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❆ ❝❤❛♥♥❡❧ s❡♥❞s ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥s t♦ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥s✱ ✐✳❡✳ ✐s ❛ ♣♦s✐t✐✈❡ ❧✐♥❡❛r ♠❛♣ ❢r♦♠ ✶ t♦ ✶ s✉❝❤ t❤❛t ❤♦❧❞s ❢♦r t❤❡ ❧✐♥❡r ❢✉♥❝t✐♦♥❛❧ ✳ ✶ ❚❤❡ ♠✐♥✐♠❛❧ ❡♥tr♦♣② ♠✐♥ ✐s ❛ ♠❡❛s✉r❡ ♦❢ ❤♦✇ ♥♦✐s② ❛ ❝❤❛♥♥❡❧ ✐s✳ ✵ ♠✐♥ ❊♥tr♦♣② ❢♦r ❝❧❛ss✐❝❛❧ ❝❤❛♥♥❡❧s ❋♦r ❛ ❞✐s❝r❡t❡ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥ � k p k = ✶ t❤❡ ❡♥tr♦♣② ✐s ❣✐✈❡♥ ❜② � H ( p ) = − p k ❧♦❣ ✷ p k k ❆❝❝♦r❞✐♥❣ t♦ ❙❤❛♥♥♦♥✱ t❤❡ ❡♥tr♦♣② ✐s t❤❡ r❛t❡ rn ♦❢ ❜✐ts ✇❤✐❝❤ ❝❛♥ ❜❡ ❡①tr❛❝t❡❞ ❢r♦♠ n ❝♦♣✐❡s ♦❢ t❤❡ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥✱ ✐✳❡✳ t❤❡ ✉♥✐❢♦r♠ ❞✐str✐❜✉t✐♦♥ ♦❢ { ✵ , ✶ } rn ✐s r❡❛❧✐③❡❞ ♦♥ ❛ s❡t ✇✐t❤ ❧❛r❣❡ ♣r♦❜❛❜✐❧✐t②✳ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

❚❤❡ ♠✐♥✐♠❛❧ ❡♥tr♦♣② ♠✐♥ ✐s ❛ ♠❡❛s✉r❡ ♦❢ ❤♦✇ ♥♦✐s② ❛ ❝❤❛♥♥❡❧ ✐s✳ ✵ ♠✐♥ ✐✳❡✳ ✐s ❛ ♣♦s✐t✐✈❡ ❧✐♥❡❛r ♠❛♣ ❢r♦♠ ✶ t♦ ✶ s✉❝❤ t❤❛t ❤♦❧❞s ❢♦r t❤❡ ❧✐♥❡r ❢✉♥❝t✐♦♥❛❧ ✳ ✶ ❊♥tr♦♣② ❢♦r ❝❧❛ss✐❝❛❧ ❝❤❛♥♥❡❧s ❋♦r ❛ ❞✐s❝r❡t❡ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥ � k p k = ✶ t❤❡ ❡♥tr♦♣② ✐s ❣✐✈❡♥ ❜② � H ( p ) = − p k ❧♦❣ ✷ p k k ❆❝❝♦r❞✐♥❣ t♦ ❙❤❛♥♥♦♥✱ t❤❡ ❡♥tr♦♣② ✐s t❤❡ r❛t❡ rn ♦❢ ❜✐ts ✇❤✐❝❤ ❝❛♥ ❜❡ ❡①tr❛❝t❡❞ ❢r♦♠ n ❝♦♣✐❡s ♦❢ t❤❡ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥✱ ✐✳❡✳ t❤❡ ✉♥✐❢♦r♠ ❞✐str✐❜✉t✐♦♥ ♦❢ { ✵ , ✶ } rn ✐s r❡❛❧✐③❡❞ ♦♥ ❛ s❡t ✇✐t❤ ❧❛r❣❡ ♣r♦❜❛❜✐❧✐t②✳ ❆ ❝❤❛♥♥❡❧ Φ s❡♥❞s ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥s t♦ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥s✱ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

❚❤❡ ♠✐♥✐♠❛❧ ❡♥tr♦♣② ♠✐♥ ✐s ❛ ♠❡❛s✉r❡ ♦❢ ❤♦✇ ♥♦✐s② ❛ ❝❤❛♥♥❡❧ ✐s✳ ✵ ♠✐♥ ✐✳❡✳ ✐s ❛ ♣♦s✐t✐✈❡ ❧✐♥❡❛r ♠❛♣ ❢r♦♠ ✶ t♦ ✶ s✉❝❤ t❤❛t ❤♦❧❞s ❢♦r t❤❡ ❧✐♥❡r ❢✉♥❝t✐♦♥❛❧ ✳ ✶ ❊♥tr♦♣② ❢♦r ❝❧❛ss✐❝❛❧ ❝❤❛♥♥❡❧s ❋♦r ❛ ❞✐s❝r❡t❡ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥ � k p k = ✶ t❤❡ ❡♥tr♦♣② ✐s ❣✐✈❡♥ ❜② � H ( p ) = − p k ❧♦❣ ✷ p k k ❆❝❝♦r❞✐♥❣ t♦ ❙❤❛♥♥♦♥✱ t❤❡ ❡♥tr♦♣② ✐s t❤❡ r❛t❡ rn ♦❢ ❜✐ts ✇❤✐❝❤ ❝❛♥ ❜❡ ❡①tr❛❝t❡❞ ❢r♦♠ n ❝♦♣✐❡s ♦❢ t❤❡ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥✱ ✐✳❡✳ t❤❡ ✉♥✐❢♦r♠ ❞✐str✐❜✉t✐♦♥ ♦❢ { ✵ , ✶ } rn ✐s r❡❛❧✐③❡❞ ♦♥ ❛ s❡t ✇✐t❤ ❧❛r❣❡ ♣r♦❜❛❜✐❧✐t②✳ ❆ ❝❤❛♥♥❡❧ Φ s❡♥❞s ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥s t♦ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥s✱ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

❚❤❡ ♠✐♥✐♠❛❧ ❡♥tr♦♣② ♠✐♥ ✐s ❛ ♠❡❛s✉r❡ ♦❢ ❤♦✇ ♥♦✐s② ❛ ❝❤❛♥♥❡❧ ✐s✳ ✵ ♠✐♥ ❊♥tr♦♣② ❢♦r ❝❧❛ss✐❝❛❧ ❝❤❛♥♥❡❧s ❋♦r ❛ ❞✐s❝r❡t❡ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥ � k p k = ✶ t❤❡ ❡♥tr♦♣② ✐s ❣✐✈❡♥ ❜② � H ( p ) = − p k ❧♦❣ ✷ p k k ❆❝❝♦r❞✐♥❣ t♦ ❙❤❛♥♥♦♥✱ t❤❡ ❡♥tr♦♣② ✐s t❤❡ r❛t❡ rn ♦❢ ❜✐ts ✇❤✐❝❤ ❝❛♥ ❜❡ ❡①tr❛❝t❡❞ ❢r♦♠ n ❝♦♣✐❡s ♦❢ t❤❡ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥✱ ✐✳❡✳ t❤❡ ✉♥✐❢♦r♠ ❞✐str✐❜✉t✐♦♥ ♦❢ { ✵ , ✶ } rn ✐s r❡❛❧✐③❡❞ ♦♥ ❛ s❡t ✇✐t❤ ❧❛r❣❡ ♣r♦❜❛❜✐❧✐t②✳ ❆ ❝❤❛♥♥❡❧ Φ s❡♥❞s ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥s t♦ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥s✱ ✐✳❡✳ Φ ✐s ❛ ♣♦s✐t✐✈❡ ❧✐♥❡❛r ♠❛♣ ❢r♦♠ ℓ n ✶ t♦ ℓ m ✶ s✉❝❤ t❤❛t Σ m ◦ Φ = Σ n ❤♦❧❞s ❢♦r t❤❡ ❧✐♥❡r ❢✉♥❝t✐♦♥❛❧ Σ n ( p ) = � n k = ✶ p k ✳ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

✵ ♠✐♥ ❊♥tr♦♣② ❢♦r ❝❧❛ss✐❝❛❧ ❝❤❛♥♥❡❧s ❋♦r ❛ ❞✐s❝r❡t❡ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥ � k p k = ✶ t❤❡ ❡♥tr♦♣② ✐s ❣✐✈❡♥ ❜② � H ( p ) = − p k ❧♦❣ ✷ p k k ❆❝❝♦r❞✐♥❣ t♦ ❙❤❛♥♥♦♥✱ t❤❡ ❡♥tr♦♣② ✐s t❤❡ r❛t❡ rn ♦❢ ❜✐ts ✇❤✐❝❤ ❝❛♥ ❜❡ ❡①tr❛❝t❡❞ ❢r♦♠ n ❝♦♣✐❡s ♦❢ t❤❡ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥✱ ✐✳❡✳ t❤❡ ✉♥✐❢♦r♠ ❞✐str✐❜✉t✐♦♥ ♦❢ { ✵ , ✶ } rn ✐s r❡❛❧✐③❡❞ ♦♥ ❛ s❡t ✇✐t❤ ❧❛r❣❡ ♣r♦❜❛❜✐❧✐t②✳ ❆ ❝❤❛♥♥❡❧ Φ s❡♥❞s ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥s t♦ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥s✱ ✐✳❡✳ Φ ✐s ❛ ♣♦s✐t✐✈❡ ❧✐♥❡❛r ♠❛♣ ❢r♦♠ ℓ n ✶ t♦ ℓ m ✶ s✉❝❤ t❤❛t Σ m ◦ Φ = Σ n ❤♦❧❞s ❢♦r t❤❡ ❧✐♥❡r ❢✉♥❝t✐♦♥❛❧ Σ n ( p ) = � n k = ✶ p k ✳ ❚❤❡ ♠✐♥✐♠❛❧ ❡♥tr♦♣② H min (Φ) = ♠✐♥ p H (Φ( p ))) ✐s ❛ ♠❡❛s✉r❡ ♦❢ ❤♦✇ ♥♦✐s② ❛ ❝❤❛♥♥❡❧ ✐s✳ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

✵ ♠✐♥ ❊♥tr♦♣② ❢♦r ❝❧❛ss✐❝❛❧ ❝❤❛♥♥❡❧s ❋♦r ❛ ❞✐s❝r❡t❡ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥ � k p k = ✶ t❤❡ ❡♥tr♦♣② ✐s ❣✐✈❡♥ ❜② � H ( p ) = − p k ❧♦❣ ✷ p k k ❆❝❝♦r❞✐♥❣ t♦ ❙❤❛♥♥♦♥✱ t❤❡ ❡♥tr♦♣② ✐s t❤❡ r❛t❡ rn ♦❢ ❜✐ts ✇❤✐❝❤ ❝❛♥ ❜❡ ❡①tr❛❝t❡❞ ❢r♦♠ n ❝♦♣✐❡s ♦❢ t❤❡ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥✱ ✐✳❡✳ t❤❡ ✉♥✐❢♦r♠ ❞✐str✐❜✉t✐♦♥ ♦❢ { ✵ , ✶ } rn ✐s r❡❛❧✐③❡❞ ♦♥ ❛ s❡t ✇✐t❤ ❧❛r❣❡ ♣r♦❜❛❜✐❧✐t②✳ ❆ ❝❤❛♥♥❡❧ Φ s❡♥❞s ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥s t♦ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥s✱ ✐✳❡✳ Φ ✐s ❛ ♣♦s✐t✐✈❡ ❧✐♥❡❛r ♠❛♣ ❢r♦♠ ℓ n ✶ t♦ ℓ m ✶ s✉❝❤ t❤❛t Σ m ◦ Φ = Σ n ❤♦❧❞s ❢♦r t❤❡ ❧✐♥❡r ❢✉♥❝t✐♦♥❛❧ Σ n ( p ) = � n k = ✶ p k ✳ ❚❤❡ ♠✐♥✐♠❛❧ ❡♥tr♦♣② H min (Φ) = ♠✐♥ p H (Φ( p ))) ✐s ❛ ♠❡❛s✉r❡ ♦❢ ❤♦✇ ♥♦✐s② ❛ ❝❤❛♥♥❡❧ ✐s✳ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

❊♥tr♦♣② ❢♦r ❝❧❛ss✐❝❛❧ ❝❤❛♥♥❡❧s ❋♦r ❛ ❞✐s❝r❡t❡ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥ � k p k = ✶ t❤❡ ❡♥tr♦♣② ✐s ❣✐✈❡♥ ❜② � H ( p ) = − p k ❧♦❣ ✷ p k k ❆❝❝♦r❞✐♥❣ t♦ ❙❤❛♥♥♦♥✱ t❤❡ ❡♥tr♦♣② ✐s t❤❡ r❛t❡ rn ♦❢ ❜✐ts ✇❤✐❝❤ ❝❛♥ ❜❡ ❡①tr❛❝t❡❞ ❢r♦♠ n ❝♦♣✐❡s ♦❢ t❤❡ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥✱ ✐✳❡✳ t❤❡ ✉♥✐❢♦r♠ ❞✐str✐❜✉t✐♦♥ ♦❢ { ✵ , ✶ } rn ✐s r❡❛❧✐③❡❞ ♦♥ ❛ s❡t ✇✐t❤ ❧❛r❣❡ ♣r♦❜❛❜✐❧✐t②✳ ❆ ❝❤❛♥♥❡❧ Φ s❡♥❞s ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥s t♦ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥s✱ ✐✳❡✳ Φ ✐s ❛ ♣♦s✐t✐✈❡ ❧✐♥❡❛r ♠❛♣ ❢r♦♠ ℓ n ✶ t♦ ℓ m ✶ s✉❝❤ t❤❛t Σ m ◦ Φ = Σ n ❤♦❧❞s ❢♦r t❤❡ ❧✐♥❡r ❢✉♥❝t✐♦♥❛❧ Σ n ( p ) = � n k = ✶ p k ✳ ❚❤❡ ♠✐♥✐♠❛❧ ❡♥tr♦♣② H min (Φ) = ♠✐♥ p H (Φ( p ))) ✐s ❛ ♠❡❛s✉r❡ ♦❢ ❤♦✇ ♥♦✐s② ❛ ❝❤❛♥♥❡❧ ✐s✳ H ♠✐♥ ( id ) = ✵ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

▲❡t ❜❡ ❛ ❞✐str✐❜✉t✐♦♥ ♦♥ ❛ ♣r♦❞✉❝t s♣❛❝❡ ❛♥❞ ✱ ❜❡ t❤❡ ♠❛r❣✐♥❛❧s✱ t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ♣r♦❜❛❜✐❧✐t②✳ ❚❤❡♥ ❧♦❣ ✵ ✐s t❤❡ r❡❧❛t✐✈❡ ❡♥tr♦♣② ❛♥❞ ✐s t❤❡ ♠✉t✉❛❧ ✐♥❢♦r♠❛t✐♦♥✳ ❘❡❧❛t✐✈❡ ❊♥tr♦♣② ❛♥❞ ♠✉t✉❛❧ ✐♥❢♦r♠❛t✐♦♥ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

❛♥❞ ✐s t❤❡ ♠✉t✉❛❧ ✐♥❢♦r♠❛t✐♦♥✳ ❚❤❡♥ ❧♦❣ ✵ ✐s t❤❡ r❡❧❛t✐✈❡ ❡♥tr♦♣② ❘❡❧❛t✐✈❡ ❊♥tr♦♣② ❛♥❞ ♠✉t✉❛❧ ✐♥❢♦r♠❛t✐♦♥ ▲❡t p XY ❜❡ ❛ ❞✐str✐❜✉t✐♦♥ ♦♥ ❛ ♣r♦❞✉❝t s♣❛❝❡ ❛♥❞ y p ( x , y ) ✱ p Y = � p X ( x ) = � x p ( x , y ) ❜❡ t❤❡ ♠❛r❣✐♥❛❧s✱ p ( x | y ) = p ( x , y ) / p Y ( y ) t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ♣r♦❜❛❜✐❧✐t②✳ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

❛♥❞ ✐s t❤❡ ♠✉t✉❛❧ ✐♥❢♦r♠❛t✐♦♥✳ ❚❤❡♥ ❧♦❣ ✵ ✐s t❤❡ r❡❧❛t✐✈❡ ❡♥tr♦♣② ❘❡❧❛t✐✈❡ ❊♥tr♦♣② ❛♥❞ ♠✉t✉❛❧ ✐♥❢♦r♠❛t✐♦♥ ▲❡t p XY ❜❡ ❛ ❞✐str✐❜✉t✐♦♥ ♦♥ ❛ ♣r♦❞✉❝t s♣❛❝❡ ❛♥❞ y p ( x , y ) ✱ p Y = � p X ( x ) = � x p ( x , y ) ❜❡ t❤❡ ♠❛r❣✐♥❛❧s✱ p ( x | y ) = p ( x , y ) / p Y ( y ) t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ♣r♦❜❛❜✐❧✐t②✳ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

❛♥❞ ✐s t❤❡ ♠✉t✉❛❧ ✐♥❢♦r♠❛t✐♦♥✳ ❘❡❧❛t✐✈❡ ❊♥tr♦♣② ❛♥❞ ♠✉t✉❛❧ ✐♥❢♦r♠❛t✐♦♥ ▲❡t p XY ❜❡ ❛ ❞✐str✐❜✉t✐♦♥ ♦♥ ❛ ♣r♦❞✉❝t s♣❛❝❡ ❛♥❞ y p ( x , y ) ✱ p Y = � p X ( x ) = � x p ( x , y ) ❜❡ t❤❡ ♠❛r❣✐♥❛❧s✱ p ( x | y ) = p ( x , y ) / p Y ( y ) t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ♣r♦❜❛❜✐❧✐t②✳ ❚❤❡♥ H ( X | Y ) p = H ( p XY ) − H ( p Y ) p Y ( y ) � � = − p ( x | y ) ❧♦❣ ( p ( x | y )) ≥ ✵ x ✐s t❤❡ r❡❧❛t✐✈❡ ❡♥tr♦♣② ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

❘❡❧❛t✐✈❡ ❊♥tr♦♣② ❛♥❞ ♠✉t✉❛❧ ✐♥❢♦r♠❛t✐♦♥ ▲❡t p XY ❜❡ ❛ ❞✐str✐❜✉t✐♦♥ ♦♥ ❛ ♣r♦❞✉❝t s♣❛❝❡ ❛♥❞ y p ( x , y ) ✱ p Y = � p X ( x ) = � x p ( x , y ) ❜❡ t❤❡ ♠❛r❣✐♥❛❧s✱ p ( x | y ) = p ( x , y ) / p Y ( y ) t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ♣r♦❜❛❜✐❧✐t②✳ ❚❤❡♥ H ( X | Y ) p = H ( p XY ) − H ( p Y ) p Y ( y ) � � = − p ( x | y ) ❧♦❣ ( p ( x | y )) ≥ ✵ x ✐s t❤❡ r❡❧❛t✐✈❡ ❡♥tr♦♣② ❛♥❞ I ( X , Y ) p = H ( X )+ H ( Y ) − H ( XY ) = H ( p X )+ H ( p Y ) − H ( p XY ) ✐s t❤❡ ♠✉t✉❛❧ ✐♥❢♦r♠❛t✐♦♥✳ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

