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❋♦❝❦ s♣❛❝❡s ♦❢ ❨❛♥❣✲❇❛①t❡r t②♣❡ ❍❡❝❦❡ ♦♣❡r❛t♦rs ❛♥❞ ♣♦s✐t✐✈✐t② ♦❢ ❚ ✲s②♠♠❡tr✐③❛t♦rs ❈♦♥♥❡❝t✐♦♥s ♦❢ ❲♦r♦♥♦✇✐❝③✲P✉s③ ♦♣❡r❛t♦rs ❚ ❈❆❘ µ ◆♦♥✲❝♦♠♠✉t❛t✐✈❡ ▲❡✈② ♣r♦❝❡ss ❢♦r ❣❡♥❡r❛❧✐③❡❞ ✧❆◆❨❖◆✧ ❉❡❢♦r♠❡❞ ❋♦❝❦ s♣❛❝❡s✱ ❍❡❝❦❡ ♦♣❡r❛t♦rs ❛♥❞ ♠♦♥♦t♦♥❡ ❋♦❝❦ s♣❛❝❡ ♦❢ ▼✉r❛❦✐ ▼❛r❡❦ ❇♦➺❡❥❦♦ ❯♥✐✈❡r✐st② ♦❢ ❲r♦❝➟❛✇✱ ■♥st✐t✉t❡ ♦❢ ▼❛t❤❡♠❛t✐❝s ▼✳ ❇♦➺❡❥❦♦ ❉❡❢♦r♠❡❞ ❋♦❝❦ s♣❛❝❡s✱ ❍❡❝❦❡ ♦♣❡r❛t♦rs✳✳✳

❋♦❝❦ s♣❛❝❡s ♦❢ ❨❛♥❣✲❇❛①t❡r t②♣❡ ❍❡❝❦❡ ♦♣❡r❛t♦rs ❛♥❞ ♣♦s✐t✐✈✐t② ♦❢ ❚ ✲s②♠♠❡tr✐③❛t♦rs ❈♦♥♥❡❝t✐♦♥s ♦❢ ❲♦r♦♥♦✇✐❝③✲P✉s③ ♦♣❡r❛t♦rs ❚ ❈❆❘ µ ◆♦♥✲❝♦♠♠✉t❛t✐✈❡ ▲❡✈② ♣r♦❝❡ss ❢♦r ❣❡♥❡r❛❧✐③❡❞ ✧❆◆❨❖◆✧ P❧❛♥ ♦❢ t❤❡ ♣r❡s❡♥t❛t✐♦♥ ❋♦❝❦ s♣❛❝❡s ♦❢ ❨❛♥❣✲❇❛①t❡r t②♣❡ ✶ ❍❡❝❦❡ ♦♣❡r❛t♦r P♦s✐t✐✈✐t② ♦❢ ❚✲s②♠♠❡tr✐③❛t♦rs ❍❡❝❦❡ ♦♣❡r❛t♦rs ❛♥❞ ♣♦s✐t✐✈✐t② ♦❢ ❚ ✲s②♠♠❡tr✐③❛t♦rs ✷ P♦s✐t✐✈✐t② ♦❢ P ( ♥ ) ❢♦r ❨❛♥❣✲❇❛①t❡r✲❍❡❝❦❡ ♦♣❡r❛t♦rs ❚ ❈♦♥♥❡❝t✐♦♥s ♦❢ ❲♦r♦♥♦✇✐❝③✲P✉s③ ♦♣❡r❛t♦rs ❚ ❈❆❘ ✸ µ ◆♦♥✲❝♦♠♠✉t❛t✐✈❡ ▲❡✈② ♣r♦❝❡ss ❢♦r ❣❡♥❡r❛❧✐③❡❞ ✧❆◆❨❖◆✧ ✹ ▼✳ ❇♦➺❡❥❦♦ ❉❡❢♦r♠❡❞ ❋♦❝❦ s♣❛❝❡s✱ ❍❡❝❦❡ ♦♣❡r❛t♦rs✳✳✳

