥ ❞♦rt✱ ✇❛s ❞❡♠ Ü❜❡r❣❛♥❣ ✈♦♠ ❇✐t ③✉♠ ❘❡✲ ❣✐st❡r ♠❛t❤❡♠❛t✐s❝❤ ❡♥ts♣r✐❝❤t❀ ✇❛s ❛❧s♦ ❞❡r ❩✉s❛♠♠❡♥❤❛♥❣ ③✇✐s❝❤❡♥ ❞❡r ❊♥t✇✐❝❦❧✉♥❣ ❡✐♥❡s ❘❡❣✐st❡rs ✉♥❞ ❞❡r ❊♥t✇✐❝❦❧✉♥❣ s❡✐♥❡r ❡✐♥③❡❧♥❡♥ ❇✐ts ✐st✳ ❉❛s ❙t✐❝❤✇♦rt ❞❛③✉ ❧❛✉t❡t ❚❡♥s♦r♣r♦❞✉❦t✳ ❩✉✈♦r ❜❡s♣r❡❝❤❡♥ ✇✐r ✇✐❝❤t✐❣❡ ❊✐❣❡♥s❝❤❛❢t❡♥ ✉♥✐tär❡r ❚r❛♥s❢♦r♠❛t✐♦♥❡♥✿ ▲♦❦❛❧✐tät ✸✷ ✷ ❱♦♠ ❇✐t ③✉♠ ◗✉❛♥t❡♥r❡❣✐st❡r A ✐st ✉♥✐tär✱ ✇❡♥♥ A−1 = A† ❊✐♥❡ ▼❛tr✐① ❣✐❧t✳ ❉❛s ✐st ❡✐♥❡ s❡❤r st❛r❦❡ ❋♦r✲ ❞❡r✉♥❣✱ ✇♦r❛✉s s✐❝❤ ❡✐♥✐❣❡ ✇✐❝❤t✐❣❡ ❊✐❣❡♥s❝❤❛❢t❡♥ ❞❡r ③✉❣❡❤ör✐❣❡♥ ✉♥✐tär❡♥ ❚r❛♥s❢♦r♠❛t✐♦♥ ❡r❣❡❜❡♥✳ ❊✐❣❡♥s❝❤❛❢t❡♥ ✉♥✐tär❡r ❚r❛♥s✲ ❢♦r♠❛t✐♦♥❡♥ ❋ür ✉♥✐tär❡ ❚r❛♥s❢♦r♠❛t✐♦♥❡♥ ❣✐❧t✿ ✶✳ ❏❡❞❡r ③✉❧äss✐❣❡ ❩✉st❛♥❞ ❡✐♥❡s ◗✉❛♥t❡♥❜✐ts ✇✐r❞ ✇✐❡❞❡r ❛✉❢ ❡✐♥❡♥ ③✉❧äss✐❣❡♥ ❩✉st❛♥❞ ❛❜❣❡❜✐❧❞❡t❀ ✉♥✐tär❡ ❚r❛♥s❢♦r♠❛t✐♦♥❡♥ s✐♥❞ ❧ä♥✲ ❣❡♥❡r❤❛❧t❡♥❞✳ ✷✳ ❯♥✐tär❡ ❚r❛♥s❢♦r♠❛t✐♦♥❡♥ ✈❡rä♥❞❡r♥ ❞❛s ❙❦❛❧❛r♣r♦❞✉❦t ③✇❡✐❡r ❩✉st❛♥❞s✈❡❦t♦r❡♥ ♥✐❝❤t✿ ✉♥✐tär❡ ❚r❛♥s❢♦r♠❛t✐♦♥❡♥ s✐♥❞ ✇✐♥❦❡❧❡r✲ ❤❛❧t❡♥❞✳ ●❡♦♠❡tr✐s❝❤ ❜❡tr❛❝❤t❡t✱ ❞r❡❤t ❡✐♥❡ ✉♥✐tär❡ ❚r❛♥s❢♦r♠❛✲ t✐♦♥ ❩✉st❛♥❞s✈❡❦t♦r❡♥ ✉♠ ❞❡♥ 0✲P✉♥❦t ♦❞❡r s♣✐❡❣❡❧t s✐❡ ❛♥ ❡✐♥❡r ❆❝❤s❡ ❞✉r❝❤ ❞❡♥ ❯rs♣r✉♥❣✳ ✸✳ ❯♥✐tär❡ ❚r❛♥s❢♦r♠❛t✐♦♥❡♥ s✐♥❞ ✉♠❦❡❤r❜❛r✳ ❉❛r❛✉s ❢♦❧❣t ❡✐♥❡ ✇✐❝❤✲ t✐❣❡ ❇❡s❝❤rä♥❦✉♥❣ ❢ür ❇❡r❡❝❤♥✉♥❣❡♥ ❡✐♥❡s ◗✉❛♥t❡♥❝♦♠♣✉t❡rs✿ ❥❡✲ ❞❡r ❙❝❤r✐tt ♠✉ss ✉♠❦❡❤r❜❛r s❡✐♥✳ ✹✳ ❯♥✐tär❡ ❚r❛♥s❢♦r♠❛t✐♦♥❡♥ s✐♥❞ ❧✐♥❡❛r✱ ❞❛ s✐❡ ❞✉r❝❤ ▼❛tr✐③❡♥ ❜❡✲ s❝❤r✐❡❜❡♥ ✇❡r❞❡♥ ❦ö♥♥❡♥✱ s✐❡❤❡ ❆✳✸✳ ❙❦❛❧❛r♣r♦❞✉❦t |u , |v ✈❡r✇❡♥❞❡♥ ✇✐r u | v ✳ ❉❛s ✐st ❞✐❡ ❇r❛❦❡t✲◆♦t❛t✐♦♥ ✭✈♦♥ ❡♥❣❧✳ ❜r❛❝❦❡t✮✳ ❋ür n ❦♦♠♣❧❡①❡♥ ❑♦♠♣♦♥❡♥t❡♥ |u = (α0 , .

A2n −1,i |2n − 1 ❛❜❣❡❜✐❧❞❡t✳ ❲✐r ❞❡♠♦♥str✐❡r❡♥ ❞✐❡ ◆✉♠♠❡r✐❡r✉♥❣ ✈♦♥ ❩❡✐❧❡♥ ✉♥❞ ❙♣❛❧t❡♥ ❛♥ ❞❡r ❖♣❡r❛t✐♦♥ CNOT✳ |00 |01 |10 |11 |00 1  0   0 0  |01 0 1 0 0 |10 |11  0 0 0 0   0 1  1 0 ✷✳✺ ◗✉❛♥t❡♥r❡❣✐st❡r ✸✶ ❙♦ s✐❡❤t ♠❛♥ s♦❢♦rt✱ ❞❛ss ❞✐❡s❡ ❖♣❡r❛t✐♦♥ ❞✐❡ ❆♠♣❧✐t✉❞❡♥ ✈♦♥ t❛✉s❝❤t✿    α0 α0  α1   α1   CNOT   α2  =  α3 α3 α2 |10 ✉♥❞ |11   .  ❊s ❤❛♥❞❡❧t s✐❝❤ ✉♠ ❡✐♥❡ P❡r♠✉t❛t✐♦♥s♠❛tr✐①✳ ❇❡✐s♣✐❡❧ ✷✳✻✿ ❲✐r ❜❡tr❛❝❤t❡♥ ▼❛tr✐③❡♥ ✈♦♥ ❞❡r ❢♦❧❣❡♥❞❡♥ ❋♦r♠✳ ■♥ ❥❡❞❡r ❙♣❛❧t❡ ✉♥❞ ✐♥ ❥❡❞❡r ❩❡✐❧❡ ❣✐❜t ❡s ❣❡♥❛✉ ❡✐♥❡ 1✱ ❛❧❧❡ ❛♥❞❡r❡♥ ❊✐♥trä❣❡ s✐♥❞ 0✳ ❉✐❡s❡ ▼❛tr✐③❡♥ ❤❡✐ÿ❡♥ P❡r♠✉t❛t✐♦♥s♠❛tr✐③❡♥✳ ❆♥❣❡✇❛♥❞t ❛✉❢ ❡✐♥❡♥ ❱❡❦t♦r✱ ✈❡rä♥❞❡rt ❡✐♥❡ s♦❧❝❤❡ ▼❛tr✐① ❞✐❡ ❘❡✐❤❡♥❢♦❧❣❡ ❞❡r ❑♦♠♣♦♥❡♥t❡♥✱ ä♥❞❡rt ❛❜❡r ✐❤r❡♥ ❲❡rt ♥✐❝❤t✳ ❆✉❢ ❡✐♥❡♥ ❩✉st❛♥❞s✈❡❦t♦r ❛♥❣❡✇❛♥❞t✱ ✇✐r❞ ❞✐❡ ❩✉♦r❞♥✉♥❣ ✈♦♥ ❆♠♣❧✐t✉❞❡♥ ③✉ ❇❛s✐s③✉stä♥❞❡♥ ❣❡ä♥❞❡rt✳ ❆✉❢❣❛❜❡ ✷✳✾✿ ❩❡✐❣❡♥ ❙✐❡✱ ❞❛ss P❡r♠✉t❛t✐♦♥s♠❛tr✐③❡♥ ✉♥✐tär s✐♥❞✳ ❉✐❡ ❢♦❧❣❡♥❞❡ ❉❡✜♥✐t✐♦♥ ✐st ✇❡s❡♥t❧✐❝❤✿ ❩✇❛r ✇✐ss❡♥ ✇✐r✱ ✇✐❡ s✐❝❤ ❡✐♥ ◗✉❛♥✲ t❡♥r❡❣✐st❡r ❡♥t✇✐❝❦❡❧t✳ ❆❜❡r ✇✐r ❤❛❜❡♥ ♥♦❝❤ ♥✐❝❤t ❣❡s❛❣t✱ ✇❛s ❡✐♥ ❡✐♥❢❛❝❤❡r ❘❡❝❤❡♥s❝❤r✐tt ✐st✳ ❊✐♥❢❛❝❤❡ ❘❡❝❤❡♥s❝❤r✐tt❡ ❛✉❢ ❡✐♥❡♠ ◗✉❛♥t❡♥r❡❣✐st❡r s✐♥❞ ❧♦❦❛❧❡ ✉♥✐tär❡ ❚r❛♥s❢♦r♠❛t✐♦♥❡♥✳ ▲♦❦❛❧ s✐♥❞ ❚r❛♥s❢♦r♠❛t✐♦♥❡♥✱ ❛♥ ❞❡♥❡♥ ❤ö❝❤st❡♥s ❞r❡✐ ❇✐ts ❜❡t❡✐❧✐❣t s✐♥❞✳ ❲❡♥♥ ✇✐r ❛❧s♦ ❡✐♥❡ ✉♥✐tär❡ ❚r❛♥s❢♦r♠❛t✐♦♥ ✈❡r✇❡♥❞❡♥✱ ❞✐❡ ❛✉❢ ❞❡♠ ❣❛♥③❡♥ ❘❡❣✐st❡r ♦♣❡r✐❡rt✱ ✐♥t❡r❡ss✐❡rt ✉♥s ❞✐❡ ❩❡r❧❡❣✉♥❣ ✐♥ ❧♦❦❛❧❡ ❚r❛♥s❢♦r♠❛t✐♦♥❡♥✳ ❉❡r ●r✉♥❞ ❞❛❢ür ✐st ❧❡✐❝❤t ❡✐♥③✉s❡❤❡♥✿ ❊✐♥❡ ❘❡❣✐st❡r♦♣❡r❛t✐♦♥✱ ❞✐❡ ✈♦♥ ❡✐♥❡r 2n × 2n ✲▼❛tr✐① ❜❡s❝❤r✐❡❜❡♥ ✇✐r❞✱ ❦❛♥♥ s❡❤r ❦♦♠♣❧✐③✐❡rt ✉♥❞ ♥✉r ♠✐t ❣r♦ÿ❡♠ ❆✉❢✇❛♥❞ ❛✉s❢ü❤r❜❛r s❡✐♥✳ ❉❛❣❡❣❡♥ ❦❛♥♥ ❡✐♥❡ ❉r❡✐✲❇✐t✲❚r❛♥s❢♦r♠❛t✐♦♥ ✭③✉✲ ♠✐♥❞❡st t❤❡♦r❡t✐s❝❤✮ ✈♦♥ ❡✐♥❢❛❝❤❡♥ ♣❤②s✐❦❛❧✐s❝❤❡♥ ❙②st❡♠❡♥ r❡❛❧✐s✐❡rt ✇❡r✲ ❞❡♥✳ ❉✐❡ ❩❛❤❧ ❞r❡✐ ❤❛❜❡♥ ✇✐r ♥❛❝❤ ♣r❛❦t✐s❝❤❡♥ ●❡s✐❝❤ts♣✉♥❦t❡♥ ❢❡st❣❡❧❡❣t✳ ■♠ Pr✐♥③✐♣ ✐st ❥❡❞❡ ✉♥✐tär❡ ❖♣❡r❛t✐♦♥ ❧♦❦❛❧✱ ❛♥ ❞❡r ❦♦♥st❛♥t ✈✐❡❧❡ ❇✐ts ❜❡t❡✐❧✐❣t s✐♥❞❀ ❞✐❡s❡ ❆♥③❛❤❧ ✇ä❝❤st ❛❧s♦ ♥✐❝❤t ♠✐t ❞❡r ❘❡❣✐st❡r❣röÿ❡✳ ●r✉♥❞sät③❧✐❝❤ ❦ä♠❡ ♠❛♥ ♠✐t ❚r❛♥s❢♦r♠❛t✐♦♥❡♥ ③✇❡✐❡r ❇✐ts ❛✉s✳ ❯♠ ❦❧❛ss✐s❝❤❡ ❘❡❝❤❡♥✈❡r✲ ❢❛❤r❡♥ ✐♥ ◗✉❛♥t❡♥❛❧❣♦r✐t❤♠❡♥ ③✉ ü❜❡r❢ü❤r❡♥✱ ✐st ❥❡❞♦❝❤ ❡✐♥ ❉r❡✐✲❇✐t✲●❛tt❡r ♥öt✐❣ ✭s✐❡❤❡ ❆❜s❝❤♥✐tt ✸✳✺✮✳ ❉❡r❛rt✐❣❡ ❩❡r❧❡❣✉♥❣❡♥ ✈♦♥ ❚r❛♥s❢♦r♠❛t✐♦♥❡♥ s✐♥❞ ❞❛s ❚❤❡♠❛ ❞❡s ❆❜✲ s❝❤♥✐tts ✷✳✼✳ ❲✐r ✉♥t❡rs✉❝❤❡♥ ❞♦rt✱ ✇❛s ❞❡♠ Ü❜❡r❣❛♥❣ ✈♦♠ ❇✐t ③✉♠ ❘❡✲ ❣✐st❡r ♠❛t❤❡♠❛t✐s❝❤ ❡♥ts♣r✐❝❤t❀ ✇❛s ❛❧s♦ ❞❡r ❩✉s❛♠♠❡♥❤❛♥❣ ③✇✐s❝❤❡♥ ❞❡r ❊♥t✇✐❝❦❧✉♥❣ ❡✐♥❡s ❘❡❣✐st❡rs ✉♥❞ ❞❡r ❊♥t✇✐❝❦❧✉♥❣ s❡✐♥❡r ❡✐♥③❡❧♥❡♥ ❇✐ts ✐st✳ ❉❛s ❙t✐❝❤✇♦rt ❞❛③✉ ❧❛✉t❡t ❚❡♥s♦r♣r♦❞✉❦t✳ ❩✉✈♦r ❜❡s♣r❡❝❤❡♥ ✇✐r ✇✐❝❤t✐❣❡ ❊✐❣❡♥s❝❤❛❢t❡♥ ✉♥✐tär❡r ❚r❛♥s❢♦r♠❛t✐♦♥❡♥✿ ▲♦❦❛❧✐tät ✸✷ ✷ ❱♦♠ ❇✐t ③✉♠ ◗✉❛♥t❡♥r❡❣✐st❡r A ✐st ✉♥✐tär✱ ✇❡♥♥ A−1 = A† ❊✐♥❡ ▼❛tr✐① ❣✐❧t✳ ❉❛s ✐st ❡✐♥❡ s❡❤r st❛r❦❡ ❋♦r✲ ❞❡r✉♥❣✱ ✇♦r❛✉s s✐❝❤ ❡✐♥✐❣❡ ✇✐❝❤t✐❣❡ ❊✐❣❡♥s❝❤❛❢t❡♥ ❞❡r ③✉❣❡❤ör✐❣❡♥ ✉♥✐tär❡♥ ❚r❛♥s❢♦r♠❛t✐♦♥ ❡r❣❡❜❡♥✳ ❊✐❣❡♥s❝❤❛❢t❡♥ ✉♥✐tär❡r ❚r❛♥s✲ ❢♦r♠❛t✐♦♥❡♥ ❋ür ✉♥✐tär❡ ❚r❛♥s❢♦r♠❛t✐♦♥❡♥ ❣✐❧t✿ ✶✳ ❏❡❞❡r ③✉❧äss✐❣❡ ❩✉st❛♥❞ ❡✐♥❡s ◗✉❛♥t❡♥❜✐ts ✇✐r❞ ✇✐❡❞❡r ❛✉❢ ❡✐♥❡♥ ③✉❧äss✐❣❡♥ ❩✉st❛♥❞ ❛❜❣❡❜✐❧❞❡t❀ ✉♥✐tär❡ ❚r❛♥s❢♦r♠❛t✐♦♥❡♥ s✐♥❞ ❧ä♥✲ ❣❡♥❡r❤❛❧t❡♥❞✳ ✷✳ ❯♥✐tär❡ ❚r❛♥s❢♦r♠❛t✐♦♥❡♥ ✈❡rä♥❞❡r♥ ❞❛s ❙❦❛❧❛r♣r♦❞✉❦t ③✇❡✐❡r ❩✉st❛♥❞s✈❡❦t♦r❡♥ ♥✐❝❤t✿ ✉♥✐tär❡ ❚r❛♥s❢♦r♠❛t✐♦♥❡♥ s✐♥❞ ✇✐♥❦❡❧❡r✲ ❤❛❧t❡♥❞✳ ●❡♦♠❡tr✐s❝❤ ❜❡tr❛❝❤t❡t✱ ❞r❡❤t ❡✐♥❡ ✉♥✐tär❡ ❚r❛♥s❢♦r♠❛✲ t✐♦♥ ❩✉st❛♥❞s✈❡❦t♦r❡♥ ✉♠ ❞❡♥ 0✲P✉♥❦t ♦❞❡r s♣✐❡❣❡❧t s✐❡ ❛♥ ❡✐♥❡r ❆❝❤s❡ ❞✉r❝❤ ❞❡♥ ❯rs♣r✉♥❣✳ ✸✳ ❯♥✐tär❡ ❚r❛♥s❢♦r♠❛t✐♦♥❡♥ s✐♥❞ ✉♠❦❡❤r❜❛r✳ ❉❛r❛✉s ❢♦❧❣t ❡✐♥❡ ✇✐❝❤✲ t✐❣❡ ❇❡s❝❤rä♥❦✉♥❣ ❢ür ❇❡r❡❝❤♥✉♥❣❡♥ ❡✐♥❡s ◗✉❛♥t❡♥❝♦♠♣✉t❡rs✿ ❥❡✲ ❞❡r ❙❝❤r✐tt ♠✉ss ✉♠❦❡❤r❜❛r s❡✐♥✳ ✹✳ ❯♥✐tär❡ ❚r❛♥s❢♦r♠❛t✐♦♥❡♥ s✐♥❞ ❧✐♥❡❛r✱ ❞❛ s✐❡ ❞✉r❝❤ ▼❛tr✐③❡♥ ❜❡✲ s❝❤r✐❡❜❡♥ ✇❡r❞❡♥ ❦ö♥♥❡♥✱ s✐❡❤❡ ❆✳✸✳ ❙❦❛❧❛r♣r♦❞✉❦t |u , |v ✈❡r✇❡♥❞❡♥ ✇✐r u | v ✳ ❉❛s ✐st ❞✐❡ ❇r❛❦❡t✲◆♦t❛t✐♦♥ ✭✈♦♥ ❡♥❣❧✳ ❜r❛❝❦❡t✮✳ ❋ür n ❦♦♠♣❧❡①❡♥ ❑♦♠♣♦♥❡♥t❡♥ |u = (α0 , .

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