❋♦r ❛ ❝❤❛♥♥❡❧ ✶ t❤❡ ❝❛♣❛❝✐t② ✐s t❤❡ r❛t❡ ♦❢ ✶ ❢❛✐t❤❢✉❧❧② tr❛♥s♠✐tt❡❞ ❜✐ts ❢♦r ✉s❡s ♦❢ t❤❡ ❝❤❛♥♥❡❧ ✶ ✶ ❡♥❝♦❞❡r✱ ❞❡❝♦r❞❡r ✷ ✷ ✶ ✶ ❙❤❛♥♥♦♥ ♣r♦✈❡❞ t❤❛t ❢♦r t❤❡ ♦♣t✐♠❛❧ ✇❡ ❤❛✈❡ ✐♥❢ ❧✐♠ ✐♥❢ s✉♣ ✐♥♣✉t ♦✉t♣✉t ▼❛t❤❡♠❛t✐❝❛❧❧② ✭❏P✮ ❧✐♠ ❧♦❣ ❧✐♠ ❧♦❣ ✳ ♠✐♥ ❈❛♣❛❝✐t② ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

❙❤❛♥♥♦♥ ♣r♦✈❡❞ t❤❛t ❢♦r t❤❡ ♦♣t✐♠❛❧ ✇❡ ❤❛✈❡ ✐♥❢ ❧✐♠ ✐♥❢ s✉♣ ✐♥♣✉t ♦✉t♣✉t ▼❛t❤❡♠❛t✐❝❛❧❧② ✭❏P✮ ❧✐♠ ❧♦❣ ❧✐♠ ❧♦❣ ✳ ♠✐♥ ❈❛♣❛❝✐t② ❋♦r ❛ ❝❤❛♥♥❡❧ N : ℓ m ✶ → ℓ m ✶ t❤❡ ❝❛♣❛❝✐t② ✐s t❤❡ r❛t❡ rn ♦❢ ❢❛✐t❤❢✉❧❧② tr❛♥s♠✐tt❡❞ ❜✐ts ❢♦r n ✉s❡s ♦❢ t❤❡ ❝❤❛♥♥❡❧ N ⊗ n ℓ m n ℓ m n → ✶ ✶ ↑ E ↓ D E , D ❡♥❝♦❞❡r✱ ❞❡❝♦r❞❡r ≈ id ℓ ✷ rn ℓ ✷ rn → ✶ ✶ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

▼❛t❤❡♠❛t✐❝❛❧❧② ✭❏P✮ ❧✐♠ ❧♦❣ ❧✐♠ ❧♦❣ ✳ ♠✐♥ ❈❛♣❛❝✐t② ❋♦r ❛ ❝❤❛♥♥❡❧ N : ℓ m ✶ → ℓ m ✶ t❤❡ ❝❛♣❛❝✐t② ✐s t❤❡ r❛t❡ rn ♦❢ ❢❛✐t❤❢✉❧❧② tr❛♥s♠✐tt❡❞ ❜✐ts ❢♦r n ✉s❡s ♦❢ t❤❡ ❝❤❛♥♥❡❧ N ⊗ n ℓ m n ℓ m n → ✶ ✶ ↑ E ↓ D E , D ❡♥❝♦❞❡r✱ ❞❡❝♦r❞❡r ≈ id ℓ ✷ rn ℓ ✷ rn → ✶ ✶ ❙❤❛♥♥♦♥ ♣r♦✈❡❞ t❤❛t ❢♦r t❤❡ ♦♣t✐♠❛❧ r n ( ε ) ✇❡ ❤❛✈❡ ✐♥❢ ε ❧✐♠ ✐♥❢ r n = C ( N ) = s✉♣ p I ( ✐♥♣✉t , ♦✉t♣✉t ) . n ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

❧✐♠ ❧♦❣ ✳ ♠✐♥ ❈❛♣❛❝✐t② ❋♦r ❛ ❝❤❛♥♥❡❧ N : ℓ m ✶ → ℓ m ✶ t❤❡ ❝❛♣❛❝✐t② ✐s t❤❡ r❛t❡ rn ♦❢ ❢❛✐t❤❢✉❧❧② tr❛♥s♠✐tt❡❞ ❜✐ts ❢♦r n ✉s❡s ♦❢ t❤❡ ❝❤❛♥♥❡❧ N ⊗ n ℓ m n ℓ m n → ✶ ✶ ↑ E ↓ D E , D ❡♥❝♦❞❡r✱ ❞❡❝♦r❞❡r ≈ id ℓ ✷ rn ℓ ✷ rn → ✶ ✶ ❙❤❛♥♥♦♥ ♣r♦✈❡❞ t❤❛t ❢♦r t❤❡ ♦♣t✐♠❛❧ r n ( ε ) ✇❡ ❤❛✈❡ ✐♥❢ ε ❧✐♠ ✐♥❢ r n = C ( N ) = s✉♣ p I ( ✐♥♣✉t , ♦✉t♣✉t ) . n ▼❛t❤❡♠❛t✐❝❛❧❧② ✭❏P✮ p →∞ p ❧♦❣ π p ( N ∗ ) C ( N ) = ❧✐♠ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

❈❛♣❛❝✐t② ❋♦r ❛ ❝❤❛♥♥❡❧ N : ℓ m ✶ → ℓ m ✶ t❤❡ ❝❛♣❛❝✐t② ✐s t❤❡ r❛t❡ rn ♦❢ ❢❛✐t❤❢✉❧❧② tr❛♥s♠✐tt❡❞ ❜✐ts ❢♦r n ✉s❡s ♦❢ t❤❡ ❝❤❛♥♥❡❧ N ⊗ n ℓ m n ℓ m n → ✶ ✶ ↑ E ↓ D E , D ❡♥❝♦❞❡r✱ ❞❡❝♦r❞❡r ≈ id ℓ ✷ rn ℓ ✷ rn → ✶ ✶ ❙❤❛♥♥♦♥ ♣r♦✈❡❞ t❤❛t ❢♦r t❤❡ ♦♣t✐♠❛❧ r n ( ε ) ✇❡ ❤❛✈❡ ✐♥❢ ε ❧✐♠ ✐♥❢ r n = C ( N ) = s✉♣ p I ( ✐♥♣✉t , ♦✉t♣✉t ) . n ▼❛t❤❡♠❛t✐❝❛❧❧② ✭❏P✮ p →∞ p ❧♦❣ π p ( N ∗ ) C ( N ) = ❧✐♠ H ♠✐♥ = ❧✐♠ p →∞ p ❧♦❣ �N ∗ : ℓ p → ℓ ∞ � ✳ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

♠✐♥ ♠✐♥ ♠✐♥ ❛♥❞ ❇✉t ♦♥❧② ❢♦r ❝❧❛ss✐❝❛❧ ❝❤❛♥♥❡❧s✦ ❆❞❞✐t✐✈✐t② ❢♦r ❝❧❛ss✐❝❛❧ ❝❤❛♥♥❡❧s ❚❤❡ ♠✐♥✐♠❛❧ ♦✉t♣✉t ❡♥tr♦♣② ❛♥❞ t❤❡ ❝❤❛♥♥❡❧ ❝❛♣❛❝✐t② ❛r❡ ❛❞❞✐t✐✈❡✿ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

❛♥❞ ❇✉t ♦♥❧② ❢♦r ❝❧❛ss✐❝❛❧ ❝❤❛♥♥❡❧s✦ ❆❞❞✐t✐✈✐t② ❢♦r ❝❧❛ss✐❝❛❧ ❝❤❛♥♥❡❧s ❚❤❡ ♠✐♥✐♠❛❧ ♦✉t♣✉t ❡♥tr♦♣② ❛♥❞ t❤❡ ❝❤❛♥♥❡❧ ❝❛♣❛❝✐t② ❛r❡ ❛❞❞✐t✐✈❡✿ H ♠✐♥ (Φ ⊗ Ψ) = H ♠✐♥ (Φ) + H ♠✐♥ (Ψ) ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

❇✉t ♦♥❧② ❢♦r ❝❧❛ss✐❝❛❧ ❝❤❛♥♥❡❧s✦ ❆❞❞✐t✐✈✐t② ❢♦r ❝❧❛ss✐❝❛❧ ❝❤❛♥♥❡❧s ❚❤❡ ♠✐♥✐♠❛❧ ♦✉t♣✉t ❡♥tr♦♣② ❛♥❞ t❤❡ ❝❤❛♥♥❡❧ ❝❛♣❛❝✐t② ❛r❡ ❛❞❞✐t✐✈❡✿ H ♠✐♥ (Φ ⊗ Ψ) = H ♠✐♥ (Φ) + H ♠✐♥ (Ψ) ❛♥❞ C (Φ ⊗ Ψ) = C (Φ) + C (Ψ) . ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

❆❞❞✐t✐✈✐t② ❢♦r ❝❧❛ss✐❝❛❧ ❝❤❛♥♥❡❧s ❚❤❡ ♠✐♥✐♠❛❧ ♦✉t♣✉t ❡♥tr♦♣② ❛♥❞ t❤❡ ❝❤❛♥♥❡❧ ❝❛♣❛❝✐t② ❛r❡ ❛❞❞✐t✐✈❡✿ H ♠✐♥ (Φ ⊗ Ψ) = H ♠✐♥ (Φ) + H ♠✐♥ (Ψ) ❛♥❞ C (Φ ⊗ Ψ) = C (Φ) + C (Ψ) . ❇✉t ♦♥❧② ❢♦r ❝❧❛ss✐❝❛❧ ❝❤❛♥♥❡❧s✦ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

❆ ❝❤❛♥♥❡❧ ✐s ❛ ❝♦♠♣❧❡t❡❧② ♣♦s✐t✐✈❡✱ tr❛❝❡ ♣r❡s❡r✈✐♥❣ ♠❛♣ ✶ ✱ ✇❤❡r❡ ✳ ✶ ✶ ❵❘❡❝❛❧❧✬ ❤❡r❡ t❤❛t ✐s ♣♦s✐t✐✈❡ ✐❢ ✵ ✐♠♣❧✐❡s ✵ ✐♥ s❡♥s❡ ♦❢ ♣♦s✐t✐✈❡ ❞❡✜♥✐t❡ ♠❛tr✐❝❡s✳ ✐s ❝♦♠♣❧❡t❡❧② ♣♦s✐t✐✈❡ ✐❢ ✐s ♣♦s✐t✐✈❡ ❢♦r ❛❧❧ ✭❝♦✉♣❧✐♥❣ ✇✐t❤ ❛♥♦t❤❡r ✶ s②st❡♠✮✳ ❧♥ ✐s t❤❡ ✈♦♥ ◆❡✉♠❛♥♥ ❡♥tr♦♣②✱ ✷ ✶ ✇❤❡r❡ ✐s t❤❡ ✉s✉❛❧ ❙❝❤❛tt❡♥ ✲♥♦r♠✳ ✐♥❢ ✳ ♠✐♥ ✶ ✵ ❍❛st✐♥❣s ✭✷✵✵✽✮ s❤♦✇❡❞ t❤❛t ♠✐♥ ♠✐♥ ♠✐♥ ✐♥ ❣❡♥❡r❛❧✳ ◗✉❛♥t✉♠ ❝❤❛♥♥❡❧ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

❵❘❡❝❛❧❧✬ ❤❡r❡ t❤❛t ✐s ♣♦s✐t✐✈❡ ✐❢ ✵ ✐♠♣❧✐❡s ✵ ✐♥ s❡♥s❡ ♦❢ ♣♦s✐t✐✈❡ ❞❡✜♥✐t❡ ♠❛tr✐❝❡s✳ ✐s ❝♦♠♣❧❡t❡❧② ♣♦s✐t✐✈❡ ✐❢ ✐s ♣♦s✐t✐✈❡ ❢♦r ❛❧❧ ✭❝♦✉♣❧✐♥❣ ✇✐t❤ ❛♥♦t❤❡r ✶ s②st❡♠✮✳ ❧♥ ✐s t❤❡ ✈♦♥ ◆❡✉♠❛♥♥ ❡♥tr♦♣②✱ ✷ ✶ ✇❤❡r❡ ✐s t❤❡ ✉s✉❛❧ ❙❝❤❛tt❡♥ ✲♥♦r♠✳ ✐♥❢ ✳ ♠✐♥ ✶ ✵ ❍❛st✐♥❣s ✭✷✵✵✽✮ s❤♦✇❡❞ t❤❛t ♠✐♥ ♠✐♥ ♠✐♥ ✐♥ ❣❡♥❡r❛❧✳ ◗✉❛♥t✉♠ ❝❤❛♥♥❡❧ ❆ ❝❤❛♥♥❡❧ ✐s ❛ ❝♦♠♣❧❡t❡❧② ♣♦s✐t✐✈❡✱ tr❛❝❡ ♣r❡s❡r✈✐♥❣ ♠❛♣ Φ : S m ✶ → S m ✶ ✱ ✇❤❡r❡ S m ✶ = ( M m ) ∗ ✳ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

❵❘❡❝❛❧❧✬ ❤❡r❡ t❤❛t ✐s ♣♦s✐t✐✈❡ ✐❢ ✵ ✐♠♣❧✐❡s ✵ ✐♥ s❡♥s❡ ♦❢ ♣♦s✐t✐✈❡ ❞❡✜♥✐t❡ ♠❛tr✐❝❡s✳ ✐s ❝♦♠♣❧❡t❡❧② ♣♦s✐t✐✈❡ ✐❢ ✐s ♣♦s✐t✐✈❡ ❢♦r ❛❧❧ ✭❝♦✉♣❧✐♥❣ ✇✐t❤ ❛♥♦t❤❡r ✶ s②st❡♠✮✳ ❧♥ ✐s t❤❡ ✈♦♥ ◆❡✉♠❛♥♥ ❡♥tr♦♣②✱ ✷ ✶ ✇❤❡r❡ ✐s t❤❡ ✉s✉❛❧ ❙❝❤❛tt❡♥ ✲♥♦r♠✳ ✐♥❢ ✳ ♠✐♥ ✶ ✵ ❍❛st✐♥❣s ✭✷✵✵✽✮ s❤♦✇❡❞ t❤❛t ♠✐♥ ♠✐♥ ♠✐♥ ✐♥ ❣❡♥❡r❛❧✳ ◗✉❛♥t✉♠ ❝❤❛♥♥❡❧ ❆ ❝❤❛♥♥❡❧ ✐s ❛ ❝♦♠♣❧❡t❡❧② ♣♦s✐t✐✈❡✱ tr❛❝❡ ♣r❡s❡r✈✐♥❣ ♠❛♣ Φ : S m ✶ → S m ✶ ✱ ✇❤❡r❡ S m ✶ = ( M m ) ∗ ✳ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

❧♥ ✐s t❤❡ ✈♦♥ ◆❡✉♠❛♥♥ ❡♥tr♦♣②✱ ✷ ✶ ✇❤❡r❡ ✐s t❤❡ ✉s✉❛❧ ❙❝❤❛tt❡♥ ✲♥♦r♠✳ ✐♥❢ ✳ ♠✐♥ ✶ ✵ ❍❛st✐♥❣s ✭✷✵✵✽✮ s❤♦✇❡❞ t❤❛t ♠✐♥ ♠✐♥ ♠✐♥ ✐♥ ❣❡♥❡r❛❧✳ ◗✉❛♥t✉♠ ❝❤❛♥♥❡❧ ❆ ❝❤❛♥♥❡❧ ✐s ❛ ❝♦♠♣❧❡t❡❧② ♣♦s✐t✐✈❡✱ tr❛❝❡ ♣r❡s❡r✈✐♥❣ ♠❛♣ Φ : S m ✶ → S m ✶ ✱ ✇❤❡r❡ S m ✶ = ( M m ) ∗ ✳ ❵❘❡❝❛❧❧✬ ❤❡r❡ t❤❛t Φ ✐s ♣♦s✐t✐✈❡ ✐❢ d ≥ ✵ ✐♠♣❧✐❡s Φ( d ) ≥ ✵ ✐♥ s❡♥s❡ ♦❢ ♣♦s✐t✐✈❡ ❞❡✜♥✐t❡ ♠❛tr✐❝❡s✳ Φ ✐s ❝♦♠♣❧❡t❡❧② ♣♦s✐t✐✈❡ ✐❢ id S n ✶ ⊗ Φ ✐s ♣♦s✐t✐✈❡ ❢♦r ❛❧❧ n ∈ N ✭❝♦✉♣❧✐♥❣ ✇✐t❤ ❛♥♦t❤❡r s②st❡♠✮✳ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

✐♥❢ ✳ ♠✐♥ ✶ ✵ ❍❛st✐♥❣s ✭✷✵✵✽✮ s❤♦✇❡❞ t❤❛t ♠✐♥ ♠✐♥ ♠✐♥ ✐♥ ❣❡♥❡r❛❧✳ ◗✉❛♥t✉♠ ❝❤❛♥♥❡❧ ❆ ❝❤❛♥♥❡❧ ✐s ❛ ❝♦♠♣❧❡t❡❧② ♣♦s✐t✐✈❡✱ tr❛❝❡ ♣r❡s❡r✈✐♥❣ ♠❛♣ Φ : S m ✶ → S m ✶ ✱ ✇❤❡r❡ S m ✶ = ( M m ) ∗ ✳ ❵❘❡❝❛❧❧✬ ❤❡r❡ t❤❛t Φ ✐s ♣♦s✐t✐✈❡ ✐❢ d ≥ ✵ ✐♠♣❧✐❡s Φ( d ) ≥ ✵ ✐♥ s❡♥s❡ ♦❢ ♣♦s✐t✐✈❡ ❞❡✜♥✐t❡ ♠❛tr✐❝❡s✳ Φ ✐s ❝♦♠♣❧❡t❡❧② ♣♦s✐t✐✈❡ ✐❢ id S n ✶ ⊗ Φ ✐s ♣♦s✐t✐✈❡ ❢♦r ❛❧❧ n ∈ N ✭❝♦✉♣❧✐♥❣ ✇✐t❤ ❛♥♦t❤❡r s②st❡♠✮✳ H ( d ) = − tr ( d ❧♥ d ) = d dp � d � p ✐s t❤❡ ✈♦♥ ◆❡✉♠❛♥♥ ❡♥tr♦♣②✱ ✇❤❡r❡ � d � p = tr (( d ∗ d ) p / ✷ ) ✶ / p ✐s t❤❡ ✉s✉❛❧ ❙❝❤❛tt❡♥ p ✲♥♦r♠✳ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