❋♦❝❦ s♣❛❝❡s ♦❢ ❨❛♥❣✲❇❛①t❡r t②♣❡ ❍❡❝❦❡ ♦♣❡r❛t♦rs ❛♥❞ ♣♦s✐t✐✈✐t② ♦❢ ❚ ✲s②♠♠❡tr✐③❛t♦rs ❈♦♥♥❡❝t✐♦♥s ♦❢ ❲♦r♦♥♦✇✐❝③✲P✉s③ ♦♣❡r❛t♦rs ❚ ❈❆❘ µ ◆♦♥✲❝♦♠♠✉t❛t✐✈❡ ▲❡✈② ♣r♦❝❡ss ❢♦r ❣❡♥❡r❛❧✐③❡❞ ✧❆◆❨❖◆✧ ■♥ t❤✐s t❛❧❦ ✇❡ ✇✐❧❧ ♣r❡s❡♥t t❤❡ ❢♦❧❧♦✇✐♥❣ s✉❜❥❡❝ts✿ ✶ ❋♦❝❦ s♣❛❝❡s ♦❢ ❨❛♥❣✲❇❛①t❡r t②♣❡✳ ❍❡❝❦❡ ♦♣❡r❛t♦rs ❛♥❞ ♣♦s✐t✐✈✐t② ♦❢ ❚✲s②♠♠❡tr✐③❛t♦rs✳ ✷ ❈♦♥♥❡❝t✐♦♥s ♦❢ ❲♦r♦♥♦✇✐❝③✲P✉s③ ♦♣❡r❛t♦rs ❚ ❈❆❘ ✇✐t❤ µ ♠♦♥♦t♦♥❡ ❋♦❝❦ s♣❛❝❡ ♦❢ ▼✉r❛❦✐✲▲✉ ✭ µ = ✵✮✳ ✸ ◆♦♥✲❝♦♠♠✉t❛t✐✈❡ ▲❡✈② ♣r♦❝❡ss ❢♦r ❣❡♥❡r❛❧✐③❡❞ ✧❆◆❨❖◆✧ st❛t✐st✐❝s✳ ▼✳ ❇♦➺❡❥❦♦ ❉❡❢♦r♠❡❞ ❋♦❝❦ s♣❛❝❡s✱ ❍❡❝❦❡ ♦♣❡r❛t♦rs✳✳✳

❋♦❝❦ s♣❛❝❡s ♦❢ ❨❛♥❣✲❇❛①t❡r t②♣❡ ❍❡❝❦❡ ♦♣❡r❛t♦rs ❛♥❞ ♣♦s✐t✐✈✐t② ♦❢ ❚ ✲s②♠♠❡tr✐③❛t♦rs ❍❡❝❦❡ ♦♣❡r❛t♦r ❈♦♥♥❡❝t✐♦♥s ♦❢ ❲♦r♦♥♦✇✐❝③✲P✉s③ ♦♣❡r❛t♦rs ❚ ❈❆❘ P♦s✐t✐✈✐t② ♦❢ ❚✲s②♠♠❡tr✐③❛t♦rs µ ◆♦♥✲❝♦♠♠✉t❛t✐✈❡ ▲❡✈② ♣r♦❝❡ss ❢♦r ❣❡♥❡r❛❧✐③❡❞ ✧❆◆❨❖◆✧ ❋♦❝❦ s♣❛❝❡s ♦❢ ❨❛♥❣✲❇❛①t❡r t②♣❡ ▲❡t ❚ : H ⊗ H → H ⊗ H ❜❡ ❛ ❨❛♥❣✲❇❛①t❡r ♦♣❡r❛t♦r✱ ✐✳❡✳ ❚ = ❚ ∗ , ❚ ✶ ❚ ✷ ❚ ✶ = ❚ ✷ ❚ ✶ ❚ ✷ , ❚ ≥ − ■ ♦♥ H ⊗ H ⊗ H ✱ ✇❤❡r❡ ❚ ✶ = ❚ ⊗ ■ ✱ ❚ ✷ = ■ ⊗ ❚ ✳ ❲❡ ❞❡✜♥❡ t❤❡ ❚✲s②♠♠❡tr✐③❛t♦r ♦♣❡r❛t♦r : H ⊗ ♥ → H ⊗ ♥ P ( ♥ ) ❚ ( ❚ ✶ , ❚ ✷ , . . . , ❚ ♥ − ✶ ) = P ( ♥ ) ❚ ❛s ❢♦❧❧♦✇s✿ P ( ♥ ) = ( ✶ + ❚ ✶ + ❚ ✷ ❚ ✶ + ❚ ✸ ❚ ✷ ❚ ✶ + . . . ❚ . . . + ❚ ♥ − ✶ . . . ❚ ✶ ) P ( ♥ − ✶ ) ( ❚ ✷ , ❚ ✸ , . . . , ❚ ♥ − ✶ ) , ❚ ▼✳ ❇♦➺❡❥❦♦ ❉❡❢♦r♠❡❞ ❋♦❝❦ s♣❛❝❡s✱ ❍❡❝❦❡ ♦♣❡r❛t♦rs✳✳✳