❍❛st✐♥❣s ✭✷✵✵✽✮ s❤♦✇❡❞ t❤❛t ♠✐♥ ♠✐♥ ♠✐♥ ✐♥ ❣❡♥❡r❛❧✳ ◗✉❛♥t✉♠ ❝❤❛♥♥❡❧ ❆ ❝❤❛♥♥❡❧ ✐s ❛ ❝♦♠♣❧❡t❡❧② ♣♦s✐t✐✈❡✱ tr❛❝❡ ♣r❡s❡r✈✐♥❣ ♠❛♣ Φ : S m ✶ → S m ✶ ✱ ✇❤❡r❡ S m ✶ = ( M m ) ∗ ✳ ❵❘❡❝❛❧❧✬ ❤❡r❡ t❤❛t Φ ✐s ♣♦s✐t✐✈❡ ✐❢ d ≥ ✵ ✐♠♣❧✐❡s Φ( d ) ≥ ✵ ✐♥ s❡♥s❡ ♦❢ ♣♦s✐t✐✈❡ ❞❡✜♥✐t❡ ♠❛tr✐❝❡s✳ Φ ✐s ❝♦♠♣❧❡t❡❧② ♣♦s✐t✐✈❡ ✐❢ id S n ✶ ⊗ Φ ✐s ♣♦s✐t✐✈❡ ❢♦r ❛❧❧ n ∈ N ✭❝♦✉♣❧✐♥❣ ✇✐t❤ ❛♥♦t❤❡r s②st❡♠✮✳ H ( d ) = − tr ( d ❧♥ d ) = d dp � d � p ✐s t❤❡ ✈♦♥ ◆❡✉♠❛♥♥ ❡♥tr♦♣②✱ ✇❤❡r❡ � d � p = tr (( d ∗ d ) p / ✷ ) ✶ / p ✐s t❤❡ ✉s✉❛❧ ❙❝❤❛tt❡♥ p ✲♥♦r♠✳ H ♠✐♥ ( N ) = ✐♥❢ tr ( ρ )= ✶ ,ρ ≥ ✵ H ( N ( ρ )) ✳ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

◗✉❛♥t✉♠ ❝❤❛♥♥❡❧ ❆ ❝❤❛♥♥❡❧ ✐s ❛ ❝♦♠♣❧❡t❡❧② ♣♦s✐t✐✈❡✱ tr❛❝❡ ♣r❡s❡r✈✐♥❣ ♠❛♣ Φ : S m ✶ → S m ✶ ✱ ✇❤❡r❡ S m ✶ = ( M m ) ∗ ✳ ❵❘❡❝❛❧❧✬ ❤❡r❡ t❤❛t Φ ✐s ♣♦s✐t✐✈❡ ✐❢ d ≥ ✵ ✐♠♣❧✐❡s Φ( d ) ≥ ✵ ✐♥ s❡♥s❡ ♦❢ ♣♦s✐t✐✈❡ ❞❡✜♥✐t❡ ♠❛tr✐❝❡s✳ Φ ✐s ❝♦♠♣❧❡t❡❧② ♣♦s✐t✐✈❡ ✐❢ id S n ✶ ⊗ Φ ✐s ♣♦s✐t✐✈❡ ❢♦r ❛❧❧ n ∈ N ✭❝♦✉♣❧✐♥❣ ✇✐t❤ ❛♥♦t❤❡r s②st❡♠✮✳ H ( d ) = − tr ( d ❧♥ d ) = d dp � d � p ✐s t❤❡ ✈♦♥ ◆❡✉♠❛♥♥ ❡♥tr♦♣②✱ ✇❤❡r❡ � d � p = tr (( d ∗ d ) p / ✷ ) ✶ / p ✐s t❤❡ ✉s✉❛❧ ❙❝❤❛tt❡♥ p ✲♥♦r♠✳ H ♠✐♥ ( N ) = ✐♥❢ tr ( ρ )= ✶ ,ρ ≥ ✵ H ( N ( ρ )) ✳ ❍❛st✐♥❣s ✭✷✵✵✽✮ s❤♦✇❡❞ t❤❛t H ♠✐♥ (Φ ⊗ Ψ) � = H ♠✐♥ (Φ) + H ♠✐♥ (Ψ) ✐♥ ❣❡♥❡r❛❧✳ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

❲❡ ✇✐❧❧ ✇r✐t❡ ❢♦r t❤❡ ❙❝❤❛tt❡♥❝❧❛ss ♦❢ ❛ ✶ ✶ ❍✐❧❜❡rt s♣❛❝❡ ✇✐t❤ ❧❛❜❡❧ ✭❜❡❧♦♥❣s t♦ ❆❧✐❝❡✮✳ ▲❡t ❜❡ ❛ ❝❤❛♥♥❡❧ ❛♥❞ ✶ ✳ ✶ ✶ ❚❤❡♥ s✉♣ ✐s t❤❡ ♦♥❡ s❤♦t ❡①♣r❡ss✐♦♥ ❢♦r ❝❛♣❛❝✐t② s❡♥❞✐♥❣ ❝❧❛ss✐❝❛❧ ✐♥❢♦r♠❛t✐♦♥ ✇✐t❤ t❤❡ ❤❡❧♣ ♦❢ ❧♦❣ q✉❜✐ts ♦❢ ❡♥t❛♥❣❧❡♠❡♥t✳ ❍❡r❡ ❛r❡ ♣✉r❡ ❜✐♣❛rt✐t❡ st❛t❡s ♦❢ r❛♥❦ ✳ ✭❏P✮ ✐s ✐♥ ❣❡♥❡r❛❧ ♥♦t ❛❞❞✐t✐✈❡✳ ❙❝❤♦r ❝♦♥s✐❞❡r❡❞ r❡❧❛t❡❞ r❡str✐❝t✐♦♥✳ ❋♦r ✶ ✇❡ ✜♥❞ ❍♦❧❡✈♦✬s ❝❛♣❛❝✐t②✳ ❙❤♦r s❤♦✇❡❞ ✭❜❡❢♦r❡ ❍❛st✐♥❣s✮ t❤❛t t❤❡ ❛❞❞✐t✐✈✐t② ♦❢ t❤❡ ♠✐♥✲❡♥tr♦♣② ✐s ❡q✉✐✈❛❧❡♥t t♦ t❤❡ ❛❞❞✐t✐✈❡ ♦❢ t❤❡ ♦♥❡ s❤♦t ❍♦❧❡✈♦ ❝❛♣❛❝✐t②✳ ❘❡str✐❝t❡❞ ❈❛♣❛❝✐t② ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

▲❡t ❜❡ ❛ ❝❤❛♥♥❡❧ ❛♥❞ ✶ ✳ ✶ ✶ ❚❤❡♥ s✉♣ ✐s t❤❡ ♦♥❡ s❤♦t ❡①♣r❡ss✐♦♥ ❢♦r ❝❛♣❛❝✐t② s❡♥❞✐♥❣ ❝❧❛ss✐❝❛❧ ✐♥❢♦r♠❛t✐♦♥ ✇✐t❤ t❤❡ ❤❡❧♣ ♦❢ ❧♦❣ q✉❜✐ts ♦❢ ❡♥t❛♥❣❧❡♠❡♥t✳ ❍❡r❡ ❛r❡ ♣✉r❡ ❜✐♣❛rt✐t❡ st❛t❡s ♦❢ r❛♥❦ ✳ ✭❏P✮ ✐s ✐♥ ❣❡♥❡r❛❧ ♥♦t ❛❞❞✐t✐✈❡✳ ❙❝❤♦r ❝♦♥s✐❞❡r❡❞ r❡❧❛t❡❞ r❡str✐❝t✐♦♥✳ ❋♦r ✶ ✇❡ ✜♥❞ ❍♦❧❡✈♦✬s ❝❛♣❛❝✐t②✳ ❙❤♦r s❤♦✇❡❞ ✭❜❡❢♦r❡ ❍❛st✐♥❣s✮ t❤❛t t❤❡ ❛❞❞✐t✐✈✐t② ♦❢ t❤❡ ♠✐♥✲❡♥tr♦♣② ✐s ❡q✉✐✈❛❧❡♥t t♦ t❤❡ ❛❞❞✐t✐✈❡ ♦❢ t❤❡ ♦♥❡ s❤♦t ❍♦❧❡✈♦ ❝❛♣❛❝✐t②✳ ❘❡str✐❝t❡❞ ❈❛♣❛❝✐t② ❲❡ ✇✐❧❧ ✇r✐t❡ S ✶ ( A ) = S ✶ ( H A ) ❢♦r t❤❡ ❙❝❤❛tt❡♥❝❧❛ss ♦❢ ❛ ❍✐❧❜❡rt s♣❛❝❡ ✇✐t❤ ❧❛❜❡❧ A ✭❜❡❧♦♥❣s t♦ ❆❧✐❝❡✮✳ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

✭❏P✮ ✐s ✐♥ ❣❡♥❡r❛❧ ♥♦t ❛❞❞✐t✐✈❡✳ ❙❝❤♦r ❝♦♥s✐❞❡r❡❞ r❡❧❛t❡❞ r❡str✐❝t✐♦♥✳ ❋♦r ✶ ✇❡ ✜♥❞ ❍♦❧❡✈♦✬s ❝❛♣❛❝✐t②✳ ❙❤♦r s❤♦✇❡❞ ✭❜❡❢♦r❡ ❍❛st✐♥❣s✮ t❤❛t t❤❡ ❛❞❞✐t✐✈✐t② ♦❢ t❤❡ ♠✐♥✲❡♥tr♦♣② ✐s ❡q✉✐✈❛❧❡♥t t♦ t❤❡ ❛❞❞✐t✐✈❡ ♦❢ t❤❡ ♦♥❡ s❤♦t ❍♦❧❡✈♦ ❝❛♣❛❝✐t②✳ ❘❡str✐❝t❡❞ ❈❛♣❛❝✐t② ❲❡ ✇✐❧❧ ✇r✐t❡ S ✶ ( A ) = S ✶ ( H A ) ❢♦r t❤❡ ❙❝❤❛tt❡♥❝❧❛ss ♦❢ ❛ ❍✐❧❜❡rt s♣❛❝❡ ✇✐t❤ ❧❛❜❡❧ A ✭❜❡❧♦♥❣s t♦ ❆❧✐❝❡✮✳ ▲❡t N : S ✶ ( H A ′ ) → S ✶ ( H B ) ❜❡ ❛ ❝❤❛♥♥❡❧ ❛♥❞ ✶ ≤ d ≤ m ✳ ❚❤❡♥ C ( d ) ( N ) = s✉♣ ρ = � H ( N ( ρ )) k p k ρ A ′ A k � � � + H ( A ) N ( ρ k ) − H ( AB ) id ⊗N ( ρ k ) p k k ✐s t❤❡ ♦♥❡ s❤♦t ❡①♣r❡ss✐♦♥ ❢♦r ❝❛♣❛❝✐t② s❡♥❞✐♥❣ ❝❧❛ss✐❝❛❧ ✐♥❢♦r♠❛t✐♦♥ ✇✐t❤ t❤❡ ❤❡❧♣ ♦❢ ❧♦❣ k q✉❜✐ts ♦❢ ❡♥t❛♥❣❧❡♠❡♥t✳ ❍❡r❡ ρ k ❛r❡ ♣✉r❡ ❜✐♣❛rt✐t❡ st❛t❡s ♦❢ r❛♥❦ k ✳ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

❋♦r ✶ ✇❡ ✜♥❞ ❍♦❧❡✈♦✬s ❝❛♣❛❝✐t②✳ ❙❤♦r s❤♦✇❡❞ ✭❜❡❢♦r❡ ❍❛st✐♥❣s✮ t❤❛t t❤❡ ❛❞❞✐t✐✈✐t② ♦❢ t❤❡ ♠✐♥✲❡♥tr♦♣② ✐s ❡q✉✐✈❛❧❡♥t t♦ t❤❡ ❛❞❞✐t✐✈❡ ♦❢ t❤❡ ♦♥❡ s❤♦t ❍♦❧❡✈♦ ❝❛♣❛❝✐t②✳ ❘❡str✐❝t❡❞ ❈❛♣❛❝✐t② ❲❡ ✇✐❧❧ ✇r✐t❡ S ✶ ( A ) = S ✶ ( H A ) ❢♦r t❤❡ ❙❝❤❛tt❡♥❝❧❛ss ♦❢ ❛ ❍✐❧❜❡rt s♣❛❝❡ ✇✐t❤ ❧❛❜❡❧ A ✭❜❡❧♦♥❣s t♦ ❆❧✐❝❡✮✳ ▲❡t N : S ✶ ( H A ′ ) → S ✶ ( H B ) ❜❡ ❛ ❝❤❛♥♥❡❧ ❛♥❞ ✶ ≤ d ≤ m ✳ ❚❤❡♥ C ( d ) ( N ) = s✉♣ ρ = � H ( N ( ρ )) k p k ρ A ′ A k � � � + H ( A ) N ( ρ k ) − H ( AB ) id ⊗N ( ρ k ) p k k ✐s t❤❡ ♦♥❡ s❤♦t ❡①♣r❡ss✐♦♥ ❢♦r ❝❛♣❛❝✐t② s❡♥❞✐♥❣ ❝❧❛ss✐❝❛❧ ✐♥❢♦r♠❛t✐♦♥ ✇✐t❤ t❤❡ ❤❡❧♣ ♦❢ ❧♦❣ k q✉❜✐ts ♦❢ ❡♥t❛♥❣❧❡♠❡♥t✳ ❍❡r❡ ρ k ❛r❡ ♣✉r❡ ❜✐♣❛rt✐t❡ st❛t❡s ♦❢ r❛♥❦ k ✳ ✭❏P✮ C ( d ) ✐s ✐♥ ❣❡♥❡r❛❧ ♥♦t ❛❞❞✐t✐✈❡✳ ❙❝❤♦r ❝♦♥s✐❞❡r❡❞ r❡❧❛t❡❞ r❡str✐❝t✐♦♥✳ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

❘❡str✐❝t❡❞ ❈❛♣❛❝✐t② ❲❡ ✇✐❧❧ ✇r✐t❡ S ✶ ( A ) = S ✶ ( H A ) ❢♦r t❤❡ ❙❝❤❛tt❡♥❝❧❛ss ♦❢ ❛ ❍✐❧❜❡rt s♣❛❝❡ ✇✐t❤ ❧❛❜❡❧ A ✭❜❡❧♦♥❣s t♦ ❆❧✐❝❡✮✳ ▲❡t N : S ✶ ( H A ′ ) → S ✶ ( H B ) ❜❡ ❛ ❝❤❛♥♥❡❧ ❛♥❞ ✶ ≤ d ≤ m ✳ ❚❤❡♥ C ( d ) ( N ) = s✉♣ ρ = � H ( N ( ρ )) k p k ρ A ′ A k � � � + H ( A ) N ( ρ k ) − H ( AB ) id ⊗N ( ρ k ) p k k ✐s t❤❡ ♦♥❡ s❤♦t ❡①♣r❡ss✐♦♥ ❢♦r ❝❛♣❛❝✐t② s❡♥❞✐♥❣ ❝❧❛ss✐❝❛❧ ✐♥❢♦r♠❛t✐♦♥ ✇✐t❤ t❤❡ ❤❡❧♣ ♦❢ ❧♦❣ k q✉❜✐ts ♦❢ ❡♥t❛♥❣❧❡♠❡♥t✳ ❍❡r❡ ρ k ❛r❡ ♣✉r❡ ❜✐♣❛rt✐t❡ st❛t❡s ♦❢ r❛♥❦ k ✳ ✭❏P✮ C ( d ) ✐s ✐♥ ❣❡♥❡r❛❧ ♥♦t ❛❞❞✐t✐✈❡✳ ❙❝❤♦r ❝♦♥s✐❞❡r❡❞ r❡❧❛t❡❞ r❡str✐❝t✐♦♥✳ ❋♦r d = ✶ ✇❡ ✜♥❞ ❍♦❧❡✈♦✬s ❝❛♣❛❝✐t②✳ ❙❤♦r s❤♦✇❡❞ ✭❜❡❢♦r❡ ❍❛st✐♥❣s✮ t❤❛t t❤❡ ❛❞❞✐t✐✈✐t② ♦❢ t❤❡ ♠✐♥✲❡♥tr♦♣② ✐s ❡q✉✐✈❛❧❡♥t t♦ t❤❡ ❛❞❞✐t✐✈❡ ♦❢ t❤❡ ♦♥❡ s❤♦t ❍♦❧❡✈♦ ❝❛♣❛❝✐t②✳ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

❋♦r ❞✐♠ ✇❡ ✜♥❞ t❤❡ ❝❧❛ss✐❝❛❧ ❝❛♣❛❝✐t② ✇✐t❤ ❛ss✐st❡❞ ❡♥t❛♥❣❧❡♠❡♥t s✉♣ ✇❤✐❝❤ ✐s ❛❞❞✐t✐✈❡ ✭ ❧✐♠ ❧♦❣ ✳✮ ❚❤❡ ❝❧❛ss✐❝❛❧ ❝❛♣❛❝✐t② ✇✐t❤ ❛ss✐st❡❞ ❡♥t❛♥❣❧❡♠❡♥t ✐s ❛ r❛t❡ ✭❡❛r❧② s✉❝❝❡s ✐♥ ◗■❚✿ ❍❙❲✱ ❙❝❤♦r✱ ❉❡✈❡t❛❦✱ ❲✐♥t❡r✱✳✳✳✮ ❈❧❛ss✐❝❛❧ ❝❛♣❛❝✐t② ✇✐t❤ ✉♥❧✐♠✐t❡❞ ❡♥t❛♥❣❧❡♠❡♥t ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

❚❤❡ ❝❧❛ss✐❝❛❧ ❝❛♣❛❝✐t② ✇✐t❤ ❛ss✐st❡❞ ❡♥t❛♥❣❧❡♠❡♥t ✐s ❛ r❛t❡ ✭❡❛r❧② s✉❝❝❡s ✐♥ ◗■❚✿ ❍❙❲✱ ❙❝❤♦r✱ ❉❡✈❡t❛❦✱ ❲✐♥t❡r✱✳✳✳✮ ❈❧❛ss✐❝❛❧ ❝❛♣❛❝✐t② ✇✐t❤ ✉♥❧✐♠✐t❡❞ ❡♥t❛♥❣❧❡♠❡♥t ❋♦r d = ❞✐♠ ( H A ′ ) = | A ′ | ✇❡ ✜♥❞ t❤❡ ❝❧❛ss✐❝❛❧ ❝❛♣❛❝✐t② ✇✐t❤ ❛ss✐st❡❞ ❡♥t❛♥❣❧❡♠❡♥t C EA ( N ) = s✉♣ I ( A , B ) id ⊗N ( ρ A ′ A ) ρ A ′ A ✇❤✐❝❤ ✐s ❛❞❞✐t✐✈❡ ✭ C EA = ❧✐♠ p ❧♦❣ π o p ( N )) ✳✮ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

❈❧❛ss✐❝❛❧ ❝❛♣❛❝✐t② ✇✐t❤ ✉♥❧✐♠✐t❡❞ ❡♥t❛♥❣❧❡♠❡♥t ❋♦r d = ❞✐♠ ( H A ′ ) = | A ′ | ✇❡ ✜♥❞ t❤❡ ❝❧❛ss✐❝❛❧ ❝❛♣❛❝✐t② ✇✐t❤ ❛ss✐st❡❞ ❡♥t❛♥❣❧❡♠❡♥t C EA ( N ) = s✉♣ I ( A , B ) id ⊗N ( ρ A ′ A ) ρ A ′ A ✇❤✐❝❤ ✐s ❛❞❞✐t✐✈❡ ✭ C EA = ❧✐♠ p ❧♦❣ π o p ( N )) ✳✮ ❚❤❡ ❝❧❛ss✐❝❛❧ ❝❛♣❛❝✐t② ✇✐t❤ ❛ss✐st❡❞ ❡♥t❛♥❣❧❡♠❡♥t ✐s ❛ r❛t❡ ✭❡❛r❧② s✉❝❝❡s ✐♥ ◗■❚✿ ❍❙❲✱ ❙❝❤♦r✱ ❉❡✈❡t❛❦✱ ❲✐♥t❡r✱✳✳✳✮ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

▲❡t ❜❡ ❛ ❝❤❛♥♥❡❧ ❛♥❞ ✶ ✶ t❤❡ ✐♠❛❣❡ ♦❢ ❛ ❜✐♣❛rt✐t❡ st❛t❡✳ ❲❡ ♠❛② ✇r✐t❡ t❤❡ ♠✉t✉❛❧ ✐♥❢♦r♠❛t✐♦♥ ❛s ❚❤❡ ♥❡❣❛t✐✈❡ r❡❧❛t✐✈❡ ❡♥tr♦♣② ✐s ❝♦♥✈❡① ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ✉♥❞❡r❧②✐♥❣ st❛t❡✳ ■♥❞❡❡❞✱ ❛❝❝♦r❞✐♥❣ t♦ ❬❉❏❑❘❙❪ ✶ ✶ ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❛s ❞❡r✐✈❛t✐✈❡ ♦❢ P✐s✐❡r✬s ✈❡❝t♦r✲✈❛❧✉❡❞ ✳ ✶ ❯s✐♥❣ t❤❡ ❝♦♥tr❛❝t✐♦♥ ✱ ✇❡ ❞❡❞✉❝❡ ❜② ✶ ✶ ❞✐✛❡r❡♥t✐❛t✐♦♥ str♦♥❣ s✉♣❡r ❛❞❞✐t✐✈✐t② ❈♦❤❡r❡♥t ✐♥❢♦r♠❛t✐♦♥ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

❚❤❡ ♥❡❣❛t✐✈❡ r❡❧❛t✐✈❡ ❡♥tr♦♣② ✐s ❝♦♥✈❡① ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ✉♥❞❡r❧②✐♥❣ st❛t❡✳ ■♥❞❡❡❞✱ ❛❝❝♦r❞✐♥❣ t♦ ❬❉❏❑❘❙❪ ✶ ✶ ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❛s ❞❡r✐✈❛t✐✈❡ ♦❢ P✐s✐❡r✬s ✈❡❝t♦r✲✈❛❧✉❡❞ ✳ ✶ ❯s✐♥❣ t❤❡ ❝♦♥tr❛❝t✐♦♥ ✱ ✇❡ ❞❡❞✉❝❡ ❜② ✶ ✶ ❞✐✛❡r❡♥t✐❛t✐♦♥ str♦♥❣ s✉♣❡r ❛❞❞✐t✐✈✐t② ❈♦❤❡r❡♥t ✐♥❢♦r♠❛t✐♦♥ ▲❡t N : S ✶ ( A ′ ) → S ✶ ( B ) ❜❡ ❛ ❝❤❛♥♥❡❧ ❛♥❞ ρ AB = id ⊗ N ( ρ A ′ A ) t❤❡ ✐♠❛❣❡ ♦❢ ❛ ❜✐♣❛rt✐t❡ st❛t❡✳ ❲❡ ♠❛② ✇r✐t❡ t❤❡ ♠✉t✉❛❧ ✐♥❢♦r♠❛t✐♦♥ ❛s I ( A , B ) = H ( A ) + H ( B ) − H ( AB ) = H ( A ) + I c ( A � B ) . ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

■♥❞❡❡❞✱ ❛❝❝♦r❞✐♥❣ t♦ ❬❉❏❑❘❙❪ ✶ ✶ ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❛s ❞❡r✐✈❛t✐✈❡ ♦❢ P✐s✐❡r✬s ✈❡❝t♦r✲✈❛❧✉❡❞ ✳ ✶ ❯s✐♥❣ t❤❡ ❝♦♥tr❛❝t✐♦♥ ✱ ✇❡ ❞❡❞✉❝❡ ❜② ✶ ✶ ❞✐✛❡r❡♥t✐❛t✐♦♥ str♦♥❣ s✉♣❡r ❛❞❞✐t✐✈✐t② ❈♦❤❡r❡♥t ✐♥❢♦r♠❛t✐♦♥ ▲❡t N : S ✶ ( A ′ ) → S ✶ ( B ) ❜❡ ❛ ❝❤❛♥♥❡❧ ❛♥❞ ρ AB = id ⊗ N ( ρ A ′ A ) t❤❡ ✐♠❛❣❡ ♦❢ ❛ ❜✐♣❛rt✐t❡ st❛t❡✳ ❲❡ ♠❛② ✇r✐t❡ t❤❡ ♠✉t✉❛❧ ✐♥❢♦r♠❛t✐♦♥ ❛s I ( A , B ) = H ( A ) + H ( B ) − H ( AB ) = H ( A ) + I c ( A � B ) . ❚❤❡ ♥❡❣❛t✐✈❡ r❡❧❛t✐✈❡ ❡♥tr♦♣② I c ( A � B ) = H ( B ) − H ( AB ) ✐s ❝♦♥✈❡① ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ✉♥❞❡r❧②✐♥❣ st❛t❡✳ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

❯s✐♥❣ t❤❡ ❝♦♥tr❛❝t✐♦♥ ✱ ✇❡ ❞❡❞✉❝❡ ❜② ✶ ✶ ❞✐✛❡r❡♥t✐❛t✐♦♥ str♦♥❣ s✉♣❡r ❛❞❞✐t✐✈✐t② ❈♦❤❡r❡♥t ✐♥❢♦r♠❛t✐♦♥ ▲❡t N : S ✶ ( A ′ ) → S ✶ ( B ) ❜❡ ❛ ❝❤❛♥♥❡❧ ❛♥❞ ρ AB = id ⊗ N ( ρ A ′ A ) t❤❡ ✐♠❛❣❡ ♦❢ ❛ ❜✐♣❛rt✐t❡ st❛t❡✳ ❲❡ ♠❛② ✇r✐t❡ t❤❡ ♠✉t✉❛❧ ✐♥❢♦r♠❛t✐♦♥ ❛s I ( A , B ) = H ( A ) + H ( B ) − H ( AB ) = H ( A ) + I c ( A � B ) . ❚❤❡ ♥❡❣❛t✐✈❡ r❡❧❛t✐✈❡ ❡♥tr♦♣② I c ( A � B ) = H ( B ) − H ( AB ) ✐s ❝♦♥✈❡① ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ✉♥❞❡r❧②✐♥❣ st❛t❡✳ ■♥❞❡❡❞✱ ❛❝❝♦r❞✐♥❣ t♦ ❬❉❏❑❘❙❪ I c ( A � B ) = c d dp � ρ BA � S ✶ ( H B ; S p ( H A )) | p = ✶ ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❛s ❞❡r✐✈❛t✐✈❡ ♦❢ P✐s✐❡r✬s ✈❡❝t♦r✲✈❛❧✉❡❞ S ✶ ( S p ) ✳ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

❈♦❤❡r❡♥t ✐♥❢♦r♠❛t✐♦♥ ▲❡t N : S ✶ ( A ′ ) → S ✶ ( B ) ❜❡ ❛ ❝❤❛♥♥❡❧ ❛♥❞ ρ AB = id ⊗ N ( ρ A ′ A ) t❤❡ ✐♠❛❣❡ ♦❢ ❛ ❜✐♣❛rt✐t❡ st❛t❡✳ ❲❡ ♠❛② ✇r✐t❡ t❤❡ ♠✉t✉❛❧ ✐♥❢♦r♠❛t✐♦♥ ❛s I ( A , B ) = H ( A ) + H ( B ) − H ( AB ) = H ( A ) + I c ( A � B ) . ❚❤❡ ♥❡❣❛t✐✈❡ r❡❧❛t✐✈❡ ❡♥tr♦♣② I c ( A � B ) = H ( B ) − H ( AB ) ✐s ❝♦♥✈❡① ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ✉♥❞❡r❧②✐♥❣ st❛t❡✳ ■♥❞❡❡❞✱ ❛❝❝♦r❞✐♥❣ t♦ ❬❉❏❑❘❙❪ I c ( A � B ) = c d dp � ρ BA � S ✶ ( H B ; S p ( H A )) | p = ✶ ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❛s ❞❡r✐✈❛t✐✈❡ ♦❢ P✐s✐❡r✬s ✈❡❝t♦r✲✈❛❧✉❡❞ S ✶ ( S p ) ✳ ❯s✐♥❣ t❤❡ ❝♦♥tr❛❝t✐♦♥ id ⊗ tr C : S ✶ ( H BC ; S p ( H A )) → S ✶ ( H B ; S p ( H A )) ✱ ✇❡ ❞❡❞✉❝❡ ❜② ❞✐✛❡r❡♥t✐❛t✐♦♥ str♦♥❣ s✉♣❡r ❛❞❞✐t✐✈✐t② H ( B ) − H ( AB ) ≤ H ( BC ) − H ( ABC ) . ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

✭❙❤♦r ❡t ❛❧✮ ❚❤❡ r❡❣✉❧❛r✐③❛t✐♦♥ ✶ ❧✐♠ s✉♣ ❣✐✈❡s t❤❡ r❛t❡ ♦❢ tr❛♥s♠✐tt✐♥❣ q✉❜✐ts ♣❡r ✉s❡s ♦❢ t❤❡ ❝❤❛♥♥❡❧ ✶ ✶ ❡♥❝♦❞❡r✱ ❞❡❝♦r❞❡r ✷ ✷ ✶ ✶ ❚❤❡ r❡❣✉❧❛r✐③❛t✐♦♥ ✐s ❣❡♥❡r❛❧ ❤❛r❞ t♦ ❝❛❧❝✉❧❛t❡✳ ❇② t❡❧❡♣♦rt❛t✐♦♥ ✇❡ ❤❛✈❡ ✷ ✷ ✳ ◗✉❛♥t✉♠ ❝❛♣❛❝✐t② ❚❤❡ ♦♥❡ s❤♦t q✉❛♥t✉♠ ❝❛♣❛❝✐t② Q ( ✶ ) ( N ) = s✉♣ ρ I c ( B � A ) ✐s ✐♥ ❣❡♥❡r❛❧ ♥♦t ❛❞❞✐t✐✈❡✳ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

❚❤❡ r❡❣✉❧❛r✐③❛t✐♦♥ ✐s ❣❡♥❡r❛❧ ❤❛r❞ t♦ ❝❛❧❝✉❧❛t❡✳ ❇② t❡❧❡♣♦rt❛t✐♦♥ ✇❡ ❤❛✈❡ ✷ ✷ ✳ ◗✉❛♥t✉♠ ❝❛♣❛❝✐t② ❚❤❡ ♦♥❡ s❤♦t q✉❛♥t✉♠ ❝❛♣❛❝✐t② Q ( ✶ ) ( N ) = s✉♣ ρ I c ( B � A ) ✐s ✐♥ ❣❡♥❡r❛❧ ♥♦t ❛❞❞✐t✐✈❡✳ ✭❙❤♦r ❡t ❛❧✮ ❚❤❡ r❡❣✉❧❛r✐③❛t✐♦♥ Q ( ✶ ) ( N ⊗ n ) Q (Φ) = ❧✐♠ s✉♣ n n ❣✐✈❡s t❤❡ r❛t❡ R ♦❢ tr❛♥s♠✐tt✐♥❣ rn q✉❜✐ts ♣❡r n ✉s❡s ♦❢ t❤❡ ❝❤❛♥♥❡❧ N ⊗ n S m n S m n → ✶ ✶ ↑ E ↓ D E , D ❡♥❝♦❞❡r✱ ❞❡❝♦r❞❡r ≈ id S ✷ rn S ✷ rn → . ✶ ✶ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

◗✉❛♥t✉♠ ❝❛♣❛❝✐t② ❚❤❡ ♦♥❡ s❤♦t q✉❛♥t✉♠ ❝❛♣❛❝✐t② Q ( ✶ ) ( N ) = s✉♣ ρ I c ( B � A ) ✐s ✐♥ ❣❡♥❡r❛❧ ♥♦t ❛❞❞✐t✐✈❡✳ ✭❙❤♦r ❡t ❛❧✮ ❚❤❡ r❡❣✉❧❛r✐③❛t✐♦♥ Q ( ✶ ) ( N ⊗ n ) Q (Φ) = ❧✐♠ s✉♣ n n ❣✐✈❡s t❤❡ r❛t❡ R ♦❢ tr❛♥s♠✐tt✐♥❣ rn q✉❜✐ts ♣❡r n ✉s❡s ♦❢ t❤❡ ❝❤❛♥♥❡❧ N ⊗ n S m n S m n → ✶ ✶ ↑ E ↓ D E , D ❡♥❝♦❞❡r✱ ❞❡❝♦r❞❡r ≈ id S ✷ rn S ✷ rn → . ✶ ✶ ❚❤❡ r❡❣✉❧❛r✐③❛t✐♦♥ ✐s ❣❡♥❡r❛❧ ❤❛r❞ t♦ ❝❛❧❝✉❧❛t❡✳ ❇② t❡❧❡♣♦rt❛t✐♦♥ ✇❡ ❤❛✈❡ ✷ Q ( N ) ≤ ✷ Q E ( N ) = C EA ( N ) ✳ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

■t ✐s ♦♣❡♥ ❢♦r ✇❤✐❝❤ ✈❛❧✉❡s ♦❢ t❤❡ ❞❡♣♦❧❛r✐③✐♥❣ ❝❤❛♥♥❡❧ ✶ ✶ s❛t✐s✜❡s ✵✳ ❖✉r ♠❛✐♥ r❡s✉❧t ✐s t❤❛t t❤❡r❡ ✐s ❛ ♥✐❝❡ ❝❧❛ss ♦❢ ❝❤❛♥♥❡❧s s✉❝❤ t❤❛t ✶ ✷ ✶ ✷ ♠❛① ♠❛① ❧♥ ❧♥ ♠❛① ❧♥ ❧♥ ❍❡r❡ ✐s t❤❡ s②♠❜♦❧ ♦❢ ❛ ❝❤❛♥♥❡❧ ❛♥❞ ✐s ❛ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ✈♦♥ ◆❡✉♠❛♥♥ ❛❧❣❡❜r❛✳ ❋♦r t❤❡s❡ ❝❤❛♥♥❡❧s ✇❡ ❛❧s♦ ❤❛✈❡ ❡st✐♠❛t❡s ❢♦r t❤❡ r❡❣✐♦♥ ✭❝❧❛ss✐❝❛❧✱ q✉❛♥t✉♠✱ ❡♥t❛♥❣❧❡♠❡♥t✮ r❡❣✐♦♥ ♦❢ t❤❡ ❝❤❛♥♥❡❧✱ ✇❤✐❝❤ ❣♦❡s ❜❛❝❦ t♦ ❙❝❤♦r✱ s❡❡ ❛❧s♦ ❲✐❧❞❡✳ ❘❡s✉❧ts ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

❖✉r ♠❛✐♥ r❡s✉❧t ✐s t❤❛t t❤❡r❡ ✐s ❛ ♥✐❝❡ ❝❧❛ss ♦❢ ❝❤❛♥♥❡❧s s✉❝❤ t❤❛t ✶ ✷ ✶ ✷ ♠❛① ♠❛① ❧♥ ❧♥ ♠❛① ❧♥ ❧♥ ❍❡r❡ ✐s t❤❡ s②♠❜♦❧ ♦❢ ❛ ❝❤❛♥♥❡❧ ❛♥❞ ✐s ❛ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ✈♦♥ ◆❡✉♠❛♥♥ ❛❧❣❡❜r❛✳ ❋♦r t❤❡s❡ ❝❤❛♥♥❡❧s ✇❡ ❛❧s♦ ❤❛✈❡ ❡st✐♠❛t❡s ❢♦r t❤❡ r❡❣✐♦♥ ✭❝❧❛ss✐❝❛❧✱ q✉❛♥t✉♠✱ ❡♥t❛♥❣❧❡♠❡♥t✮ r❡❣✐♦♥ ♦❢ t❤❡ ❝❤❛♥♥❡❧✱ ✇❤✐❝❤ ❣♦❡s ❜❛❝❦ t♦ ❙❝❤♦r✱ s❡❡ ❛❧s♦ ❲✐❧❞❡✳ ❘❡s✉❧ts ■t ✐s ♦♣❡♥ ❢♦r ✇❤✐❝❤ ✈❛❧✉❡s ♦❢ λ t❤❡ ❞❡♣♦❧❛r✐③✐♥❣ ❝❤❛♥♥❡❧ Φ λ ( ✶ − λ ) id + λ tr d () ✶ d s❛t✐s✜❡s Q (Φ λ )) > ✵✳ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