❋♦❝❦ s♣❛❝❡s ♦❢ ❨❛♥❣✲❇❛①t❡r t②♣❡ ❍❡❝❦❡ ♦♣❡r❛t♦rs ❛♥❞ ♣♦s✐t✐✈✐t② ♦❢ ❚ ✲s②♠♠❡tr✐③❛t♦rs ❍❡❝❦❡ ♦♣❡r❛t♦r ❈♦♥♥❡❝t✐♦♥s ♦❢ ❲♦r♦♥♦✇✐❝③✲P✉s③ ♦♣❡r❛t♦rs ❚ ❈❆❘ P♦s✐t✐✈✐t② ♦❢ ❚✲s②♠♠❡tr✐③❛t♦rs µ ◆♦♥✲❝♦♠♠✉t❛t✐✈❡ ▲❡✈② ♣r♦❝❡ss ❢♦r ❣❡♥❡r❛❧✐③❡❞ ✧❆◆❨❖◆✧ ✇❤❡r❡ P ( ✶ ) = ✶✱ P ( ✷ ) = ✶ + ❚ ✶ ❛♥❞ ❚ ❚ : H ⊗ ♥ → H ⊗ ♥ . ❚ ✐ = ✶ ⊗ · · · ⊗ ✶ ⊗ ❚ ⊗ ✶ ⊗ · · · ⊗ ✶ � �� � � �� � ✐ − ✶ t✐♠❡s ♥ − ✐ − ✶ t✐♠❡s ❯♥❞❡r t❤❡ ❛ss✉♠♣t✐♦♥ t❤❛t � ❚ � ≤ ✶ ✇❡ ♣r♦✈❡❞ ✭s❡❡ ❇♦➺❡❥❦♦✱ ❙♣❡✐❝❤❡r ✶✾✾✶✱ ✶✾✾✹ ✮ ✱ t❤❛t P ( ♥ ) ≥ ✵ ❢♦r ❡❛❝❤ ♥ ❛♥❞ ❤❡♥❝❡ ✇❡ ❝❛♥ ❢♦r♠ ❛ ♥❡✇ ♣r❡✲s❝❛❧❛r ❚ ♣r♦❞✉❝t ♦♥ H ⊗ ♥ ❛s ❢♦❧❧♦✇s✿ ❢♦r ξ, η ∈ H ⊗ ♥ � ξ | η � ❚ := � P ( ♥ ) ❚ ξ | η � , ✇❤❡r❡ �·|·� ✐s t❤❡ ♥❛t✉r❛❧ s❝❛❧❛r ♣r♦❞✉❝t ♦♥ H ⊗ ♥ ✳ ▼✳ ❇♦➺❡❥❦♦ ❉❡❢♦r♠❡❞ ❋♦❝❦ s♣❛❝❡s✱ ❍❡❝❦❡ ♦♣❡r❛t♦rs✳✳✳

❋♦❝❦ s♣❛❝❡s ♦❢ ❨❛♥❣✲❇❛①t❡r t②♣❡ ❍❡❝❦❡ ♦♣❡r❛t♦rs ❛♥❞ ♣♦s✐t✐✈✐t② ♦❢ ❚ ✲s②♠♠❡tr✐③❛t♦rs ❍❡❝❦❡ ♦♣❡r❛t♦r ❈♦♥♥❡❝t✐♦♥s ♦❢ ❲♦r♦♥♦✇✐❝③✲P✉s③ ♦♣❡r❛t♦rs ❚ ❈❆❘ P♦s✐t✐✈✐t② ♦❢ ❚✲s②♠♠❡tr✐③❛t♦rs µ ◆♦♥✲❝♦♠♠✉t❛t✐✈❡ ▲❡✈② ♣r♦❝❡ss ❢♦r ❣❡♥❡r❛❧✐③❡❞ ✧❆◆❨❖◆✧ ❚❤❡♥ ✇❡ ❝❛♥ ❢♦r♠ t❤❡ ❝r❡❛t✐♦♥ ♦♣❡r❛t♦r ❛ + ( ❢ ) ξ = ❢ ⊗ ξ ❛♥❞ t❤❡ ❛♥♥✐❤✐❧❛t✐♦♥ ♦♣❡r❛t♦r ❢♦r ξ ∈ H ⊗ ♥ , ❛ ( ❢ ) ξ = ❧ ( ❢ )( ✶ + ❚ ✶ + ❚ ✶ ❚ ✷ + . . . + ❚ ✶ ❚ ✷ . . . ❚ ♥ − ✶ ) ξ ✇❤❡r❡ ❧ ( ❢ ) ✐s t❤❡ ❢r❡❡ ❛♥♥✐❤✐❧❛t✐♦♥ ♦♣❡r❛t♦r ❞❡✜♥❡❞ ❛s ❧ ( ❢ )( ① ✶ ⊗ . . . ⊗ ① ♥ ) = � ❢ | ① ✶ � ① ✷ ⊗ . . . ⊗ ① ♥ . ❚❤❡ ♠❛✐♥ ♦❜❥❡❝t ♦❢ t❤✐s ♥♦t❡ ✐s t❤❡ str✉❝t✉r❡ ♦❢ t❤❡ ✈♦♥ ◆❡✉♠❛♥♥ ❛❧❣❡❜r❛ Γ ❚ ( H ) = { ● ❚ ( ❢ ) : ❢ ∈ H R } ′′ ❣❡♥❡r❛t❡❞ ❜② t❤❡ ❚ ✲●❛✉ss✐❛♥ ✜❡❧❞ ● ❚ ( ❢ ) = ❛ + ( ❢ ) + ❛ ( ❢ ) ✱ ✇❤❡r❡ H R ❞❡♥♦t❡s t❤❡ r❡❛❧ ♣❛rt ♦❢ H ✳ ▼✳ ❇♦➺❡❥❦♦ ❉❡❢♦r♠❡❞ ❋♦❝❦ s♣❛❝❡s✱ ❍❡❝❦❡ ♦♣❡r❛t♦rs✳✳✳

❋♦❝❦ s♣❛❝❡s ♦❢ ❨❛♥❣✲❇❛①t❡r t②♣❡ ❍❡❝❦❡ ♦♣❡r❛t♦rs ❛♥❞ ♣♦s✐t✐✈✐t② ♦❢ ❚ ✲s②♠♠❡tr✐③❛t♦rs ❍❡❝❦❡ ♦♣❡r❛t♦r ❈♦♥♥❡❝t✐♦♥s ♦❢ ❲♦r♦♥♦✇✐❝③✲P✉s③ ♦♣❡r❛t♦rs ❚ ❈❆❘ P♦s✐t✐✈✐t② ♦❢ ❚✲s②♠♠❡tr✐③❛t♦rs µ ◆♦♥✲❝♦♠♠✉t❛t✐✈❡ ▲❡✈② ♣r♦❝❡ss ❢♦r ❣❡♥❡r❛❧✐③❡❞ ✧❆◆❨❖◆✧ ❆s ✇❛s s❤♦✇♥ ✐♥ ♠❛♥② ♣❛♣❡rs✲ ❱♦✐❝✉❧❡s❝✉✱ ❇♦➺❡❥❦♦✲❙♣❡✐❝❤❡r✱ ❘✐❝❛r❞✱ ◆♦✉ t❤❛t t❤✐s ✈♦♥ ◆❡✉♠❛♥♥ ❛❧❣❡❜r❛ ✐s ❛ ♥♦♥✲✐♥❥❡❝t✐✈❡ ■■ ✶ ✲❢❛❝t♦r✳ ❚❤❡ ❧✐♥❡❛r s♣❛♥ ♦❢ ❚ ✲●❛✉ss✐❛♥ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s ✐s ❝♦♠♣❧❡t❡❧② ✐s♦♠♦r♣❤✐❝ t♦ t❤❡ ♦♣❡r❛t♦r s♣❛❝❡ ❝❛❧❧❡❞ r♦✇ ❛♥❞ ❝♦❧✉♠♥✱ ❛s ✇❡ ✇✐❧❧ s❤♦✇ ❧❛t❡r✳ ❚❤✐s ✐s ❛♥ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ r❡s✉❧ts ♦❢ ❍❛❛❣❡r✉♣ ❛♥❞ P✐s✐❡r ✱ ❆✳ ❇✉❝❤❤♦❧③ ❛♥❞ ♦❢ ♦✉r r❡s✉❧ts ✇✐t❤ ❘✳❙♣❡✐❝❤❡r✳ ▼✳ ❇♦➺❡❥❦♦ ❉❡❢♦r♠❡❞ ❋♦❝❦ s♣❛❝❡s✱ ❍❡❝❦❡ ♦♣❡r❛t♦rs✳✳✳