❋♦r t❤❡s❡ ❝❤❛♥♥❡❧s ✇❡ ❛❧s♦ ❤❛✈❡ ❡st✐♠❛t❡s ❢♦r t❤❡ r❡❣✐♦♥ ✭❝❧❛ss✐❝❛❧✱ q✉❛♥t✉♠✱ ❡♥t❛♥❣❧❡♠❡♥t✮ r❡❣✐♦♥ ♦❢ t❤❡ ❝❤❛♥♥❡❧✱ ✇❤✐❝❤ ❣♦❡s ❜❛❝❦ t♦ ❙❝❤♦r✱ s❡❡ ❛❧s♦ ❲✐❧❞❡✳ ❘❡s✉❧ts ■t ✐s ♦♣❡♥ ❢♦r ✇❤✐❝❤ ✈❛❧✉❡s ♦❢ λ t❤❡ ❞❡♣♦❧❛r✐③✐♥❣ ❝❤❛♥♥❡❧ Φ λ ( ✶ − λ ) id + λ tr d () ✶ d s❛t✐s✜❡s Q (Φ λ )) > ✵✳ ❖✉r ♠❛✐♥ r❡s✉❧t ✐s t❤❛t t❤❡r❡ ✐s ❛ ♥✐❝❡ ❝❧❛ss ♦❢ ❝❤❛♥♥❡❧s θ f : S ✶ ( L ✷ ( M )) → S ✶ ( L ✷ ( M )) s✉❝❤ t❤❛t ❧♥ n k , τ ( f ❧♥ f ) } ≤ Q ( θ f ) ≤ Q pot ( θ f ) ♠❛① { ♠❛① k ≤ ♠❛① ❧♥ n k + τ ( f ❧♥ f ) . k ❍❡r❡ f ∈ N ✐s t❤❡ s②♠❜♦❧ ♦❢ ❛ ❝❤❛♥♥❡❧ ❛♥❞ M = ⊕ k M n k ✐s ❛ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ✈♦♥ ◆❡✉♠❛♥♥ ❛❧❣❡❜r❛✳ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

❘❡s✉❧ts ■t ✐s ♦♣❡♥ ❢♦r ✇❤✐❝❤ ✈❛❧✉❡s ♦❢ λ t❤❡ ❞❡♣♦❧❛r✐③✐♥❣ ❝❤❛♥♥❡❧ Φ λ ( ✶ − λ ) id + λ tr d () ✶ d s❛t✐s✜❡s Q (Φ λ )) > ✵✳ ❖✉r ♠❛✐♥ r❡s✉❧t ✐s t❤❛t t❤❡r❡ ✐s ❛ ♥✐❝❡ ❝❧❛ss ♦❢ ❝❤❛♥♥❡❧s θ f : S ✶ ( L ✷ ( M )) → S ✶ ( L ✷ ( M )) s✉❝❤ t❤❛t ❧♥ n k , τ ( f ❧♥ f ) } ≤ Q ( θ f ) ≤ Q pot ( θ f ) ♠❛① { ♠❛① k ≤ ♠❛① ❧♥ n k + τ ( f ❧♥ f ) . k ❍❡r❡ f ∈ N ✐s t❤❡ s②♠❜♦❧ ♦❢ ❛ ❝❤❛♥♥❡❧ ❛♥❞ M = ⊕ k M n k ✐s ❛ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ✈♦♥ ◆❡✉♠❛♥♥ ❛❧❣❡❜r❛✳ ❋♦r t❤❡s❡ ❝❤❛♥♥❡❧s ✇❡ ❛❧s♦ ❤❛✈❡ ❡st✐♠❛t❡s ❢♦r t❤❡ ( CQE ) r❡❣✐♦♥ ✭❝❧❛ss✐❝❛❧✱ q✉❛♥t✉♠✱ ❡♥t❛♥❣❧❡♠❡♥t✮ r❡❣✐♦♥ ♦❢ t❤❡ ❝❤❛♥♥❡❧✱ ✇❤✐❝❤ ❣♦❡s ❜❛❝❦ t♦ ❙❝❤♦r✱ s❡❡ ❛❧s♦ ❲✐❧❞❡✳ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

▲❡t ❜❡ ❛ ✜♥✐t❡ ❣r♦✉♣✱ t❤❡ ❧❡❢t r❡❣✉❧❛r r❡♣r❡s❡♥t❛t✐♦♥ ♦♥ ✳ ▲❡t ❛♥❞ ✷ ✷ ✶ ❚❤❡♥ ✶ ✐s t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ❡①♣❡❝t❛t✐♦♥ ♦♥t♦ r✐❣❤t r❡❣✉❧❛r r❡♣r❡s❡♥t❛t✐♦♥ ✳ ❲❡ ❤❛✈❡ ✶ ♠❛① ♠❛① ❧♥ ❧♥ ❧♥ ♠❛① ❧♥ ❇♦t❤ ❧♦✇❡r ❜♦✉♥❞s ❛r❡ ❛❝❤✐❡✈❛❜❧❡ ❛s ✇❡❧❧ ❛s t❤❡ ✉♣♣❡r ❜♦✉♥❞ ❢♦r ✳ ❋♦r ✶ ✇❡ ✜♥❞ t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ❡①♣❡❝t✐♦♥ ♦♥t♦ ✳ ❜❡❧♦♥❣s t♦ t❤❡ ✜①♣♦✐♥t ❛❧❣❡❜r❛ ♦❢ ❡✈❡r② ✱ t❤✐s ❣✐✈❡s ♦♥❡ ♦❢ t❤❡ ❧♦✇❡r ❡st✐♠❛t❡s✳ ▼♦t✐✈❛t✐♥❣ ❡①❛♠♣❧❡s ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

❜❡❧♦♥❣s t♦ t❤❡ ✜①♣♦✐♥t ❛❧❣❡❜r❛ ♦❢ ❡✈❡r② ✱ t❤✐s ❣✐✈❡s ♦♥❡ ♦❢ t❤❡ ❧♦✇❡r ❡st✐♠❛t❡s✳ ❚❤❡♥ ✶ ✐s t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ❡①♣❡❝t❛t✐♦♥ ♦♥t♦ r✐❣❤t r❡❣✉❧❛r r❡♣r❡s❡♥t❛t✐♦♥ ✳ ❲❡ ❤❛✈❡ ✶ ♠❛① ♠❛① ❧♥ ❧♥ ❧♥ ♠❛① ❧♥ ❇♦t❤ ❧♦✇❡r ❜♦✉♥❞s ❛r❡ ❛❝❤✐❡✈❛❜❧❡ ❛s ✇❡❧❧ ❛s t❤❡ ✉♣♣❡r ❜♦✉♥❞ ❢♦r ✳ ❋♦r ✶ ✇❡ ✜♥❞ t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ❡①♣❡❝t✐♦♥ ♦♥t♦ ✳ ▼♦t✐✈❛t✐♥❣ ❡①❛♠♣❧❡s ❅ ▲❡t G ❜❡ ❛ ✜♥✐t❡ ❣r♦✉♣✱ λ ( g ) e h = e gh t❤❡ ❧❡❢t r❡❣✉❧❛r r❡♣r❡s❡♥t❛t✐♦♥ ♦♥ ℓ ✷ ( G ) = L ✷ ( L ( G )) ✳ ▲❡t f : G → R + ❛♥❞ ✶ � f ( g ) λ ( g ) ∗ ρλ ( g ) . θ f ( ρ ) = | G | g ∈ G ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

❜❡❧♦♥❣s t♦ t❤❡ ✜①♣♦✐♥t ❛❧❣❡❜r❛ ♦❢ ❡✈❡r② ✱ t❤✐s ❣✐✈❡s ♦♥❡ ♦❢ t❤❡ ❧♦✇❡r ❡st✐♠❛t❡s✳ ❋♦r ✶ ✇❡ ✜♥❞ t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ❡①♣❡❝t✐♦♥ ♦♥t♦ ✳ ▼♦t✐✈❛t✐♥❣ ❡①❛♠♣❧❡s ❅ ▲❡t G ❜❡ ❛ ✜♥✐t❡ ❣r♦✉♣✱ λ ( g ) e h = e gh t❤❡ ❧❡❢t r❡❣✉❧❛r r❡♣r❡s❡♥t❛t✐♦♥ ♦♥ ℓ ✷ ( G ) = L ✷ ( L ( G )) ✳ ▲❡t f : G → R + ❛♥❞ ✶ � f ( g ) λ ( g ) ∗ ρλ ( g ) . θ f ( ρ ) = | G | g ∈ G ❅ ❚❤❡♥ θ ✶ ✐s t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ❡①♣❡❝t❛t✐♦♥ ♦♥t♦ r✐❣❤t r❡❣✉❧❛r r❡♣r❡s❡♥t❛t✐♦♥ L ( G ) ′ = R ( G ) = � k M n k = M ✳ ❲❡ ❤❛✈❡ ❧♥ n k , τ ( f ❧♥ f )) ≤ Q ( ✶ ) ( θ f ) ≤ τ ( f ❧♥ f ) + ♠❛① ♠❛① ( ♠❛① ❧♥ n k . k k ❇♦t❤ ❧♦✇❡r ❜♦✉♥❞s ❛r❡ ❛❝❤✐❡✈❛❜❧❡ ❛s ✇❡❧❧ ❛s t❤❡ ✉♣♣❡r ❜♦✉♥❞ ❢♦r G = Z n ✳ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

❜❡❧♦♥❣s t♦ t❤❡ ✜①♣♦✐♥t ❛❧❣❡❜r❛ ♦❢ ❡✈❡r② ✱ t❤✐s ❣✐✈❡s ♦♥❡ ♦❢ t❤❡ ❧♦✇❡r ❡st✐♠❛t❡s✳ ▼♦t✐✈❛t✐♥❣ ❡①❛♠♣❧❡s ❅ ▲❡t G ❜❡ ❛ ✜♥✐t❡ ❣r♦✉♣✱ λ ( g ) e h = e gh t❤❡ ❧❡❢t r❡❣✉❧❛r r❡♣r❡s❡♥t❛t✐♦♥ ♦♥ ℓ ✷ ( G ) = L ✷ ( L ( G )) ✳ ▲❡t f : G → R + ❛♥❞ ✶ � f ( g ) λ ( g ) ∗ ρλ ( g ) . θ f ( ρ ) = | G | g ∈ G ❅ ❚❤❡♥ θ ✶ ✐s t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ❡①♣❡❝t❛t✐♦♥ ♦♥t♦ r✐❣❤t r❡❣✉❧❛r r❡♣r❡s❡♥t❛t✐♦♥ L ( G ) ′ = R ( G ) = � k M n k = M ✳ ❲❡ ❤❛✈❡ ❧♥ n k , τ ( f ❧♥ f )) ≤ Q ( ✶ ) ( θ f ) ≤ τ ( f ❧♥ f ) + ♠❛① ♠❛① ( ♠❛① ❧♥ n k . k k ❇♦t❤ ❧♦✇❡r ❜♦✉♥❞s ❛r❡ ❛❝❤✐❡✈❛❜❧❡ ❛s ✇❡❧❧ ❛s t❤❡ ✉♣♣❡r ❜♦✉♥❞ ❢♦r G = Z n ✳ ❅ ❋♦r f = ✶ ✇❡ ✜♥❞ t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ❡①♣❡❝t✐♦♥ ♦♥t♦ R ( G ) ✳ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

❜❡❧♦♥❣s t♦ t❤❡ ✜①♣♦✐♥t ❛❧❣❡❜r❛ ♦❢ ❡✈❡r② ✱ t❤✐s ❣✐✈❡s ♦♥❡ ♦❢ t❤❡ ❧♦✇❡r ❡st✐♠❛t❡s✳ ▼♦t✐✈❛t✐♥❣ ❡①❛♠♣❧❡s ❅ ▲❡t G ❜❡ ❛ ✜♥✐t❡ ❣r♦✉♣✱ λ ( g ) e h = e gh t❤❡ ❧❡❢t r❡❣✉❧❛r r❡♣r❡s❡♥t❛t✐♦♥ ♦♥ ℓ ✷ ( G ) = L ✷ ( L ( G )) ✳ ▲❡t f : G → R + ❛♥❞ ✶ � f ( g ) λ ( g ) ∗ ρλ ( g ) . θ f ( ρ ) = | G | g ∈ G ❅ ❚❤❡♥ θ ✶ ✐s t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ❡①♣❡❝t❛t✐♦♥ ♦♥t♦ r✐❣❤t r❡❣✉❧❛r r❡♣r❡s❡♥t❛t✐♦♥ L ( G ) ′ = R ( G ) = � k M n k = M ✳ ❲❡ ❤❛✈❡ ❧♥ n k , τ ( f ❧♥ f )) ≤ Q ( ✶ ) ( θ f ) ≤ τ ( f ❧♥ f ) + ♠❛① ♠❛① ( ♠❛① ❧♥ n k . k k ❇♦t❤ ❧♦✇❡r ❜♦✉♥❞s ❛r❡ ❛❝❤✐❡✈❛❜❧❡ ❛s ✇❡❧❧ ❛s t❤❡ ✉♣♣❡r ❜♦✉♥❞ ❢♦r G = Z n ✳ ❅ ❋♦r f = ✶ ✇❡ ✜♥❞ t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ❡①♣❡❝t✐♦♥ ♦♥t♦ R ( G ) ✳ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

▼♦t✐✈❛t✐♥❣ ❡①❛♠♣❧❡s ❅ ▲❡t G ❜❡ ❛ ✜♥✐t❡ ❣r♦✉♣✱ λ ( g ) e h = e gh t❤❡ ❧❡❢t r❡❣✉❧❛r r❡♣r❡s❡♥t❛t✐♦♥ ♦♥ ℓ ✷ ( G ) = L ✷ ( L ( G )) ✳ ▲❡t f : G → R + ❛♥❞ ✶ � f ( g ) λ ( g ) ∗ ρλ ( g ) . θ f ( ρ ) = | G | g ∈ G ❅ ❚❤❡♥ θ ✶ ✐s t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ❡①♣❡❝t❛t✐♦♥ ♦♥t♦ r✐❣❤t r❡❣✉❧❛r r❡♣r❡s❡♥t❛t✐♦♥ L ( G ) ′ = R ( G ) = � k M n k = M ✳ ❲❡ ❤❛✈❡ ❧♥ n k , τ ( f ❧♥ f )) ≤ Q ( ✶ ) ( θ f ) ≤ τ ( f ❧♥ f ) + ♠❛① ♠❛① ( ♠❛① ❧♥ n k . k k ❇♦t❤ ❧♦✇❡r ❜♦✉♥❞s ❛r❡ ❛❝❤✐❡✈❛❜❧❡ ❛s ✇❡❧❧ ❛s t❤❡ ✉♣♣❡r ❜♦✉♥❞ ❢♦r G = Z n ✳ ❅ ❋♦r f = ✶ ✇❡ ✜♥❞ t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ❡①♣❡❝t✐♦♥ ♦♥t♦ R ( G ) ✳ R ( G ) ❜❡❧♦♥❣s t♦ t❤❡ ✜①♣♦✐♥t ❛❧❣❡❜r❛ ♦❢ ❡✈❡r② θ f ✱ t❤✐s ❣✐✈❡s ♦♥❡ ♦❢ t❤❡ ❧♦✇❡r ❡st✐♠❛t❡s✳ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

❋♦r t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ❡①♣❡❝t❛t✐♦♥ ✶ ✇❡ ❤❛✈❡ ❡q✉❛❧✐t② ♠❛① ❧♥ ✳ ❚❡❧❡♣♦rt❛t✐♦♥ ✷ ❧♥ ✶ ✶ ❝♦♥✜r♠s t❤❛t ❢♦r ❣r♦✉♣s t❤❡ ♠❛①✐♠❛❧ ❞✐♠❡♥s✐♦♥ ♦❢ ❛♥ ✐rr❡❞✉❝✐❜❧❡ r❡♣r❡s❡♥t❛t✐♦♥ s❛t✐s✜❡s ✶ ✷ ❞✐♠ ❆❝❝♦r❞✐♥❣ t♦ ❱❡rs❤✐❦ ❡t ❛❧❧✱ t❤❡ s②♠♠❡tr✐❝ ❣r♦✉♣ s❛t✐s✜❡s ♠❛① ❧♥ ❛♥❞ ❤❡♥❝❡ ✇❡ ✜♥❞ ❡①❛♠♣❧❡s ✇❤❡r❡ ❡♥t❛❣❧❡♠❡♥t ✐♠♣r♦✈❡s t❤❡ r❛t❡ ♦♥❧② ❜② ❛ ✈❡r② s♠❛❧❧ ❛♠♦✉♥t ✇❤✐❝❤ ✐s s♠❛❧❧ ❝♦♠♣❛r❡❞ t♦ t❤❡ ✈❛❧✉❡ ✳ ●♦♦❞ ❡①❛♠♣❧❡s ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

❆❝❝♦r❞✐♥❣ t♦ ❱❡rs❤✐❦ ❡t ❛❧❧✱ t❤❡ s②♠♠❡tr✐❝ ❣r♦✉♣ s❛t✐s✜❡s ♠❛① ❧♥ ❛♥❞ ❤❡♥❝❡ ✇❡ ✜♥❞ ❡①❛♠♣❧❡s ✇❤❡r❡ ❡♥t❛❣❧❡♠❡♥t ✐♠♣r♦✈❡s t❤❡ r❛t❡ ♦♥❧② ❜② ❛ ✈❡r② s♠❛❧❧ ❛♠♦✉♥t ✇❤✐❝❤ ✐s s♠❛❧❧ ❝♦♠♣❛r❡❞ t♦ t❤❡ ✈❛❧✉❡ ✳ ❚❡❧❡♣♦rt❛t✐♦♥ ✷ ❧♥ ✶ ✶ ❝♦♥✜r♠s t❤❛t ❢♦r ❣r♦✉♣s t❤❡ ♠❛①✐♠❛❧ ❞✐♠❡♥s✐♦♥ ♦❢ ❛♥ ✐rr❡❞✉❝✐❜❧❡ r❡♣r❡s❡♥t❛t✐♦♥ s❛t✐s✜❡s ✶ ✷ ❞✐♠ ●♦♦❞ ❡①❛♠♣❧❡s ❅ ❋♦r t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ❡①♣❡❝t❛t✐♦♥ θ f = θ ✶ ✇❡ ❤❛✈❡ ❡q✉❛❧✐t② Q ( θ f ) = ♠❛① k ❧♥ n k ✳ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

❆❝❝♦r❞✐♥❣ t♦ ❱❡rs❤✐❦ ❡t ❛❧❧✱ t❤❡ s②♠♠❡tr✐❝ ❣r♦✉♣ s❛t✐s✜❡s ♠❛① ❧♥ ❛♥❞ ❤❡♥❝❡ ✇❡ ✜♥❞ ❡①❛♠♣❧❡s ✇❤❡r❡ ❡♥t❛❣❧❡♠❡♥t ✐♠♣r♦✈❡s t❤❡ r❛t❡ ♦♥❧② ❜② ❛ ✈❡r② s♠❛❧❧ ❛♠♦✉♥t ✇❤✐❝❤ ✐s s♠❛❧❧ ❝♦♠♣❛r❡❞ t♦ t❤❡ ✈❛❧✉❡ ✳ ❚❡❧❡♣♦rt❛t✐♦♥ ✷ ❧♥ ✶ ✶ ❝♦♥✜r♠s t❤❛t ❢♦r ❣r♦✉♣s t❤❡ ♠❛①✐♠❛❧ ❞✐♠❡♥s✐♦♥ ♦❢ ❛♥ ✐rr❡❞✉❝✐❜❧❡ r❡♣r❡s❡♥t❛t✐♦♥ s❛t✐s✜❡s ✶ ✷ ❞✐♠ ●♦♦❞ ❡①❛♠♣❧❡s ❅ ❋♦r t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ❡①♣❡❝t❛t✐♦♥ θ f = θ ✶ ✇❡ ❤❛✈❡ ❡q✉❛❧✐t② Q ( θ f ) = ♠❛① k ❧♥ n k ✳ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

●♦♦❞ ❡①❛♠♣❧❡s ❅ ❋♦r t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ❡①♣❡❝t❛t✐♦♥ θ f = θ ✶ ✇❡ ❤❛✈❡ ❡q✉❛❧✐t② Q ( θ f ) = ♠❛① k ❧♥ n k ✳ ❚❡❧❡♣♦rt❛t✐♦♥ Q ( θ ✶ ) ≤ ✷ C EA ( θ ✶ ) = ❧♥ n ❝♦♥✜r♠s t❤❛t ❢♦r ❣r♦✉♣s t❤❡ ♠❛①✐♠❛❧ ❞✐♠❡♥s✐♦♥ ♦❢ ❛♥ ✐rr❡❞✉❝✐❜❧❡ r❡♣r❡s❡♥t❛t✐♦♥ π s❛t✐s✜❡s | ❞✐♠ ( π ) | ≤ | G | ✶ / ✷ . ❅ ❆❝❝♦r❞✐♥❣ t♦ ❱❡rs❤✐❦ ❡t ❛❧❧✱ t❤❡ s②♠♠❡tr✐❝ ❣r♦✉♣ n = m ! s❛t✐s✜❡s ♠❛① k ❧♥ n k ≥ √ n − c √ m ❛♥❞ ❤❡♥❝❡ ✇❡ ✜♥❞ ❡①❛♠♣❧❡s ✇❤❡r❡ ❡♥t❛❣❧❡♠❡♥t ✐♠♣r♦✈❡s t❤❡ r❛t❡ ♦♥❧② ❜② ❛ ✈❡r② √ s♠❛❧❧ ❛♠♦✉♥t √ m ✇❤✐❝❤ ✐s s♠❛❧❧ ❝♦♠♣❛r❡❞ t♦ t❤❡ ✈❛❧✉❡ m ! ✳ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

❋♦r t❤❡ ♠❛①✐♠❛❧ ❞✐♠❡♥s✐♦♥ ♦❢ ❛♥ ✐rr❡❞✉❝✐❜❧❡ r❡♣r❡s❡♥t❛t✐♦♥ ✐s s♠❛❧❧ ❝♦♠♣❛r❡❞ t♦ ❛♥❞ ❤❡♥❝❡ ♦✉r ♥❡✇ ❡st✐♠❛t❡s ♦✉t♣❡r❢♦r♠ t❤❡ ♠♦r❡ ❝❧❛ss✐❝❛❧ ✉♣♣❡r ❡st✐♠❛t❡s ❧♥ ❧♥ ❈♦✐♥❝✐❞❡♥t❛❧❧②✱ t❤✐s ❡st✐♠❛t❡ ✐s ❛❧s♦ ♥❡✇ ✭❜✉t r❡❧❛t❡❞ t♦ ♣r❡✈✐♦✉s ✇♦r❦ ✇✐t❤ ❘✉❛♥ ❛♥❞ ◆❡✉❢❛♥❣✮✳ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

❅ ❋♦r G = Z m d ⋊ Z m t❤❡ ♠❛①✐♠❛❧ ❞✐♠❡♥s✐♦♥ ♦❢ ❛♥ ✐rr❡❞✉❝✐❜❧❡ r❡♣r❡s❡♥t❛t✐♦♥ m ✐s s♠❛❧❧ ❝♦♠♣❛r❡❞ t♦ n = | G | = md m ❛♥❞ ❤❡♥❝❡ ♦✉r ♥❡✇ ❡st✐♠❛t❡s ♦✉t♣❡r❢♦r♠ t❤❡ ♠♦r❡ ❝❧❛ss✐❝❛❧ ✉♣♣❡r ❡st✐♠❛t❡s C EA ( θ f ) = ❧♥ n + τ ( f ❧♥ f ) ❈♦✐♥❝✐❞❡♥t❛❧❧②✱ t❤✐s ❡st✐♠❛t❡ ✐s ❛❧s♦ ♥❡✇ ✭❜✉t r❡❧❛t❡❞ t♦ ♣r❡✈✐♦✉s ✇♦r❦ ✇✐t❤ ❘✉❛♥ ❛♥❞ ◆❡✉❢❛♥❣✮✳ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

■t ✇❛s ✐♥❞❡♣❡♥❞t❧② ♣r♦✈❡❞ ❜② ◆❡✉❢❛♥❣ ❛♥❞ ❈r❛♥♥ t❤❛t ❧♥ ❙❝❤✉r ♠✉t❧✐♣❧✐❡rs ❛r❡ s♦✲❝❛❧❧❡❞ ❞❡❣r❛❞❛❜❧❡ ❝❤❛♥♥❡❧s ❛♥❞ t❤♦s❡ ❞♦♥✬t r❡q✉✐r❡ r❡❣✉❧❛r✐③❛t✐♦♥✦ ❖✉r ♣r❡✈✐♦✉s ❣r♦✉♣ ❝❤❛♥♥❡❧s✱ ❤♦✇❡✈❡r✱ ❛r❡ ✐♥ ❣❡♥❡r❛❧ ♥♦t ❞❡❣r❛❞❛❜❧❡✳ ❙❝❤✉r ♠✉❧t✐♣❧✐❡rs ❆ ❙t❛rt✐♥❣ ❢r♦♠ ❛ ✜♥✐t❡ ❣r♦✉♣ ✇❡ ❝❛♥ ❛❧s♦ ❞❡✜♥❡ t❤❡ ❙❝❤✉r ♠✉❧t✐♣❧✐❡r � f ( g − ✶ h ) x gh θ f ([ x gh ]) = gh ❣✐✈❡♥ ❜② ♣♦✐♥t✇✐s❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ t❤❡ ♠❛tr✐① x = [ x gh ] ✳ ❲❡ ♠❛② ❛ss✉♠❡ t❤❛t f ( g ) = τ ( d λ ( g )) ❣✐✈❡♥ ❜② t❤❡ ♥♦r♠❛❧✐③❡❞ tr❛❝❡ ♦♥ L ( G ) ✳ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

❙❝❤✉r ♠✉t❧✐♣❧✐❡rs ❛r❡ s♦✲❝❛❧❧❡❞ ❞❡❣r❛❞❛❜❧❡ ❝❤❛♥♥❡❧s ❛♥❞ t❤♦s❡ ❞♦♥✬t r❡q✉✐r❡ r❡❣✉❧❛r✐③❛t✐♦♥✦ ❖✉r ♣r❡✈✐♦✉s ❣r♦✉♣ ❝❤❛♥♥❡❧s✱ ❤♦✇❡✈❡r✱ ❛r❡ ✐♥ ❣❡♥❡r❛❧ ♥♦t ❞❡❣r❛❞❛❜❧❡✳ ❙❝❤✉r ♠✉❧t✐♣❧✐❡rs ❆ ❙t❛rt✐♥❣ ❢r♦♠ ❛ ✜♥✐t❡ ❣r♦✉♣ ✇❡ ❝❛♥ ❛❧s♦ ❞❡✜♥❡ t❤❡ ❙❝❤✉r ♠✉❧t✐♣❧✐❡r � f ( g − ✶ h ) x gh θ f ([ x gh ]) = gh ❣✐✈❡♥ ❜② ♣♦✐♥t✇✐s❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ t❤❡ ♠❛tr✐① x = [ x gh ] ✳ ❲❡ ♠❛② ❛ss✉♠❡ t❤❛t f ( g ) = τ ( d λ ( g )) ❣✐✈❡♥ ❜② t❤❡ ♥♦r♠❛❧✐③❡❞ tr❛❝❡ ♦♥ L ( G ) ✳ ❆ ■t ✇❛s ✐♥❞❡♣❡♥❞t❧② ♣r♦✈❡❞ ❜② ◆❡✉❢❛♥❣ ❛♥❞ ❈r❛♥♥ t❤❛t Q ( θ f ) = τ ( f ❧♥ f ) . ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

❙❝❤✉r ♠✉❧t✐♣❧✐❡rs ❆ ❙t❛rt✐♥❣ ❢r♦♠ ❛ ✜♥✐t❡ ❣r♦✉♣ ✇❡ ❝❛♥ ❛❧s♦ ❞❡✜♥❡ t❤❡ ❙❝❤✉r ♠✉❧t✐♣❧✐❡r � f ( g − ✶ h ) x gh θ f ([ x gh ]) = gh ❣✐✈❡♥ ❜② ♣♦✐♥t✇✐s❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ t❤❡ ♠❛tr✐① x = [ x gh ] ✳ ❲❡ ♠❛② ❛ss✉♠❡ t❤❛t f ( g ) = τ ( d λ ( g )) ❣✐✈❡♥ ❜② t❤❡ ♥♦r♠❛❧✐③❡❞ tr❛❝❡ ♦♥ L ( G ) ✳ ❆ ■t ✇❛s ✐♥❞❡♣❡♥❞t❧② ♣r♦✈❡❞ ❜② ◆❡✉❢❛♥❣ ❛♥❞ ❈r❛♥♥ t❤❛t Q ( θ f ) = τ ( f ❧♥ f ) . ❆ ❙❝❤✉r ♠✉t❧✐♣❧✐❡rs ❛r❡ s♦✲❝❛❧❧❡❞ ❞❡❣r❛❞❛❜❧❡ ❝❤❛♥♥❡❧s ❛♥❞ t❤♦s❡ ❞♦♥✬t r❡q✉✐r❡ r❡❣✉❧❛r✐③❛t✐♦♥✦ ❖✉r ♣r❡✈✐♦✉s ❣r♦✉♣ ❝❤❛♥♥❡❧s✱ ❤♦✇❡✈❡r✱ ❛r❡ ✐♥ ❣❡♥❡r❛❧ ♥♦t ❞❡❣r❛❞❛❜❧❡✳ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

❲✐♥t❡r ❡t ❛❧ ✐♥tr♦❞✉❝❡❞ t❤❡ ♣♦t❡♥t✐❛❧ ❝❛♣❛❝✐t② ✶ ✶ s✉♣ ❚❤✐s ❡①♣r❡ss✐♦♥ ✐s ❛❞❞✐t✐✈❡ ❛♥❞ ❧❛r❣❡r t❤❛♥ ❧✐♠ s✉♣ ✳ ❖✉r ✉♣♣❡r ❡st✐♠❛t❡s ❛❧s♦ ❤♦❧❞ ❢♦r ✳ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

❖✉r ✉♣♣❡r ❡st✐♠❛t❡s ❛❧s♦ ❤♦❧❞ ❢♦r ✳ ❆ ❲✐♥t❡r ❡t ❛❧ ✐♥tr♦❞✉❝❡❞ t❤❡ ♣♦t❡♥t✐❛❧ ❝❛♣❛❝✐t② Q ( pot ) ( N ) = s✉♣ N ′ Q ( ✶ ) ( N ⊗ N ′ ) − Q ( ✶ ) ( N ) . ❚❤✐s ❡①♣r❡ss✐♦♥ ✐s ❛❞❞✐t✐✈❡ ❛♥❞ ❧❛r❣❡r t❤❛♥ Q pot ( N ) = ❧✐♠ s✉♣ n Q ( pot ) ( N ⊗ n ) / n ✳ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

❆ ❲✐♥t❡r ❡t ❛❧ ✐♥tr♦❞✉❝❡❞ t❤❡ ♣♦t❡♥t✐❛❧ ❝❛♣❛❝✐t② Q ( pot ) ( N ) = s✉♣ N ′ Q ( ✶ ) ( N ⊗ N ′ ) − Q ( ✶ ) ( N ) . ❚❤✐s ❡①♣r❡ss✐♦♥ ✐s ❛❞❞✐t✐✈❡ ❛♥❞ ❧❛r❣❡r t❤❛♥ Q pot ( N ) = ❧✐♠ s✉♣ n Q ( pot ) ( N ⊗ n ) / n ✳ ❆ ❖✉r ✉♣♣❡r ❡st✐♠❛t❡s ❛❧s♦ ❤♦❧❞ ❢♦r Q ( pot ) ✳ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

❖✉r ❡①❛♠♣❧❡s ❛❧s♦ ✐♥❝❧✉❞❡ r❛♥❞♦♠ ✉♥✐t❛r✐❡s ❣✐✈❡♥ ❜② ♣r♦❞✉❝ts ♦❢ t❤❡ ✭▼❛❥♦r❛♥❛✮ ❈❧✐✛♦r❞ ❣❡♥❡r❛t♦rs✳ ❍♦✇❡✈❡r✱ ✐♥ t❤❡s❡ ♣❛rt✐❝✉❧❛r ❝❛s❡s ♦♥❧② t❤❡ ❡st✐♠❛t❡s ❢♦r ❛r❡ ❵♥❡✇✬✳ ❆s ♦❢ ♥♦✇ ❜❡tt❡r ❡st✐♠❛t❡s ❛r❡ ♦❜t❛✐♥❡❞ ❜② ❥✉st ❛❞❞✐♥❣ ❡♥t❛♥❣❧❡♠❡♥t ❛♥❞ ✉s❡ t❡❧❡♣♦rt❛t✐♦♥✳ ▼♦r❡ ❖✉r ❝❧❛ss ♦❢ ❝❤❛♥♥❡❧s ✐♥❝❧✉❞❡ q✉❛♥t✉♠ ❣r♦✉♣ ❝❤❛♥♥❡❧s✱ ✐♥ ♣❛rt✐❝✉❧❛r ❉r✐♥❢❡❧❞ ❞♦✉❜❧❡s✱ ✇❤✐❝❤ ❤❛✈❡ ❑r❛✉s ♦♣❡r❛t♦rs ✇❤✐❝❤ ❛r❡ ♥❡✐t❤❡r ✉♥✐t❛r✐❡s ♦r ♣r♦❥❡❝t✐♦♥s✳ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

▼♦r❡ ❖✉r ❝❧❛ss ♦❢ ❝❤❛♥♥❡❧s ✐♥❝❧✉❞❡ q✉❛♥t✉♠ ❣r♦✉♣ ❝❤❛♥♥❡❧s✱ ✐♥ ♣❛rt✐❝✉❧❛r ❉r✐♥❢❡❧❞ ❞♦✉❜❧❡s✱ ✇❤✐❝❤ ❤❛✈❡ ❑r❛✉s ♦♣❡r❛t♦rs ✇❤✐❝❤ ❛r❡ ♥❡✐t❤❡r ✉♥✐t❛r✐❡s ♦r ♣r♦❥❡❝t✐♦♥s✳ ❖✉r ❡①❛♠♣❧❡s ❛❧s♦ ✐♥❝❧✉❞❡ r❛♥❞♦♠ ✉♥✐t❛r✐❡s ❣✐✈❡♥ ❜② ♣r♦❞✉❝ts ♦❢ t❤❡ ✭▼❛❥♦r❛♥❛✮ ❈❧✐✛♦r❞ ❣❡♥❡r❛t♦rs✳ ❍♦✇❡✈❡r✱ ✐♥ t❤❡s❡ ♣❛rt✐❝✉❧❛r ❝❛s❡s ♦♥❧② t❤❡ ❡st✐♠❛t❡s ❢♦r Q ( pot ) ❛r❡ ❵♥❡✇✬✳ ❆s ♦❢ ♥♦✇ ❜❡tt❡r ❡st✐♠❛t❡s ❛r❡ ♦❜t❛✐♥❡❞ ❜② ❥✉st ❛❞❞✐♥❣ ❡♥t❛♥❣❧❡♠❡♥t ❛♥❞ ✉s❡ t❡❧❡♣♦rt❛t✐♦♥✳ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

■♥❞❡❡❞✱ ✶ ✷ s✉♣ r❝ ♣ ❞ st ❲❡ ♠❛② ♣✉t ❞✐✛❡r❡♥t ♥♦r♠s ❛♥❞ ✭s✐❝✮ ❞✐✛❡r❡♥t ♦♣❡r❛t♦r s♣❛❝❡ str✉❝t✉r❡s ♦♥ t❤✐s ❙t✐♥s♣r✐♥❣ s♣❛❝❡ ✳ st ❍❡r❡ ✐s ♦❜t❛✐♥❡❞ ❜② ❝♦♠♣❧❡s ✐♥t❡r♣♦❛t✐♦♥✳ ✶ ❚❤❡ ♦♥❡ s❤♦t ❝❛♣❛❝✐t② ❝♦rr❡s♣♦♥❞s t♦ ❛ ❝♦♥✈❡①✐t② ♣r♦♣❡rt② r❝ ✷ ✇❤❡r❡ ❛♥❞ ✳ ❚♦♦❧s ✭❙t✐♥❡s♣r✐♥❣✲❑r❛✉s✲❍❛②❞♦♥✴❲✐♥t❡r✮ ❊✈❡r② ❝❤❛♥♥❡❧ ❝♦♠❡s ✇✐t❤ ❛ ♣❛rt✐❛❧ ✐s♦♠❡tr② V : H A → H B ⊗ H E ❛♥❞ ❤❡♥❝❡ ❛ s✉❜s♣❛❝❡ st ( N ) ⊂ H B ⊗ H E ♦❢ ❛ t❡♥s♦r ♣r♦❞✉❝t ♦❢ t✇♦ ❍✐❧❜❡rt s♣❛❝❡s✳ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

■♥❞❡❡❞✱ ✶ ✷ s✉♣ r❝ ♣ ❞ st ❍❡r❡ ✐s ♦❜t❛✐♥❡❞ ❜② ❝♦♠♣❧❡s ✐♥t❡r♣♦❛t✐♦♥✳ ✶ ❚❤❡ ♦♥❡ s❤♦t ❝❛♣❛❝✐t② ❝♦rr❡s♣♦♥❞s t♦ ❛ ❝♦♥✈❡①✐t② ♣r♦♣❡rt② r❝ ✷ ✇❤❡r❡ ❛♥❞ ✳ ❚♦♦❧s ✭❙t✐♥❡s♣r✐♥❣✲❑r❛✉s✲❍❛②❞♦♥✴❲✐♥t❡r✮ ❊✈❡r② ❝❤❛♥♥❡❧ ❝♦♠❡s ✇✐t❤ ❛ ♣❛rt✐❛❧ ✐s♦♠❡tr② V : H A → H B ⊗ H E ❛♥❞ ❤❡♥❝❡ ❛ s✉❜s♣❛❝❡ st ( N ) ⊂ H B ⊗ H E ♦❢ ❛ t❡♥s♦r ♣r♦❞✉❝t ♦❢ t✇♦ ❍✐❧❜❡rt s♣❛❝❡s✳ ❲❡ ♠❛② ♣✉t ❞✐✛❡r❡♥t ♥♦r♠s ❛♥❞ ✭s✐❝✮ ❞✐✛❡r❡♥t ♦♣❡r❛t♦r s♣❛❝❡ str✉❝t✉r❡s H c p B ⊗ h H r E ♦♥ t❤✐s ❙t✐♥s♣r✐♥❣ s♣❛❝❡ X = st p ( N ) ✳ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

■♥❞❡❡❞✱ ✶ ✷ s✉♣ r❝ ♣ ❞ st ❚❤❡ ♦♥❡ s❤♦t ❝❛♣❛❝✐t② ❝♦rr❡s♣♦♥❞s t♦ ❛ ❝♦♥✈❡①✐t② ♣r♦♣❡rt② r❝ ✷ ✇❤❡r❡ ❛♥❞ ✳ ❚♦♦❧s ✭❙t✐♥❡s♣r✐♥❣✲❑r❛✉s✲❍❛②❞♦♥✴❲✐♥t❡r✮ ❊✈❡r② ❝❤❛♥♥❡❧ ❝♦♠❡s ✇✐t❤ ❛ ♣❛rt✐❛❧ ✐s♦♠❡tr② V : H A → H B ⊗ H E ❛♥❞ ❤❡♥❝❡ ❛ s✉❜s♣❛❝❡ st ( N ) ⊂ H B ⊗ H E ♦❢ ❛ t❡♥s♦r ♣r♦❞✉❝t ♦❢ t✇♦ ❍✐❧❜❡rt s♣❛❝❡s✳ ❲❡ ♠❛② ♣✉t ❞✐✛❡r❡♥t ♥♦r♠s ❛♥❞ ✭s✐❝✮ ❞✐✛❡r❡♥t ♦♣❡r❛t♦r s♣❛❝❡ str✉❝t✉r❡s H c p B ⊗ h H r E ♦♥ t❤✐s ❙t✐♥s♣r✐♥❣ s♣❛❝❡ X = st p ( N ) ✳ ❍❡r❡ H c p = [ H c , H r ] ✶ / p ✐s ♦❜t❛✐♥❡❞ ❜② ❝♦♠♣❧❡s ✐♥t❡r♣♦❛t✐♦♥✳ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

■♥❞❡❡❞✱ ✶ ✷ s✉♣ r❝ ♣ ❞ st ❚♦♦❧s ✭❙t✐♥❡s♣r✐♥❣✲❑r❛✉s✲❍❛②❞♦♥✴❲✐♥t❡r✮ ❊✈❡r② ❝❤❛♥♥❡❧ ❝♦♠❡s ✇✐t❤ ❛ ♣❛rt✐❛❧ ✐s♦♠❡tr② V : H A → H B ⊗ H E ❛♥❞ ❤❡♥❝❡ ❛ s✉❜s♣❛❝❡ st ( N ) ⊂ H B ⊗ H E ♦❢ ❛ t❡♥s♦r ♣r♦❞✉❝t ♦❢ t✇♦ ❍✐❧❜❡rt s♣❛❝❡s✳ ❲❡ ♠❛② ♣✉t ❞✐✛❡r❡♥t ♥♦r♠s ❛♥❞ ✭s✐❝✮ ❞✐✛❡r❡♥t ♦♣❡r❛t♦r s♣❛❝❡ str✉❝t✉r❡s H c p B ⊗ h H r E ♦♥ t❤✐s ❙t✐♥s♣r✐♥❣ s♣❛❝❡ X = st p ( N ) ✳ ❍❡r❡ H c p = [ H c , H r ] ✶ / p ✐s ♦❜t❛✐♥❡❞ ❜② ❝♦♠♣❧❡s ✐♥t❡r♣♦❛t✐♦♥✳ ❚❤❡ ♦♥❡ s❤♦t ❝❛♣❛❝✐t② ❝♦rr❡s♣♦♥❞s t♦ ❛ ❝♦♥✈❡①✐t② ♣r♦♣❡rt② ✷ ⊗ id X : R d ( X ) → C d r❝ p , d ( X ) = � id ℓ d p ( X ) � , ✇❤❡r❡ R d ( X ) = X ⊗ h R d ❛♥❞ C d p ( X ) = C p ⊗ h X ✳ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

■♥❞❡❡❞✱ ✶ ✷ s✉♣ r❝ ♣ ❞ st ❚♦♦❧s ✭❙t✐♥❡s♣r✐♥❣✲❑r❛✉s✲❍❛②❞♦♥✴❲✐♥t❡r✮ ❊✈❡r② ❝❤❛♥♥❡❧ ❝♦♠❡s ✇✐t❤ ❛ ♣❛rt✐❛❧ ✐s♦♠❡tr② V : H A → H B ⊗ H E ❛♥❞ ❤❡♥❝❡ ❛ s✉❜s♣❛❝❡ st ( N ) ⊂ H B ⊗ H E ♦❢ ❛ t❡♥s♦r ♣r♦❞✉❝t ♦❢ t✇♦ ❍✐❧❜❡rt s♣❛❝❡s✳ ❲❡ ♠❛② ♣✉t ❞✐✛❡r❡♥t ♥♦r♠s ❛♥❞ ✭s✐❝✮ ❞✐✛❡r❡♥t ♦♣❡r❛t♦r s♣❛❝❡ str✉❝t✉r❡s H c p B ⊗ h H r E ♦♥ t❤✐s ❙t✐♥s♣r✐♥❣ s♣❛❝❡ X = st p ( N ) ✳ ❍❡r❡ H c p = [ H c , H r ] ✶ / p ✐s ♦❜t❛✐♥❡❞ ❜② ❝♦♠♣❧❡s ✐♥t❡r♣♦❛t✐♦♥✳ ❚❤❡ ♦♥❡ s❤♦t ❝❛♣❛❝✐t② ❝♦rr❡s♣♦♥❞s t♦ ❛ ❝♦♥✈❡①✐t② ♣r♦♣❡rt② ✷ ⊗ id X : R d ( X ) → C d r❝ p , d ( X ) = � id ℓ d p ( X ) � , ✇❤❡r❡ R d ( X ) = X ⊗ h R d ❛♥❞ C d p ( X ) = C p ⊗ h X ✳ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

❚♦♦❧s ✭❙t✐♥❡s♣r✐♥❣✲❑r❛✉s✲❍❛②❞♦♥✴❲✐♥t❡r✮ ❊✈❡r② ❝❤❛♥♥❡❧ ❝♦♠❡s ✇✐t❤ ❛ ♣❛rt✐❛❧ ✐s♦♠❡tr② V : H A → H B ⊗ H E ❛♥❞ ❤❡♥❝❡ ❛ s✉❜s♣❛❝❡ st ( N ) ⊂ H B ⊗ H E ♦❢ ❛ t❡♥s♦r ♣r♦❞✉❝t ♦❢ t✇♦ ❍✐❧❜❡rt s♣❛❝❡s✳ ❲❡ ♠❛② ♣✉t ❞✐✛❡r❡♥t ♥♦r♠s ❛♥❞ ✭s✐❝✮ ❞✐✛❡r❡♥t ♦♣❡r❛t♦r s♣❛❝❡ str✉❝t✉r❡s H c p B ⊗ h H r E ♦♥ t❤✐s ❙t✐♥s♣r✐♥❣ s♣❛❝❡ X = st p ( N ) ✳ ❍❡r❡ H c p = [ H c , H r ] ✶ / p ✐s ♦❜t❛✐♥❡❞ ❜② ❝♦♠♣❧❡s ✐♥t❡r♣♦❛t✐♦♥✳ ❚❤❡ ♦♥❡ s❤♦t ❝❛♣❛❝✐t② ❝♦rr❡s♣♦♥❞s t♦ ❛ ❝♦♥✈❡①✐t② ♣r♦♣❡rt② ✷ ⊗ id X : R d ( X ) → C d r❝ p , d ( X ) = � id ℓ d p ( X ) � , ✇❤❡r❡ R d ( X ) = X ⊗ h R d ❛♥❞ C d p ( X ) = C p ⊗ h X ✳ ■♥❞❡❡❞✱ d dp r❝ ♣ , ❞ ( st p ( N )) ✷ . Q ( ✶ ) ( N ) = s✉♣ d ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

❯♥❞❡r ♦✉r ❛ss✉♠♣t✐♦♥s t❤❡ s♣❛❝❡ st ✶ ❝♦rr❡s♣♦♥❞s t♦ ❛ s♣❛❝❡ ♦❢ t❤❡ ❢♦r♠ ✇❤❡r❡ ✐s ❛ ✈♦♥ ◆❡✉♠❛♥♥ ❛❧❣❡❜r❛✳ ❘❡❝❛❧❧ t❤❛t t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ❡①♣❡❝t❛t✐♦♥ ♦♥t♦ ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s ✱ ❛♥ ❤❡♥❝❡ ✐s ♥✐❝❡❧② ❝♦♠♣❧❡♠❡♥t❡❞ ✐♥ ✳ ❯s✐♥❣ ❝♦♠♣❧❡① ✐♥t❡r♣♦❧❛t✐♦♥ ✇❡ s❤♦✇ t❤❛t ✶ ❚❤✐s ✐s ❡♥♦✉❣❤ t♦ s❤♦✇ ♦✉r ✉♣♣❡r ❜♦✉♥❞s✳ ❯s✐♥❣ s✐♠✐❧❛r t❡❝❤♥✐q✉❡s ✇❡ ✇✐❧❧ ♣r♦✈❡ ✶ ✶ ✶ ✶ ❚❤✐s ❛❧❧♦✇s ✉s t♦ ♣r♦✈❡ ❡st✐♠❛t❡s ❢♦r t❤❡ ♣r✐✈❛t❡ ❝❛♣❛❝✐t②✳ ▲♦❝❛❧ ❝♦♠♣❛r✐s♦♥ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

❯s✐♥❣ ❝♦♠♣❧❡① ✐♥t❡r♣♦❧❛t✐♦♥ ✇❡ s❤♦✇ t❤❛t ✶ ❚❤✐s ✐s ❡♥♦✉❣❤ t♦ s❤♦✇ ♦✉r ✉♣♣❡r ❜♦✉♥❞s✳ ❯s✐♥❣ s✐♠✐❧❛r t❡❝❤♥✐q✉❡s ✇❡ ✇✐❧❧ ♣r♦✈❡ ✶ ✶ ✶ ✶ ❚❤✐s ❛❧❧♦✇s ✉s t♦ ♣r♦✈❡ ❡st✐♠❛t❡s ❢♦r t❤❡ ♣r✐✈❛t❡ ❝❛♣❛❝✐t②✳ ❘❡❝❛❧❧ t❤❛t t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ❡①♣❡❝t❛t✐♦♥ ♦♥t♦ ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s ✱ ❛♥ ❤❡♥❝❡ ✐s ♥✐❝❡❧② ❝♦♠♣❧❡♠❡♥t❡❞ ✐♥ ✳ ▲♦❝❛❧ ❝♦♠♣❛r✐s♦♥ ❯♥❞❡r ♦✉r ❛ss✉♠♣t✐♦♥s t❤❡ s♣❛❝❡ st p ( θ ✶ ) ❝♦rr❡s♣♦♥❞s t♦ ❛ s♣❛❝❡ ♦❢ t❤❡ ❢♦r♠ Mv ✇❤❡r❡ M ✐s ❛ ✈♦♥ ◆❡✉♠❛♥♥ ❛❧❣❡❜r❛✳ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

❯s✐♥❣ ❝♦♠♣❧❡① ✐♥t❡r♣♦❧❛t✐♦♥ ✇❡ s❤♦✇ t❤❛t ✶ ❚❤✐s ✐s ❡♥♦✉❣❤ t♦ s❤♦✇ ♦✉r ✉♣♣❡r ❜♦✉♥❞s✳ ❯s✐♥❣ s✐♠✐❧❛r t❡❝❤♥✐q✉❡s ✇❡ ✇✐❧❧ ♣r♦✈❡ ✶ ✶ ✶ ✶ ❚❤✐s ❛❧❧♦✇s ✉s t♦ ♣r♦✈❡ ❡st✐♠❛t❡s ❢♦r t❤❡ ♣r✐✈❛t❡ ❝❛♣❛❝✐t②✳ ▲♦❝❛❧ ❝♦♠♣❛r✐s♦♥ ❯♥❞❡r ♦✉r ❛ss✉♠♣t✐♦♥s t❤❡ s♣❛❝❡ st p ( θ ✶ ) ❝♦rr❡s♣♦♥❞s t♦ ❛ s♣❛❝❡ ♦❢ t❤❡ ❢♦r♠ Mv ✇❤❡r❡ M ✐s ❛ ✈♦♥ ◆❡✉♠❛♥♥ ❛❧❣❡❜r❛✳ ❘❡❝❛❧❧ t❤❛t t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ❡①♣❡❝t❛t✐♦♥ ♦♥t♦ M ❝❛♥ ❜❡ ✇r✐tt❡♥ U ( M ′ ) u ∗ ( ) ud µ ( u ) ✱ ❛♥ ❤❡♥❝❡ ✐s ♥✐❝❡❧② ❝♦♠♣❧❡♠❡♥t❡❞ ✐♥ � ❛s H c p B ⊗ h H E ✳ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

❯s✐♥❣ ❝♦♠♣❧❡① ✐♥t❡r♣♦❧❛t✐♦♥ ✇❡ s❤♦✇ t❤❛t ✶ ❚❤✐s ✐s ❡♥♦✉❣❤ t♦ s❤♦✇ ♦✉r ✉♣♣❡r ❜♦✉♥❞s✳ ❯s✐♥❣ s✐♠✐❧❛r t❡❝❤♥✐q✉❡s ✇❡ ✇✐❧❧ ♣r♦✈❡ ✶ ✶ ✶ ✶ ❚❤✐s ❛❧❧♦✇s ✉s t♦ ♣r♦✈❡ ❡st✐♠❛t❡s ❢♦r t❤❡ ♣r✐✈❛t❡ ❝❛♣❛❝✐t②✳ ▲♦❝❛❧ ❝♦♠♣❛r✐s♦♥ ❯♥❞❡r ♦✉r ❛ss✉♠♣t✐♦♥s t❤❡ s♣❛❝❡ st p ( θ ✶ ) ❝♦rr❡s♣♦♥❞s t♦ ❛ s♣❛❝❡ ♦❢ t❤❡ ❢♦r♠ Mv ✇❤❡r❡ M ✐s ❛ ✈♦♥ ◆❡✉♠❛♥♥ ❛❧❣❡❜r❛✳ ❘❡❝❛❧❧ t❤❛t t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ❡①♣❡❝t❛t✐♦♥ ♦♥t♦ M ❝❛♥ ❜❡ ✇r✐tt❡♥ U ( M ′ ) u ∗ ( ) ud µ ( u ) ✱ ❛♥ ❤❡♥❝❡ ✐s ♥✐❝❡❧② ❝♦♠♣❧❡♠❡♥t❡❞ ✐♥ � ❛s H c p B ⊗ h H E ✳ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

❯s✐♥❣ s✐♠✐❧❛r t❡❝❤♥✐q✉❡s ✇❡ ✇✐❧❧ ♣r♦✈❡ ✶ ✶ ✶ ✶ ❚❤✐s ❛❧❧♦✇s ✉s t♦ ♣r♦✈❡ ❡st✐♠❛t❡s ❢♦r t❤❡ ♣r✐✈❛t❡ ❝❛♣❛❝✐t②✳ ▲♦❝❛❧ ❝♦♠♣❛r✐s♦♥ ❯♥❞❡r ♦✉r ❛ss✉♠♣t✐♦♥s t❤❡ s♣❛❝❡ st p ( θ ✶ ) ❝♦rr❡s♣♦♥❞s t♦ ❛ s♣❛❝❡ ♦❢ t❤❡ ❢♦r♠ Mv ✇❤❡r❡ M ✐s ❛ ✈♦♥ ◆❡✉♠❛♥♥ ❛❧❣❡❜r❛✳ ❘❡❝❛❧❧ t❤❛t t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ❡①♣❡❝t❛t✐♦♥ ♦♥t♦ M ❝❛♥ ❜❡ ✇r✐tt❡♥ U ( M ′ ) u ∗ ( ) ud µ ( u ) ✱ ❛♥ ❤❡♥❝❡ ✐s ♥✐❝❡❧② ❝♦♠♣❧❡♠❡♥t❡❞ ✐♥ � ❛s H c p B ⊗ h H E ✳ ❯s✐♥❣ ❝♦♠♣❧❡① ✐♥t❡r♣♦❧❛t✐♦♥ ✇❡ s❤♦✇ t❤❛t � id ⊗ θ f ( ρ ) � p ≤ � f � p � id ⊗ θ ✶ ( ρ ) � p ≤ � f � p � id ⊗ θ f ( ρ ) � p . ❚❤✐s ✐s ❡♥♦✉❣❤ t♦ s❤♦✇ ♦✉r ✉♣♣❡r ❜♦✉♥❞s✳ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

▲♦❝❛❧ ❝♦♠♣❛r✐s♦♥ ❯♥❞❡r ♦✉r ❛ss✉♠♣t✐♦♥s t❤❡ s♣❛❝❡ st p ( θ ✶ ) ❝♦rr❡s♣♦♥❞s t♦ ❛ s♣❛❝❡ ♦❢ t❤❡ ❢♦r♠ Mv ✇❤❡r❡ M ✐s ❛ ✈♦♥ ◆❡✉♠❛♥♥ ❛❧❣❡❜r❛✳ ❘❡❝❛❧❧ t❤❛t t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ❡①♣❡❝t❛t✐♦♥ ♦♥t♦ M ❝❛♥ ❜❡ ✇r✐tt❡♥ U ( M ′ ) u ∗ ( ) ud µ ( u ) ✱ ❛♥ ❤❡♥❝❡ ✐s ♥✐❝❡❧② ❝♦♠♣❧❡♠❡♥t❡❞ ✐♥ � ❛s H c p B ⊗ h H E ✳ ❯s✐♥❣ ❝♦♠♣❧❡① ✐♥t❡r♣♦❧❛t✐♦♥ ✇❡ s❤♦✇ t❤❛t � id ⊗ θ f ( ρ ) � p ≤ � f � p � id ⊗ θ ✶ ( ρ ) � p ≤ � f � p � id ⊗ θ f ( ρ ) � p . ❚❤✐s ✐s ❡♥♦✉❣❤ t♦ s❤♦✇ ♦✉r ✉♣♣❡r ❜♦✉♥❞s✳ ❯s✐♥❣ s✐♠✐❧❛r t❡❝❤♥✐q✉❡s ✇❡ ✇✐❧❧ ♣r♦✈❡ � θ f ⊗ id ( ρ ) � S ✶ ( H B ; S p ( H A )) ≤ � f � p � θ ✶ ⊗ id ( ρ ) � S ✶ ( H B ; S p ( H A )) ≤ � f � p � θ f ⊗ id ( ρ ) � S ✶ ( H B ; S p ( H A )) . ❚❤✐s ❛❧❧♦✇s ✉s t♦ ♣r♦✈❡ ❡st✐♠❛t❡s ❢♦r t❤❡ ♣r✐✈❛t❡ ❝❛♣❛❝✐t②✳ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

▲♦✇❡r ❜♦✉♥❞s ❛r❡ ♦❜t❛✐♥❡❞ ❜② ❝❛❧❝✉❧❛t✐♥❣ t❤❡ ♠✐♥✲❝❜✲❡♥tr♦♣② ♦r s♦✲❝❛❧❧❡❞ r❡✈❡rs❡❞ ❝♦❤❡r❡♥t ✐♥❢♦r♠❛t✐♦♥ ✶ ✶ ♠✐♥ ■♥❞❡❡❞✱ ❧♥ ❀ ♠✐♥ ❖✉r ❝❧❛ss❡s ♦❢ ❝❤❛♥♥❡❧s ❛r❡ ❛✉t♦♠❛t✐❝❛❧❧② ❝❧♦s❡❞ ✉♥❞❡r t❛❦✐♥❣ t❡♥s♦r ♣r♦❞✉❝ts ❛♥❞ ❤❡♥❝❡ r❡❣✉❧❛r✐③❛t✐♦♥ ✐s ♥♦t ❛ ❜✐❣ ✐ss✉❡ ❤❡r❡✳ ❚❤✐s s❡❡♠s t♦ ❜❡ ❛ ✜rst s②st❡♠❛t✐❝ st✉❞② ❢♦r ♥♦♥ ❞❡❣r❛❞❛❜❧❡ ❝❤❛♥♥❡❧s✳ ❙❝❤✉r ♠✉t❧✐♣❧✐❡rs ❛r❡ ❞❡❣r❛❞❛❜❧❡✱ ✐✳❡✳ t❤❡r❡ ❡①✐sts ❛ ❝❤❛♥♥❡❧ s✉❝❤ t❤❛t ❤♦❧❞s ❢♦r t❤❡ ❝♦♠♣❧❡♠❡♥t❛r② ❝❤❛♥♥❡❧ ✭tr❛❝❡ ♦✉t ❇♦❜ ✐♥st❡❛❞ ♦❢ ❡♥✈✐r♦♥♠❡♥t✮✳ ❆♥ ✐♠♣♦rt❛♥t t♦♦❧ ❢♦r ♦✉r ❛♥❛❧②s✐s ❢❡❛t✉r❡ ✐s t❤❡ st❛♥❞❛r❞ ❢♦r♠ ♦❢ ❛ ✜♥✐t❡ ✈♦♥ ◆❡✉♠❛♥♥ ❛❧❣❡❜r❛✳ ▲♦✇❡r ❜♦✉♥❞s ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

❚❤✐s s❡❡♠s t♦ ❜❡ ❛ ✜rst s②st❡♠❛t✐❝ st✉❞② ❢♦r ♥♦♥ ❞❡❣r❛❞❛❜❧❡ ❝❤❛♥♥❡❧s✳ ❙❝❤✉r ♠✉t❧✐♣❧✐❡rs ❛r❡ ❞❡❣r❛❞❛❜❧❡✱ ✐✳❡✳ t❤❡r❡ ❡①✐sts ❛ ❝❤❛♥♥❡❧ s✉❝❤ t❤❛t ❤♦❧❞s ❢♦r t❤❡ ❝♦♠♣❧❡♠❡♥t❛r② ❝❤❛♥♥❡❧ ✭tr❛❝❡ ♦✉t ❇♦❜ ✐♥st❡❛❞ ♦❢ ❡♥✈✐r♦♥♠❡♥t✮✳ ❆♥ ✐♠♣♦rt❛♥t t♦♦❧ ❢♦r ♦✉r ❛♥❛❧②s✐s ❢❡❛t✉r❡ ✐s t❤❡ st❛♥❞❛r❞ ❢♦r♠ ♦❢ ❛ ✜♥✐t❡ ✈♦♥ ◆❡✉♠❛♥♥ ❛❧❣❡❜r❛✳ ■♥❞❡❡❞✱ ❧♥ ❀ ♠✐♥ ❖✉r ❝❧❛ss❡s ♦❢ ❝❤❛♥♥❡❧s ❛r❡ ❛✉t♦♠❛t✐❝❛❧❧② ❝❧♦s❡❞ ✉♥❞❡r t❛❦✐♥❣ t❡♥s♦r ♣r♦❞✉❝ts ❛♥❞ ❤❡♥❝❡ r❡❣✉❧❛r✐③❛t✐♦♥ ✐s ♥♦t ❛ ❜✐❣ ✐ss✉❡ ❤❡r❡✳ ▲♦✇❡r ❜♦✉♥❞s ▲♦✇❡r ❜♦✉♥❞s ❛r❡ ♦❜t❛✐♥❡❞ ❜② ❝❛❧❝✉❧❛t✐♥❣ t❤❡ ♠✐♥✲❝❜✲❡♥tr♦♣② ♦r s♦✲❝❛❧❧❡❞ r❡✈❡rs❡❞ ❝♦❤❡r❡♥t ✐♥❢♦r♠❛t✐♦♥ d H cb ♠✐♥ ( N ) = dp �N : S ✶ → S p � cb | p = ✶ . ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

❚❤✐s s❡❡♠s t♦ ❜❡ ❛ ✜rst s②st❡♠❛t✐❝ st✉❞② ❢♦r ♥♦♥ ❞❡❣r❛❞❛❜❧❡ ❝❤❛♥♥❡❧s✳ ❙❝❤✉r ♠✉t❧✐♣❧✐❡rs ❛r❡ ❞❡❣r❛❞❛❜❧❡✱ ✐✳❡✳ t❤❡r❡ ❡①✐sts ❛ ❝❤❛♥♥❡❧ s✉❝❤ t❤❛t ❤♦❧❞s ❢♦r t❤❡ ❝♦♠♣❧❡♠❡♥t❛r② ❝❤❛♥♥❡❧ ✭tr❛❝❡ ♦✉t ❇♦❜ ✐♥st❡❛❞ ♦❢ ❡♥✈✐r♦♥♠❡♥t✮✳ ❆♥ ✐♠♣♦rt❛♥t t♦♦❧ ❢♦r ♦✉r ❛♥❛❧②s✐s ❢❡❛t✉r❡ ✐s t❤❡ st❛♥❞❛r❞ ❢♦r♠ ♦❢ ❛ ✜♥✐t❡ ✈♦♥ ◆❡✉♠❛♥♥ ❛❧❣❡❜r❛✳ ❖✉r ❝❧❛ss❡s ♦❢ ❝❤❛♥♥❡❧s ❛r❡ ❛✉t♦♠❛t✐❝❛❧❧② ❝❧♦s❡❞ ✉♥❞❡r t❛❦✐♥❣ t❡♥s♦r ♣r♦❞✉❝ts ❛♥❞ ❤❡♥❝❡ r❡❣✉❧❛r✐③❛t✐♦♥ ✐s ♥♦t ❛ ❜✐❣ ✐ss✉❡ ❤❡r❡✳ ▲♦✇❡r ❜♦✉♥❞s ▲♦✇❡r ❜♦✉♥❞s ❛r❡ ♦❜t❛✐♥❡❞ ❜② ❝❛❧❝✉❧❛t✐♥❣ t❤❡ ♠✐♥✲❝❜✲❡♥tr♦♣② ♦r s♦✲❝❛❧❧❡❞ r❡✈❡rs❡❞ ❝♦❤❡r❡♥t ✐♥❢♦r♠❛t✐♦♥ d H cb ♠✐♥ ( N ) = dp �N : S ✶ → S p � cb | p = ✶ . ■♥❞❡❡❞✱ H cb ♠✐♥ ( θ f ) = τ ( f ❧♥ f ) ❀ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

❙❝❤✉r ♠✉t❧✐♣❧✐❡rs ❛r❡ ❞❡❣r❛❞❛❜❧❡✱ ✐✳❡✳ t❤❡r❡ ❡①✐sts ❛ ❝❤❛♥♥❡❧ s✉❝❤ t❤❛t ❤♦❧❞s ❢♦r t❤❡ ❝♦♠♣❧❡♠❡♥t❛r② ❝❤❛♥♥❡❧ ✭tr❛❝❡ ♦✉t ❇♦❜ ✐♥st❡❛❞ ♦❢ ❡♥✈✐r♦♥♠❡♥t✮✳ ❆♥ ✐♠♣♦rt❛♥t t♦♦❧ ❢♦r ♦✉r ❛♥❛❧②s✐s ❢❡❛t✉r❡ ✐s t❤❡ st❛♥❞❛r❞ ❢♦r♠ ♦❢ ❛ ✜♥✐t❡ ✈♦♥ ◆❡✉♠❛♥♥ ❛❧❣❡❜r❛✳ ❚❤✐s s❡❡♠s t♦ ❜❡ ❛ ✜rst s②st❡♠❛t✐❝ st✉❞② ❢♦r ♥♦♥ ❞❡❣r❛❞❛❜❧❡ ❝❤❛♥♥❡❧s✳ ▲♦✇❡r ❜♦✉♥❞s ▲♦✇❡r ❜♦✉♥❞s ❛r❡ ♦❜t❛✐♥❡❞ ❜② ❝❛❧❝✉❧❛t✐♥❣ t❤❡ ♠✐♥✲❝❜✲❡♥tr♦♣② ♦r s♦✲❝❛❧❧❡❞ r❡✈❡rs❡❞ ❝♦❤❡r❡♥t ✐♥❢♦r♠❛t✐♦♥ d H cb ♠✐♥ ( N ) = dp �N : S ✶ → S p � cb | p = ✶ . ■♥❞❡❡❞✱ H cb ♠✐♥ ( θ f ) = τ ( f ❧♥ f ) ❀ ❖✉r ❝❧❛ss❡s ♦❢ ❝❤❛♥♥❡❧s ❛r❡ ❛✉t♦♠❛t✐❝❛❧❧② ❝❧♦s❡❞ ✉♥❞❡r t❛❦✐♥❣ t❡♥s♦r ♣r♦❞✉❝ts ❛♥❞ ❤❡♥❝❡ r❡❣✉❧❛r✐③❛t✐♦♥ ✐s ♥♦t ❛ ❜✐❣ ✐ss✉❡ ❤❡r❡✳ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

❙❝❤✉r ♠✉t❧✐♣❧✐❡rs ❛r❡ ❞❡❣r❛❞❛❜❧❡✱ ✐✳❡✳ t❤❡r❡ ❡①✐sts ❛ ❝❤❛♥♥❡❧ s✉❝❤ t❤❛t ❤♦❧❞s ❢♦r t❤❡ ❝♦♠♣❧❡♠❡♥t❛r② ❝❤❛♥♥❡❧ ✭tr❛❝❡ ♦✉t ❇♦❜ ✐♥st❡❛❞ ♦❢ ❡♥✈✐r♦♥♠❡♥t✮✳ ❆♥ ✐♠♣♦rt❛♥t t♦♦❧ ❢♦r ♦✉r ❛♥❛❧②s✐s ❢❡❛t✉r❡ ✐s t❤❡ st❛♥❞❛r❞ ❢♦r♠ ♦❢ ❛ ✜♥✐t❡ ✈♦♥ ◆❡✉♠❛♥♥ ❛❧❣❡❜r❛✳ ❚❤✐s s❡❡♠s t♦ ❜❡ ❛ ✜rst s②st❡♠❛t✐❝ st✉❞② ❢♦r ♥♦♥ ❞❡❣r❛❞❛❜❧❡ ❝❤❛♥♥❡❧s✳ ▲♦✇❡r ❜♦✉♥❞s ▲♦✇❡r ❜♦✉♥❞s ❛r❡ ♦❜t❛✐♥❡❞ ❜② ❝❛❧❝✉❧❛t✐♥❣ t❤❡ ♠✐♥✲❝❜✲❡♥tr♦♣② ♦r s♦✲❝❛❧❧❡❞ r❡✈❡rs❡❞ ❝♦❤❡r❡♥t ✐♥❢♦r♠❛t✐♦♥ d H cb ♠✐♥ ( N ) = dp �N : S ✶ → S p � cb | p = ✶ . ■♥❞❡❡❞✱ H cb ♠✐♥ ( θ f ) = τ ( f ❧♥ f ) ❀ ❖✉r ❝❧❛ss❡s ♦❢ ❝❤❛♥♥❡❧s ❛r❡ ❛✉t♦♠❛t✐❝❛❧❧② ❝❧♦s❡❞ ✉♥❞❡r t❛❦✐♥❣ t❡♥s♦r ♣r♦❞✉❝ts ❛♥❞ ❤❡♥❝❡ r❡❣✉❧❛r✐③❛t✐♦♥ ✐s ♥♦t ❛ ❜✐❣ ✐ss✉❡ ❤❡r❡✳ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

❆♥ ✐♠♣♦rt❛♥t t♦♦❧ ❢♦r ♦✉r ❛♥❛❧②s✐s ❢❡❛t✉r❡ ✐s t❤❡ st❛♥❞❛r❞ ❢♦r♠ ♦❢ ❛ ✜♥✐t❡ ✈♦♥ ◆❡✉♠❛♥♥ ❛❧❣❡❜r❛✳ ▲♦✇❡r ❜♦✉♥❞s ▲♦✇❡r ❜♦✉♥❞s ❛r❡ ♦❜t❛✐♥❡❞ ❜② ❝❛❧❝✉❧❛t✐♥❣ t❤❡ ♠✐♥✲❝❜✲❡♥tr♦♣② ♦r s♦✲❝❛❧❧❡❞ r❡✈❡rs❡❞ ❝♦❤❡r❡♥t ✐♥❢♦r♠❛t✐♦♥ d H cb ♠✐♥ ( N ) = dp �N : S ✶ → S p � cb | p = ✶ . ■♥❞❡❡❞✱ H cb ♠✐♥ ( θ f ) = τ ( f ❧♥ f ) ❀ ❖✉r ❝❧❛ss❡s ♦❢ ❝❤❛♥♥❡❧s ❛r❡ ❛✉t♦♠❛t✐❝❛❧❧② ❝❧♦s❡❞ ✉♥❞❡r t❛❦✐♥❣ t❡♥s♦r ♣r♦❞✉❝ts ❛♥❞ ❤❡♥❝❡ r❡❣✉❧❛r✐③❛t✐♦♥ ✐s ♥♦t ❛ ❜✐❣ ✐ss✉❡ ❤❡r❡✳ ❚❤✐s s❡❡♠s t♦ ❜❡ ❛ ✜rst s②st❡♠❛t✐❝ st✉❞② ❢♦r ♥♦♥ ❞❡❣r❛❞❛❜❧❡ ❝❤❛♥♥❡❧s✳ ❙❝❤✉r ♠✉t❧✐♣❧✐❡rs N ❛r❡ ❞❡❣r❛❞❛❜❧❡✱ ✐✳❡✳ t❤❡r❡ ❡①✐sts ❛ ❝❤❛♥♥❡❧ Ψ s✉❝❤ t❤❛t N c = Ψ N ❤♦❧❞s ❢♦r t❤❡ ❝♦♠♣❧❡♠❡♥t❛r② ❝❤❛♥♥❡❧ ✭tr❛❝❡ ♦✉t ❇♦❜ ✐♥st❡❛❞ ♦❢ ❡♥✈✐r♦♥♠❡♥t✮✳ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

▲♦✇❡r ❜♦✉♥❞s ▲♦✇❡r ❜♦✉♥❞s ❛r❡ ♦❜t❛✐♥❡❞ ❜② ❝❛❧❝✉❧❛t✐♥❣ t❤❡ ♠✐♥✲❝❜✲❡♥tr♦♣② ♦r s♦✲❝❛❧❧❡❞ r❡✈❡rs❡❞ ❝♦❤❡r❡♥t ✐♥❢♦r♠❛t✐♦♥ d H cb ♠✐♥ ( N ) = dp �N : S ✶ → S p � cb | p = ✶ . ■♥❞❡❡❞✱ H cb ♠✐♥ ( θ f ) = τ ( f ❧♥ f ) ❀ ❖✉r ❝❧❛ss❡s ♦❢ ❝❤❛♥♥❡❧s ❛r❡ ❛✉t♦♠❛t✐❝❛❧❧② ❝❧♦s❡❞ ✉♥❞❡r t❛❦✐♥❣ t❡♥s♦r ♣r♦❞✉❝ts ❛♥❞ ❤❡♥❝❡ r❡❣✉❧❛r✐③❛t✐♦♥ ✐s ♥♦t ❛ ❜✐❣ ✐ss✉❡ ❤❡r❡✳ ❚❤✐s s❡❡♠s t♦ ❜❡ ❛ ✜rst s②st❡♠❛t✐❝ st✉❞② ❢♦r ♥♦♥ ❞❡❣r❛❞❛❜❧❡ ❝❤❛♥♥❡❧s✳ ❙❝❤✉r ♠✉t❧✐♣❧✐❡rs N ❛r❡ ❞❡❣r❛❞❛❜❧❡✱ ✐✳❡✳ t❤❡r❡ ❡①✐sts ❛ ❝❤❛♥♥❡❧ Ψ s✉❝❤ t❤❛t N c = Ψ N ❤♦❧❞s ❢♦r t❤❡ ❝♦♠♣❧❡♠❡♥t❛r② ❝❤❛♥♥❡❧ ✭tr❛❝❡ ♦✉t ❇♦❜ ✐♥st❡❛❞ ♦❢ ❡♥✈✐r♦♥♠❡♥t✮✳ ❆♥ ✐♠♣♦rt❛♥t t♦♦❧ ❢♦r ♦✉r ❛♥❛❧②s✐s ❢❡❛t✉r❡ ✐s t❤❡ st❛♥❞❛r❞ ❢♦r♠ ♦❢ ❛ ✜♥✐t❡ ✈♦♥ ◆❡✉♠❛♥♥ ❛❧❣❡❜r❛✳ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②

❚❤❛♥❦s ❢♦r ❧✐st❡♥✐♥❣ ▼❛r✐✉s ❏✉♥❣❡ ◗✉❛♥t✉♠ ❈❛♣❛❝✐t②